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# Lowest order constrained variational calculation of polarized neutron matter
at finite temperature
G.H. Bordbar 1,3 111Corresponding author. E-mail: bordbar@physics.susc.ac.ir
and M. Bigdeli2,3222E-mail: m bigdeli@znu.ac.ir 1Department of Physics, Shiraz
University, Shiraz 71454, Iran333Permanent address,
2Department of Physics, Zanjan University, P.O. Box 45195-313, Zanjan, Iran
and
3Research Institute for Astronomy and Astrophysics of Maragha,
P.O. Box 55134-441, Maragha, Iran
###### Abstract
Some properties of the polarized neutron matter at finite temperature has been
studied using the lowest order constrained variational (LOCV) method with the
$AV_{18}$ potential. Our results indicate that spontaneous transition to the
ferromagnetic phase does not occur. Effective mass, free energy, magnetic
susceptibility, entropy and the equation of state of the polarized neutron
matter at finite temperature are also calculated. A comparison is also made
between our results and those of other many-body techniques.
###### pacs:
21.65.-f, 26.60.-c, 64.70.-p
## I INTRODUCTION
The spontaneous phase transition to a ferromagnetic state in the neutron
matter is of particular interest in astrophysics. Specially, this transition
could have important consequences for the physical origin of magnetic field of
pulsars, that are believed to be rapidly rotating neutron stars with strong
surface magnetic fields in the range of $10^{12}-10^{13}$ Gauss shap ; paci ;
gold ; navarro , and also the evolution of a protoneutron star. A hot neutron
star is born within a short time after supernovae explosion. In this stage
(protoneutron star), the interior temperature of a neutron star is of the
order 20-50 MeV burro . Therefore, the study of magnetic properties of
polarized neutron matter at finite temperature is of special interest in the
description of protoneutron stars.
There exist several possibilities of the generation of the magnetic field in a
neutron star. From the nuclear physics point of view, one is the possible
existence of a phase transition to a ferromagnetic state at densities
corresponding to the theoretically stable neutron stars and, therefore, of a
ferromagnetic core in the liquid interior of such compact objects. Such a
possibility has been studied by several authors using different theoretical
approaches [6-29], but the results are still contradictory. In most
calculations, neutron star matter is approximated by pure neutron matter, as
proposed just after the discovery of pulsars. Whereas some calculations, like
those of based on hard sphere gas model brown ; rice , Skyrmelike interactions
vida , Reid soft-core potential pandh and relativistic Dirac-Hartree-Fock
approximation with an effective nucleon-meson Lagrangian marcos showed that
neutron matter becomes ferromagnetic at some densities. Others, like recent
Monte Carlo fanto and Brueckner-Hartree-Fock calculations [21-23] using
modern two-body and three-body realistic interactions show no indication of
ferromagnetic transition at any density for neutron matter and asymmetrical
nuclear matter.
Most of the studies of the ferromagnetic transition in neutron matter and
nuclear matter have been done at zero temperature. The properties of polarized
neutron matter both at zero and finite temperature have been studied by
several authors apv ; dapv ; bprrv . Bombaci et al. (BPRRV) bprrv have
studied the properties of polarized neutron matter within the framework of the
Brueckner-Hartree-Fock formalism using the $AV_{18}$ nucleon-nucleon
interaction. Their results show no indication of a ferromagnetic transition at
any density and temperature. Lopez-Val et al. dapv have used the D1 and D1P
parameterization of the Gogny interaction and the results of their calculation
show two different behaviors: whereas the D1P force exhibits a ferromagnetic
transition at density of $\rho\sim 1.31fm^{-3}$ whose onset increases with
temperature, no sign of such a transition is found for D1 at any density and
temperature. Rios and Polls apv have used Skyrme-like interactions and their
results indicate the occurrence of a ferromagnetic phase of neutron matter.
Recently, we have computed the properties of polarized neutron matter bordbig
, polarized symmetrical nuclear matter bordbig2 , such as total energy,
magnetic susceptibility, pressure, etc at zero temperature using the
microscopic calculations employing the lowest order constrained variational
(LOCV) method with the $AV_{18}$ potential. We have also calculated the
properties of spin polarized asymmetrical nuclear matter and neutron star
matter bordbig3 using the LOCV method employing the $AV_{18}$ wiring ,
$Reid93$ R93 , $UV_{14}$ UV14 and $AV_{14}$ AV14 potentials. We have
concluded that the spontaneous phase transition to a ferromagnetic state in
the neutron matter, symmetrical and asymmetrical nuclear matter and neutron
star matter does not occur.
In the present work, we study the properties of polarized neutron matter at
finite temperature using the LOCV technique employing the $AV_{18}$ potential.
## II LOCV FORMALISM FOR POLARIZED HOT NEUTRON MATTER
We consider a system of $N$ interacting neutrons with $N_{1}$ spin up and
$N_{2}$ spin down neutrons. The total number density ($\rho$) and spin
asymmetry parameter ($\delta$) are defined as
$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\rho_{1}+\rho_{2},$
$\displaystyle\delta$ $\displaystyle=$ $\displaystyle\frac{N_{1}-N_{2}}{N}.$
(1)
$\delta$ shows the spin ordering of matter which can have a value in the range
of $\delta=0.0$ (unpolarized matter) to $\delta=1.0$ (fully polarized matter).
Now, we consider a trial many-body wave function of the form
$\displaystyle\psi=F\phi,$ (2)
where $\phi$ is the uncorrelated ground state wave function (simply the Slater
determinant of plane waves) of $N$ independent neutrons and $F=F(1\cdots N)$
is an appropriate N-body correlation operator which can be replaced by a
Jastrow form i.e.,
$\displaystyle F=\textsf{S}\prod_{i>j}f(ij),$ (3)
in which S is a symmetrizing operator. We consider a cluster expansion of the
energy functional up to the two-body term,
$\displaystyle
E([f])=\frac{1}{N}\frac{\langle\psi|H\psi\rangle}{\langle\psi|\psi\rangle}=E_{1}+E_{2}\cdot$
(4)
For hot neutron matter, the one-body term $E_{1}$ is
$\displaystyle E_{1}=\sum_{i=1,2}\varepsilon_{i}\cdot$ (5)
Labels 1 and 2 are used instead of spin up and spin down neutrons,
respectively, and $\varepsilon_{i}$ is
$\displaystyle\varepsilon_{i}=\sum_{k}\frac{\hbar^{2}{k^{2}}}{2m}n_{i}(k,T,\rho_{i}),$
(6)
where $n_{i}(k,T,\rho_{i})$ is the Fermi-Dirac distribution function,
$\displaystyle
n_{i}(k,T,\rho_{i})=\frac{1}{e^{\beta[\epsilon_{i}(k,T,\rho_{i})-\mu_{i}(T,\rho_{i})]}+1}\cdot$
(7)
In the above equation $\beta=\frac{1}{T}$ and $\mu_{i}$ being the chemical
potential, $\rho_{i}$ is the number density and $\epsilon_{i}$ is the single
particle energy of a neutron with spin projection $i$.
In our formalism, the single particle energy, $\epsilon_{i}$, of a neutron
with momentum $k$ and spin projection $i$ is approximately written in terms of
effective mass as apv ; dapv
$\displaystyle\epsilon_{i}(k,T,\rho_{i})=\frac{\hbar^{2}{k^{2}}}{2m^{*}_{i}(\rho,T)}+U_{i}(T,\rho_{i})\cdot$
(8)
In fact, we use a quadratic approximation for single particle potential,
incorporated in the single particle energy as a momentum independent effective
mass. $U_{i}(T,\rho_{i})$ is the momentum independent single particle
potential. The effective mass, $m^{*}_{i}$, is determined variationally mod ;
mod95 ; mod97 ; modbord ; fp .
The chemical potentials, $\mu_{i}$, at any adopted values of the temperature
($T$), number density ($\rho_{i}$) and spin polarization ($\delta$), are
determined by applying the constraint,
$\displaystyle\sum_{k}n_{i}(k,T,\rho_{i})=N_{i}\cdot$ (9)
This is an implicit equation which can be solved numerically.
The two-body energy $E_{2}$ is
$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2A}\sum_{ij}\langle ij\left|\nu(12)\right|ij-
ji\rangle,$ (10)
where
$\nu(12)=-\frac{\hbar^{2}}{2m}[f(12),[\nabla_{12}^{2},f(12)]]+f(12)V(12)f(12)$,
$f(12)$ and $V(12)$ are the two-body correlation and potential. For the two-
body correlation function, $f(12)$, we consider the following form borda ;
bordb :
$\displaystyle f(12)$ $\displaystyle=$
$\displaystyle\sum^{3}_{k=1}f^{(k)}(12)O^{(k)}(12),$ (11)
where, the operators $O^{(k)}(12)$ are given by
$\displaystyle O^{(k=1-3)}(12)$ $\displaystyle=$ $\displaystyle 1,\
(\frac{2}{3}+\frac{1}{6}S_{12}),\ (\frac{1}{3}-\frac{1}{6}S_{12}),$ (12)
and $S_{12}$ is the tensor operator. After doing some algebra, we find the
following equation for the two-body energy:
$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\frac{2}{\pi^{4}\rho}\left(\frac{h^{2}}{2m}\right)\sum_{JLSS_{z}}\frac{(2J+1)}{2(2S+1)}[1-(-1)^{L+S+1}]\left|\left\langle\frac{1}{2}\sigma_{z1}\frac{1}{2}\sigma_{z2}\mid
SS_{z}\right\rangle\right|^{2}\int
dr\left\\{\left[{f_{\alpha}^{(1)^{{}^{\prime}}}}^{2}{a_{\alpha}^{(1)}}^{2}(k_{f}r)\right.\right.$
(13)
$\displaystyle\left.\left.+\frac{2m}{h^{2}}(\\{V_{c}-3V_{\sigma}+V_{\tau}-3V_{\sigma\tau}+2(V_{T}-3V_{\sigma\tau})+2V_{\tau
z}\\}{a_{\alpha}^{(1)}}^{2}(k_{f}r)\right.\right.$
$\displaystyle\left.\left.+[V_{l2}-3V_{l2\sigma}+V_{l2\tau}-3V_{l2\sigma\tau}]{c_{\alpha}^{(1)}}^{2}(k_{f}r))(f_{\alpha}^{(1)})^{2}\right]+\sum_{k=2,3}\left[{f_{\alpha}^{(k)^{{}^{\prime}}}}^{2}{a_{\alpha}^{(k)}}^{2}(k_{f}r)\right.\right.$
$\displaystyle\left.\left.+\frac{2m}{h^{2}}(\\{V_{c}+V_{\sigma}+V_{\tau}+V_{\sigma\tau}+(-6k+14)(V_{tz}+V_{t})-(k-1)(V_{ls\tau}+V_{ls})\right.\right.$
$\displaystyle\left.\left.+[V_{T}+V_{\sigma\tau}+(-6k+14)V_{tT}][2+2V_{\tau
z}]\\}{a_{\alpha}^{(k)}}^{2}(k_{f}r)\right.\right.$
$\displaystyle\left.\left.+[V_{l2}+V_{l2\sigma}+V_{l2\tau}+V_{l2\sigma\tau}]{c_{\alpha}^{(k)}}^{2}(k_{f}r)+[V_{ls2}+V_{ls2\tau}]{d_{\alpha}^{(k)}}^{2}(k_{f}r)){f_{\alpha}^{(k)}}^{2}\right]\right.$
$\displaystyle\left.+\frac{2m}{h^{2}}\\{V_{ls}+V_{ls\tau}-2(V_{l2}+V_{l2\sigma}+V_{l2\sigma\tau}+V_{l2\tau})-3(V_{ls2}+V_{ls2\tau})\\}b_{\alpha}^{2}(k_{f}r)f_{\alpha}^{(2)}f_{\alpha}^{(3)}\right.$
$\displaystyle\left.+\frac{1}{r^{2}}(f_{\alpha}^{(2)}-f_{\alpha}^{(3)})^{2}b_{\alpha}^{2}(k_{f}r)\right\\},$
where $\alpha=\\{J,L,S,S_{z}\\}$ and the coefficient ${a_{\alpha}^{(1)}}^{2}$,
etc., are defined as
$\displaystyle{a_{\alpha}^{(1)}}^{2}(r,\rho,T)=r^{2}I_{L,S_{z}}(r,\rho,T),$
(14) $\displaystyle{a_{\alpha}^{(2)}}^{2}(r,\rho,T)=r^{2}[\beta
I_{J-1,S_{z}}(r,\rho,T)+\gamma I_{J+1,S_{z}}(r,\rho,T)],$ (15)
$\displaystyle{a_{\alpha}^{(3)}}^{2}(r,\rho,T)=r^{2}[\gamma
I_{J-1,S_{z}}(r,\rho,T)+\beta I_{J+1,S_{z}}(r,\rho,T)],$ (16) $\displaystyle
b_{\alpha}^{(2)}(r,\rho,T)=r^{2}[\beta_{23}I_{J-1,S_{z}}(r,\rho,T)-\beta_{23}I_{J+1,S_{z}}(r,\rho,T)],$
(17)
$\displaystyle{c_{\alpha}^{(1)}}^{2}(r,\rho,T)=r^{2}\nu_{1}I_{L,S_{z}}(r,\rho,T),$
(18)
$\displaystyle{c_{\alpha}^{(2)}}^{2}(r,\rho,T)=r^{2}[\eta_{2}I_{J-1,S_{z}}(r,\rho,T)+\nu_{2}I_{J+1,S_{z}}(r,\rho,T)],$
(19)
$\displaystyle{c_{\alpha}^{(3)}}^{2}(r,\rho,T)=r^{2}[\eta_{3}I_{J-1,S_{z}}(r,\rho,T)+\nu_{3}I_{J+1,S_{z}}(r,\rho,T)],$
(20)
$\displaystyle{d_{\alpha}^{(2)}}^{2}(r,\rho,T)=r^{2}[\xi_{2}I_{J-1,S_{z}}(r,\rho,T)+\lambda_{2}I_{J+1,S_{z}}(r,\rho,T)],$
(21)
$\displaystyle{d_{\alpha}^{(3)}}^{2}(r,\rho,T)=r^{2}[\xi_{3}I_{J-1,S_{z}}(r,\rho,T)+\lambda_{3}I_{J+1,S_{z}}(r,\rho,T)],$
(22)
with
$\displaystyle\beta=\frac{J+1}{2J+1},\ \gamma=\frac{J}{2J+1},\
\beta_{23}=\frac{2J(J+1)}{2J+1},$ (23) $\displaystyle\nu_{1}=L(L+1),\
\nu_{2}=\frac{J^{2}(J+1)}{2J+1},\ \nu_{3}=\frac{J^{3}+2J^{2}+3J+2}{2J+1},$
(24) $\displaystyle\eta_{2}=\frac{J(J^{2}+2J+1)}{2J+1},\
\eta_{3}=\frac{J(J^{2}+J+2)}{2J+1},$ (25)
$\displaystyle\xi_{2}=\frac{J^{3}+2J^{2}+2J+1}{2J+1},\
\xi_{3}=\frac{J(J^{2}+J+4)}{2J+1},$ (26)
$\displaystyle\lambda_{2}=\frac{J(J^{2}+J+1)}{2J+1},\
\lambda_{3}=\frac{J^{3}+2J^{2}+5J+4}{2J+1},$ (27)
and
$\displaystyle I_{J,S_{z}}(r,\rho,T)=\frac{1}{2\pi^{6}\rho^{2}}\int
dk_{1}dk_{2}n_{i}(k_{1},T,\rho_{i})n_{j}(k_{2},T,\rho_{j})J_{J}^{2}(|k_{2}-k_{1}|r)\cdot$
(28)
In the above equation $J_{J}(x)$ is the Bessel’s function .
Now, we minimize the two-body energy Eq.(13), with respect to the variations
in the functions ${f_{\alpha}}^{(i)}$, but subject to the normalization
constraint bordb ,
$\displaystyle\frac{1}{A}\sum_{ij}\langle
ij\left|h_{S_{z}}^{2}-f^{2}(12)\right|ij\rangle_{a}=0\cdot$ (29)
In the case of spin polarized neutron matter, the function $h_{S_{z}}(r)$ is
defined as
$\displaystyle h_{S_{z}}(r)$ $\displaystyle=$
$\displaystyle\left[1-\left(\frac{\gamma_{i}(r)}{\rho}\right)^{2}\right]^{-1/2};\
S_{z}=\pm 1$ (30) $\displaystyle=$ $\displaystyle 1\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ;\ S_{z}=0$
where
$\displaystyle\gamma_{i}(r)=\frac{1}{2\pi^{2}}\int
n_{i}(k,T,\rho_{i})J_{0}(kr)k^{2}dk\cdot$ (31)
From the minimization of the two-body cluster energy, we get a set of coupled
and uncoupled differential equations which are the same as presented in Ref.
bordb , with the coefficients replaced by those indicated in equations
(14)-(22). We can obtain correlation functions by solving the differential
equations and then the two-body energy is computed.
Finally, we must calculate the total free energy per particle, $F$, to get the
macroscopic properties of hot neutron matter,
$\displaystyle F=E-TS,$ (32)
where $S$ is the entropy per neutron,
$\displaystyle
S(\rho,T)=-\frac{1}{N}\sum_{i=1,2}\sum_{k}\\{[1-n_{i}(k,T,\rho_{i})]\textrm{ln}[1-n_{i}(k,T,\rho_{i})]+n_{i}(k,T,\rho_{i})\textrm{ln}n_{i}(k,T,\rho_{i})\\}.$
(33)
We introduce the effective masses, $m^{*}_{i}$, as variational parameters mod
; mod95 ; mod97 ; modbord ; fp . In fact, we minimize the free energy with
respect to the variations in the effective masses and then, we obtain the
chemical potentials and the effective masses of spin-up and spin-down neutrons
at the minimum point of the free energy. This minimization is done
numerically.
## III RESULTS
In Fig. 1, we have plotted the effective mass of spin-up and spin-down
neutrons versus spin polarization($\delta$) for fixed density
$\rho=0.16fm^{-3}$ at $T=10$ and $T=20$ MeV. We see that the effective masses
of the above components have the same values at $\delta=0$ and get different
values by increasing the spin polarization. It can be seen that the effective
mass of spin-up neutrons is larger than spin-down neutrons for $\delta>0$ and
the effective mass of spin-down (spin-up) neutrons decreases (increases) by
increasing the polarization. In Fig. 1, we have also included the results of
BPRRV calculations bprrv for comparison. It is seen that the behavior of the
effective mass versus $\delta$ obtained from our method shows the same
properties as BPRRV calculations bprrv .
Our results indicate that the effective masses of spin-up and spin-down
neutrons increase by increasing the temperature. Whereas, the results of BPRRV
bprrv and Isayev Isayev show that the effective masses of nucleons in the
spin polarized neutron matter and nuclear matter are decreasing functions of
the temperature. However, in the references bprrv and Isayev , the authors
determine the momentum independent effective masses through the derivatives of
the self-consistently calculated single particle potentials at the
corresponding Fermi momenta while, in our calculations, the effective masses
are introduced in the single particle energies and considered as variational
parameters for the free energy. We note that increasing the effective mass by
increasing the temperature can be also seen in the results of others where the
procedures for finding the effective mass are the same as ours mod ; mod95 ;
mod97 ; fp .
The behavior of the free energy per particle of the polarized hot neutron
matter versus total number density ($\rho$) for different values of the spin
polarization ($\delta$) at $T=10$ and $T=20$ MeV is shown in Fig. 2. This
figure shows that the free energy increases with increasing both density and
polarization. It is seen that the minimum value of free energy occurs for
unpolarized case ($\delta=0.0$) at any density and at relevant considered
temperature. By comparison, in the two panels of Fig. 2, we see that the free
energy decreases by increasing temperature. It is also seen that there is no
crossing of the free energy curves for different polarizations. Conversely, by
increasing the density, the difference between the free energy of neutron
matter at different polarizations becomes more appreciable. According to this
result, the spontaneous phase transition to a ferromagnetic state in the hot
neutron matter does not occur. If such a transition existed, a crossing of the
energies of different polarizations would have been observed at some density,
indicating that the ground state of the system would be ferromagnetic from
that density on.
In Fig. 3, we have presented the free energy per particle as a function of the
quadratic spin polarization ($\delta^{2}$) for fixed density
$\rho=0.36fm^{-3}$ at $T=10$ and $T=20$ MeV. It can be seen that the free
energy per particle increases by increasing the polarization. We see that the
variation of the free energy of hot neutron matter versus $\delta^{2}$ is
nearly linear. This indicates that the ground state of hot neutron matter is
paramagnetic. In Fig. 3, we have compared our results with those of BPRRV
calculations bprrv . It is seen that there is an overall agreement between our
results and the results of BPRRV bprrv .
The magnetic susceptibility, $\chi$, which characterizes the response of a
system to the magnetic field is given by
$\displaystyle\chi=\frac{\mu^{2}\rho}{\left(\frac{\partial^{2}F}{\partial\delta^{2}}\right)_{\delta=0}},$
(34)
where $\mu$ is the magnetic moment of neutron. In Fig. 4, we have shown the
ratio ${{\chi_{F}}/{\chi}}$ as a function of the total number density at two
values of temperature $T=10$ and $T=20$ MeV. $\chi_{F}$ is the magnetic
susceptibility for a free Fermi gas. As can be seen from Fig. 4, even at high
densities, the ratio ${{\chi_{F}}/{\chi}}$ increases monotonically and
continuously as the density increases for any temperature. This shows that
there is no magnetic instability in hot neutron matter. In Fig. 4, the results
of BPRRV calculations bprrv at $T=20$ MeV are also presented for comparison.
We see that our results and BPRRV results bprrv have an agreement at low
densities.
In Fig. 5, the difference of the entropy per particle of fully polarized and
unpolarized cases is plotted as a function of the total number density at
$T=10$ and $T=20$ MeV. Fig. 5 shows that for all relevant densities, this
difference has negative values. According to this result, we can conclude that
the fully polarized case is more ordered than the unpolarized case. In Fig. 5,
the results of BPRRV calculations bprrv are also given for comparison. There
is an agreement between our results and the BPRRV bprrv results at low
densities.
In Fig. 6, we have plotted the entropy per particle of hot neutron matter
versus spin polarization for fixed density $\rho=0.32fm^{-3}$ and temperature
$T=20MeV$. It is seen that the entropy decreases as polarization increases
with its highest value occurring for the unpolarized case. To prevent
anomalous behavior of entropy as a function of spin polarization, for a given
density and temperature, a condition for effective masses can be derived apv ,
$\displaystyle\frac{m^{*}(\rho,\delta=1.0)}{m^{*}(\rho,\delta=0.0)}<2^{2/3},$
(35)
where $m^{*}(\rho,\delta=1.0)$ and $m^{*}(\rho,\delta=0.0)$ are the effective
masses of the fully polarized and unpolarized neutron matter, respectively.
From Fig. 1, we have found out that for density $\rho=0.16fm^{-3}$, the above
ratio at $T=10$ MeV is
$\frac{m_{1}^{*}(\delta=1.0)}{m_{1,2}^{*}(\delta=0.0)}=1.22$ and at $T=20MeV$
is $\frac{m_{1}^{*}(\delta=1.0)}{m_{1,2}^{*}(\delta=0.0)}=1.18$ which are
smaller than the indicated limit. This condition is satisfied for all other
densities explored in this work. Therefor, we can see that the entropy of
polarized case is always smaller than the entropy of unpolarized case.
The equation of state of hot polarized neutron matter, $P(\rho,T,\delta)$, can
be simply obtained using
$\displaystyle P(\rho,T,\delta)=\rho^{2}\frac{\partial
F(\rho,T,\delta)}{\partial\rho}$ (36)
In Fig. 7, we have presented the pressure of neutron matter as a function of
the total number density ($\rho$) for different polarizations at $T=10$ and
$T=20$ MeV. We see that the equation of state becomes stiffer by increasing
the polarization.
## IV Summary and Conclusions
Some thermodynamic properties of the polarized neutron matter at finite
temperature were reexamined using the lowest order constrained variational
(LOCV) method employing the $AV_{18}$ nucleon-nucleon potential. Our main goal
was to check the occurrence of the spontaneous transition to the ferromagnetic
state. We found no indication for the occurrence the ferromagnetic phase, in
agreement with the results of others who used the different many-body
techniques. Effective mass, free energy per particle, magnetic susceptibility,
entropy per particle, and the equation of state for the polarized neutron
matter at finite temperature were calculated and the effect of polarization on
these properties were examined.
###### Acknowledgements.
This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha. We wish to thank Shiraz University and Zanjan
University Research Councils. We also wish to thank A. Poostforush for various
useful discussion.
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Figure 1: Our results for the effective mass of spin-up and spin-down neutrons
versus spin polarization ($\delta$) for density $\rho=0.16fm^{-3}$ at $T=10$
and $T=20$ MeV. The results of BPRV calculations [31] are also given for
comparison.
Figure 2: The free energy per particle of polarized hot neutron matter as a
function of the total number density ($\rho$) for different values of the spin
polarization ($\delta$) at $T=10$ (a) and $T=20$ MeV (b).
Figure 3: Our results for the free energy as a function of the quadratic spin
polarization ($\delta^{2}$) at $T=10$ MeV (full curve) and $T=20$ MeV (dashed
curve) for $\rho=0.36fm^{-3}$. The results of BPRV [31] (dashed curve) are
also given for comparison.
Figure 4: The magnetic susceptibility of the hot neutron matter versus total
number density ($\rho$) at $T=10$ MeV (full curve) and $T=20$ MeV (dashed
curve). The results of BPRV [31] (dotted curve) are also given for comparison
.
Figure 5: As Fig. 4 but for the entropy difference of fully polarized and
unpolarized cases .
Figure 6: Our results (full curve) for the entropy per particle as a function
of the spin polarization ($\delta$) at $T=20$ MeV and $\rho=0.32fm^{-3}$. The
results of BPRV [31] (dashed curve) are also given for comparison.
Figure 7: The equation of state of the hot neutron matter for different values
of the spin polarization ($\delta$) at $T=10$ (a) and $T=20$ MeV (b).
|
arxiv-papers
| 2008-10-18T11:29:51 |
2024-09-04T02:48:58.312775
|
{
"license": "Public Domain",
"authors": "G.H. Bordbar and M. Bigdeli",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/0810.3310"
}
|
0810.3385
|
# Formation of atomic nanoclusters on graphene sheets
M. Neek-Amal School of Physics, Institute for research in fundamental
sciences, IPM 19395-5531 Tehran, Iran Department of physics, Shahid Rajaei
University , Lavizan, Tehran 16788, Iran Reza Asgari 111Corresponding author:
asgari@theory.ipm.ac.ir School of Physics, Institute for research in
fundamental sciences, IPM 19395-5531 Tehran, Iran M. R. Rahimi Tabar
Department of Physics, Sharif University of Technology, 11365-9161, Tehran,
Iran Institute of Physics, Carl von Ossietzky University, D-26111 Oldenburg,
Germany
###### Abstract
The formation of atomic nanoclusters on suspended graphene sheets have been
investigated by employing a Molecular dynamics simulation at finite
temperature. Our systematic study is based on temperature dependent Molecular
dynamics simulations of some transition and alkali atoms on suspended graphene
sheets. We find that the transition atoms aggregate and make various size
nanoclusters distributed randomly on graphene surface. We also report that
most alkali atoms make one atomic layer on graphene sheets. Interestingly, the
potassium atoms almost deposit regularly on the surface at low temperature. We
expect from this behavior that the electrical conductivity of a suspended
graphene doped by potassium atoms would be much higher than the case doped by
the other atoms at low temperature.
###### pacs:
61.46.+w, 73.25.+i, 71.15.Pd
## 1 Introduction
Graphene is a newly realized two-dimensional electron system novoselov ; geim
which has produced a great deal of interest because of the new physics which
it exhibits and because of its potential as a new material for electronic
technology. The agent responsible for many of the interesting electronic
properties of graphene sheets is the non-Bravais honeycomb-lattice arrangement
of carbon atoms, which leads to a gapless semiconductor with valence and
conduction $\pi$-bands. States near the Fermi energy of a graphene sheet are
described by a massless Dirac equation which have chiral band states in which
the honeycomb-sublattice pseudospin is aligned either parallel to or opposite
to the envelope function momentum. polini The linear energy-momentum
dispersion has been confirmed by recent observations novoselov1 ; bostwich .
There are significant efforts to grow graphene epitaxially berger by thermal
decomposition of Silicon Carbide (SiC), deposition of graphene sheets on solid
banerjee or by vapor deposition of hydrocarbons on catalytic metallic surfaces
which could later be etched away leaving graphene on an insulating substrate.
An unusual feature of the single-atom-thick layer of carbon atoms is the
absence of strongly localization meyer ; ishigami when charge impurities, the
short range ripples and surface roughness exist on graphene sheets. The issue
of localization in graphene has recently attracted some attention and the
chiral nature of electron behavior has been discussed in the literature.
suzuura ; mccann Suzuura and Ando suzuura claimed that the quantum
correction to the conductivity in graphene can differ from what is observed in
normal two-dimensional electron gas due to the nature of elastic scattering in
graphene. This is possibly because of changing the sign of the localization
correction and turn weak localization into weak antilocalization for the
region when intervalley scattering time is much larger than the phase
coherence time. Further consideration of the behavior of the quantum
correction to the conductivity in graphene mccann conclude that this behavior
is entirely suppressed due to time-reversal symmetry breaking of electronic
states around each degenerate valley.
Another special feature of the graphene is the capacity for using as a gas
sensor because of special characteristics of graphene. schedin ; wehling ;
gierz ; meyer1 ; leenaerts Graphene-based gas sensors enable sensitivity to
detect individual events when a gas molecule is absorbed by a graphene sheet.
The absorbed molecules change the local carrier concentration in graphene.
Constant mobility of charge carriers in graphene with increasing chemical
doping of NO2 has been observed in experiment. schedin Note that doping
molecules add some charge carriers but also induces charged impurities. The
later effect results decreasing of the mobility. Different possible scenarios
have been discussed to prove the constant mobility of charge carriers. schedin
Chen et al. chen have reported a systematic theoretical and experimental
studies of the charged impurities mechanism by monitoring a reduction of the
charge carrier mobility. The density of the charged impurities is induced by
adding potassium atoms onto a graphene surface placed on SiO${}_{2}/$Si
substrate in ultrahigh vacuum at low temperature. They have reported that the
addition of charged impurities produces a more linear behavior of conductance,
reduces the mobility and moreover indicated that the minimum conductance
depends on charged impurities density.
In addition, the enhancement of electrical conductivity in a single-wall
carbon nanotubes which are doped by alkali atoms has been studied both
experimentally and theoretically RSLeenature ; RSLeePRB ; Gao . Gao et al. Gao
carried out a Molecular dynamics simulation to find optimum structure of the
doped potassium atoms for different number of the potassium atoms in two
different packing schemes.
A series of measurements on suspended graphene have been performed by
experimental groups Giem2008 ; lee and showed that graphene has an
extraordinary stiffness which can support an additional weight of many
crystalline copper nanoparticles. The results from STEM micrograph of graphene
sheets incorporated copper atoms showed an aggregation shape that
nanoparticles with different sizes distributed on graphene randomly. On the
other hand, the pinning of size-selected the gold and nickel atoms on graphite
has been investigated smith by using the MD simulation and it is shown that
the atoms aggregated on the surface and gold cluster are shown to be flatter
and more spread out the nickel cluster which are more compact. Moreover, they
calculated the pinning energy thresholds and showed that there is a good
agreement with those measured in experiment. praton
Particle aggregation in materials science is a direct consequence of mutual
attraction between particles, atoms or molecules via van der Waals forces or
chemical bonding. elimelech When there are collisions between particles,
there are chances that particles will attach to each other and become a larger
particle. There are three major physical mechanisms to form aggregate;
Brownian motion, fluid shear or motion and differential settling. Because of
interparticle forces, not all collisions may be successful in producing
aggregates. If there is strong repulsion between particles then practically no
collision gives an aggregate. It is known that collisions bring out by
Brownian motion do not generally lead to the rapid formation of very large
aggregates. In addition, the differential settling becomes more important when
the particles are large and dense and this mechanism can promoting
aggregation.
In general, there are two main methods for investigating the physical
behaviors of the absorbed particles on a surface. A Monte Carlo method which
is a sophisticate approach and the other one is a Molecular dynamics
simulation. The purpose of this paper is the study of deposition and
aggregation of atoms on graphene sheets by using the Molecular dynamics
simulations. Eventually, we generally report that the potassium atoms arrange
regularly and deposit on a suspended graphene sheet at low temperature. This
is an evidence that the electrical conductivity of the doped potassium atoms
on a suspended graphene should be higher than the electrical conductivity of
other examined atoms.
It is worthwhile to note that since adding atoms are placed in the setback
distance from graphene sheet, the effect of charged impurities can be
negligible for the suspended graphene due to large screening effects. There
are at least two reasons to explain why the effect of charge impurities should
be important in graphene placed on substrate. The first effect comes from the
fact that the averaged background dielectric constant value of graphene placed
on a substrate is more or less three times larger than the vacuum dielectric
constant. The second effect is related to the image of charged impurity khanh
induces in another side of the dielectric substrate when there is a charged
impurity above graphene. In the latter case, an induced electric dipole in the
system increases the effect of electron scattering. We consequently expect
that the difference between the carrier mobility for a suspended graphene and
a graphene placed on substrate incorporating both the potassium atoms at low
temperature.
The contents of the paper are described briefly as follows. In Sec. II, we
introduce the models and theory. Section III contains our numerical
calculations. Finally, we conclude in Sec. IV with a brief summary.
## 2 Model and Theory
We have used the empirical inter-atomic interaction potential, carbon-carbon
interaction in graphite brenner , which contains three-body interaction for
Molecular dynamics simulation of a suspended graphene sheet at finite
temperature. The two-body potential gives a description of the formation of a
chemical bond between two atoms. Moreover, the three-body potential favors
structures in which the angle between two bonds is made by the same atom.
Many-body effects of electron system, in average, is considered in the Brenner
potential brenner , through the bond-order and furthermore, the potential
depends on the local environment.
We have considered a graphene sheet including $280\times 160$ atoms with
periodic boundary condition. The area of a graphene sheet is $A_{0}=1173.5$
nm2. Considering the canonical ensemble (NVT), we have employed Nosé-Hoover
thermostat to control temperature. Our simulation time step is $1~{}fs$ in all
cases and the thermostat’s parameter is $5~{}fs$. Therefore, we have found a
stable two-dimensional graphene sheet in our simulation.
First of all, we have simulated the stable graphene sheet at finite
temperature, and then extra atoms have been distributed randomly on graphene.
The number of adatoms is equal to $2500$ and distributed in a portion of area,
$A=\frac{4}{9}A_{0}$ in order to avoid boundary effects. We let system
achieves to its equilibrium condition where the atoms above the surface are
mostly fluctuating around their equilibrium positions. At the beginning of
simulation, the atoms are initially located on the height equal to
$3.65\dot{A}$ above the graphene sheet. To further proceed, we do need
implement the interparticle interactions. We have used van der Waals
potentials between atom-atom and carbon-atom denoted by $X-X$ and $C-X$,
respectively. The parameters of two-component interactions between two types
of atoms can be estimated by simple average expressions proposed by Steel et
al, Steel
$\displaystyle\sigma_{C-X}$ $\displaystyle=$
$\displaystyle\frac{\sigma_{X-X}+\sigma_{C-C}}{2}$
$\displaystyle\varepsilon_{C-X}$ $\displaystyle=$
$\displaystyle\sqrt{\varepsilon_{X-X}\cdot\varepsilon_{C-C}}~{},$ (1)
where $\sigma_{i-j}$ is the collision diameter and $\varepsilon_{i-j}$ is the
depth of primary energy well between $i$th and $j$th atoms. The parameters
which have been used in our simulations are given in Table.1 adopted from Ref.
[erkoc2001, ]. It is worthwhile to note that our final results are independent
of the explicit values of parameters appearing in von der Waals potentials. We
have used new set of parameters guan ; ulbricht ; hu and our final results
did not change.
element | $\sigma$ | $\varepsilon$ | Atomic mass
---|---|---|---
| (Å) | (eV) | (gr/mol)
Cu | 2.338 | 0.40933 | 63.546
Ag | 2.644 | 0.3447 | 107.8682
Au | 2.637 | 0.44147 | 196.96
Li | 2.839 | 0.2053 | 6.941
Na | 3.475 | 0.1378 | 22.98977
K | 4.285 | 0.11444 | 39.09
C | 3.369 | 0.00263 | 12.01
Cu-C | 2.853 | 0.0328 | -
Ag-C | 3.006 | 0.0301 | -
Au-C | 3.003 | 0.0341 | -
Li-C | 3.104 | 0.0232 | -
Na-C | 3.422 | 0.0190 | -
K-C | 3.827 | 0.0173 | -
Table 1: Physical parameters and Lennard-Jones $12-6$ parameters for some
atoms and molecules.
## 3 Numerical Results
In this section we present our numerical results based on the method described
above. To this purpose, we have considered a system incorporating the
transition atoms like copper, silver and gold atoms or the alkali atoms like
lithium, sodium and potassium atoms on graphene sheets. Note that the
gravitational force is very small in comparison to interparticle forces and
tends to go unnoticed. Our main results have been summarized in Fig. 1. The
copper atoms (Fig. 1a) are aggregated and the potassium atoms are deposited
(Fig. 1b) on graphene sheets at $T=50$K.
Here, a graphene sheet plays the role of a substrate for external absorbed
atoms. Fig. 2 shows three snapshots of distributed nanoclusters on graphene at
$T=50$K. The transition atoms show aggregation configurations with producing
different nanoclusters size. Apparently each nanocluster trapped around the
height of out-of-plane because of graphene roughness and preferably made a
nanocluster. Furthermore, since $\varepsilon_{cu-C}/\varepsilon_{cu-cu}$ is
smaller than $\varepsilon_{Au-C}/\varepsilon_{Au-Au}$, the copper atoms thus
distribute laterally on the surface much more than the gold atoms. In case,
the gold atoms prefer to grow more vertically in comparison to the copper
atoms. Our numerical simulations confirm such behaviors. Furthermore, our
finding for the copper atoms is quite similar to those observed from STEM
micrograph of graphene sheets incorporated copper atoms by experimental group.
Giem2008 To be sure about the independency of interparticle interaction
potentials, we have examined the Morse potential erkoc2001 and we entirely
get the same results.
We have seen, on the other hand, new physics by adding the alkali atoms on
graphene sheets at $T=50$K which the results are depicted in Fig. 3. The
lithium atoms form a few layers of nanoclusters on graphene apparently having
regular shape. Importantly by using the sodium atoms, an atomic layer thick
arranged above the surface. This layer shows a sort of percolation
configuration. Surprisingly, the potassium atoms arranged regularly and
noticeably wet the graphene’s surface. There are some small vacant islands
where the potassium atoms could not occupy because they are bounded locally by
bumps on the graphene surface. For graphene the doping is usually realized by
surface transfer doping.
The electronic properties that result from absorption depend on the ionic and
covalent character of the bonds formed between carbon and the metal atoms. As
alkali atoms easily release their valance electron, they may effectively
induce n-type doping. bostwich Since the alkali atoms play the role as a
cation, it can give electrons to the system easily. The amount of charge
transfer to graphene by potassium atoms is a challenge problem and has been
studied for many years in potassium deposited graphite surface. Caragiu The
charge transfer between a paramagnetic molecule $NO_{2}$ and a graphene layer
has been recently calculated peeters and found charge transfer of $1$e per
molecule. Furthermore, The effect of potassium doping on the electric
properties of graphene by using density functional theory (DFT$+U$) has been
recently studied. uchoa They predicted a charge transfer of $0.51$e per
potassium atoms at $U=0$. As $U$ increases, the charge transfer is linearly
suppressed.
The regular arrangement distributions of potassium (K) atoms on top of the
surface indicate that the value of opening gap due to breaking symmetry would
be much smaller than those values induced by the other metallic adatoms.
Moreover, It is shown alireza that the conductance of charge carriers
decreases by increasing the value of gap. Therefore, the K atoms give rise to
increase the electrical conductivity, however the transition atoms evidently
play a role as the central scatterers and result in decreasing the electrical
conductivity of system. Note that the effect of charged impurities from
chemical doping are negligible for suspended graphene wehling . Our results
regarding to the regular arrangement of K atoms on graphene are similar to the
alkali atoms doped on the single wall carbon nanotube. Gao
It is well known elimelech that for the alkali metal cations, the critical
coagulation concentration values which estimated by assuming that no energy
barrier exists, decreases in the order Li${}^{+}>$Na${}^{+}>$K+ shows that the
most hydrated ion, (Li+) is the least effective in reducing repulsion term in
particle-particle interactions. Consequently, the K atoms deposit easily on
the surface respect to Li and Na atoms.
We expect physically that the atoms escape from graphene sheets when
temperature increases because of increasing the kinetic energy of atoms. The
number and sizes of nanoclusters decrease by increasing temperature up to
$T=300$K and at higher temperature atoms escape from the surface. In Fig. 4,
we have shown the aggregation behavior of transition atoms at room
temperature. The number of nanoclusters decrease by increasing temperature.
Similar to the results shown in Fig. 2, the nanoclusters have not any regular
crystalline configuration at higher temperature and moreover the distribution
of potassium atoms changes and they show a percolation on top of graphene
sheets as shown in Fig. 5.
For estimating the typical size of nanoclusters at different temperature, we
have calculated the density number of copper atoms, $P(N)$ as a function of
$N$, where $N$ is the number of copper atoms in a nanocluster as depicted in
Fig. 6. The number of copper atoms in each nanocluster decreases by increasing
temperature and some of the atoms escape from the surface.
One of the interesting subject for studying alkali metals absorption on the
graphite is the phase transition in the structure shapes of overlayer. Caragiu
Most studied alkali metal on the graphite is potassium. A common convention
for overlayer structure is $2\times 2$ phase in which 8 carbon atoms are
surrounded by one K atom Caragiu which is most condensed phase of alkali atom
absorbtion on the graphite sheet in temperature lower than room temperature.
There are other less condensed structure such as $7\times 7$ phase where 98
carbon atoms are surrounded by one K atom. Two relevant parameters for change
in the coverage type in the graphite sheet are temperature and density of
adatoms. Increasing the number of K atoms on the graphite at low temperatures
($\sim 90K$) closed pack islands ($2\times 2$) covers all the entire surfaces
but also it may be the existence of the other low condensed phase. After
saturation of K atoms over the graphene it starts to make other layers over
the first one. Caragiu In the case of Li atom most observation indicates on
the intercalation into the graphite even at $100$ K 26 . For Na atoms growing
some multilayered non ordered islands over each other have both experimentally
and computationally been observed 38 .
In our calculation, quite interestingly, the atoms pattern ($2\times 2$), one
K atom per eight carbon, for overlayer in high density for K atoms and a
snapshot of the structure is shown in Fig. 7. This pattern is similar to that
predicted for K atoms on graphite. In the low density, on the other hand, we
did not observe low dense phases, ($7\times 7$) whereas small islands of the
$2\times 2$ structure have been observed. Absence of low dense structures
might be understood from the dominant of thermal vibration which is larger in
a strictly two dimensional system with respect to three dimensional structure
in graphite.
Furthermore, the type of ordering of sodium atoms on the $C_{60}$ molecules
has been investigated by Roques _et al_ Roques . Their results showed that the
ordering of deposited atoms is temperature and concentration dependence. They
found that up to eight atoms on the surface of $C_{60}$ molecule there is no
homogenous deposition and they begin to form nano-clusters on the surface.
This is because of saturation of charges transfer from alkali atom to the
substrate when the substrate has no enough surface ( limiting by curvature)
for accepting other charges. In comparison with our calculations, crudely
deducing, a graphene sheet with the very large radius of carbon cage, has
enough available surface for adding more alkali atoms and those alkali atoms
enable to wet the surface more and more.
We are also interested in calculating the out-of-plan carbon atoms in the
presence of external atoms above graphene. The common procedure for measuring
the roughness exponents of a rough surface is to use a surface structure
function fasolino ; nima
$S_{x}(\delta)=<|h(x+\delta,y)-h(x,y)|^{2}>~{},$ (2)
where the average is taken over some different $y$ values. The variation of
$S_{x}(\delta)$ for different systems at $T=50$ K are shown in Fig. 8. The
characteristic length is defined by the position that curves are bended. From
our finding, the characteristic length in all cases are less than the case of
graphene without absorbed atoms. Moreover, the characteristic length of the
system by using the transition atoms are smaller than when the alkalis atoms
are used. The reason is as follow, the nanocluster made by transition atoms
distribute randomly over the graphene sheets increase the surface randomness.
The $S_{x}(\delta)$ refereing to the situation that the potassium atoms added
on graphene is very similar to one which graphene is itself alone because
correlation between the potassium atoms and the carbon atoms are strong at low
temperature. Consequently, this sort of atoms added on graphene does not
change the morphology of surface at low temperature.
Another interesting quantity is normal-normal correlation for a surface. The
normal-normal correlation in the $x-$ direction is defined as
$c(\delta,T)=<\hat{n}(x,y).\hat{n}(x+\delta,y)>~{}.$ (3)
The persistence length, $l_{p}(T)$ which is a criterion of surface stiffness
is expressed as followsafran
$l_{p}(T)=\frac{\delta}{\ln c(0,T)-\ln c(\delta,T)}~{}.$ (4)
A surface is rigid in a length smaller than persistence length and it behaves
as a soft membrane for longer length. To calculate the adatom dependence of
persistence length of graphene sheets, we have calculated this parameter for
different used atoms at three temperatures which are listed in Table 2.
Moreover, by using the Morse potential for the transition atoms, the same
persistence length values have been obtained.
element | $l_{p}$(Å) at T=50 K | $l_{p}$(Å) at T=200 K | $l_{p}$(Å) at T=300 K
---|---|---|---
g & Cu | 88.1 | 60.0 | 52.0
g & Ag | 82.5 | 58.0 | 52.0
g & Au | 87.2 | 58.5 | 52.1
g & Li | 95.3 | 60.0 | 54.3
g & Na | 94.5 | 60.5 | 53.5
g & K | 102.1 | 67.2 | 54.0
g | 112.0 | 69.0 | 55.5
Table 2: Persistence length of graphene (denoted by g) and the presence of
adatoms on graphene at different temperatures.
As it is clear from the Table 2, the persistence length decreases by
incorporating atoms and importantly the persistence length of the potassium
atoms is close to $l_{p}$ of graphene itself. In all cases, $l_{p}$ decreases
by increasing temperature.
## 4 Conclusion
We have studied the formation of atomic nanoclusters on suspended graphene
sheets by using a Molecular dynamics simulation at finite temperature. We have
used the model of interparticle potentials. The Brenner potential for carbon-
carbon interactions potential and van der Waals model potentials for the
interactions between atoms and moreover the interactions between atoms and the
carbon atoms. We have shown that the transition atoms aggregated on the
surface with different nanoclusters sizes, however the sodium and the
potassium atoms produced one atomic layer on graphene. Interestingly, the
potassium atoms arranged regularly on graphene sheets at low temperature and
it indicates that the value of opening gap due to breaking symmetry would be
much smaller than those values induced by the other metallic adatoms.
As a consequence, the charge carriers electrical conductivity of a suspended
graphene doped by the potassium atoms would be higher than a suspended
graphene doped by the other atoms. We qualitatively expect that the electrical
conductivity of system would increase by adding the potassium atoms on
suspended graphene sheets at low temperature. Note that since the induced
charged impurities are set far from the suspended graphene sheet, the effect
of charged impurities can be ignored due to large screening effects. Our
finding at low temperature would be verified by experiments.
We remark that a model going beyond the Molecular dynamics simulation is
necessary to account quantitatively the effect of the potassium atoms on
electrical conductivity for a suspended graphene sheet at low temperature. One
approach would be the density-functional theory together with the Molecular
dynamics simulation where the transport properties of charge carriers in the
presence of extra atoms are considered.
###### Acknowledgements.
We thank Andre Geim for pointing this problem out to us. We are grateful to N.
Abedpour for discussions and comments. R. A. would like to thank the
International Center for Theoretical Physics, Trieste for its hospitality
during the period when part of this work was carried out.
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* (24) Pratontep S, Preece P, Xirouchaki C, Palmer R. E, Sanz- Navarro C. F, Kenny S. D, and Smith R, Phys. Rev. Lett. 90 (2003) 055503 .
* (25) Elimelech M, Gregory J, Jia X and Williams R A 1995 Particle deposition and aggregation, Butterworth-Heinemann .
* (26) In the contex of conventinal two-dimensional electron gas see: Khanh N Q 2001, physica status solidi b 225 89 .
* (27) Brenner D W 1990 Phys. Rev. B 42 9458 .
* (28) Steel H A 1974 The Introduction of Gases with Solid Surfaces, Oxford, Pergamon .
* (29) Ercok S 2001 Ann. Rev. Comp. Phys. IX, 1 101 .
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* (32) Hu N, Sun X and Hsu A 2005 J. Chem. Phys. 123 044708 .
* (33) Caragiu M and Finberg S 2005 J. Phys.: Condens. Matter 17 R995 and references therein .
* (34) Leenaerts O, Partoens B, and Peeters F. M 2008 Appl. Phys. Lett. 92 243125 .
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* (38) Breitholtz M, Kihlgren T, Lindgren S- and Walld en L 2003 Phys. Rev. B 67 235416, Rytkonen K, Akola J and Manninen M 2004 Phys. Rev. B 69 205404 .
* (39) Roques J, Calvo F, Spiegelman F, and Mijoule C 2003 Phys. Rev. Lett 90 075505 .
* (40) Fasolino A, Los J H and Katsnelson M I 2007 Nature Materials 6 858 .
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* (42) Safran S 1984 Statistical Thermodynamics Of Surfaces, Interfaces, And Membranes, Westview press .
Figure 1: (Color online) Distribution of copper atoms (top) and potassium
atoms (bottom) on graphene sheets at $T=50$ K. The copper and potassium atoms
denoted by red spheres and graphene sheets denoted by white spheres as a
background. The number of atoms added on graphene is $2500$ for both cases.
Figure 2: (Color online) Aggregation of transition atoms on graphene sheets at
$T=50$ K. Nanoclusters are distributed randomly above the surface. Figure 3:
(Color online) Distribution of alkali atoms on graphene sheets at $T=50$ K.
Potassium atoms deposit on graphene. Figure 4: (Color online) Aggregation of
transition atoms on graphene sheets at $T=300$ K. Figure 5: (Color online)
Distribution of alkali atoms on graphene sheets at $T=300$ K.
Figure 6: (Color online) Density number of copper atoms as a function of
number of atoms at $T=50$K (a) and $T=300$ K (b) . Figure 7: (Color online)
Pattern of potassium atoms distribution, (2 $\times$ 2) on a graphene sheet at
$T=50$ K. The big circles represent the potassium atoms and the small dots
represent the $C$ atoms. Figure 8: (Color online) Surface structure function,
$S_{x}(\delta)$ as a function of $\delta$ at $T=50$K. Graphene is denoted by
$g$.
|
arxiv-papers
| 2008-10-19T11:34:38 |
2024-09-04T02:48:58.318431
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Neek-Amal, Reza Asgari, M. R. Rahimi Tabar",
"submitter": "Reza Asgari",
"url": "https://arxiv.org/abs/0810.3385"
}
|
0810.3419
|
This paper has been withdrawn by the authors.
|
arxiv-papers
| 2008-10-19T18:14:36 |
2024-09-04T02:48:58.323842
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Inkang Kim, Georg Schumacher",
"submitter": "Georg Schumacher",
"url": "https://arxiv.org/abs/0810.3419"
}
|
0810.3428
|
Top quark properties at ATLAS
Dilip Jana (_on behalf of the ATLAS Collaboration_)
_University of Oklahoma, Norman, OK. 73019, USA_
###### Abstract
The ATLAS potential for the study of the top quark properties and physics
beyond the Standard Model in the top quark sector is described. The
measurements of the top quark charge, the spin and spin correlations, the
Standard Model decay ($t\rightarrow bW$), rare top quark decays associated to
flavour changing neutral currents ($t\rightarrow qX$ with $X$ = gluon, $Z$,
photon) and $t\bar{t}$ resonances are discussed. The sensitivity of the ATLAS
experiment is estimated for an expected luminosity of $1$ fb-1 at the LHC,
using full simulation Monte Carlo samples. For the Standard Model measurements
the expected precision is presented. For the tests of physics beyond the
Standard Model, the 5$\sigma$ discovery potential (in the presence of a
signal) and the 95% Confidence Level (C.L.) limit (in the absence of a signal)
are given.
## I Introduction
The top quark was discovered in 1995 at Fermilab in pair production mode
($t\bar{t}$ events) through strong interactions[1]. Several properties of the
top quark such as the mass, charge, lifetime, production cross-section and
rare decay through flavour changing neutral currents (FCNC) have been explored
in Fermilab, but most of these studies are limited by low statistics. Due to
the high event rates at LHC (one $t\bar{t}$ event per second at a luminosity
of $10^{33}$ cm-2s-1) at ATLAS, these top quark properties can be studied
extensively at ATLAS giving the possibility to discover physics beyond the
Standard Model. Due to the very short life time, the top quark decays before
it has time to hadronise. But its spin properties are not washed out by
hadronization, rather the top quark spin information propagates to its decay
products. This unique feature allows direct top quark spin studies. The top
quark spin can be reconstructed by measuring the angular distributions of its
decay products in the top quark rest frame. Measurements of $W$-boson
polarization complement top quark spin studies which can disentangle the
origin of new physics. In the Standard Model, flavour changing neutral
currents (FCNC) are strongly suppressed at the tree level due to the Glashow-
Iliopoulos-Maiani mechanism. At the one loop level, small FCNC contributions
are expected due to the CKM mixing matrix. The existence of $q\bar{q}$ bound
states (mesons) of all other quarks encourages us to look for $t\bar{t}$ bound
states. New resonances and gauge bosons strongly coupled to the top quark are
expected in several theoretical models which can decay into $t\bar{t}$ pairs,
leading to deviations from Standard Model $t\bar{t}$ production cross-section
and top quark kinematics[2]. These new particles can reveal themselves in the
$t\bar{t}$ invariant mass distribution.
## II Basic Event Selection
We have used semileptonic ($t\bar{t}\rightarrow WWb\bar{b}\rightarrow l\nu
j_{1}j_{2}b\bar{b}$ with $l=e,\mu$) and dileptonic ($t\bar{t}\rightarrow
WWb\bar{b}\rightarrow l\nu l^{{}^{\prime}}\nu^{{}^{\prime}}b\bar{b}$ with
$l,l^{{}^{\prime}}=e,\mu$) decays of $t\bar{t}$ events for the top quark
charge reconstruction. We have used only the semileptonic decay channel for
$Wtb$ anomalous coupling, $t\bar{t}$ spin and spin correlation and $t\bar{t}$
resonance. For the semileptonic topology, we require exactly one isolated
electron (muon) with $|\eta|<2.5$ and $p_{\mathrm{T}}>25{\mathrm{\
Ge\kern-1.00006ptV}}$ ($p_{\mathrm{T}}>20{\mathrm{\ Ge\kern-1.00006ptV}}$), at
least 4 jets with $|\eta|<2.5$ and $p_{\mathrm{T}}>30{\mathrm{\
Ge\kern-1.00006ptV}}$, at least 2 jets tagged as $b$-jets and missing
transverse energy above 20 GeV[3]. For the dileptonic topology, we require
exactly two isolated electrons (muons) with $|\eta|<2.5$ and
$p_{\mathrm{T}}>25{\mathrm{\ Ge\kern-1.00006ptV}}$
($p_{\mathrm{T}}>20{\mathrm{\ Ge\kern-1.00006ptV}}$), at least 2 jets with
$|\eta|<2.5$ and $p_{\mathrm{T}}>30{\mathrm{\ Ge\kern-1.00006ptV}}$, at least
2 jets tagged as $b$-jets and missing transverse energy above 20 GeV[3]. Since
the final state topology for the rare top quark decays via FCNC are different
from semileptonic and dileptonic topologies, we have used different selection
criteria which will be described in section 2.3.
### II.1 Top quark charge measurement
We have presented the measurement of the top quark charge based on the
reconstruction of the charge of the top quark decay products. The $W$ boson
charge can be directly measured easily using its leptonic decay modes. Due to
quark confinement inside hadrons, we cannot measure the $b$ quark charge
directly. We have used $b$-jet charge weighting (weighted sum of all the
tracks in the jet) and semileptonic $b$-decay approaches to measure $b$ quark
charge. By using the weighting technique, it is possible to distinguish
between the $b$-jet charges associated with leptons of opposite charges with a
$5\sigma$ significance with only $0.1$ fb-1 of data ( $1$ fb-1 for
semileptonic $b$-decay approach) which allows the Standard Model
(${t\rightarrow W^{+}b}$) and exotic (${t^{\prime}\rightarrow W^{-}b}$)
scenarios to be distinguished. Reconstruction of the magnitude of the top
quark charge seems to be possible with $\simeq 1$ fb-1 using the weighting
technique, but it is necessary to check the performance of the method with
real data. The reconstructed $b$ quark and top quark charge are shown in
Figure 1 with $1$ fb-1 of simulated data. The resulting top quark charge is
$Q_{t}=0.67\pm 0.06\;(stat)\pm 0.08\;(syst)$.
Figure 1: Left: the $b$-jet charge ($Q_{\mathrm{{b}}}$) distribution; right:
the reconstructed top quark charge ($Q\mathrm{{}_{t}}$).
### II.2 Top quark spin and spin correlations and _Wtb_ anomalous couplings
In the Standard Model, the top quarks are produced unpolarised in $t\bar{t}$
events, but their spins are correlated[4]. The production asymmetry ($A$ and
$A_{\mathrm{D}}$) can be obtained from the angular distribution of the top
quark decay products. In addition to the $t\bar{t}$ spin correlation, we can
measure the $W$ polarization. The $W$-boson can be produced with
right($F_{\mathrm{R}}$), left($F_{\mathrm{L}}$) or longitudinal
polarizations($F_{\mathrm{0}}$) with
$F_{\mathrm{0}}+F_{\mathrm{L}}+F_{\mathrm{R}}=1$. The expected measurement
results, using 1 fb-1 of simulated data, are shown in Table 1. It is also
possible to parameterise new physics in the $Wtb$ vertex using anomalous
coupling parameters $V_{\mathrm{L}}$, $V_{\mathrm{R}}$, $g_{\mathrm{L}}$ and
$g_{\mathrm{R}}$. Figure 2 shows the expected 68% C.L. allowed regions on the
$Wtb$ anomalous couplings for 1 fb-1.
Table 1: $W$-boson polarization and top quark spin correlation parameters with statistical and systematic errors. $W$-boson polarization | $F_{\mathrm{L}}$ | $F_{\mathrm{0}}$ | $F_{\mathrm{R}}$
---|---|---|---
| 0.29 $\pm$0.02 $\pm$0.03 | 0.70 $\pm$0.04 $\pm$0.02 | 0.01 $\pm$0.02 $\pm$0.02
$t\bar{t}$ spin correlation | $A$ | $A_{\mathrm{D}}$ |
| 0.67 $\pm$0.17$\pm$0.18 | -0.40 $\pm$0.11 $\pm$0.09 |
Figure 2: The expected 68% C.L. allowed regions on the $Wtb$ anomalous
couplings for $L=1$ fb-1
Figure 3: 95% C.L. expected limits on the $BR(t\to q\gamma)$ vs $BR(t\to qZ)$
Figure 4: 5$\sigma$ discovery potential of a generic narrow $t\bar{t}$
resonance as a function of the integrated luminosity.
### II.3 ATLAS sensitivity to FCNC top quark decays
We have studied the rare top quark decays via FCNC ($t\rightarrow qX$,
$X=\gamma,Z,g$) using $t\bar{t}$ events in $1$ fb-1 of simulated LHC data. One
of the top quarks is assumed to decay through its dominant decay mode ($t\to
bW$), while the other top quark decays via one of the FCNC modes ($t\to qZ$,
$t\to q\gamma$, $t\to qg$). Due to the large QCD background, it is very
difficult to search for FCNC signal using modes where $W$ or $Z$ decay
hadronically. Due to this reason, only leptonic decays of both $W$ and $Z$
were taken into account. For signal events, we have used $t\bar{t}\to b\ell\nu
qX$, where $X=\gamma,Z\to\ell\ell,g$ and $\ell=\mathrm{e},\mu$ and taken into
account the expected Standard Model backgrounds.
For $t\bar{t}\to bWq\gamma$, we require exactly one lepton with
$p_{\mathrm{T}}>25$ GeV, at least two jets with $p_{\mathrm{T}}>20$ GeV, one
$\gamma$ with $p_{\mathrm{T}}>25$ GeV and $\not\\!p_{\mathrm{T}}>20$ GeV. For
$t\bar{t}\to bWqg$, we require exactly one lepton with $p_{\mathrm{T}}>25$
GeV, exactly three jets with $p_{\mathrm{T}}>40,20,20$ GeV and
$\not\\!p_{\mathrm{T}}>20$ GeV. For $t\bar{t}\to bWqZ$, we require exactly
three leptons with $p_{\mathrm{T}}>25,15,15$ GeV, at least two jets with
$p_{\mathrm{T}}>30,20$ GeV and $\not\\!p_{\mathrm{T}}>20$ GeV. The neutrino
four momentum was estimated using a kinematic fit[3]. The expected 95% C.L.
upper limits on the branching ratios for $t\to qZ,t\to q\gamma,t\to qg$ are
10-3, 10-3, 10-2 respectively using $1$ fb-1 simulated data. Figure 3 shows
the expected 95% C.L. for the first $1$ fb-1 in the absence of signal for the
$t\to q\gamma$ and $t\to qZ$ channels.
### II.4 $t\bar{t}$ resonances
The discovery potential for generic $t\bar{t}$ resonances with the ATLAS
detector has been explored as a function of the resonance mass for the
semileptonic $t\bar{t}$ channels[5]. $t\bar{t}$ resonances were produced with
Pythia for $Z^{\prime}\to t\bar{t}$ channel. The common selection criteria
have been applied for event reconstruction. The main source of background for
$t\bar{t}$ resonances is the Standard Model $t\bar{t}$ events (other
backgrounds like $W$+jets are negligible). It is possible to discover a 700
GeV $Z^{\prime}$ resonance produced with a $\sigma\times Br(Z^{\prime}\to
t\bar{t})$ of 11 pb with a 5$\sigma$ significance with 1 fb-1 of data [Figure
4]. Using a model-independent approach, ATLAS can exclude Kaluza-Klein gluon
resonances upto 1.5 TeV with only 1 fb-1 data[3].
## III References
[1] F. Abe $et$ $al.$ (CDF Collaboration), Phys. Rev. Lett. 74, 2626 (1995);
S. Abachi $et$ $al.$ (D0 Collaboration, $ibid.$ 74, 2632 (1995).
[2] Lillie, B and Randall, L. and Wang, L.-T., hep-ph/0701166v1 (2007)
[3] ATLAS Collaboration, CERN-OPEN–2008-020, Geneva, 2008 to appear.
[4] W. Bernreuther, Nucl. Phys. B 690 81 (2004).
[5] E. Cogneras and D. Pallin, ATL-PHYS-PUB-2006-033 (2006).
## IV Acknowledgements
The author would like to thank the organizers of ICHEP08 conference for
creating fruitful collaborative environment. My sincere thanks to Antonio
Onofre, Patrick Skubic and Martine Bosman for valuable suggestions.
|
arxiv-papers
| 2008-10-20T19:59:08 |
2024-09-04T02:48:58.327120
|
{
"license": "Public Domain",
"authors": "Dilip Jana (for the ATLAS Collaboration)",
"submitter": "Dilip Jana",
"url": "https://arxiv.org/abs/0810.3428"
}
|
0810.3465
|
# Temperature effect on the power spectrum in inflation
Shaoyu Yin1111051019008@fudan.edu.cn, Bin Wang1222wangb@fudan.edu.cn, Ru-Keng
Su1,2333rksu@fudan.ac.cn 1\. Department of Physics, Fudan University,Shanghai
200433, China
2\. CCAST(World Laboratory), P.O.Box 8730, Beijing 100080, China
###### Abstract
We examine the effect of the thermal vacuum on the power spectrum of inflation
by using the thermal field dynamics. We find that the thermal effect
influences the CMB anisotropy at large length scale. After removing the
divergence by using the holographic cutoff, we observe that the thermal vacuum
explains well the observational CMB result at low multipoles. This shows that
the temperature dependent factor should be considered in the study of power
spectrum in inflation, especially at large length scale.
###### pacs:
98.80.Cq, 11.10.Wx
The scenario of the inflation provides an excellent solution to the horizon
and flatness problems Guth:1981 . During inflation the quantum fluctuations in
the inflaton field can freeze out and become the seed for large scale
structure Linde:1990 ; Liddle:2000 . These fluctuations can translate into
perturbations in energy density and curvature in the universe and can be
imprinted in the anisotropy of cosmic microwave background (CMB), which can be
measured with higher and higher precision nowadays. Recently an intriguing
possibility has been discovered in the literature that inflation might provide
a window towards physics beyond the Planck scale Easther:2001 ; Easther:2002 ;
Danielsson:2002a ; Martin:2004 ; Martin:2005&2006 ; Groeneboom:2008 ; Ren:2006
; Wang:2003 ; Goldstein:2003 . Chances to detect such trans-Planckian effect
in the CMB observation have been addressed in Refs.Martin:2004 ;
Martin:2005&2006 ; Groeneboom:2008 ; Ren:2006 .
In the standard inflationary scenario, initial conditions for the inflaton
field are imposed in the infinite past with an infinitely short wavelength
when the effect of the inflationary horizon and the expansion of the universe
can be ignored. The spacetime is essentially Minkowskian and there is a unique
vacuum, the Bunch-Davis vacuum, for the inflaton field. To encode the new
physics near the Planck scale, a simple modification in the standard scenario
has been focused on different choices of the vacuum Goldstein:2003 ;
Danielsson:2002b ; Brandenberger:2002 ; WangX:2003 ; Burgess:2003 ;
Chernikov:1968 ; Mottola:1985 ; Allen:1985 ; Floreanini:1987 ; Bousso:2002 ;
Spradlin:2002 . It has been shown that based on some choices of vacua
discussions of many trans-Planckian effects can be carried out and qualitative
correction due to Planck scale can be obtained without detailed knowledge of
trans-Planckian physics. However, all these vacua chosen to discuss the
inflation are of zero temperature.
Inflation started just shortly after the big bang when the universe was
extremely hot at that moment. Before the inflation it would be reasonable to
consider that the universe was in thermal equilibrium. After the inflation,
the thermal equilibrium can be restored. In the process of the inflation, the
inflaton coupled extremely weak to thermal fields so that the inflaton itself
is out of the thermal equilibrium during inflation. However, considering that
the inflation process is extremely short, one can suppose that this process
can be described with the comoving temperature $\tilde{T}=Ta$ to relate the
initial and final states Guth:1985 ; Bhattacharya:2006 , where $a$ is the
scale factor, and use the thermal ground state for the inflaton. In our work
we will use this assumption and employ the thermo field dynamics to discuss
the problem in detail.
In thermo field dynamics Umezawa:1982 , suppose the temperature of the form
$T=1/\beta$, the Bogoliubov transformations of the usual boson operators
$\displaystyle\hat{a}_{k}(\beta)$ $\displaystyle=$
$\displaystyle\cosh\theta_{k}\hat{a}_{k}-\sinh\theta_{-k}\hat{\tilde{a}}^{\dagger}_{-k};$
$\displaystyle\hat{\tilde{a}}_{k}(\beta)$ $\displaystyle=$
$\displaystyle\cosh\theta_{-k}\hat{\tilde{a}}_{-k}-\sinh\theta_{k}\hat{a}^{\dagger}_{k},$
(1)
can be reversed into
$\displaystyle\hat{a}_{k}$ $\displaystyle=$
$\displaystyle\cosh\theta_{k}\hat{a}_{k}(\beta)+\sinh\theta_{k}\hat{\tilde{a}}^{\dagger}_{k}(\beta);$
$\displaystyle\hat{\tilde{a}}_{k}$ $\displaystyle=$
$\displaystyle\cosh\theta_{-k}\hat{\tilde{a}}_{-k}(\beta)+\sinh\theta_{-k}\hat{a}^{\dagger}_{-k}(\beta),$
(2)
where $\cosh\theta_{k}=\frac{1}{\sqrt{1-e^{-\beta\omega(k)}}}$,
$\sinh\theta_{k}=\frac{e^{-\beta\omega(k)}}{\sqrt{1-e^{-\beta\omega(k)}}}$ and
spectrum $\omega(k)=\sqrt{p^{2}+m^{2}}=\sqrt{k^{2}/a^{2}+m^{2}}$. Obviously
$\cosh\theta_{k}=\cosh\theta_{-k}$ and $\sinh\theta_{k}=\sinh\theta_{-k}$,
then in the following we will simply use $\theta_{k}$.
In above definition we introduced the tilde operators $\hat{\tilde{a}}_{k}$
and $\hat{\tilde{a}}^{\dagger}_{k}$ in the duplicated space in additional to
the normal operators. Correspondingly, the state space should also be doubled,
which means that we will not only have $|n\rangle$, but also
$|\widetilde{n}\rangle$ ($n=0,1,2,\cdots$). The tilde operators operate on
tilde space vector $|\widetilde{n}\rangle$ while the normal operators operate
on the normal state $|n\rangle$. Thermal operators satisfy the commuting
relations, $[\hat{a}_{k}(\beta),\hat{a}^{\dagger}_{k}(\beta)]=1$,
$[\hat{\tilde{a}}_{k}(\beta),\hat{\tilde{a}}^{\dagger}_{k}(\beta)]=1$,
$[\hat{a}_{k}(\beta),\hat{\tilde{a}}_{k}(\beta)]=0$, which can be verified
directly from the commutators of the operators $\hat{a}_{k}$ and
$\hat{\tilde{a}}_{k}$.
Now we express the thermal vacuum as $|\beta 0\rangle$. From the thermo field
dynamics, we learn that thermal vacuum state of a boson field is the infinite
linear composition of the state $|n,\widetilde{n}\rangle$:
$|\beta
0\rangle=\sqrt{1-e^{-\beta\omega}}\sum_{n=0}^{\infty}e^{-n\beta\omega/2}|n,\widetilde{n}\rangle,$
(3)
which satisfies
$\hat{a}_{k}(\beta)|\beta 0\rangle=\hat{\tilde{a}}(\beta)|\beta
0\rangle=0;\qquad\langle\beta 0|\hat{a}^{\dagger}_{k}(\beta)=\langle\beta
0|\hat{\tilde{a}}^{\dagger}(\beta)=0.$ (4)
With these formalism, we are in a position to examine the influence of the
thermal vacuum on the power spectrum of the inflation.
In zero temperature case, given a statefunction $\chi_{k}$ solved from field
equation in momentum space, we can build up the field operator as
$\hat{\chi}=\sum_{k}(\chi_{k}\hat{a}_{k}+\chi^{*}_{k}\hat{a}^{\dagger}_{-k}).$
(5)
In the scalar perturbation, the power spectrum is defined as Liddle:2000
$\displaystyle
P_{k}=\left(\frac{H}{\dot{\chi}}\right)^{2}\frac{k^{3}}{2\pi^{2}}|\chi_{k}|^{2},$
(6)
where the last factor arises from the expectation value of the field operator
in the vacuum state,
$\langle 0|\hat{\chi}^{\dagger}\hat{\chi}|0\rangle=\sum_{k}|\chi_{k}|^{2}.$
(7)
In the thermal vacuum, the field operator $\hat{\chi}$ can be expressed as
$\displaystyle\hat{\chi}$ $\displaystyle=$
$\displaystyle\sum_{k}\\{\chi_{k}(\eta)[\cosh\theta_{k}\hat{a}_{k}(\beta)+\sinh\theta_{k}\hat{\tilde{a}}^{\dagger}_{k}(\beta)]+\chi^{*}_{k}(\eta)[\cosh\theta_{k}\hat{a}^{\dagger}_{-k}(\beta)+\sinh\theta_{k}\hat{\tilde{a}}_{-k}(\beta)]\\},$
(8) $\displaystyle=$
$\displaystyle\sum_{k}\\{\cosh\theta_{k}[\chi_{k}(\eta)\hat{a}_{k}(\beta)+\chi^{*}_{k}(\eta)\hat{a}^{\dagger}_{-k}(\beta)]+\sinh\theta_{k}[\chi^{*}_{k}(\eta)\hat{\tilde{a}}_{-k}(\beta)+\chi_{k}(\eta)\hat{\tilde{a}}^{\dagger}_{k}(\beta)]\\}$
by substituting Eq.(2) into Eq.(5). We see now the field space is composed of
two parts, the normal space and the duplicate space, with weights
$\cosh\theta_{k}$ and $\sinh\theta_{k}$, respectively.
The particle number can be calculated as
$\displaystyle n_{k}$ $\displaystyle=$ $\displaystyle\langle\beta
0|\hat{a}^{\dagger}_{k}\hat{a}_{k}|\beta 0\rangle$ (9) $\displaystyle=$
$\displaystyle(1-e^{-\beta\omega})\sum_{n}e^{-n\beta\omega}\langle
n,\widetilde{n}|\hat{a}^{\dagger}_{k}\hat{a}_{k}|n,\widetilde{n}\rangle$
$\displaystyle=$
$\displaystyle(1-e^{-\beta\omega})\sum_{n}e^{-n\beta\omega}\langle
n,\widetilde{n}|\hat{a}^{\dagger}_{k}\sqrt{n}|n-1,\widetilde{n}\rangle$
$\displaystyle=$ $\displaystyle(1-e^{-\beta\omega})\sum_{n}e^{-n\beta\omega}n$
$\displaystyle=$ $\displaystyle\frac{1}{e^{\beta\omega}-1},$
which is just the distribution of thermal equilibrium states.
The power spectrum of perturbation in the thermal vacuum becomes
$\displaystyle P_{k}(\beta)$ $\displaystyle=$
$\displaystyle\left(\frac{H}{\dot{\chi}}\right)^{2}\frac{k^{3}}{2\pi^{2}}\langle\beta
0|\hat{\chi}_{k}^{\dagger}\hat{\chi}_{k}|\beta 0\rangle$ (10) $\displaystyle=$
$\displaystyle\left(\frac{H}{\dot{\chi}}\right)^{2}\frac{k^{3}}{2\pi^{2}}\left(\\{\langle\beta
0|\cosh\theta_{k}^{2}[\chi_{k}^{*}(\eta)\hat{a}^{\dagger}_{k}(\beta)+\chi_{k}(\eta)\hat{a}_{-k}(\beta)][\chi_{k}(\eta)\hat{a}_{k}(\beta)+\chi^{*}_{k}(\eta)\hat{a}^{\dagger}_{-k}(\beta)]|\beta
0\rangle\\}\right.$ $\displaystyle\left.+\\{\langle\beta
0|\sinh\theta_{k}^{2}[\chi_{k}(\eta)\hat{\tilde{a}}^{\dagger}_{-k}(\beta)+\chi_{k}(\eta)\hat{\tilde{a}}_{k}(\beta)][\chi_{k}^{*}(\eta)\hat{\tilde{a}}_{-k}(\beta)+\chi_{k}(\eta)\hat{\tilde{a}}^{\dagger}_{k}(\beta)]|\beta
0\rangle\\}\right)$ $\displaystyle=$
$\displaystyle\left(\frac{H}{\dot{\chi}}\right)^{2}\frac{k^{3}}{2\pi^{2}}|\chi_{k}\cosh\theta_{k}|^{2}+\left(\frac{H}{\dot{\chi}}\right)^{2}\frac{k^{3}}{2\pi^{2}}|\chi_{k}\sinh\theta_{k}|^{2}$
$\displaystyle=$ $\displaystyle
P_{k}(\cosh^{2}\theta_{k}+\sinh^{2}\theta_{k})=P_{k}\coth\frac{\beta\omega}{2},$
where $P_{k}$ is the power spectrum obtained by considering the zero
temperature vacuum. For the massless scalar field, $\omega=p=k/a$. The
temperature effect appears in the factor $\coth\frac{k}{2aT}$, which shows
that the power spectrum gets modified due to the thermal effect. When
$T\rightarrow 0$, Eq.(10) reduces to the usual result of Eq.(6). In the above
derivation, we use the finite temperature field theory which clearly shows the
temperature effect in the vacuum. This approach is general and does not depend
on the choice of the vacuum.
Adding the thermal effect in the adiabatic vacuum, we can generalize the power
spectrum in Danielsson:2002a by including the thermal factor as
$P_{k}(\beta)=P_{k}\coth\frac{\beta\omega}{2}=\left(\frac{H}{\dot{\chi}}\right)^{2}\left(\frac{H}{2\pi}\right)^{2}\left[1-\frac{H}{\Lambda}\sin\left(\frac{2\Lambda}{H}\right)\right]\coth\frac{k}{2\tilde{T}},$
(11)
where $\tilde{T}=aT$ is the comoving temperature and $\Lambda$ is the trans-
Planckian energy level. The second term in the bracket represents the effect
brought by the trans-Planckian physics. At zero temperature the modulation of
the power spectrum of primordial density fluctuation predicted in the trans-
Planckian model has been studied by considering the change of the Hubble
parameter and slow-roll condition Bergstrom:2002
$\varepsilon\equiv
2M^{2}_{PL}\left(\frac{1}{H(\chi)}\frac{dH(\chi)}{d\chi}\right)^{2},$ (12)
where $M_{PL}^{-2}=8\pi G$ is the reduced Planck mass. In Ref.Bergstrom:2002 ,
adopting the scale parameter $\gamma=\Lambda/M_{PL}$ and
$H/\Lambda=\xi(k/k_{0})^{-\varepsilon}$, the trans-Planckian power spectrum is
expressed into
$P_{k}=\left(\frac{H}{\dot{\chi}}\right)^{2}\left(\frac{H}{2\pi}\right)^{2}\left\\{1-\xi\left(\frac{k}{k_{0}}\right)^{-\varepsilon}\sin\left[\frac{2}{\xi}\left(\frac{k}{k_{0}}\right)^{\varepsilon}\right]\right\\},$
(13)
where $\xi\approx 4\times 10^{-4}\sqrt{\varepsilon}/\gamma$ and the pivot
scale $k_{0}=0.05$ Mpc${}^{-1}\approx 213.8H_{0}$, based on the value
$H_{0}=70.1$ km s-1 Mpc-1 from five-year WMAP result Hinshaw:2008 . The power
spectra $P(k)$ for different values of $\gamma$ and $\varepsilon$ were shown
in Ref.Bergstrom:2002 . In Fig.1 we illustrate their result of zero
temperature case in the solid line by taking $\varepsilon=\gamma=0.01$.
Figure 1: The modulation of the power spectrum of primordial density
fluctuations predicted by the trans-Planckian and the thermal effects. The
solid line is for zero temperature while the dashed line shows the thermal
influence. The parameters are $\varepsilon=0.01$ and $\gamma=0.01$. Figure 2:
The angular power spectrum at low $l$ and its comparison with WMAP data. The
dashed line is for zero temperature case, while the solid line shows the
thermal effect. The best fitting of solid line is made with $c=2.3$ and
$\tilde{T}=0.9H_{0}$, while for the dashed line $c=3.1$.
Considering the thermal effect with additional term
$\coth\frac{k}{2\tilde{T}}$, we have the power spectrum
$P_{k}(\beta)=\left(\frac{H}{\dot{\chi}}\right)^{2}\left(\frac{H}{2\pi}\right)^{2}\left\\{1-\xi\left(\frac{k}{k_{0}}\right)^{-\varepsilon}\sin\left[\frac{2}{\xi}\left(\frac{k}{k_{0}}\right)^{\varepsilon}\right]\right\\}\coth\frac{k}{2\tilde{T}}.$
(14)
The temperature influence on the power spectrum $P_{k}$ is shown by the dashed
line in Fig.1. We took $\tilde{T}=H_{0}$ in our plot and noticed that
different values of $\tilde{T}$ will not change the qualitative behavior of
the dashed line. The thermal influence becomes significant especially at small
$k$, while its influence can be neglected for big enough $k$. In other words
the change in the spectrum due to the thermal effect is large at large angles
and small at small angles. This brings an interesting possibility that the
thermal effect might influence the CMB at small $l$.
When $k\rightarrow 0$, because of the factor $\coth\frac{k}{2\tilde{T}}$, we
will meet the divergence problem in the power spectrum, which is unphysical.
To tackle this problem we resort to the idea of the holographic cutoff
employed in Enqvist:2004 ; Shen:2005 . Relating the ultraviolet and infrared
cutoff Cohen:1999 , the quantum zero-point energy density $\rho_{\Lambda}$
which can be related to the cosmological constant or dark energy density in a
flat universe can be expressed by
$\rho_{\Lambda}=3c^{2}M_{PL}^{2}L^{-2},$ (15)
where $L$ is the infrared cutoff, and $c$ is a constant parameter. From this
relation, we can write $L=c/(\sqrt{\Omega_{\Lambda 0}}H_{0})$, where
$\Omega_{\Lambda 0}$ is the present value of the dark energy ratio to the
critical density
$\Omega_{\Lambda}=\rho_{\Lambda}/\rho_{cr}=\rho_{\Lambda}/(3M_{PL}^{2}H^{2})$.
Consequently this introduces the cutoff at the wave number Shen:2005
$k_{c}=\frac{\pi}{c}\sqrt{\Omega_{\Lambda 0}}H_{0}.$ (16)
This holographic cutoff can help to remove the divergency brought by $k=0$.
Now we move on to examine the thermal effect on the CMB at small $l$ by
comparing with the WMAP data. The CMB angular spectrum is expressed as
Giovannini:2005
$C_{l}=\frac{4\pi}{9}\int_{k_{c}}^{\infty}\frac{dk}{k}j_{l}^{2}(k\Delta\tau)P_{k},$
(17)
where $j_{l}$ is the spherical Bessel function and $\Delta\tau$ is the
comoving distance to the last scattering surface,
$\Delta\tau=\int_{0}^{z_{0}}\frac{dz}{H(z)},$ (18)
and $z_{0}$ is the redshift of decoupling usually taken as $z_{0}=1100$. We
choose the Hubble parameter
$H(z)=H_{0}\sqrt{(1-\Omega_{\Lambda 0})(1+z)^{3}+\Omega_{\Lambda
0}(1+z)^{3(1+w)}},$ (19)
where $w=p/\rho$ is the equation of state of dark energy and is assumed as a
constant for simplicity. In our numerical calculation, we will set
$\Omega_{\Lambda 0}=0.721$ and $w=-0.972$ Hinshaw:2008 . For the zero
temperature case, it has been shown that the relation between ultraviolet and
infrared cutoff can help to explain the small $l$ CMB suppression Shen:2005 ,
which is exhibited in the dashed curve in Fig.2. However, compared with the
WMAP data, it cannot explain well the wriggle observed at $l=3,4,6$.
Notice that the thermal effect may play the role at large angle to modify the
power spectrum, we now turn to the temperature dependent spectrum in Eq.(14).
To do the data fitting with WMAP result at small $l$, we have parameters
$\tilde{T}$, $c$, $\varepsilon$ and $\xi$ or $\gamma$ now. We observed that in
the reasonable range of slow-roll condition, values of $\varepsilon$ and
$\gamma$ do not significantly change the total power spectrum, thus we fix
them to be $\varepsilon=0.03$ and $\gamma=0.003$ to ensure observable trans-
Planckian effect to be found in CMB Bergstrom:2002 . Now we have two
parameters $c$ and $\tilde{T}$ to be determined. Calculating the angular power
spectrum $l(l+1)C_{l}$ and comparing with WMAP data by doing $\chi^{2}$
fitting for $0<l\leq 20$,
$\chi^{2}=\sum_{i}\frac{[l(l+1)C_{l}^{i}|_{theory}-l(l+1)C_{l}^{i}|_{data}]^{2}}{(\sigma^{i})^{2}}$
(20)
where $\sigma^{i}$ is the observational error of each data, we present the
best fitting result in solid line in Fig.2, where $c=2.5$ and
$\tilde{T}=0.9H_{0}\approx 1.3\times 10^{-33}$ eV $\approx 1.6\times 10^{-29}$
K. This result keeps almost the same when we shift $\varepsilon$ and $\gamma$
in the reasonable range of the slow-roll inflation condition. It can be seen
from Fig.2 that the thermal effect can well explain the WMAP data at small
$l$, especially the wriggle at $l=3,4,6$.
In summary, we have employed the thermo field dynamics to investigate the
thermal vacuum effect on the power spectrum of the inflation. Comparing with
the spectrum of the zero temperature case, we have observed that the thermal
effect plays the role essentially in the low multipoles or large length scale.
Resorting to the idea of holographic cutoff, we have removed the divergence
problem when $k=0$. Comparing with the WMAP data at small $l$, we have found
that the thermal effect explains well the data at small $l$ than the zero
temperature case. When $l=3,4,6$, the spectrum got enhanced due to the
temperature effect. This suggests that the thermal effect should be considered
in studying the inflation, especially when we want to study the CMB anisotropy
at low multipoles. The temperature was very high at the beginning of the
inflation, so it actually influences the fluctuation spectrum which escaped
earlier from the horizon and reentered later so that the large scale (small
$l$) CMB spectrum got corrected. When the inflation started, the temperature
dropped fast, and the temperature effect has less influence on the fluctuation
spectrum. These fluctuations left the horizon later and reentered earlier
which affect the small scale CMB spectrum. Since the temperature effect is
low, most CMB spectrum ($l>10$) has little difference compared with the zero
temperature case. This actually gives the reason why we see the temperature
correction is important in the small $l$ CMB spectrum, which is consistent
with the observation, while for big $l$, the temperature effect is negligible.
## Acknowledgments
This work was partially supported by the NNSF of China, Shanghai Education
Commission, Shanghai Science and Technology Commission. We would like to
acknowledge helpful discussions with R.G.Cai and Y.G.Gong. S. Yin’s work was
also partially supported by the graduate renovation foundation of Fudan
university.
## References
* (1) A. H. Guth, Phys. Rev. D 23, 347 (1981).
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* (24) K. Bhattacharya, S. Mohanty, and R. Rangarajan, Phys. Rev. Lett. 96, 121302 (2006); ibid. 97, 231301 (2006).
* (25) H. Umezawa, H. Matsumoto, and M. Tachiki. Thermo field dynamics and condensed states (North-Holland, Amsterdam, 1982).
* (26) L. Bergström and U. F. Danielsson, J. High Eenergy Phys. 12 (2002) 038.
* (27) G. Hinshaw, et al., arXiv:0803.0732.
* (28) K. Enqvist, M. S. Sloth, Phys. Lett. 93 (2004) 221302; K. Enqvist, S. Hannestad, and M. S. Sloth, astro-ph/0409275.
* (29) J. Shen, B. Wang, E. Abdalla, and R.-K. Su, Phys. Lett. B609 (2005) 200; Z.Y. Huang, B. Wang, E. Abdalla, R.K. Su, JCAP 0605 (2006) 013.
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|
arxiv-papers
| 2008-10-20T05:59:21 |
2024-09-04T02:48:58.331670
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shaoyu Yin, Bin Wang, Ru-Keng Su",
"submitter": "Shaoyu Yin",
"url": "https://arxiv.org/abs/0810.3465"
}
|
0810.3511
|
# Spin polarized liquid ${}^{3}{\rm He}$
G.H. Bordbar111Corresponding author 222E-Mail: bordbar@physics.susc.ac.ir,
S.M. Zebarjad, M.R. Vahdani and M. Bigdeli Department of Physics, Shiraz
University, Shiraz 71454, Iran
###### Abstract
We have employed the constrained variational method to study the influence of
spin polarization on the ground state properties of liquid ${}^{3}{\rm He}$.
The spin polarized phase, we have found, has stronger correlation with respect
to the unpolarized phase. It is shown that the internal energy of liquid
${}^{3}{\rm He}$ increases by increasing polarization with no crossing point
between polarized and unpolarized energy curves over the liquid density range.
The obtained internal energy curves show a bound state, even in the case of
fully spin polarized matter. We have also investigated the validity of using a
parabolic formula for calculating the energy of spin polarized liquid
${}^{3}{\rm He}$. Finally, we have compared our results with other
calculations.
## I Introduction
The spin polarized liquid ${}^{3}{\rm He}$ is an interesting quantum many-body
system which can be experimentally examined. In fact, it is expected that this
phase of liquid ${}^{3}{\rm He}$ has a large life time to be observed in a
quasi-thermodynamic equilibrium 1 ; 1p ; 1p1 ; 1p2 ; 2 ; 3 .
Some theoretical investigations have been done for spin polarized liquid
${}^{3}{\rm He}$ using different approaches such as Green’s function Monte
Carlo (GFMC), Fermi hyper-netted chain (FHNC), correlated basis functions
(CBF) and transport theory 4 ; 4p ; 5 ; 6 ; 6p ; 7 ; 8 . Recently, we have
studied the unpolarized liquid ${}^{3}{\rm He}$ and calculated some of its
thermodynamic properties at finite temperature. A good agreement between our
results and corresponding empirical values has been shown 9 ; 10 . In these
calculations, constrained variational method based on the cluster expansion of
the energy functional has been used. This method is a powerful microscopic
technique used in many-body calculations of dense matter 11 ; 12 ; 13 ; 14 ;
15 ; 16 ; 17 . In this article, we employ this method to investigate the
ground state of spin polarized liquid ${}^{3}{\rm He}$.
## II Constrained Variational Calculation of Spin Polarized Liquid
${}^{3}{\rm He}$
To calculate the ground state energy of spin polarized liquid ${}^{3}{\rm
He}$, consisting of $N$ interacting atoms with $N^{(+)}$ spin up and $N^{(-)}$
spin down atoms, $N=N^{(+)}+N^{(-)}$, we use the variational calculation based
on the cluster expansion of the energy functional 18 . We consider up to the
two-body energy term in the cluster expansion,
$E=E_{1}+E_{2}.$ (1)
The one-body energy per particle for the spin polarized matter is given by
$\displaystyle E_{1}$ $\displaystyle=$ $\displaystyle E_{1}^{(+)}+E_{1}^{(-)}$
(2) $\displaystyle=$
$\displaystyle\frac{3}{10}\bigg{(}\frac{\hbar^{2}}{2m}\bigg{)}(3\pi^{2})^{2/3}\bigg{[}(1+\xi)^{5/3}+(1-\xi)^{5/3}\bigg{]}\rho^{2/3},$
where $\rho$ is the total number density,
$\rho=\rho^{(+)}+\rho^{(-)},$ (3)
and the spin asymmetry parameter, $\xi$, is defined to be
$\xi=\frac{N^{(+)}-N^{(-)}}{N}.$ (4)
$\xi$ shows the spin ordering of the matter which can get a value in the range
of $\xi=0.0$ (unpolarized matter) to $\xi=1.0$ (fully polarized matter).
To obtain the two-body energy term, $E_{2}$, we can start from the known
equation
$E_{2}=\frac{1}{2}\sum_{ij}<ij|W(12)|ij>_{a},$ (5)
where
$W(12)=\frac{\hbar^{2}}{m}\bigg{(}\nabla f(12)\bigg{)}^{2}+f^{2}(12)V(12),$
(6)
and $V(12)$ is the two-body potential between the helium atoms (we use the
Lennard-Jones potential with $\epsilon=10.22~{}K$ and $\sigma=2.556~{}A$) and
$f(12)$ is two-body correlation function. By considering the $|i>$ as a plane
wave and performing some algebra, we have derived the following relation for
the spin polarized matter,
$E_{2}=\frac{1}{2}\rho^{2}\int dr_{1}\int
dr_{2}\bigg{\\{}1-\frac{1}{4}\bigg{[}(1+\xi)^{2}\ell^{2}(k_{F}^{(+)}r_{12})+(1-\xi)^{2}\ell^{2}(k_{F}^{(-)}r_{12})\bigg{]}\bigg{\\}}W(r_{12}),$
(7)
where
$\ell(x^{(i)})=\frac{3}{(x^{(i)})^{3}}\bigg{[}\sin(x^{(i)})-x^{(i)}\cos(x^{(i)})\bigg{]},$
(8)
and $x^{(i)}$ is $k_{F}^{(+)}r_{12}$ or $k_{F}^{(-)}r_{12}$.
$k_{F}^{(+)}=(6\pi^{2}\rho^{(+)})^{1/3}$ and
$k_{F}^{(-)}=(6\pi^{2}\rho^{(-)})^{1/3}$ are the Fermi momentum of spin up and
spin down states, respectively. By minimizing the Eq. (7) with respect to the
$f(12)$, the following differential equation is obtained:
$\frac{d}{dr}\bigg{[}L(r)f^{\prime}(r)\bigg{]}-\frac{m}{\hbar^{2}}\bigg{[}V(r)+\lambda\bigg{]}L(r)f(r)=0,$
(9)
where
$L(r)=1-\frac{1}{4}\bigg{[}(1+\xi)^{2}\ell^{2}(k_{F}^{(+)}r_{12})+(1-\xi)^{2}\ell^{2}(k_{F}^{(-)}r_{12})\bigg{]},$
(10)
and $\lambda$ is the lagrange multiplier which imposes the constraint of the
two-body wave function normalization. Eq. (9) can be solved numerically and
the correlation function $f(12)$ and finally the internal energy of system are
obtained.
## III Results
The numerical results for the two-body correlation function of liquid
${\rm{}^{3}He}$ are given in Fig. 1. This shows that for the higher value of
spin asymmetry parameter, the correlation function increases more rapidly and
reaches to the limiting value ($f(r)=1$) at smaller value of $r$. From Fig. 1,
we can see that in the case of higher polarization, the helium atoms have a
stronger correlation at short relative distance.
The numerical values of kinetic, potential and internal energy of liquid
${\rm{}^{3}He}$ for the different polarizations versus total number density
are presented in Tables 1-4. Our results of internal energy for different spin
asymmetry parameters are compared in Fig. 2. This figure indicates that as
polarization increases, the internal energy gets the higher values over the
liquid density range and there is no crossing point between the internal
energy curves of polarized and unpolarized cases in this region. This behavior
is in agreement with experiment 6 ; 7 ; 8 . A comparison between Tables 1, 2,
3 and 4 shows that for each density, the kinetic energy (potential energy)
increases (decreases) by increasing spin asymmetry parameter. Over the liquid
density range, the increasing of kinetic energy dominates which leads to the
higher internal energies. From Fig. 2, it can be also seen that for all values
of $\xi$, the energy curve has a minimum which shows the existence of a bound
state for this system, even for the fully polarized matter ($\xi=1.0$). We see
that by increasing the spin asymmetry parameter, this minimum point of energy
curve shifts to the higher densities.
To compare our method with the well-known many-body techniques, we present the
results of Green’s function Monte Carlo (GFMC) and Fermi hyper-netted chain
(FHNC) calculations in Figs. 3-6. In Fig. 3, the internal energy of fully
polarized and unpolarized liquid ${\rm{}^{3}He}$ calculated with GFMC method 7
; 8 have been compared with our results. There are two points that we can
mention from this figure. First, the crossing point problem exist and
secondly, for all densities approximately grater than $0.014A^{-3}$, the
energy of polarized case is lower than unpolarized case which are not
acceptable from the experiment. However, these problems do not exist in our
method, although we should include the three-body correlation effect to obtain
a better result. In Figs. 4-6, the results of FHNC method for different
choices of wave function 6 are presented. We can see from these figures that
before considering the backflow effect (momentum dependent two-body
correlation), the polarized curve is always lower than the unpolarized curve.
However after performing extra calculations, the appropriate results have been
obtained in Fig. 6. This shows that our constrained variational method can do
the job much simpler, although we still need to add the three-body cluster
energy to obtain a better result 17 ; 19 .
There is a similarity between spin asymmetry parameter in our calculations and
isospin asymmetry parameter in nuclear matter calculations in which the energy
of asymmetrical nuclear matter can be calculated using the parabolic
approximation. In this approximation, one considers only the quadratic term in
asymmetry parameter as well as the energy of symmetric matter 16 . In a
similar way, we can define the following relation for the internal energy of
spin polarized liquid $\rm{{}^{3}He}$:
$E(\rho,~{}\xi)=E(\rho,~{}\xi=0.0)+a_{asym.}(\rho)~{}\xi^{2},$ (11)
where the above equation gives the definition of spin asymmetry energy
$a_{asym.}$ as:
$a_{asym.}(\rho)=E(\rho,~{}\xi=1.0)-E(\rho,~{}\xi=0.0).$ (12)
$E(\rho,~{}\xi=0.0)$ is the internal energy of unpolarized matter which has a
symmetric configuration in spin state and $E(\rho,~{}\xi=1.0)$ is the internal
energy of fully polarized matter. It is now interesting to see if this
approximation, Eq. (11), agrees with our microscopic calculations. For this
purpose, we have compared the internal energy of spin polarized liquid
${}^{3}{\rm He}$ at different $\xi$ for both microscopic calculation and using
parabolic approximation in Tables 5 and 6. These tables indicate an agreement
between these two approaches, specially at high densities.
## IV Summary and Conclusion
We have considered a system consisting of $N$ Helium atoms (${}^{3}{\rm He}$)
with an asymmetrical spin configuration and derived the two-body term in the
the cluster expansion of the energy functional. Then, we have minimized the
two-body energy term under the normalization constraint and obtained the
differential equation. The numerical results of internal energy of this system
have been presented for different values of spin asymmetry parameter and
density. It is found that as the polarization of liquid ${}^{3}{\rm He}$
increases, the two-body correlation becomes stronger. Over the liquid density
range, our results show that the internal energy of liquid ${}^{3}{\rm He}$
increases by increasing spin asymmetry parameter with no crossing point
between polarized and unpolarized energy curves. This shows an agreement with
experimental results. It is also seen that there is a bound state for all
values of polarization. The validity of using parabolic approximation in
calculating the energy of spin polarized matter is shown. Therefore, by using
this approximation one can preform calculations for the spin polarized matter
much simpler. we have also compared our results with other calculations to
show that why of our constrained variational method is a powerful technique.
###### Acknowledgements.
Financial support from Shiraz University research council is gratefully
acknowledged.
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Table 1: Kinetic energy (KE), potential energy (PE) and internal energy of unpolarized liquid ${}^{3}{\rm He}$ ($\xi=0.0$) versus total number density. Density (${\rm A^{-3}}$) | KE (K) | PE (K) | Internal Energy (K)
---|---|---|---
0.003 | 0.960 | -0.832 | 0.128
0.005 | 1.349 | -1.578 | -0.228
0.007 | 1.689 | -2.341 | -0.651
0.009 | 1.997 | -3.176 | -1.178
0.011 | 2.283 | -3.864 | -1.581
0.013 | 2.552 | -4.049 | -1.496
0.015 | 2.807 | -3.386 | -0.578
0.017 | 3.052 | -1.422 | 1.629
0.019 | 3.287 | 2.321 | 5.608
Table 2: As Table 1, but for spin polarized liquid ${}^{3}{\rm He}$ at $\xi=1/3$. Density (${\rm A^{-3}}$) | KE (K) | PE (K) | Internal energy (K)
---|---|---|---
0.003 | 1.019 | -0.922 | 0.096
0.005 | 1.433 | -1.701 | -0.268
0.007 | 1.794 | -2.482 | -0.688
0.009 | 2.121 | -3.256 | -1.135
0.011 | 2.424 | -3.957 | -1.532
0.013 | 2.710 | -4.154 | -1.444
0.015 | 2.981 | -3.505 | -0.523
0.017 | 3.241 | -1.555 | 1.685
0.019 | 3.491 | 2.176 | 5.667
Table 3: As Table 1, but for spin polarized liquid ${}^{3}{\rm He}$ at $\xi=2/3$. Density (${\rm A^{-3}}$) | KE (K) | PE (K) | Internal energy (K)
---|---|---|---
0.003 | 1.202 | -1.133 | 0.069
0.005 | 1.689 | -1.968 | -0.279
0.007 | 2.114 | -2.766 | -0.652
0.009 | 2.499 | -3.518 | -1.018
0.011 | 2.857 | -4.240 | -1.382
0.013 | 3.194 | -4.477 | -1.283
0.015 | 3.514 | -3.863 | -0.349
0.017 | 3.819 | -1.945 | 1.874
0.019 | 4.114 | 1.754 | 5.868
Table 4: As Table 1, but for fully polarized liquid ${}^{3}{\rm He}$ ($\xi=1.0$). Density (${\rm A^{-3}}$) | KE (K) | PE (K) | Internal energy (K)
---|---|---|---
0.003 | 1.524 | -1.403 | 0.120
0.005 | 2.142 | -2.227 | -0.085
0.007 | 2.681 | -2.967 | -0.286
0.009 | 3.170 | -3.648 | -0.478
0.011 | 3.624 | -4.268 | -0.644
0.013 | 4.051 | -4.850 | -0.799
0.015 | 4.457 | -4.478 | -0.021
0.017 | 4.845 | -2.615 | 2.229
0.019 | 5.218 | 1.033 | 6.251
Table 5: Internal energy (K) of spin polarized liquid ${}^{3}{\rm He}$ versus total number density at $\xi=1/3$ computed with both microscopic calculation and using parabolic approximation. Density (${\rm A^{-3}}$) | Microscopic Calculation | Parabolic Approximation
---|---|---
0.003 | 0.096 | 0.127
0.005 | -0.268 | -0.212
0.007 | -0.688 | -0.611
0.009 | -1.135 | -1.100
0.011 | -1.532 | -1.477
0.013 | -1.444 | -1.420
0.015 | -0.523 | -0.517
0.017 | 1.685 | 1.695
0.019 | 5.667 | 5.697
Table 6: As Table 5, but for spin polarized liquid ${}^{3}{\rm He}$ at $\xi=2/3$. Density (${\rm A^{-3}}$) | Microscopic Calculation | Parabolic Approximation
---|---|---
0.003 | 0.069 | 0.124
0.005 | -0.279 | -0.164
0.007 | -0.652 | -0.488
0.009 | -1.018 | -0.867
0.011 | -1.382 | -1.164
0.013 | -1.283 | -1.186
0.015 | -0.349 | -0.331
0.017 | 1.874 | 1.895
0.019 | 5.868 | 5.891
Figure 1: Two-body correlation function of liquid ${\rm{}^{3}He}$, $f(r)$, as
a function of interatomic distance ($r$) at different values of spin asymmetry
parameter $\xi=$ 0.0, 1/3, 2/3 and 1.0 for $\rho=0.01~{}\rm{A^{-3}}$.
Figure 2: Internal energy of liquid ${\rm{}^{3}He}$ versus total number
density at $\xi=$ 0.0 (full curve), 1/3 (dashed curve), 2/3 (dotted curve) and
1.0 (heavy dotted curve).
Figure 3: Comparison of our results for the unpolarized (UnPol) and fully
polarized (Pol) cases with the results of GFMC for different choices of the
wave function. J refers to Jastrow, and J+T refers to Jastrow plus three-body
wave functions 7 .
Figure 4: Comparison of our results for the unpolarized (UnPol) and fully
polarized (Pol) cases with the results of FHNC 6 . J refers to Jastrow wave
function.
Figure 5: Comparison of our results for the unpolarized (UnPol) and fully
polarized (Pol) cases with the results of FHNC 6 . J+T refers to Jastrow plus
three-body wave function.
Figure 6: Comparison of our results for the unpolarized (UnPol) and fully
polarized (Pol) cases with the results of FHNC 6 . J+T+B refers to Jastrow
plus three-body plus backflow wave function.
|
arxiv-papers
| 2008-10-20T10:10:17 |
2024-09-04T02:48:58.337599
|
{
"license": "Public Domain",
"authors": "G.H. Bordbar, S.M. Zebarjad, M.R. Vahdani and M. Bigdeli",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/0810.3511"
}
|
0810.3544
|
# Critical Behavior of Liquid ${}^{3}He$
G.H. Bordbar, S.M. Zebarjad and F. Shojaei Department of Physics, Shiraz
University, Shiraz 71454, Iran
and
Institute for Studies in Theoretical Physics and Mathematics (IPM),
Tehran, P. O. Box 19395-5531, Iran
###### Abstract
We investigate the liquid-gas second-order phase transition in liquid
${}^{3}He$ using the variational calculations based on the cluster expansion
of the energy functional. We also compute the critical point exponents of
liquid ${}^{3}He$ which are in agreement with experimental data.
## I Introduction
The liquids ${}^{3}He$ and ${}^{4}He$ are the only quantum liquids which exist
naturally. The word “quantum liquid” comes from the fact that for these
systems, the interatomic distance is at the order of their De Broglie
wavelenght. ${}^{3}He$ is a liquid of strongly interacting fermionic atoms
which behaves quite differently than the normal liquids at low temperature
(Wilks, 1970; Kent, 1993). The properties of liquid ${}^{3}He$ have been
studied using different many-body techniques (Clark and Westhaus, 1966; Nafari
and Doroudi, 1995; Luijen and Meyer, 2000; Kindermann and Wetterich, 2001;
Takano and Yamada, 1994; Viviani et al., 1988; Pricaupenko and Treiner, 1995;
Fantoni et al., 1982; Krotscheck and Smith, 1983; Friman and Krotscheck,
1982). Recently, the behavior of liquid ${}^{3}He$ near its critical point has
been investigated using path-integral molecular dynamics and quantum virial
expansion (Müer and Luijten, 2002).
One of the most powerful techniques in many-body calculations is the
variational method which is based on the cluster expansion of the energy
functional (lowest order constrained variational method) (Owen et al., 1977;
Bordbar and Modarres, 1997, 1998; Modarres and Bordbar, 1998; Bordbar and
Riazi, 2001, 2002; Bordbar, 2002a, 2002b, 2003, 2004; Bordbar and Hashemi,
2002). This is a fully self-consistent method and does not introduce any free
parameter to the calculations. The crucial point in this method is the
functional minimization with respect to the two-body correlation function
subjected to the normalization constraint which finally leads to a Euler-
Lagrange differential equation. The convergence of its results has been shown
by computing the three-body cluster energy term (Bordbar and Modarres, 1997).
The liquid-gas phase transition near the critical point (second-order phase
transition) is an interesting subject in statistical mechanics. This behavior,
critical phenomena, is caused by the existence of singularity in thermodynamic
functions of the system at the transition point. The nature of these
singularities in various measurable quantities at the critical point is
described by the critical exponents.
In our previous paper, we have calculated some thermodynamic properties of
liquid ${}^{3}He$ using the variational method shows a nice agreement between
experimental data and calculated results especially for free energy and
entropy (Bordbar and Hashemi, 2002). In this article, we present the critical
behavior of the liquid ${}^{3}He$. We organize the paper as follows: In
section II, we obtain the critical properties of liquid ${}^{3}He$ by
calculating the critical isotherm. We investigate the critical behavior of
liquid ${}^{3}He$ by computing the critical point exponents in section III.
## II Critical Isothermal Equation of State
The equation of state is the key point for investigating the second-order
phase transition in a hydrostatic system. The isothermal equation of state can
be calculated from the Helmholtz free energy, $F$:
$P=\rho^{2}\frac{\partial F}{\partial\rho}|_{T},$ (1)
where $P$, $T$ and $\rho=\frac{N}{V}$ are the pressure, temperature and number
density, respectively. $N$ and $V$ are the total number of particles and
volume. To obtain the free energy of liquid ${}^{3}He$, we use the variational
method explained in Appendix. The result for the equation of state at the
critical temperature (critical isotherm) is shown in Fig. (1).
Figure 1: The critical equation of state for liquid ${}^{3}He$.
As seen from the Fig. (1), at the critical point, the isotherm curve shows an
inflection point which satisfies:
$\frac{\partial
P}{\partial\rho}|_{T_{c}}=\frac{\partial^{2}P}{\partial\rho^{2}}|_{T_{c}}=0,$
(2)
where $T_{c}$ is the critical temperature. The calculated critical
temperature, density ($\rho_{c}$) and pressure ($P_{c}$) of liquid ${}^{3}He$
are presented in Table 1. The experimental results (Heller, 1967; Fisher,
1967; Pittman et al., 1979) are also given for comparison. We can see a good
agreement between these results.
Table 1: Critical point properties of liquid ${}^{3}He$. | $T_{c}(K)$ | $\rho_{c}(A^{-3})$ | $P_{c}(KA^{-3})$
---|---|---|---
Our results | 4.36 | 0.0054 | 0.0139
Exp. results (Heller, 1967) | 3.324 | 0.00834 | 0.00844
Exp. results (Pittman et al., 1979) | 3.317 | 0.00827 | 0.00846
## III Critical Exponents
For a hydrostatic system, the two-phase coexistence conditions are
$\displaystyle P_{liquid}$ $\displaystyle=$ $\displaystyle P_{gas}$
$\displaystyle\mu_{liquid}$ $\displaystyle=$ $\displaystyle\mu_{gas},$ (3)
where the $\mu_{liquid}$ and $\mu_{gas}$ are the chemical potential of liquid
and gas phases respectively. As the temperature increases, the liquid density
decreases and the gas density increases. At the critical temperature these
densities become equal to each other. This behavior for ${}^{3}He$ is shown in
Fig. (2).
Figure 2: The liquid and gas densities versus temperature for ${}^{3}He$.
The order parameter $\rho_{liquid}-\rho_{gas}$ which is defined to investigate
the critical behavior of this system vanishes at the critical point. However
other thermodynamic properties diverge at this point. The critical point
exponents are defined to study the asymptotic behavior of singular
thermodynamic functions near the critical point. For this purpose, the
following functions for the thermodynamic quantities are introduced (Garrod,
1995):
* •
Order parameter
We can define the exponent $\beta$ for this parameter as follows:
$\rho_{liquid}-\rho_{gas}\sim(-\epsilon)^{\beta};\,\,\,\,\,\,\,\,\,\,\,\epsilon\longrightarrow
0^{-},$ (4)
where
$\epsilon=\frac{T-T_{c}}{T_{c}}.$ (5)
The critical exponent $\beta$ characterizes the behavior of the order
parameter and of course, the above function is meaningful only below the
critical point in the region where the order parameter is not zero. To obtain
$\beta$, we draw the order parameter as a function of $\epsilon$ on the log-
log scale in Fig. (3).
Figure 3: The order parameter versus $\epsilon$ on log-log scale for
${}^{3}He$.
The slope of this figure yields the value of $\beta=0.56239\pm 0.01386$.
* •
Pressure
By defining the exponent $\delta$, we can describe the critical isotherm
$P-P_{c}\sim(\rho-\rho_{c})^{\delta};\,\,\,\,\,\,\,\,\,\,\,\rho\longrightarrow\rho_{c},$
(6)
where $\epsilon=0$ ($T=T_{c}$). In Fig. (4), $P-P_{c}$ as a function of
$\rho-\rho_{c}$ is shown.
Figure 4: The $P-P_{c}$ versus $\rho-\rho_{c}$ at critical temperature
($T_{c}$) for ${}^{3}He$.
The value of $\delta$ obtained from this figure is $3.31032\pm 0.08192$.
* •
Heat Capacity
The exponent $\alpha^{\prime}$ and $\alpha$ characterize the behavior of
specific heat ($C_{V}$) below and above the critical temperature respectively
along the critical isochore ($V=V_{c}$)
$\displaystyle C_{V_{c}}$ $\displaystyle=$
$\displaystyle(-\epsilon)^{-\alpha^{\prime}};\,\,\,\,\,\,\,\,\,\,\,\epsilon\longrightarrow
0^{-},$ $\displaystyle C_{V_{c}}$ $\displaystyle=$
$\displaystyle(\epsilon)^{-\alpha};\,\,\,\,\,\,\,\,\,\,\,\epsilon\longrightarrow
0^{+}.$ (7)
In Fig. (5), the specific heat along the critical isochore versus $\epsilon$
is shown.
Figure 5: Specific heat along the critical isochore as a function of
$\epsilon$ above (full curve) and below (dashed curve) critical temperature
for ${}^{3}He$.
The values $\alpha^{\prime}=0.1018\pm 0.0001$ and $\alpha=0.10609\pm 0.0014$
are extracted from the Fig. (5).
* •
Isothermal Compressibility
For describing the behavior of isothermal compressibility ($K$) near the
critical point, the exponents $\gamma$ and $\gamma^{\prime}$ are defined to be
$\displaystyle K$ $\displaystyle=$
$\displaystyle(-\epsilon)^{-\gamma^{\prime}};\,\,\,\,\,\,\,\,\,\,\,\epsilon\longrightarrow
0^{-},$ $\displaystyle K$ $\displaystyle=$
$\displaystyle(\epsilon)^{-\gamma};\,\,\,\,\,\,\,\,\,\,\,\epsilon\longrightarrow
0^{+}.$ (8)
The calculated values of isothermal compressibility shown in Fig. (6) leads to
$\gamma=1.05343\pm 0.01077$ and $\gamma^{\prime}=1.05343\pm 0.01077$.
Figure 6: Isothermal compressibility as a function of $\epsilon$ above (full
curve) and below (dashed curve) critical temperature for ${}^{3}He$. $K_{I}$
is the ideal fermi gas compressibility at $\rho=\rho_{c}$ and $T=T_{c}$.
We have presented the whole critical exponents for the ${}^{3}He$ in Table 2.
Table 2: Critical exponents for ${}^{3}He$. | $\beta$ | $\delta$ | $\alpha^{\prime}$ | $\alpha$ | $\gamma$ | $\gamma^{\prime}$
---|---|---|---|---|---|---
Our results | $0.5624$ | $3.3103$ | $0.1018$ | $0.1061$ | $1.0534$ | $1.0560$
| $\pm 0.0139$ | $\pm 0.0819$ | $\pm 0.0001$ | $\pm 0.0014$ | $\pm 0.0108$ | $\pm 0.0093$
Exp. results (Heller, 1967) | $\sim 0.361$ | $\sim 4.21$ | $\sim 0.105$ | $\sim 0.105$ | $\sim 1.17$ | $\sim 1.17$
Exp. results (Pittman et al., 1979) | $0.322\pm 0.002$ | — | — | — | $1.19\pm 0.01$ | —
The experimental results (Heller, 1967; Fisher, 1967; Pittman et al., 1979)
are also given for the comparison in Table 2. There is a good agreement
between our calculations for the critical exponents and the experimental
results. From Table 2, it can be seen that the Griffiths and Rushbrooke
inequalities (Huang, 1987; Griffiths, 1965) are satisfied by our results for
the critical exponents of ${}^{3}He$,
$\displaystyle\alpha+2\beta+\gamma$ $\displaystyle\geq$ $\displaystyle 2$
$\displaystyle\alpha+\beta(1+\delta)$ $\displaystyle\geq$ $\displaystyle 2.$
(9)
## IV Summary and Conclusion
The liquid-gas phase transition near the critical point is of special interest
in statistical mechanics. In this work, we have computed the critical equation
of state for liquid ${}^{3}He$ which led to critical density, temperature and
pressure of this system. The critical exponents, $\beta$, $\delta$, $\alpha$
and $\gamma$ for this system are computed. The calculated critical exponents
satisfies the Griffiths and Rushbrooke inequalities. A comparison between our
results and experimental data is made which shows a good agreement between
theoretical calculation and experimental results.
###### Acknowledgements.
Financial support from Shiraz University research council and IPM is
gratefully acknowledged.
## Appendix
In this appendix, we give a brief review to obtain the free energy of liquid
${}^{3}He$ using the lowest order constrained variational method based on the
cluster expansion of the energy functional (Owen et al., 1977; Bordbar and
Modarres, 1997, 1998; Modarres and Bordbar, 1998; Bordbar and Riazi, 2001,
2002; Bordbar, 2002a, 2002b, 2003, 2004; Bordbar and Hashemi, 2002). In this
method, We choose a trial many-body wavefunction as
$\Psi=\\{\prod_{i<j}f(ij)\\}\Phi,$ (10)
where $f(ij)$ is the two-body correlation function and $\Phi$ is the Slater
determinant of noninteracting particles wave-functions (plane waves). We then
apply the cluster expansion to the energy per particle (Clark, 1979) and keep
one and two-body energy terms,
$E=\frac{1}{N}\frac{\langle\Psi|H|\Psi\rangle}{\langle\Psi|\Psi\rangle}=E_{1}+E_{2},$
(11)
where
$\displaystyle E_{1}$ $\displaystyle=$
$\displaystyle\sum_{i}\frac{\hbar^{2}k_{i}^{2}}{2m}n(k_{i}),$ (12)
$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2N}\sum_{ij}<ij\mid w(12)\mid ij-ji>.$ (13)
In the above equations, $n(k_{i})$ is the Fermi-Dirac distribution function
and
$w(12)=\frac{\hbar^{2}}{m}\bigg{(}\nabla_{12}f(12)\bigg{)}^{2}+f^{2}(12)V(12),$
(14)
where $V(12)$ is the interatomic potential. In the thermodynamic limit, Eqs.
(12) and (13) read:
$\displaystyle E_{1}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2m\rho\pi^{2}}\int_{0}^{\infty}n(k)k^{4}dk,$
(15) $\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\frac{2\pi\rho\hbar^{2}}{m}\int_{0}^{\infty}\bigg{[}f^{\prime
2}(r)+\frac{m}{\hbar^{2}}f^{2}(r)V(r)\bigg{]}\bigg{[}1-\frac{1}{2}\bigg{(}\frac{\gamma(r)}{\rho}\bigg{)}^{2}\bigg{]}r^{2}dr,$
(16)
where $\rho$ is the number density,
$\rho=\frac{1}{\pi^{2}}\int_{0}^{\infty}n(k)k^{2}dk,$ (17)
$\gamma(r)=\frac{1}{\pi^{2}}\int_{0}^{\infty}\frac{\sin kr}{kr}n(k)k^{2}dk,$
(18)
and $f^{\prime}(r)=\frac{\partial f(r)}{\partial r}$.
At this point, we minimize the energy functional with respect to the two-body
correlation function, $f(r)$, to obtain the following Euler-Lagrange
differential equation
$\bigg{\\{}\frac{2m}{\hbar^{2}}f(r)V(r)+2\lambda
f(r)\bigg{\\}}\bigg{[}1-\frac{1}{2}\bigg{(}\frac{\gamma(r)}{\rho}\bigg{)}^{2}\bigg{]}-\frac{\partial}{\partial
r}\bigg{[}2f^{\prime}(r)\bigg{[}1-\frac{1}{2}\bigg{(}\frac{\gamma(r)}{\rho}\bigg{)}^{2}\bigg{]}\bigg{]}=0,$
(19)
where $\lambda$ is the Lagrange multiplier which imposed the normalization
condition $\langle\Psi|\Psi\rangle=1$. By solving Eq. (19), using the
numerical technique, the two-body correlation function, $f(r)$ and therefore
the energy of the system are obtained. This finally leads to the free energy
function of the system
$F=E-TS,$ (20)
where $T$ and $S$ are the temperature and entropy per particle of the systems
(Fetter and Walecka, 1971).
To calculate the free energy of liquid ${}^{3}He$, we use the Aziz interatomic
potential (Aziz et al., 1979) in Eqs. (16) and (19)
$V(r)=\epsilon\left\\{Ae^{-\alpha
r/r_{m}}-\left[C_{6}\left(\frac{r_{m}}{r}\right)^{6}+C_{8}\left(\frac{r_{m}}{r}\right)^{8}+C_{10}\left(\frac{r_{m}}{r}\right)^{10}\right]F(r)\right\\},$
(21)
where
$\displaystyle F(r)$ $\displaystyle=$
$\displaystyle\left\\{\begin{tabular}[]{lll}$e^{-(\frac{Dr_{m}}{r}-1)^{2}}$&;&$\frac{r}{r_{m}}\leq
D$\\\ 1&;&$\frac{r}{r_{m}}>D$,\end{tabular}\right.$ (24)
and
(29)
The realistic Aziz Potential agrees with the He-He scattering experimental
data which satisfies the following criteria:
* •
It has a short-range repulsive part which described by exponential form
* •
It has also a long range attractive tail includes the multipole interactions.
A realistic potential between Helium atoms must have the criteria. Our results
for the free energy calculations of the liquid ${}^{3}He$ are given in Fig.
(7) (Bordbar and Hashemi, 2002).
Figure 7: The free energy of liquid ${}^{3}He$ as a function of number density
at different temperatures.
## References
* (1) Aziz, R.A., et al. (1979). Journal of Chemical Physics 70 , 4330.
* (2) Bordbar, G. H. and Modarres, M. (1997). Journal of Physics G: Nuclear and Particle Physics 23, 1631.
* (3) Bordbar, G. H. and Modarres, M. (1998). Physical Review C 57, 714.
* (4) Bordbar, G. H. and Riazi, N. (2001). International Journal of Theoretical Physics 40, 1671.
* (5) Bordbar, G. H. and Riazi, N. (2002). Astrophysics and Space Science 282, 563.
* (6) Bordbar, G. H. (2002a). International Journal of Theoretical Physics 41, 309.
* (7) Bordbar, G. H. (2002b). International Journal of Theoretical Physics 41, 1135.
* (8) Bordbar, G. H. and Hashemi, M. (2002). International Journal of Theoretical Physics, Group Theory, and Nonlinear Optics 8, 251.
* (9) Bordbar, G. H. (2003). International Journal of Modern Physics A18, 2629.
* (10) Bordbar, G. H. (2004). International Journal of Theoretical Physics 43, in press.
* (11) Clark, J.W. and Westhaus, P. (1966). Physical Review 141, 833.
* (12) Clark, J.W. (1979). Progress in Particle and Nuclear Physics 2, 89.
* (13) Fantoni, S., Pandharipande, V. R. and Schmidt, K. E. (1982). Physical Review Letter 48, 878.
* (14) Fetter, A.L. and Walecka, J.D. (1971). _Quantum Theory of Many-Body Systems_ (McGraw-Hill, New York).
* (15) Fisher, M.E. (1967). Report Progress on Physics 39, 395.
* (16) Friman, B. L. and Krotscheck, E. (1982). Physical Review Letter 49, 1705.
* (17) Garrod, C. (1995). _Statistical Mechanics and Thermodynamics_ (Oxford University Press, USA).
* (18) Griffiths, R.B. (1965). Journal of Chemical Physics 43, 1958.
* (19) Heller, P. (1967). Report Progress on Physics 30, 731\.
* (20) Huang, K. (1987). _Statistical Mechanics_ (John wiley).
* (21) Kents, A. (1993). _Experimental Low Temperature Physics_ (Macmillan Physical Science Series).
* (22) Kindermann, M. and Wetterich, C. (2001). Physical Review Letter 86, 1034.
* (23) Krotscheck, E. and Smith, R. A. (1983). Physical Review B27, 4222.
* (24) Luijten, E. and Meyer, H. (2000). Physical Review E62, 3257\.
* (25) Modarres, M. and Bordbar, G. H. (1998). Physical Review C 58, 2781.
* (26) Müer, M.H. and Luijten, E. (2002). Journal of Chemical Physics 116, 1621.
* (27) Nafari, N. and Doroudi, A. (1995). Physical Review B51, 9019.
* (28) Owen, J.C., Bishop, R.F. and Irvine, J.M. (1977). Nuclear Physics A277, 45.
* (29) Pittman, C., Doiron, T. and Meyer, H. (1979). Physical Review B20, 3678.
* (30) Pricaupenko, L. and Treiner, J. (1995). Physical Review Letter 74, 430.
* (31) Takano, M. and Yamada, M. (1994). Progress in Theoretical Physics 91, 1149.
* (32) Viviani, M., Buendia, E., Fantoni, S. and Rosati, S. (1988). Physical Review B38, 4523.
* (33) Willks, J. (1970). _An Introduction to Liquid Helium_ (Clarendon Press, Oxford).
|
arxiv-papers
| 2008-10-20T12:49:29 |
2024-09-04T02:48:58.341468
|
{
"license": "Public Domain",
"authors": "G.H. Bordbar, S.M. Zebarjad and F. Shojaei",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/0810.3544"
}
|
0810.3552
|
# Lowest Order Constrained Variational calculation for Polarized Liquid
${}^{3}\mathrm{He}$ at Finite Temperature
G.H. Bordbar111Corresponding author 222E-Mail: bordbar@physics.susc.ac.ir,
M.J. Karimi and J. Vahedi Department of Physics, Shiraz University, Shiraz
71454, Iran
###### Abstract
We have investigated some of the thermodynamic properties of spin polarized
liquid ${}^{3}\mathrm{He}$ at finite temperature using the lowest order
constrained variational method. For this system, the free energy, entropy and
pressure are calculated for different values of the density, temperature and
polarization. We have also presented the dependence of specific heat,
saturation density and incompressibility on the temperature and polarization.
## I Introduction
The spin polarized liquid ${}^{3}\mathrm{He}$ at finite temperature is an
interesting many-body system which obeys the Fermi-Dirac statistics. Several
theoretical techniques and experimental method have been used for
investigating the properties of unpolarized liquid $\mathrm{{}^{3}He}$ Wi ; Ke
; Vbf ; Nd ; Lt ; Ty ; Er ; Sv . Recently, we have studied the unpolarized
liquid $\mathrm{{}^{3}He}$ at finite temperature and the spin polarized liquid
$\mathrm{{}^{3}He}$ at zero temperature Gh1 ; Gh2 ; Gh3 . In this
calculations, we have applied the lowest order constrained variational method
based on the cluster expansion of the energy functional. This method is a
powerful microscopic technique used in the many-body calculations of dense
matter Gh1 ; Gh2 ; Gh3 ; Gh4 ; Gh5 ; Gh6 ; Gh7 ; Gh8 ; Gh9 ; Gh10 .
In the present work, we use the lowest order constrained variational method to
investigate some of the thermodynamic properties of spin polarized liquid
$\mathrm{{}^{3}He}$ at finite temperature. In our calculations, we employ the
Lennard-Jones LJ and Aziz Aziz potentials.
## II Method
We consider a system consisting of $N$ interacting $\mathrm{{}^{3}He}$ atoms
with $N^{+}$ spin up and $N^{-}$ spin down atoms. These atoms have a different
chemical potential ($\mu$) and different Fermi-Dirac distribution function Fet
as follows,
$n^{\pm}(k)=\frac{1}{exp[\beta(\frac{\hbar^{2}k^{2}}{2m}-\mu^{\pm})]+1},$ (1)
where $\beta=\frac{1}{k_{B}T}$ and $T$ is the temperature. We define the total
number density by $\rho$ and the spin asymmetry parameter by $\xi$ ($\xi=0$ is
unpolarized and $\xi=1$ is fully polarized cases),
$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\rho^{+}+\rho^{-},$
$\displaystyle\xi$ $\displaystyle=$ $\displaystyle\frac{N^{+}-N^{-}}{N}.$ (2)
Thus, to determine the chemical potential for every $\rho$, $T$ and $\xi$, we
solve the following equation by numerical method,
$\rho^{\pm}=\frac{1}{2\pi^{2}}\int_{0}^{\infty}n^{\pm}(k)k^{2}dk.$ (3)
We calculate the energy using the lowest order constrained variational method
based on the cluster expansion of the energy. We consider up to the two-body
energy term in cluster expansion,
$E=E_{1}+E_{2}.$ (4)
For this system, the one-body energy per particle, $E_{1}$, is given by
$\displaystyle E_{1}$ $\displaystyle=$ $\displaystyle E_{1}^{+}+E_{1}^{-}$ (5)
$\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{4m\pi^{2}\rho}\left[\int_{0}^{\infty}n^{+}(k)k^{4}dk+\int_{0}^{\infty}n^{-}(k)k^{4}dk\right].$
The two-body energy per particle, $E_{2}$, is defined
$E_{2}=\frac{1}{2N}\sum_{i,j}\langle ij\mid W(12)\mid ij-ji\rangle,$ (6)
where
$W(r_{12})=\frac{\hbar^{2}}{m}(\nabla{f}(r_{12}))^{2}+f^{2}(r_{12})V(r_{12}).$
(7)
In the above equation, $V(r_{12})$ is the two body potential between helium
atoms and $f(r_{12})$ is the two-body correlation function. By considering the
$|i\rangle$ as a plane wave, we have derived the following relation for the
two-body energy per particle,
$E_{2}=\frac{1}{2}\rho\int
d\vec{r}\left[1-\frac{1}{4\pi^{4}\rho^{2}}\left[(\gamma^{+}(r))^{2}+(\gamma^{-}(r))^{2}\right]\right]W(r),$
(8)
where
$\gamma^{\pm}(r)=\int_{0}^{\infty}\frac{\sin(kr)}{kr}n^{\pm}(k)k^{2}dk.$ (9)
Now, we minimize the two-body energy Eq. (8) with respect to the variations in
the two-body correlation function subject to the normalization constraint
Clark ; Feen ,
$\frac{1}{N}\sum_{i,j}\langle ij|h^{2}(r_{12})-f^{2}(r_{12})|ij-ji\rangle=1.$
(10)
The normalization constraint is conveniently re-written in the integral form
as
$\rho\int d\vec{r}[h^{2}(r)-f^{2}(r)]\Gamma(r)=1,$ (11)
where
$\Gamma(r)=\left[1-\frac{1}{4\pi^{4}\rho^{2}}\left[(\gamma^{+}(r))^{2}+(\gamma^{-}(r))^{2}\right]\right].$
(12)
and $h(r)$ is the Pauli function,
$h(r)=[\Gamma(r)]^{-\frac{1}{2}}.$ (13)
The minimization of the two-body Energy $E_{2}$ gives the following Euler-
Lagrange differential equation for the two-body correlation function
$f^{{}^{\prime\prime}}(r)+f^{{}^{\prime}}(r)\frac{\Gamma^{{}^{\prime}}(r)}{\Gamma(r)}-\frac{m}{\hbar^{2}}\left(\lambda+V(r)\right)f(r)=0.$
(14)
The lagrange multiplier $\lambda$ imposes by normalization constraint. The
two-body correlation function calculated by numerically integrating Eq. (14)
and then the energy per particle of system can be obtained. Finally the
Helmholtz free energy per particle can be determined using the following
equation,
$F=E-TS.$ (15)
$S$ is the entropy per particle Fet which is given by
$S=S^{+}+S^{-},$ (16)
where
$S^{\pm}=-\frac{1}{2\pi^{2}\rho}\int_{0}^{\infty}\left[n^{\pm}(k)\ln(n^{\pm}(k))+(1-n^{\pm}(k)\ln(1-n^{\pm}(k))\right]k^{2}dk.$
(17)
By calculating $E$, $S$ and $F$, we can obtain the thermodynamic properties of
the spin polarized liquid $\mathrm{{}^{3}He}$ at finite temperature.
## III Results
The two-body correlation function at $T=1.0K$ and $T=4.0K$ for different
values of the spin asymmetry parameter ($\xi$) are shown in Fig. 1. This
figure shows that at low temperatures, the correlation function increases more
rapidly and reaches the limiting value ($f(r)=1$) at larger interatomic
distances ($r$). It is shown that by increasing $\xi$, the correlation
function reaches the limiting value at lower $r$. It is seen that our results
for the correlation function with the Lennard-Jones and Aziz potentials are
identical.
Our results for the free energy of liquid $\mathrm{{}^{3}He}$ with the
Lennard-Jones and Aziz potentials at different spin asymmetry parameters
($\xi$) are presented in Fig. 2. We can see that at low densities
($\rho<0.011A^{-3}$), the free energies with the Lennard-Jones and Aziz
potentials are similar. However, at higher densities, the difference between
these free energies are noticeable. Fig. 2 shows that the free energy
increases by increasing spin asymmetry parameter and this variation becomes
higher at high temperatures. We see that the free energy doesn’t show a bound
state above a certain value of temperature (called flashing temperature,
$T_{f}$ ). We have found $T_{f}$ for different values of $\xi$ which given in
Table 1. From Fig. 2, it is also seen that for each value of spin asymmetry
parameter, the density of saturation point (minimum point of the free energy)
decreases by increasing temperature. We see that the saturation density
($\rho_{0}$) increases by increasing spin asymmetry parameter for each value
of temperature. Our results for the saturation density are given in Table 2.
Table 1 (Table 2) shows that for all values of $\xi$ the flashing temperature
(the saturation density) with the Aziz potential is lower than that of the
Lennard-Jones potential.
The entropy of liquid $\mathrm{{}^{3}He}$ is shown as a function of density
for two different temperatures and various asymmetry parameters in Fig. 3. It
is seen that the entropy decreases by increasing both density and spin
asymmetry parameter and increases by increasing temperature. We have found
that for each temperature, the difference between entropy of fully polarized
and unpolarized liquid $\mathrm{{}^{3}He}$ decreases for higher values of
density, especially at low temperatures.
In Figs. 4-6, we have shown the free energy, entropy and the specific heat of
liquid $\mathrm{{}^{3}He}$ as a function of temperature for different values
of $\xi$ at $\rho=0.0166A^{-3}$. In these figures, the experimental results Wi
for the unpolarized liquid $\mathrm{{}^{3}He}$ are also shown for comparison.
Fig. 4 indicates a good agreement between our results for the free energy with
the Lennard-Jones potential and the experimental results. From Figs. 4-6, it
is seen that the free energy (entropy and specific heat) decreases (increase)
by increasing temperature. In addition, we see that for the free energy,
entropy and the specific heat of liquid $\mathrm{{}^{3}He}$, the differences
between fully polarized and unpolarized cases increase by increasing
temperature.
The isothermal pressure is obtained from the free energy using the following
equation,
$P(\rho,T)=\rho^{2}\frac{\partial{F(\rho,T)}}{\partial{\rho}}.$ (18)
In Fig. 7, we have plotted the isothermal pressure as a function of density
for the various temperatures and $\xi$. It is seen that the equation of state
with the Aziz potential is stiffer than that of the Lennard-Jones potential.
This figure shows that at low temperatures we have a liquid-gas phase
equilibrium for the liquid $\mathrm{{}^{3}He}$. We know that at the critical
temperature $(T_{c})$ there is no liquid-gas phase equilibrium. Our results of
the critical temperature for different values of $\xi$ are presented in Table
3. We see that the critical temperature increases by increasing $\xi$. For all
$\xi$ it is seen that the $(T_{c})$ with the Aziz potential is lower than that
of the Leonard-Jones potential.
For liquid $\mathrm{{}^{3}He}$, the saturation incompressibility at saturation
density is given by,
$K_{0}(T)=9\left(\rho^{2}\frac{\partial^{2}F(\rho,T)}{\partial{\rho^{2}}}\right)_{\rho_{0}}.$
(19)
In Table 2, our calculated values of the saturation incompressibility are
presented for different values of $\xi$ and $T$. It is seen that for each
value of $\xi$, the incompressibility decreases by increasing temperature and
at each temperature, it increases by increasing $\xi$. From Table 2 we see
that for all values of spin asymmetry parameter our results for the $K_{0}$
with the Aziz potential are lower than those of the Leonard-Jones potential.
## IV Summary and Conclusion
We have considered a system consisting of the Helium atoms
(${}^{3}\mathrm{He}$) with an asymmetrical spin configuration. For this
system, we have calculated some of the thermodynamic properties using the
lowest order constrained variational method with the Lennard-Jones and Aziz
potentials. Our calculations lead to the following conclusions for the liquid
${}^{3}\mathrm{He}$:
* •
At high temperatures, the correlation function reaches the limiting value for
the smaller values of the interatomic distances.
* •
The results of the correlation function with the Lennard-Jones and Aziz
potentials are very similar.
* •
The free energy with the the Lennard-Jones and Aziz potentials are identical
for small values of the density.
* •
The free energy, saturation density and incompressibility decrease by
increasing temperature.
* •
The free energy, saturation density, incompressibility and critical
temperature increase by increasing spin asymmetry parameter.
* •
The entropy, specific heat and flashing temperature decrease by increasing
spin asymmetry parameter.
* •
The entropy decreases by increasing density and increases by increasing
temperature.
* •
The equation of state with the Aziz potential is stiffer than that of the
Lennard-Jones potential.
* •
For all values of spin asymmetry parameter our results for the flashing
temperature, saturation density, critical temperature and saturation
incompressibility with the Aziz potential are lower than those of the Leonard-
Jones potential.
###### Acknowledgements.
Financial support from the Shiraz University research council is gratefully
acknowledged.
## References
* (1) J. Wilks, _The Properties of Liquid and Solid Helium_ , (Clarendon, Oxford, 1967).
* (2) A. Kenet, _Exprimental Low-Temperature Physics_ , (Macmillan Physical Science Series, 1993).
* (3) M. Viviani, E. Buendia, S. Fantoni and S. Rosati, _Phys. Rev._ B38, 4523 (1988)
* (4) N. Nafari and A. Doroudi, _Phys. Rev_. B51, 9019 (1995).
* (5) M. Takano and M. Yamada, _Prog. Theor. Phys_. 91, 1149 (1994).
* (6) L. Pricaupenko and J. Treiner, _Phys. Rev. Lett._ 74, 430 (1995).
* (7) E. Krotescheck and R. A. Smith, _Phys. Rev_. B27, 4222 (1983).
* (8) S. Fantoni, V. R. Pandharipande and K. E. Schmidt, _Phys. Rev. Lett._ 48, 878 (1982).
* (9) G. H. Bordbar and M. Hashemi, _Int. J. Theor. Phys.,Group Theory and Nonlinear Optics_ 8, 251 (2002).
* (10) G. H. Bordbar, S. M. Zebarjad and F. Shojaei, _Int. J. Theor. Phys._ 43, 1863 (2004).
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* (12) G. H. Bordbar, _Int. J. Theor. Phys._ 43, 399 (2004).
* (13) G. H. Bordbar, _Int. J. Mod. Phys._ A18, 2629 (2003).
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* (15) G. H. Bordbar, _Int. J. Theor. Phys._ 41, 1135 (2002).
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* (17) G. H. Bordbar and M. Modarres, _Phys. Rev_ C57, 714 (1998).
* (18) G. H. Bordbar and M. Modarres, _J. Phys. G: Nucl. Part. Phys._ 23, 1631 (1997).
* (19) J. de Boer and A. Michels, _physica_ 6, 409 (1939).
* (20) R. A. Aziz, F. R. W. McCourt and C. C. K. Wong, _Mol. Phys._ 61, 1487 (1987).
* (21) J. W. Clark, _Prog. Part. Nucl. Phys._ 2, 89 (1979).
* (22) E. Feenberg, _Theory of Quantum Fluids_ , ( Academic Press, New York 1969).
* (23) A. L. Fetter and J. D. Walecka, _Quantum Theoty of Many-Body System_ , (McGraw-Hill, New York, 1971).
Table 1: The flashing temperature ($T_{f}$) for different values of $\xi$.
Lennard- Jones | Aziz |
---|---|---
$\xi$ | $T_{f}(K)$ | $T_{f}(K)$
---|---|---
0.0 | 2.8 | 2.6
0.4 | 2.7 | 2.5
0.8 | 2.6 | 2.3
1.0 | 2.2 | 1.9
Table 2: The saturation density ($\rho_{0}$) and the saturation
incompressibility ($K_{0}$) of liquid $\mathrm{{}^{3}He}$ for different values
of $\xi$ and temperature.
Lenard- Jones | Aziz |
---|---|---
$\xi$ | $T(K)$ | $\rho_{0}(A^{-3})$ | $K_{0}(K)$ | $\rho_{0}(A^{-3})$ | $K_{0}(K)$
---|---|---|---|---|---
0.0 | 0.5 | 0.0117 | 185 | 0.011 | 174
0.0 | 1.0 | 0.0116 | 163 | 0.0109 | 157
0.0 | 1.5 | 0.0113 | 135 | 0.0106 | 121
0.4 | 0.5 | 0.0117 | 187 | 0.011 | 183
0.4 | 1.0 | 0.0116 | 172 | 0.0109 | 162
0.4 | 1.5 | 0.0114 | 145 | 0.0107 | 132
0.8 | 0.5 | 0.0120 | 237 | 0.0113 | 212
0.8 | 1.0 | 0.0120 | 232 | 0.0113 | 206
0.8 | 1.5 | 0.0119 | 203 | 0.0112 | 179
1.0 | 0.5 | 0.0128 | 275 | 0.0119 | 260
1.0 | 1.0 | 0.0128 | 263 | 0.0119 | 246
1.0 | 1.5 | 0.0127 | 212 | 0.0118 | 204
Table 3: The critical temperature ($T_{c}$) for different values of $\xi$.
Lennard- Jones | Aziz |
---|---|---
$\xi$ | $T_{c}(K)$ | $T_{c}(K)$
---|---|---
0.0 | 5.3 | 5.1
0.4 | 5.5 | 5.3
0.8 | 6.2 | 5.8
1.0 | 6.5 | 6.1
Figure 1: The correlation function versus the interatomic distance ($r$) with
the Leonard-Jones (full curves) and Aziz (dotted curves) potentials at
$\rho=0.0166A^{-3}$ for $\xi=0.0$ (a), $\xi=0.4$ (b), $\xi=0.8$ (c) and
$\xi=1.0$ (d).
Figure 2: The free energy of liquid ${}^{3}\mathrm{He}$ versus density at
different values of temperature ($T=1.0,2.0,3.0,4.0,5.0K$)with the Leonard-
Jones (full curves) and Aziz (dotted curves) potentials for $\xi=0.0$ (a),
$\xi=0.4$ (b), $\xi=0.8$ (c) and $\xi=1.0$ (d). Figure 3: The entropy of
liquid ${}^{3}\mathrm{He}$ versus density at $T=1.0K$ and $T=4.0K$ for
different values of $\xi$. Figure 4: The free energy of liquid
${}^{3}\mathrm{He}$ versus temperature for $\rho=0.0166A^{-3}$ at different
values of $\xi$ with the Leonard-Jones (full curves) and Aziz (dotted curves)
potentials. The experimental results (dashed curve) for unpolarized liquid
${}^{3}\mathrm{He}$ Wi are also shown for comparison. Figure 5: The entropy
of liquid ${}^{3}\mathrm{He}$ versus temperature for $\rho=0.0166A^{-3}$ at
different values of $\xi$. Figure 6: As Fig. 4, but for the specific heat.
Figure 7: The pressure versus density at different values of temperature
($T=1.0,3.0,5.0,6.0K$) with the Leonard-Jones (full curves) and Aziz (dotted
curves) potentials for $\xi=0.0$ (a), $\xi=0.4$ (b), $\xi=0.8$ (c) and
$\xi=1.0$ (d).
|
arxiv-papers
| 2008-10-20T13:39:08 |
2024-09-04T02:48:58.345322
|
{
"license": "Public Domain",
"authors": "G.H. Bordbar, M.J. Karimi and J. Vahedi",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/0810.3552"
}
|
0810.3589
|
# Transverse Force on Quarks in DIS
Matthias Burkardt Department of Physics, New Mexico State University, Las
Cruces, NM 88003-0001, U.S.A. Thomas Jefferson National Accelerator Facility,
Newport News, VA 23606, U.S.A.
###### Abstract
The $x^{2}$-moment of the twist-3 polarized parton distribution $g_{2}(x)$ is
related to the transverse force acting on the active quark in deep-inelastic
scattering off a transversely polarized nucleon immediately after it has
absorbed the virtual photon. Lattice calculations of the corresponding matrix
element as well as experimental measurements of $g_{2}(x)$ are used to
estimate sign and magnitude of this force. Similarly, the $x^{2}$-moment of
the chirally odd twist-3 unpolarized parton distribution $e(x)$ can be related
to the transverse force experienced by a transversely polarized quark ejected
from a transversely polarized nucleon.
## I Introduction
In the analysis of high-energy scattering processes, leading twist effects are
often both easier to isolate (e.g. by increasing $Q^{2}$ until higher twist
effects are negligible) and have a more direct physical interpretation than
contributions from higher twist operators which are often intermingled with
$\frac{1}{Q^{n}}$ corrections to leading twist operators. A notable exception
is is the twist-3 polarized structure function $g_{2}(x,Q^{2})$, which can be
cleanly separated from its twist-2 counterpart $g_{1}(x,Q^{2})$ when
scattering longitudinally polarized leptons from a transversely polarized
target. While in general, the contribution from $g_{2}$ to the polarized cross
section is suppressed by powers of $\frac{1}{{Q^{2}}}$, at a target
polarization angle of $90\,^{\circ}$ the leading contribution vanishes and the
contribution from $g_{1}$ is exposed without kinematical suppression. This
property of the polarized DIS cross section thus allows a clean extraction of
higher twist matrix elements, without the need for fitting and subtracting a
leading twist effect, and thus makes polarized deep-inelastic scattering (DIS)
a rare opportunity for studying higher twist effects (for an overview, see
Ref. Jaffe:Erice ).
As $g_{2}(x,Q^{2})$ involves higher twist, it does not have a parton
interpretation as a single particle density. Indeed, the twist-3 part of
$g_{2}$ is related to quark-gluon correlations whose intuitive interpretation
may not be immediately clear. Since $g_{2}(x,Q^{2})$ is related to
(electromagnetic) polarizabilities at low $Q^{2}$, these twist-3 matrix
elements have been called color polarizabilities in the literature Ji .
However, at high $Q^{2}$, the twist-3 piece of $g_{2}(x,Q^{2})$ is described
by a local correlator and the physical interpretation as a polarizability no
longer applies. Indeed, while nucleons need to be polarized in order to study
$g_{2}(x,Q^{2})$, the nucleons are not distorted, but only ‘spin-alligned’.
The quark-gluon correlations embodied in the twist-3 part of $g_{2}(x,Q^{2})$
are then obtained as a matrix elements of a certain operator in a spin-
alligned, but undeformed, nucleon. This is very different from the usual use
of the term ‘polarizability’ as the tendency of a charge or magnetization
distribution to be distorted from its normal shape by an external field. Of
course, one could broaden the notion of ‘polarizability’ to encompass matrix
elements that are only non-zero when the nucleon is polarized, but within such
a broadened definition, other spin-dependent observables, such as the
polarized parton distribution $\Delta q(x)$ or even the magnetic moment of the
nucleon, would then also become ‘polarizabilities’ in the broader sense.
The main purpose of this paper is to explore an alternative physical
interpretation of these particular twist-3 matrix elements as a force. First
we summarize the connection between the $x^{2}$ moment of $g_{2}(x,Q^{2})$ and
quark-gluon correlations. After discussing the connection between these
correlations and the transverse force on the active quark in DIS, we then
estimate sign and magnitude of that force based on DIS data, lattice
calculations and heuristic pictures.
## II $x^{2}$ moments and quark-gluon correlations
The chirally even spin-dependent twist-3 parton distribution
$g_{2}(x)=g_{T}(x)-g_{1}(x)$ is defined as
$\displaystyle\int\frac{d\lambda}{2\pi}e^{i\lambda x}\langle
PS|\bar{\psi}(0)\gamma^{\mu}\gamma_{5}\psi(\lambda
n)|_{Q^{2}}|PS\rangle=2\left[g_{1}(x,Q^{2})p^{\mu}(S\cdot
n)+g_{T}(x,Q^{2})S_{\perp}^{\mu}+M^{2}g_{3}(x,Q^{2})n^{\mu}(S\cdot n)\right].$
(1)
where $p^{\mu}$ and $n^{\mu}$ are light-like vectors along the $-$ and $+$
light-cone direction with $p\cdot n=1$. Using the equations of motion
$g_{2}(x)$ can be expressed as a sum of a piece that is entirely determined in
terms of $g_{1}(x)$ plus an interaction dependent twist-3 part that involves
quark gluon correlations WW
$\displaystyle g_{2}(x)$ $\displaystyle=$ $\displaystyle
g_{2}^{WW}(x)+\bar{g}_{2}(x)$ (2) $\displaystyle g_{2}^{WW}(x)$
$\displaystyle=$ $\displaystyle-g_{1}(x)+\int_{x}^{1}\frac{dy}{y}g_{1}(y).$
Here we have neglected $m_{q}$ for simplicity. For example, the $x^{2}$ moment
yields Shuryak ; Jaffe
$\displaystyle\int dxx^{2}\bar{g}_{2}(x)=\frac{d_{2}}{3}$ (3)
with
$\displaystyle g\left\langle
P,S\left|\bar{q}(0)G^{+y}(0)\gamma^{+}q(0)\right|P,S\right\rangle=2M{P^{+}}P^{+}S^{x}d_{2}.$
(4)
Note that conventions exist where $\frac{d_{2}}{6}$ appears on the r.h.s. of
(3) in which case there is no factor $2$ on the r.h.s. of (4).
In the limit where $Q^{2}$ is so low that the virtual photon wavelength is
larger than the nucleon size, the electro-magnetic field associated with the
two virtual photons appearing in the forward Compton amplitude corresponding
to the structure function is nearly homogenous accross the nucleon and the
spin-dependent structure function $g_{2}(x,Q^{2})$ can be related to spin-
dependent polarizabilities. In contradistinction, in the Bjorken limit, the
matrix elements describing the moments of $g_{2}(x,Q^{2})$ are given by local
correlation functions, such as (4). Nevertheless, because of the
abovementioned low $Q^{2}$ interpretation of $g_{2}$, the local matrix
elements appearing in (4)
$\displaystyle\chi_{E}2M^{2}{\vec{S}}=\left\langle
P,S\right|q^{\dagger}{\vec{\alpha}}\times
g{\vec{E}}q\left|P,S\right\rangle\quad\quad\quad\quad\quad\quad\quad\quad\chi_{B}2M^{2}{\vec{S}}=\left\langle
P,S\right|q^{\dagger}g{\vec{B}}q\left|P,S\right\rangle,$ (5)
where
$\displaystyle d_{2}=\frac{1}{4}\left(\chi_{E}-2\chi_{M}\right),$ (6)
(note that $\sqrt{2}G^{+y}=B^{x}-E^{y}$) are sometimes called color electric
and magnetic polarizabilities Ji . In the following we will discuss why, at
high $Q^{2}$, a better interpretation for these matrix elements is that of a
‘force’.
In electro-magnetism, the $\hat{y}$-component of the Lorentz force $F^{y}$
acting on a particle with charge $e$ moving, with (nearly) the speed of light
along the $-\hat{z}$ direction, ${\vec{v}}\approx(0,0,-1)$, reads
$\displaystyle
F^{y}=e\left[{\vec{E}}+{\vec{v}}\times{\vec{B}}\right]^{y}=e\left(E^{y}-B^{x}\right)=-e\sqrt{2}F^{+y},$
(7)
which involves the same linear combination of Lorentz components that also
appears in the gluon field strength tensor in (4). This simple observation
already suggests a connection between $d_{2}$ and the color Lorentz force on a
quark that moves (in a DIS experiment) with ${\vec{v}}\approx(0,0,-1)$.
In order to explore this connection further we compare the matrix element
defining $d_{2}$ with that describing the average transverse momentum of
quarks in semi-inclusive DIS (SIDIS) sivers . The average intrinsic transverse
momentum of quarks bound in a nucleon vanishes and therefore any net
transverse momentum of quarks in a SIDIS experiment must come from the final
state interactions (FSI) collins . The average transverse momentum of the
ejected quark (also averaged over the momentum fraction $x$ carried by the
active quark in order to render the matrix element local in the position of
the quark field operator) in a SIDIS experiment can thus be represented by the
matrix element QS
$\displaystyle\langle k_{\perp}^{y}\rangle=-\frac{1}{2P^{+}}\left\langle
P,S\left|\bar{q}(0)\int_{0}^{\infty}dx^{-}gG^{+y}(x^{+}=0,x^{-})\gamma^{+}q(0)\right|P,S\right\rangle,$
(8)
where Wilson-line gauge links along $x^{-}$ are implicitly understood, but not
written out explicitly. One way to derive this expression is to start from
gauge invariantly defined quark momentum distributions with Wilson line gauge
links extending to ligh-cone infinity. The integral over the gauge field is
then obtained by acting with the transverse derivative (when averaging over
$k_{\perp}^{y}$) on the gauge field appearing in the Wilson line JiYuan ;
Pijlman .
The matrix element appearing in (8) thus has a simple physical interpretation
as the transverse impulse obtained by intergrating the color Lorentz force
along the trajectory of the active quark — which is an almost light-like
trajectory along the $-\hat{z}$ direction, with $z=-t$.
In order to make the correspondence more explicit, we now rewrite Eq. (8) as
an integral over time
$\displaystyle\langle
k_{\perp}^{y}\rangle=-\frac{\sqrt{2}}{2P^{+}}\left\langle
P,S\right|\bar{q}(0)\int_{0}^{\infty}dtG^{+y}(t,z=-t)\gamma^{+}q(0)\left|P,S\right\rangle$
(9)
in which the physical interpretation of $-\frac{\sqrt{2}}{2P^{+}}\left\langle
P,S\right|\bar{q}(0)G^{+y}(t,z=-t)\gamma^{+}q(0)\left|P,S\right\rangle$ as
being the (ensemble averaged) transverse force acting on the struck quark at
time $t$ after being struck by the virtual photon becomes more apparent. In
particular,
$\displaystyle F^{y}(0)$ $\displaystyle\equiv$
$\displaystyle-\frac{\sqrt{2}}{2P^{+}}\left\langle
P,S\right|\bar{q}(0)G^{+y}(0)\gamma^{+}q(0)\left|P,S\right\rangle$
$\displaystyle=$ $\displaystyle-{\sqrt{2}}MP^{+}S^{x}d_{2}=-{M^{2}}d_{2},$
where the last equality holds only in the rest frame
($p^{+}=\frac{1}{\sqrt{2}}M$) and for $S^{x}=1$, can be interpreted as the
averaged transverse force acting on the active quark in the instant right
after it has been struck by the virtual photon.
Although the identification of $\langle p|\bar{q}\gamma^{+}G^{+y}q|p\rangle$
as a color Lorentz force may be intuitively evident after the above
discussion, it is also instructive to provide a more formal justification. For
this purpose, we consider the time dependence of the transverse momentum of
the ‘good’ component of the quark fields (the component relevant for DIS in
the Bjorken limit) ${q}_{+}\equiv\frac{1}{2}\gamma^{-}\gamma^{+}q$
$\displaystyle 2p^{+}\frac{d}{dt}\langle{p}^{y}\rangle$ $\displaystyle\equiv$
$\displaystyle\frac{d}{dt}\left\langle
PS\right|\bar{q}\gamma^{+}\left(p^{y}-gA^{y}\right)q\left|PS\right\rangle=\frac{1}{\sqrt{2}}\frac{d}{dt}\left\langle
PS\right|{q}_{+}^{\dagger}\left(p^{y}-gA^{y}\right)q_{+}\left|PS\right\rangle$
$\displaystyle=$ $\displaystyle 2p^{+}\left\langle
PS\right|\left[\dot{\bar{q}}\gamma^{+}\left(p^{y}-gA^{y}\right)q+\bar{q}\gamma^{+}\left(p^{y}-gA^{y}\right)\dot{q}-\bar{q}\gamma^{+}g\dot{A}^{y}q\right]\left|PS\right\rangle.$
Using the QCD equations of motion
$\displaystyle\dot{q}=\left(igA^{0}+\gamma^{0}{\vec{\gamma}}\cdot{\vec{D}}\right)q,$
(12)
where $-iD^{\mu}=p^{\mu}-gA^{\mu}$, yields
$\displaystyle 2p^{+}\frac{d}{dt}\langle{\bf p}^{y}\rangle$ $\displaystyle=$
$\displaystyle\left\langle
PS\right|\bar{q}\gamma^{+}g\left(G^{y0}+G^{yz}\right)q\left|PS\right\rangle+`\left\langle
PS\right|\bar{q}\gamma^{+}\gamma^{-}\gamma^{i}D^{i}D^{j}q\left|PS\right\rangle^{\prime}$
(13) $\displaystyle=$ $\displaystyle\sqrt{2}\left\langle
PS\right|\bar{q}\gamma^{+}gG^{y+}q\left|PS\right\rangle+`\left\langle
PS\right|\bar{q}\gamma^{+}\gamma^{-}\gamma^{i}D^{i}D^{j}q\left|PS\right\rangle^{\prime},$
(14)
where $`\left\langle
PS\right|\bar{q}\gamma^{+}\gamma^{-}\gamma^{i}D^{i}D^{j}q\left|PS\right\rangle^{\prime}$
stands symbolically for all terms that involve a product of
$\gamma^{+}\gamma^{-}$ as well as a $\gamma^{\perp}$ and that also involve
only transverse derivatives $D^{i}$.
Now it is important to keep in mind that we are not interested in the average
force on the ‘original’ quark fields (before the quark is struck by the
virtual photon), but after absorbing the virtual photon and moving with
(nearly) the speed of light in the $-\hat{z}$ direction. In this limit, the
first term on the r.h.s. of (14) dominates, as it contains the largest number
of ‘$+$’ Lorentz indices. Dropping the other terms yields (II).
A measurement of the $x^{2}$-moment $f_{2}$ of the twist-4 distribution
$g_{3}(x)$ allows determination of the expectation value of a different linear
combination of Lorentz/Dirac components of the quark-gluon correlator
appearing in (4) f2
$\displaystyle f_{2}M^{2}S^{\mu}=\frac{1}{2}\left\langle
p,S\right|\bar{q}g\tilde{G}^{\mu\nu}\gamma_{\nu}q\left|p,S\right\rangle.$ (15)
Using rotational invariance, to relate various Lorentz components one thus
finds a linear combination of the matrix elements of electric and magnetic
quark-gluon correlators (5)
$\displaystyle f_{2}=\chi_{E}-\chi_{M},$ (16)
that differs from that in (6). In combination with (4) this allows a
decomposition of the force into electric and magnetic components
$F^{y}=F^{y}_{E}+F^{y}_{M}$, using
$\displaystyle F_{E}^{y}(0)=-\frac{M^{2}}{4}\chi_{E}\quad\quad\quad\quad
F_{B}^{y}(0)=-\frac{M^{2}}{2}\chi_{B}$ (17)
for a target nucleon polarized in the $+\hat{x}$ direction, where Ji ; color
$\displaystyle\chi_{E}=\frac{2}{3}\left(2d_{2}+f_{2}\right)\quad\quad\quad\quad\chi_{M}=\frac{1}{3}\left(4d_{2}-f_{2}\right).$
(18)
A relation similar to (II) can be derived for the $x^{2}$ moment of the
twist-3 scalar PDF $e(x)$. For its interaction dependent twist-3 part
$\bar{e}(x)$ one finds for an unpolarized target Yuji
$\displaystyle 4MP^{+}P^{+}e_{2}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}g\left\langle
p\right|\bar{q}\sigma^{+i}G^{+i}q\left|P\right\rangle,$ (19)
where $e_{2}\equiv\int_{0}^{1}dxx^{2}\bar{e}(x)$. The matrix element on the
r.h.s. of Eq. (19) can be related to the average transverse force acting on a
transversely polarized quark in an unpolarized target right after being struck
by the virtual photon. Indeed, for the average transverse momentum in the
$+\hat{y}$ direction, for a quark polarized in the $+\hat{x}$ direction (${\bf
k}_{\perp}^{2}$ moment of the Boer-Mulders function $h_{1}^{\perp}$ BM
integrated also over $x$), one finds
$\displaystyle\langle
k^{y}\rangle=\frac{1}{4P^{+}}\int_{0}^{\infty}dx^{-}g\left\langle
p\right|\bar{q}(0)\sigma^{+y}G^{+y}(x^{-})q(0)\left|p\right\rangle.$ (20)
A comparison with Eq. (19) shows that the average transverse force at $t=0$
(right after being struck) on a quark polarized in the $+\hat{x}$ direction
reads
$\displaystyle F^{y}(0)=\frac{1}{2\sqrt{2}p^{+}}g\left\langle
p\right|\bar{q}\sigma^{+y}G^{+y}q\left|p\right\rangle=\frac{1}{\sqrt{2}}MP^{+}S^{x}e_{2}=\frac{M^{2}}{2}e_{2},$
(21)
where the last identify holds only in the rest frame of the target nucleon and
for $S^{x}=1$. In the physical interpretation of (21) it is important to keep
in mind that, for a given flavor, the number of quarks on which the force in
(21) is only half that in (II) as only half the quarks in an unpolarized
nucleon will be polarized in the $+\hat{x}$ direction.
## III Heuristic Pictures and Numerical Studies
When the target nucleon is transversely polarized, e.g. in the $+\hat{x}$
direction the axial symmetry in the transverse plane is broken. In particular,
the quark distribution (more precisely the distribution of the
$\gamma^{+}$-density that dominates in DIS in the Bjorken limit) in the
transverse plane is deformed IJMPA . The average deformations can be related
to the contribution from each quark flavor to the anomalous magnetic moment of
the nucleon and was predicted to be quite substantial IJMPA and has also been
observed in lattice QCD hagler .
Figure 1: Distribution of the $j^{+}$ density for $u$ and $d$ quarks in the
$\perp$ plane ($x_{Bj}=0.3$ is fixed) for a nucleon that is polarized in the
$x$ direction in the model from Ref. IJMPA . For other values of $x$ the
distortion looks similar.
Given the fact that, for a nucleon polarized in the $+\hat{x}$ direction the
$\gamma^{+}$-distribution for $u$ ($d$) is shifted towards the $\pm\hat{y}$
direction suggests that these quarks also ‘feel’ a nonzero color-electric
force pointing on average in the $\mp\hat{y}$ direction, i.e. one would expect
that $d_{2}$ is positive (negative) for $u$ ($d$) quarks. This is also
consistent with a negative (positive) sign for the Sivers on the proton
function as observed by the HERMES collaboration hermes and the vanishing
Sivers function for deuterium in the COMPASS experiment compass . In fact, in
the large $N_{C}$ limit, one would expect that $d_{2}$ for $u$ ($d$) quarks
are equal and opposite. Note that while $d_{2}$ for $u$ ($d$) quarks being
exactly equal and opposite would imply the same for protons (neutrons), any
deviation from being exactly equal and opposite is enhanced for proton
(neutron) since there is a significant cancellation between the two quark
flavors in the nucleon.
If all spectators in the FSI were to ‘pull’ in the same direction, the force
on the active quark would be of the order of the QCD string tension
$\sigma\approx(450MeV)^{2}$, which would translate into a value $d_{2}\sim
0.2$. However, it is more natural to expect a significant cancellation between
forces from spectators pulling the active quark in different directions, the
actual value of $d_{2}$ is probably about one order of magnitude smaller, i.e.
$d_{2}\sim 0.02$ appears to be more natural. Instanton based models have
suggested an even smaller value Weiss .
Heuristic arguments/lattice calculations mb:brian ; hagler also suggest that
the deformation of (the $\gamma^{+}$-distribution for) transversely polarized
quarks in an unpolarized nucleon is more significant than that of unpolarized
quarks in a transversely polarized nucleon. When applied to the final state
interactions, this observation suggests $|e_{2}|>|d_{2}|$ (the fact that in an
unpolarized nucleon only half the quarks are polarized in the
$+\hat{x}$-direction is compensated by the factor $\frac{1}{2}$ in (21).
Lattice calculations of the twist-3 matrix element yield latticed2
$\displaystyle d_{2}^{(u)}=0.020\pm 0.024\quad\quad\quad\quad
d_{2}^{(d)}=-0.011\pm 0.010$ (22)
renormalized at a scale of $Q^{2}=5$ GeV2 for the smallest lattice spacing in
Ref. latticed2 . Note that we have multiplied the numerical results from
latticed2 by a factor of $2$ to account for the different convention for
$d_{2}$ being used.
Here the identity $M^{2}\approx 5$GeV/fm is useful to better visualize the
magnitude of the force.
$\displaystyle F_{(u)}=-25\pm 30{\rm MeV/fm}\quad\quad\quad\quad F_{(d)}=14\pm
13{\rm MeV/fm}.$ (23)
In the chromodynamic lensing picture, one would have expected that $F_{(u)}$
and $F_{(d)}$ are of about the same magnitude and with opposite sign. The same
holds in the large $N_{C}$ limit. A vanishing Sivers effect for an isoscalar
target would be more consistent with equal and opposite average forces.
However, since the error bars for $d_{2}$ include only statistical errors, the
lattice result may not be inconsistent with $d_{2}^{(d)}\sim-d_{2}^{(u)}$.
The average transverse momentum from the Sivers effect is obtained by
integrating the transverse force to infinity (along a light-like trajectory)
$\langle k^{y}\rangle=\int_{0}^{\infty}dtF^{y}(t)$ (9). This motivates us to
define an ‘effective range’
$\displaystyle R_{eff}\equiv\frac{\langle k^{y}\rangle}{F^{y}(0)}.$ (24)
Note that $R_{eff}$ depends on how rapidly the correlations fall off along a
light-like direction and it may thus be larger than the (spacelike) radius of
a hadron. Of cource, unless the functional form of the integrand is known,
$R_{eff}$ cannot really tell us about the range of the FSI, but if the
integrand does not oscillate
Fits of the Sivers function to SIDIS data yield Mauro one finds about
$|\langle k^{y}\rangle|\sim 100$ MeV Mauro . Together with the (average) value
for $|d_{2}|$ from the littice this translates into an effective range
$R_{eff}$ of several fm. It would be interesting to compare $R_{eff}$ for
different quark flavors and as a function of $Q^{2}$, but this requires more
precise values for $d_{2}$ as well as the Sivers function.
Note that a complementary approach to the effective range was chosen in Ref.
Stein , where the twist-3 matrix element appearing in Eq. (II) was, due to the
lack of lattice QCD results, estimated using QCD sum rule techniques.
Moreover, the ‘range’ was taken as a model input parameter to estimate the
magnitude of the Sivers function.
The impact parameter distribution for quarks polarized in the $+\hat{x}$
direction was found to be shifted in the $+\hat{y}$ direction DH ; hagler ;
mb:brian . Applying the chromodynamic lensing model implies a force in the
negative $-\hat{y}$ direction for these quarks and one thus expects $e_{2}<0$
for both $u$ and $d$ quarks. Magnitude: since $\kappa_{\perp}>\kappa$, expect
odd force larger than even force and thus $|e_{2}|>|d_{2}|$.
It would be interesting to study not only whether the effective range is
flavor dependent, but also whether there is a difference between the chirally
even and odd cases. It would also be very interesting to learn more about the
time dependence of the FSI by calculating matrix elements of
$\bar{q}\gamma^{+}\left(D^{+}G^{+\perp}\right)q$, or even higher derivatives,
in lattice QCD. Knowledge of not only the value of the integrand at the
origin, but also its slope and curvature at that point, would be very useful
for estimating the integral in Eq. (8).
## IV Discussion
The quark-gluon correlations in the $x^{2}$-moment $d_{2}$ of the twist-3
polarized PDF $g_{2}$ can be identified with the transverse component of the
color-Lorentz force acting on the struck quark in the instant after absorbing
the virtual photon. The direction of the the force for $u$ and $d$ quarks can
be understood in terms of the transverse deformation of parton distributions
for a transversely polarized target. In combination with a measurement of the
$x^{2}$ moment of the twist-4 polarized PDF $g_{3}$ one can even decompose
this force into color-electric and magnetic components. Although still quite
uncertain, first eperimental/lattice results suggest values around
$25-50$MeV/fm for the net force. This should be compared with the net
transverse momentum due to the Sivers effect which is on the order of
$100$MeV.
The $x^{2}$ moment $e_{2}$ of the chirally odd twist-3 (scalar) PDF $e(x)$ can
be related to the transverse force acting on transversely polarized quarks in
an unpolarized target. Therefore, $e_{2}$ is to the Boer-Mulders function
$h_{1}^{\perp}$, what is $d_{2}$ to the Sivers function $f_{1T}^{\perp}$.
Acknowledgements: I would like to thank A.Bacchetta, D. Boer, J.P. Chen,
Y.Koike, and Z.-E. Mezziani for useful discussions. This work was supported by
the DOE under grant numbers DE-FG03-95ER40965 and DE-AC05-06OR23177 (under
which Jefferson Science Associates, LLC, operates Jefferson Lab).
## References
* (1) R.L. Jaffe, Lecture Notes, Int. School of Nucleon Structure, Erice, Aug. 1995; hep-ph/9602236.
* (2) B.W. Filippone and X. Ji, Adv. Nucl. Phys. 26, 1 (2001).
* (3) S. Wandzura and F. Wilczek, Phys. Lett. B 72, 195 (1977).
* (4) E. Shuryak and A.I. Vainshtein, Nucl. Phys. B 201, 141 (1982).
* (5) R.L. Jaffe, Comm. Nucl. Part. Phys. 19, 239 (1990).
* (6) D.W. Sivers, Phys. Rev. D 43, 261 (1991).
* (7) J.C. Collins, Phys. Lett. B 536, 43 (2002).
* (8) J. Qiu and G. Sterman, Phys. Rev. Lett. 67, 2264 (1991).
* (9) X. Ji and F. Yuan, Phys. Lett. B 543, 66 (2002).
* (10) D. Boer, P.J. Mulders, and F. Pijlman, Nucl. Phys. B 667, 201 (2003).
* (11) E.V. Shuryak and A.I. Vainshtein, Nucl. Phys. B199, 451 (1982); B201, 141 (1982); B. Ehrnsperger, L. Mankiewicz, and A. Schäfer, Phys. Lett. B 323, 439 (1994); X. Ji and P. Unrau, Phys. Lett. 333, 228 (1994).
* (12) Z.-E. Mezziani et al., hep-ph/0404066.
* (13) Y. Koike and K. Tanaka, Phys. Rev. D 51, 6125 (1995).
* (14) D. Boer and P.J. Mulders, Phys. Rev. D 57, 5780 (1998); M. Anselmino, M. Boglione, and F. Murgia, Phys. Rev. D 60, 054027 (1999).
* (15) M. Burkardt, Int. J. Mod. Phys. A 18, 173 (2003).
* (16) M. Göckeler et al. (QCDSF collaboration), Phys. Rev. Lett. 98, 222001 (2007); Ph. Hägler et al. (LHPC collaboration), hep-lat/0705.4295.
* (17) A. Airapetian et al. (Hermes collaboration), Phys. Rev. Lett. 94, 012002 (2005).
* (18) A. Martin (Compass collaboration), Czech. J. Phys. 56, F33 (2006).
* (19) J. Balla, M.V. Polyakov, and C. Weiss, Nucl. Phys. B 510, 327 (1998).
* (20) M. Burkardt and B. Hannafious, Phys. Lett. B 658, 130 (2008).
* (21) M. Göckeler et al., Phys. Rev. D 72, 054507 (2005).
* (22) M. Anselmino et al., hep-ph/0805.2677
* (23) A. Schäfer et al., Phys. Rev. D 47, R1 (1993); E. Stein et al., hep-ph/9409212
* (24) M. Diehl and P.Hägler, Eur. Phys. J. C44, 87 (2005).
|
arxiv-papers
| 2008-10-20T15:39:45 |
2024-09-04T02:48:58.349642
|
{
"license": "Public Domain",
"authors": "Matthias Burkardt (New Mexico State University / Jefferson Lab)",
"submitter": "Matthias Burkardt",
"url": "https://arxiv.org/abs/0810.3589"
}
|
0810.3607
|
# Leading twist distribution amplitudes of $P$-wave nonrelativistic mesons.
V.V. Braguta braguta@mail.ru A.K. Likhoded Likhoded@ihep.ru A.V. Luchinsky
Alexey.Luchinsky@ihep.ru Institute for High Energy Physics, Protvino, Russia
###### Abstract
This paper is devoted to the study of the leading twist distribution
amplitudes of $P$-wave nonrelativistic mesons. It is shown that at the leading
order approximation in relative velocity of quark-antiquark pair inside the
mesons these distribution amplitudes can be expressed through one universal
function. As an example, the distribution amplitudes of $P$-wave charmonia
mesons are considered. Within QCD sum rules the model for the universal
function of $P$-wave charmonia mesons is built. In addition, it is found the
relations between the moments of the universal function and the
nonrelativistic QCD matrix elements that control relativistic corrections to
any amplitude involving $P$-wave charmonia. Our calculation shows that
characteristic size of these corrections is of order of $\sim 30\%$.
###### pacs:
12.38.-t, 12.38.Bx, 13.66.Bc, 13.25.Gv
## I Introduction
Hard exclusive processes are very interesting both from theoretical and
experimental points of view. Commonly, theoretical approach to the description
of such processes is based on the factorization theorem Lepage:1980fj ;
Chernyak:1983ej . Within this theorem the amplitude of hard exclusive process
can be separated into two parts. The first part is partons production at very
small distances, which can be treated within perturbative QCD. The second part
is hardronization of the partons at larger distances. This part contains
information about nonperturbative dynamic of strong interaction. For hard
exclusive processes it can be parameterized by process independent
distribution amplitudes (DA), which can be considered as hadrons’ wave
functions at light like separation between the partons in the hadron. It
should stressed that DAs are very important for the calculation of the
amplitude of any hard exclusive process.
Recently, the leading twist DAs of $S$-wave nonrelativistic mesons have become
the object of intensive study Bodwin:2006dm ; Ma:2006hc ; Braguta:2006wr ;
Braguta:2007fh ; Braguta:2007tq ; Choi:2007ze ; Feldmann:2007id ; Bell:2008er
. Knowledge about these DAs allowed one to build some models for $S$-wave
charmonia DAs, that can be used in practical calculations. In this paper
general properties of the leading twist DAs of $P$-wave nonrelativistc mesons
will be studied. The results of this study will be used to build the model for
the DAs of $P$-wave charmonia mesons, that can be used in calculations.
This paper is organized as follows. In the next section all definitions of the
DAs of $P$-wave mesons will be given. These DAs will be studied in section III
at the leading order approximation in relative velocity of quark-antiquark
pair inside $P$-wave meson. In section IV QCD sum rules will be applied to the
calculation of the moments of the $P$-wave charmonia DAs. Using the results of
this study the model of the $P$-wave charmonia DAs will be build in section V.
In the last section the results of this paper will be summarized.
## II Definitions of the distribution amplitudes.
In this section the definitions of the leading twist distribution amplitudes
(DA) of $P$-wave nonrelativistic mesons will be given. In the conventional
quark model nonrelativistic mesons are quark-antiquark($Q\bar{Q}$) bound
states. In these mesons the quark-antiquark pair can be in the spin singlet or
spin triplet states. Since orbital momentum of the quark-antiquark pair is
unity one can conclude that there are four $P$-wave mesons:
$\chi_{0}(^{3}P_{0}),\chi_{1}(^{3}P_{1}),\chi_{2}(^{3}P_{2}),h(^{1}P_{1})$.
The leading twist DAs of these mesons can be defined as follows.
for the $\chi_{0}$ meson:
$\displaystyle{\langle\chi_{0}(p)|{\bar{Q}}(z)\gamma_{\mu}[z,-z]Q(-z)|0\rangle}$
$\displaystyle=$ $\displaystyle
f_{\chi_{0}}p_{\mu}\int^{1}_{-1}d\xi\,e^{i(pz)\xi}\phi_{\chi_{0}}(\xi,\mu),$
(1)
for the $\chi_{1}$ meson:
$\displaystyle{\langle\chi_{1}(p,\epsilon_{\lambda=0})|{\bar{Q}}(z)\gamma_{\mu}\gamma_{5}[z,-z]Q(-z)|0\rangle}$
$\displaystyle=$ $\displaystyle
f^{\prime}_{\chi_{1}}p_{\mu}\int^{1}_{-1}d\xi\,e^{i(pz)\xi}\phi^{\prime}_{\chi_{1}}(\xi,\mu),$
$\displaystyle{\langle\chi_{1}(p,\epsilon_{\lambda=\pm
1})|{\bar{Q}}(z)\sigma_{\mu\nu}[z,-z]Q(-z)|0\rangle}$ $\displaystyle=$
$\displaystyle
f_{\chi_{1}}e_{\mu\nu\alpha\beta}\epsilon^{\alpha}p^{\beta}\int^{1}_{-1}d\xi\,e^{i(pz)\xi}\phi_{\chi_{1}}(\xi,\mu),$
(2)
for the $h$ meson:
$\displaystyle{\langle
h(p,\epsilon_{\lambda=0})|{\bar{Q}}(z)\gamma_{\mu}\gamma_{5}[z,-z]Q(-z)|0\rangle}$
$\displaystyle=$ $\displaystyle
f_{h}p_{\mu}\int^{1}_{-1}d\xi\,e^{i(pz)\xi}\phi_{h}(\xi,\mu),$
$\displaystyle{\langle h(p,\epsilon_{\lambda=\pm
1})|{\bar{Q}}(z)\sigma_{\mu\nu}[z,-z]Q(-z)|0\rangle}$ $\displaystyle=$
$\displaystyle
f^{\prime}_{h}e_{\mu\nu\alpha\beta}\epsilon^{\alpha}p^{\beta}\int^{1}_{-1}d\xi\,e^{i(pz)\xi}\phi^{\prime}_{h}(\xi,\mu),$
(3)
for the $\chi_{2}$ meson:
$\displaystyle{\langle\chi_{2}(p,\epsilon_{\lambda=0})|{\bar{Q}}(z)\gamma_{\mu}[z,-z]Q(-z)|0\rangle}$
$\displaystyle=$ $\displaystyle
f_{\chi_{2}}p_{\mu}\int^{1}_{-1}d\xi\,e^{i(pz)\xi}\phi_{\chi_{2}}(\xi,\mu),$
$\displaystyle{\langle\chi_{2}(p,\epsilon_{\lambda=\pm
1})|{\bar{Q}}(z)\sigma_{\mu\nu}[z,-z]Q(-z)|0\rangle}$ $\displaystyle=$
$\displaystyle\tilde{f}_{\chi_{2}}M_{\chi_{2}}(\rho_{\mu}p_{\nu}-\rho_{\nu}p_{\mu})\int^{1}_{-1}d\xi\,e^{i(pz)\xi}\tilde{\phi}_{\chi_{2}}(\xi,\mu),~{}~{}~{}\rho_{\mu}=\frac{\epsilon_{\mu\sigma}z^{\sigma}}{pz},$
(4)
where the following designations are used: $x_{1},x_{2}$ are the fractions of
momentum of meson carried by quark and antiquark correspondingly,
$\xi=x_{1}-x_{2}$, $p,~{}\epsilon$ are the momentum and polarizations of
$P$-wave mesons. For the mesons $\chi_{1},h$ the polarization $\epsilon$ is
described by the four vector $\epsilon_{\mu}$, for the $\chi_{2}$ meson the
polarization $\epsilon$ is described by the tensor $\epsilon_{\mu\nu}$. The
factor $[z,-z]$, that makes matrix elements (1)-(4) gauge invariant, is
defined as
$\displaystyle[z,-z]=P\exp[ig\int_{-z}^{z}dx^{\mu}A_{\mu}(x)].$ (5)
In applications it is useful to rewrite the four-vector $\rho_{\mu}$ in the
following way. Evidently, one can write the polarization of the $\chi_{2}$
meson in terms of the polarization of two vector mesons. Thus for the
transverse polarization of the meson $\chi_{2}$ one has
$\epsilon^{\mu\nu}_{\lambda=\pm 1}=(\epsilon_{\lambda=\pm
1}^{\mu}\cdot\epsilon_{\lambda=0}^{\nu}+\epsilon_{\lambda=0}^{\nu}\cdot\epsilon_{\lambda=\pm
1}^{\mu})/\sqrt{2}$ ($\epsilon^{+}_{\mu\nu}\epsilon^{\mu\nu}=1$). If we
further contract the polarization tensor $\epsilon_{\mu\nu}$ with lightlike
four-vector $z$, to the leading twist accuracy we will get
$\epsilon^{\mu\nu}z_{\nu}=\epsilon_{\lambda=\pm
1}^{\mu}(pz)/(\sqrt{2}M_{\chi_{2}})$ or $\rho^{\mu}=\epsilon_{\lambda=\pm
1}^{\mu}/(\sqrt{2}M_{\chi_{2}})$. This form of the vector $\rho$ can be used
in the calculation with the leading twist accuracy. It should be noted that
the states of the $\chi_{2}$ meson with the polarizations $\lambda=\pm 2$ give
contribution only to higher twist DAs. Since, this paper is devoted to the
study of the leading twist DAs we don’t consider these states.
The functions without primes $\phi_{\chi 0}(\xi),\phi_{\chi
1}(\xi),\phi_{h}(\xi),\phi_{\chi 2}(\xi),\tilde{\phi}_{\chi 2}(\xi)$ are $\xi$
odd and they are normalized as
$\displaystyle\int_{-1}^{1}d\xi~{}\xi\phi(\xi)=1.$ (6)
The functions with primes $\phi^{\prime}_{\chi 1}(\xi),\phi^{\prime}_{h}(\xi)$
are $\xi$ even and they are normalized as
$\displaystyle\int_{-1}^{1}d\xi~{}\phi^{\prime}(\xi)=1.$ (7)
In this paper all DAs will be parameterized by their moments
$\displaystyle\langle\xi^{n}\rangle=\int_{-1}^{1}d\xi~{}\xi^{n}\phi(\xi).$ (8)
Evidently, for the DAs with primes all odd moments are zero. For the DAs
without primes all even moments are zero. To separate the moments of different
DAs of the $\chi_{2}$ meson, below we are going to use the following
designations: $\langle\xi^{n}\rangle$ for the moments of the function
$\phi_{\chi_{2}}(\xi)$, $\langle\tilde{\xi}^{n}\rangle$ for the moments of the
function $\tilde{\phi}_{\chi_{2}}(\xi)$.
The distribution amplitudes $\phi(\xi)$ and the constants $f$ that
parameterizes corresponding currents (1)-(4) are scale dependent objects. For
applications it is useful to write how they depend on the scale
Chernyak:1983ej . To do this we expand DA in the series
$\displaystyle\phi(\xi,\mu)=\frac{3}{4}(1-\xi^{2})\biggl{[}\sum_{n=0}^{\infty}a_{n}(\mu)C_{n}^{3/2}(\xi)\biggr{]},$
(9)
where $C_{n}^{3/2}(\xi)$ are Gegenbauer polynomials. At the leading
logarithmic accuracy the coefficients $a_{n}$ are renormalized
multiplicatively
$\displaystyle
a_{n}(\mu)=\biggl{(}\frac{\alpha_{s}(\mu)}{\alpha_{s}(\mu_{0})}\biggr{)}^{\gamma_{n}/{b_{0}}}a_{n}^{L,T}(\mu_{0}),$
(10)
where $\gamma_{n}$ are the anomalous dimensions. For the current
$\bar{Q}\sigma_{\mu\nu}[z,-z]Q$ the anomalous dimentions are
$\displaystyle\gamma_{n}$ $\displaystyle=$ $\displaystyle
C_{f}\biggl{(}1+4\sum_{j=2}^{n+1}\frac{1}{j}\biggr{)},~{}~{}b_{0}=11-\frac{2}{3}n_{\rm
fl},~{}~{}C_{f}=\frac{4}{3},$ (11)
for the other currents
$\displaystyle\gamma_{n}$ $\displaystyle=$ $\displaystyle
C_{f}\biggl{(}1-\frac{2}{(n+1)(n+2)}+4\sum_{j=2}^{n+1}\frac{1}{j}\biggr{)}.$
It is clear that the DAs without primes contain only n-odd terms in series
(9). DAs with primes contain n-even terms in the series.
The constants defined in equations (1)-(4) are multiplicatively
renormalizable. Using formulas (9)-(LABEL:an_dimVA) one can determine the
evolution of these constants
$\displaystyle
f(\mu)=\biggl{(}\frac{\alpha_{s}(\mu)}{\alpha_{s}(\mu_{0})}\biggr{)}^{\gamma/{b_{0}}}f(\mu_{0}).$
(13)
For the constants $f_{\chi_{0}},f_{h},f_{\chi_{2}}$ the anomalous dimentions
$\gamma$ is equal to $8/3C_{f}$, for the constants
$f_{\chi_{1}},\tilde{f}_{\chi_{2}}$ $\gamma=3C_{f}$, for the constant
$f^{\prime}_{h}$ $\gamma=C_{f}$, for the constant $f^{\prime}_{\chi_{1}}$
$\gamma=0$. It should be noted that if the anomalous dimensions $\gamma$ of
the constants $f$ are factored from sum (9), the anomalous dimensions of the
remaining terms equal to the difference $\gamma_{n}-\gamma$.
## III The distribution amplitudes at the leading order approximation in
relative velocity.
In this section the DAs under study will be considered at the leading order
approximation in relative velocity of quark-antiquark pairs inside the mesons.
First let us consider the DA of the $\chi_{0}$ meson. The moments of this DA
can be represented as follows
$\displaystyle
f_{\chi_{0}}(pz)^{n+1}<\xi^{n}>_{\chi_{0}}=\langle\chi_{0}|\bar{Q}\hat{z}(-iz{\overset{\leftrightarrow}{D}})^{n}Q|0\rangle.$
(14)
To get the expressions for the constant $f_{\chi_{0}}$ and the moment
$<\xi^{n}>_{\chi_{0}}$ one needs to calculate the matrix element in the right
hand side. At the leading order approximation this calculation can be done
using projector Bodwin:2002hg ; Braaten:2002fi
$\displaystyle\bar{Q}(\bar{p})Q(p)\to\int
dq\frac{\varphi(-q^{2})}{\sqrt{3m_{Q}}}\frac{1}{4\sqrt{2}E(E+m_{Q})}(\hat{\bar{p}}-m_{Q})\Gamma(\hat{P}+2E)(\hat{p}+m_{Q}),$
(15)
where $P,q$ are the total and relative momentum of the $Q\bar{Q}$ pair,
$m_{Q}$ is the mass of the quark $Q$, $p=P/2+q,\bar{p}=P/2-q$,
$E^{2}=P^{2}/4=m_{Q}^{2}-q^{2}$. The matrix $\Gamma=\gamma_{5},\hat{e}_{S}$
for the spin singlet and spin triplet quark-antiquark pair correspondingly,
where $e_{S}$ is the spin polarization of this pair. The scalar products
$P\cdot e_{S}=0,P\cdot q=0$. In the center mass frame the $dq$ is reduced to
the $d^{3}{\bf q}/(2\pi)^{3}$ and the function $\varphi(-q^{2})$ is reduced to
the usual nonrelativistic wave function $\phi(\bf{q^{2}})$. For the $\chi_{0}$
meson the wave function $\varphi(\bf{q^{2}})$ can be written in the form
$\displaystyle\varphi({\bf{q^{2}}})=\frac{{\bf e_{S}\cdot
q}}{\sqrt{3}}\psi({\bf q}).$ (16)
At the leading order approximation in relative velocity the function $\psi(\bf
q)$ is universal function for all $P$-wave mesons. It is normalized as
$\displaystyle\int\frac{d^{3}{\bf q}}{(2\pi)^{3}}{\bf q^{i}}{\bf
q^{j}}|\psi({\bf q})|^{2}=\delta^{ij}.$ (17)
With this normalization of the function $\psi({\bf q})$, the function
$\varphi({\bf q})$ is normalized as
$\displaystyle\int\frac{d^{3}{\bf q}}{(2\pi)^{3}}|\varphi(\bf{q^{2}})|^{2}=1.$
(18)
The same normalization condition will be used for the wave functions of all
mesons under consideration. Using equations (14), (15) and (16) one gets the
result
$\displaystyle
f_{\chi_{0}}<\xi^{n+1}>_{\chi_{0}}=-2^{n+1}\sqrt{\frac{2}{m}_{Q}}\frac{A_{n+2}}{M^{n+1}_{\chi_{0}}}\frac{1}{n+3},$
(19)
where $M_{\chi_{0}}$ is the mass of the $\chi_{0}$ meson, $A_{n}$ equals to
$\displaystyle A_{n}=\int\frac{d^{3}{\bf q}}{(2\pi)^{3}}{|\bf q|}^{n}\psi({\bf
q}).$ (20)
It should be noted that because of the Coulombic part of the nonrelativistic
QCD potential the right hand side of equation (19) is ultraviolet divergent
Bodwin:1994jh ; Bodwin:2006dn . The moments of the DA in the left hand side
are QCD operators (see equation (14) ). In full QCD the Coulombic part of the
nonrelativistic potential corresponds to the rescattering of the quark-
antiquark pair of the QCD operators which is also ultraviolet divergent
Chernyak:1983ej . To control the divergences in the right and left hand sides
of equation (19) it is assumed that both sides are regularized within
dimensional regularization.
It is interesting to note that relation (19) is closely connected with
Brodsky-Huang-Lepage (BHL) Brodsky:1981jv procedure. This fact can be seen as
follows. Let us rewrite this relation as follows:
$\displaystyle\int
d\xi\xi^{n+1}\phi_{\chi_{0}}(\xi)\sim\frac{1}{n+3}\frac{A_{n+2}}{m_{Q}^{n+1}}\sim\frac{1}{n+3}\int{\bf
q^{2}}d{\bf q}\frac{{\bf q}^{n+2}}{m_{Q}^{n+1}}\psi({\bf q})\sim\int d^{3}{\bf
q}\biggl{(}\frac{q_{z}}{m_{Q}}\biggr{)}^{n+1}q_{z}\psi({\bf q}).$ (21)
Note that $q_{z}\psi({\bf q})$ is the $L_{z}=0$ component of the wave function
$\varphi({\bf q^{2}})$ which is the only component important for the leading
twist DA. So, the last relation can be written as follows
$\displaystyle\int d\xi\xi^{n+1}\phi_{\chi_{0}}(\xi)\sim\int
dq_{z}\biggl{(}\frac{q_{z}}{m_{Q}}\biggr{)}^{n+1}\int d^{2}{\bf
q}_{\perp}\varphi_{L_{z}=0}({\bf q^{2}}).$ (22)
Further, we change the variables in the right side of this equation
$\displaystyle{\bf q}_{\perp}\to{\bf q}_{\perp},\quad q_{z}\to\xi{M_{0}},\quad
M_{0}^{2}=\frac{M_{Q}^{2}+{\bf q}_{\perp}^{2}}{1-\xi^{2}}.$ (23)
Note also that $\xi\ll 1,{\bf q}_{\perp}\ll M_{Q}$ and at the leading order
approximation in relative velocity of the quark-antiquark pair in the meson
relation (22) can be written as follows
$\displaystyle\int d\xi\xi^{n+1}\times\phi_{\chi_{0}}(\xi)\sim\int
d\xi\xi^{n+1}\times\int d^{2}{\bf
q}_{\perp}~{}\varphi_{L_{z}=0}\biggl{(}\frac{M_{Q}^{2}\xi^{2}+{\bf
q}_{\perp}^{2}}{1-\xi^{2}}\biggr{)}.$ (24)
So, at this level of accuracy the DA is just
$\displaystyle\phi_{\chi_{0}}(\xi)\sim\int d^{2}{\bf
q}_{\perp}~{}\varphi_{L_{z}=0}\biggl{(}\frac{M_{Q}^{2}\xi^{2}+{\bf
q}_{\perp}^{2}}{1-\xi^{2}}\biggr{)},$ (25)
what coincides with BHL procedure.
It is not difficult to get the relations for the other DAs and mesons
$\displaystyle{\chi_{1}~{}\mbox{meson:}}~{}~{}~{}~{}\phi({\bf{q^{2}}})$
$\displaystyle=$ $\displaystyle\frac{\epsilon_{ijk}{{\bf e}^{i}{\bf
e}_{S}^{j}{\bf q}^{k}}}{\sqrt{2}}\psi({\bf q}),$
$\displaystyle{h~{}\mbox{meson:}}~{}~{}~{}~{}\phi({\bf{q^{2}}})$
$\displaystyle=$ $\displaystyle\frac{{\bf e\cdot q}}{\sqrt{2}}\psi({\bf q}),$
$\displaystyle{\chi_{2}~{}\mbox{meson:}}~{}~{}~{}~{}\phi({\bf{q^{2}}})$
$\displaystyle=$ $\displaystyle{\bf e}_{ij}\bf{e_{S}}^{i}{\bf q}^{j}\psi({\bf
q}),$ (26)
where $\bf e$ is the polarization three vector of the $\chi_{1}$ and $h$
mesons, ${\bf e}_{ij}$ is the polarization tensor of the $\chi_{2}$ meson.
Using these expressions one can get
$\displaystyle
f_{\chi_{1}}<\xi^{n+1}>_{\chi_{1}}=-2^{n+1}\sqrt{\frac{3}{m_{Q}}}\frac{A_{n+2}}{M^{n+1}_{\chi_{1}}}\frac{1}{n+3},$
$\displaystyle
f^{\prime}_{\chi_{1}}<\xi^{n}>^{\prime}_{\chi_{1}}=2^{n+1}i\sqrt{\frac{12}{m_{Q}}}\frac{A_{n+2}}{M^{n+1}_{\chi_{1}}}\frac{1}{(n+1)(n+3)},$
$\displaystyle
f_{h}<\xi^{n+1}>_{h}=-2^{n+1}\sqrt{\frac{6}{m_{Q}}}\frac{A_{n+2}}{M^{n+1}_{h}}\frac{1}{n+3},$
$\displaystyle
f^{\prime}_{h}<\xi^{n}>^{\prime}_{h}=2^{n+1}i\sqrt{\frac{6}{m_{Q}}}\frac{A_{n+2}}{M^{n+1}_{h}}\frac{1}{(n+1)(n+3)},$
$\displaystyle
f_{\chi_{2}}<\xi^{n+1}>_{\chi_{2}}=-2^{n+1}\sqrt{\frac{4}{m_{Q}}}\frac{A_{n+2}}{M^{n+1}_{\chi_{2}}}\frac{1}{n+3},$
$\displaystyle\tilde{f}_{\chi_{2}}<\tilde{\xi}^{n+1}>_{\chi_{2}}=-2^{n+1}\sqrt{\frac{6}{m_{Q}}}\frac{A_{n+2}}{M^{n+1}_{\chi_{2}}}\frac{1}{n+3}.$
(27)
The moments with primes correspond to the DAs with primes. As it was noted
above, since all the DAs without primes are $\xi$-odd only odd moments of
these DAs are nonzero. Similarly, for the DAs with primes only even moments
different from zero. For these reason we took $(n+1)$th moments of DAs without
primes and $n$th moments with primes, where n is assumed to be even. It should
be noted that since the function $\psi({\bf q})$ is universal for the $P$-wave
mesons, the $A_{n}$ is the same for all mesons under study. From relation (27)
one can conclude that all the constants $f_{i}$ can be expressed through the
only constant which will be designated as $F$
$\displaystyle
F={\sqrt{3}}{f_{\chi_{0}}}={\sqrt{2}}{f_{\chi_{1}}}=i\frac{f^{\prime}_{\chi_{1}}}{\sqrt{2}}=f_{h}=if^{\prime}_{h}=\sqrt{\frac{3}{2}}f_{\chi_{2}}={\tilde{f}_{\chi_{2}}}.$
(28)
From equations (27) one can also find that at the leading order approximation
the $n$th moments of all functions without primes coincide. This means that
these DAs are equal to one universal DA which will be designated as
$\Phi(\xi)$
$\displaystyle\Phi(\xi)=\phi_{\chi_{0}}(\xi)=\phi_{\chi_{1}}(\xi)=\phi_{h}(\xi)=\phi_{\chi_{2}}(\xi)=\tilde{\phi}_{\chi_{2}}(\xi).$
(29)
The same is true for all DAs with primes, which are equal to one function
which will be designated below as $\Psi(\xi)$
$\displaystyle\Psi(\xi)=\phi^{\prime}_{\chi_{1}}(\xi)=\phi^{\prime}_{h}(\xi).$
(30)
In addition, one can relate the moments of the $\Phi(\xi)$ to the moments of
the $\Psi(\xi)$ as follows
$\displaystyle<\xi^{n}>_{\Psi}=\frac{<\xi^{n+1}>_{\Phi}}{n+1}.$ (31)
It should be noted that the constants and DAs in relations (28)-(31) depend on
scale in a different way. This means that relations (28)-(31), which are valid
at not to large scale, will be violated at sufficiently large scale.
Recursive relation (31) determines the function $\Psi(\xi)$ through the
function $\Phi(\xi)$. One can guess the solution of this relation:
$\displaystyle\Psi(\xi)=-\int_{-1}^{\xi}dt~{}\Phi(t).$ (32)
To prove that (32) is the solution of relations (31) one should put this
function to the definition of the n-th moment (8) and integrate the resulting
expression by parts. It is seen from equations (29), (30) and (32) that all
DAs of the $P$-wave mesons are defined through the universal function
$\Psi(\xi)$. It should be noted that this fact results from the
nonrelativistic spin-symmetry, which holds at leading order in the heavy-quark
velocity. Below equations (29) and (32) will be used to build the models for
the function $\Phi(\xi)$ and $\Psi(\xi)$ of $P$-wave charmonia.
At the end of this section it is interesting to discuss the question about
relativistic corrections to the matrix elements involving $P$-wave quarkonia.
If we ignore the contribution coming from the higher Fock states, relativistic
corrections to the matrix involving, for instance, the $\chi_{0}$ meson is
given by the matrix element
$\displaystyle\langle
v^{n}\rangle_{P}=\frac{1}{(m_{Q}^{*})^{2}}\frac{\langle\chi_{0}|\chi^{+}\bigl{(}{\bf\sigma
D}\bigr{)}{\bf D^{2}}\psi|0\rangle}{\langle\chi_{0}|\chi^{+}\bigl{(}{\bf\sigma
D}\bigr{)}\psi|0\rangle},$ (33)
where $\psi$ and $\chi^{+}$ are Pauli spinor fields that annihilate a quark
and an antiquark respectively, ${\bf\sigma}$ are Pauli matrixes, $m_{Q}^{*}$
is the quark pole mass. Using equation (19) it is not difficult to obtain
general formula that connects the moment $\langle\xi^{n+1}\rangle$ with the
matrix element $\langle v^{n}\rangle_{P}$ at the leading order approximation
in relative velocity
$\displaystyle\langle
v^{n}\rangle_{P}=\frac{A_{n+2}}{A_{2}}+O(v^{n+2})=\frac{n+3}{3}\langle\xi^{n+1}\rangle+O(v^{n+2}).$
(34)
Although the derivation was done for the $\chi_{0}$ meson, at the leading
order approximation in relative velocity the matrix element $\langle
v^{n}\rangle_{P}$ is universal for all $P$-wave mesons.
## IV Charmonia distribution amplitudes within QCD sum rules.
In this section QCD sum rules approach Shifman:1978bx ; Shifman:1978by will
be applied to the calculation of the moments of the $P$-wave charmonia DAs.
The problem which can be met if one tries to apply QCD sum rules to the
calculation of these moments is that it is not possible to write QCD sum rules
for one DA. Commonly, the contribution of different mesons and different DAs
mix in QCD sum rules, what does not allow us to calculate the constants and
the moments of DAs (1)-(4) separately. Only some combinations of different
constants and moments can be extracted from QCD sum rules. For instance, two
point QCD sum rules for the currents
$\bar{Q}(x)\gamma_{\mu}(z{\overset{\leftrightarrow}{D}})Q(x)\cdot\bar{Q}(0)\gamma_{\mu}(z{\overset{\leftrightarrow}{D}})^{n+1}Q(0)$
can be written as follows
$\displaystyle\frac{f_{\chi_{c0}}^{2}\langle\xi^{n+1}\rangle_{\chi_{c0}}}{(m_{\chi_{c0}}^{2}+Q^{2})^{m+1}}+\frac{f_{\chi_{c2}}^{2}\langle\xi^{n+1}\rangle_{\chi_{c2}}}{(m_{\chi_{c2}}^{2}+Q^{2})^{m+1}}=\frac{1}{\pi}\int_{4m_{c}^{2}}^{s_{0}}ds~{}\frac{\mbox{Im}~{}\Pi_{\rm
pert}(s,n)}{(s+Q^{2})^{m+1}}+\Pi^{(m)}_{\rm npert}(Q^{2},n).$ (35)
The expressions for the $\mbox{Im}~{}\Pi_{\rm pert}(s,n)$, $\Pi^{(m)}_{\rm
npert}(Q^{2},n)$ will be given below (equations (37),(38)). It is seen from
this example that it is not possible to extract the constants
$f_{\chi_{c0}},f_{\chi_{c2}}$ or the moments
$\langle\xi^{n+1}\rangle_{\chi_{c0}},\langle\xi^{n+1}\rangle_{\chi_{c2}}$ from
(35) separately. Evidently, this strongly restricts the accuracy of the
calculation. The only QCD sum rules which are free from this problem are two
point sum rules for the currents
$\bar{Q}(x)\gamma_{\mu}\gamma_{5}(z{\overset{\leftrightarrow}{D}})Q(x)\cdot\bar{Q}(0)\gamma_{\mu}\gamma_{5}(z{\overset{\leftrightarrow}{D}})^{n+1}Q(0)$
$\displaystyle\frac{f_{h_{c}}^{2}\langle\xi^{n+1}\rangle_{h_{c}}}{(m_{h_{c}}^{2}+Q^{2})^{m+1}}=\frac{1}{\pi}\int_{4m_{c}^{2}}^{s_{0}}ds~{}\frac{\mbox{Im}~{}\Pi_{\rm
pert}(s,n)}{(s+Q^{2})^{m+1}}+\Pi^{(m)}_{\rm npert}(Q^{2},n),$ (36)
where $\mbox{Im}~{}\Pi_{\rm pert}(s,n)$, $\Pi^{(m)}_{\rm npert}(Q^{2},n)$ can
be written as
$\displaystyle\mbox{Im}~{}\Pi_{\rm
pert}(s,n)=\frac{3}{8\pi}v^{n+3}(\frac{1}{n+3}-\frac{v^{2}}{n+5}),~{}~{}~{}~{}v^{2}=1-\frac{4m_{c}^{2}}{s},$
(37) $\displaystyle\Pi^{(m)}_{\rm npert}(Q^{2},n)$ $\displaystyle=$
$\displaystyle\Pi_{1}^{(m)}(Q^{2},n)+\Pi_{2}^{(m)}(Q^{2},n)+\Pi_{3}^{(m)}(Q^{2},n),$
(38) $\displaystyle\Pi_{1}^{(m)}(Q^{2},n)$ $\displaystyle=$
$\displaystyle\frac{\langle\alpha_{s}G^{2}\rangle}{24\pi}(m+1)\int_{-1}^{1}d\xi~{}\biggl{(}\xi^{n+2}+\frac{n(n+1)}{4}\xi^{n}(1-\xi^{2})\biggr{)}\frac{(1-\xi^{2})^{m+2}}{\bigl{(}4m_{c}^{2}+Q^{2}(1-\xi^{2})\bigr{)}^{m+2}},$
$\displaystyle\Pi_{2}^{(m)}(Q^{2},n)$ $\displaystyle=$
$\displaystyle-\frac{\langle\alpha_{s}G^{2}\rangle}{6\pi}m_{c}^{2}(m^{2}+3m+2)\int_{-1}^{1}d\xi~{}\xi^{n+2}\bigl{(}1+3\xi^{2}\bigr{)}\frac{(1-\xi^{2})^{m+1}}{\bigl{(}4m_{c}^{2}+Q^{2}(1-\xi^{2})\bigr{)}^{m+3}},$
$\displaystyle\Pi_{3}^{(m)}(Q^{2},n)$ $\displaystyle=$
$\displaystyle\frac{\langle\alpha_{s}G^{2}\rangle}{384\pi}n(n+1)(m+1)\int_{-1}^{1}d\xi~{}\xi^{n}\frac{(1-\xi^{2})^{m+3}}{\bigl{(}4m_{c}^{2}+Q^{2}(1-\xi^{2})\bigr{)}^{m+2}}.$
In the calculation we take $Q^{2}=4m_{c}^{2}$ Reinders:1984sr . In the
numerical analysis of QCD sum rules the values of parameters $m_{c}$ and
$\langle\alpha_{s}G^{2}/\pi\rangle$ will be taken from paper Reinders:1984sr :
$\displaystyle m_{c}=1.24\pm
0.02~{}\mbox{GeV},~{}~{}\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle=0.012\pm
30\%~{}\mbox{GeV}^{4}.$ (39)
First sum rules (36) will be applied to the calculation of the constant
$f_{h_{c}}^{2}$. It is not difficult to express the constant $f_{h_{c}}^{2}$
from equation (36) at $n=0$ as a function of $m$. For too small values of $m$
($m<m_{1}$) there is large contributions from higher resonances and continuum
which spoil sum rules (36). Although for $m\gg m_{1}$ these contributions are
strongly suppressed, it is not possible to apply sum rules for too large $m$
($m>m_{2}$) since the contribution arising from higher dimensional vacuum
condensates rapidly grows with $m$ what invalidates our approximation. If
$m_{1}<m_{2}$ there is some region of applicability of sum rules (36)
$[m_{1},m_{2}]$ where the resonance and the higher dimensional vacuum
condensates contributions are not too large. Within this region
$f_{h_{c}}^{2}$ as a function of $m$ varies slowly and one can determine the
value of this constant. The value of the continuum threshold $s_{0}$ must be
taken so that to appear stability region $[m_{1},m_{2}]$. Our calculation
shows that for the central values of parameters (39) there exists stability
region for $s_{0}>(4.3$ GeV$)^{2}$. If the value of the continuum threshold
$s_{0}$ is varied in the region $s_{0}\in(4.3^{2},\infty)$ GeV2, the value of
the constant $f_{h_{c}}^{2}$ can be written as $f_{h_{c}}^{2}=(0.037\pm
0.005)$ GeV2. In addition to the uncertainty due to the variation of the value
of $s_{0}$, there are uncertainties due to the variation of the values of the
$m_{c}$ (which is $\pm 0.004$) and $\langle{\alpha_{s}}/{\pi}G^{2}\rangle$
(which is $\pm 0.001$). The last source of uncertainty is the radiative
corrections to the perturbative density $\mbox{Im}\Pi_{\rm pert}(s,n)$, which
will be estimated as $\alpha_{s}(m_{c})/\pi\sim 13\%$. Adding these
uncertainties in quadrature, one gets
$\displaystyle f_{h_{c}}^{2}(\mu\sim m_{c})=(0.037\pm 0.007)\mbox{GeV}^{2}.$
(40)
As it was noted above, the value of the constant $f_{h_{c}}^{2}$ is scale
dependent quantity. The characteristic scale of QCD sum rules is $\sim m_{c}$.
This means that the value of the constant $f_{h_{c}}^{2}$ is determined at the
scale $\sim m_{c}$, as it is shown in (40).
Next let us consider the moments
$\langle\xi^{3}\rangle_{h_{c}},\langle\xi^{5}\rangle_{h_{c}},\langle\xi^{7}\rangle_{h_{c}}$.
However, instead of considering QCD sum rules for $n=2,4,6$ we will consider
the ratios of sum rules at $n=2,4,6$ and the sum rules at $n=0$. Such approach
improves the accuracy of the calculation (see paper Braguta:2006wr for
details). The analisys similar to that for the constant $f_{h_{c}}^{2}$ gives
$\displaystyle\langle\xi^{3}\rangle$ $\displaystyle=$ $\displaystyle 0.18\pm
0.03,$ $\displaystyle\langle\xi^{5}\rangle$ $\displaystyle=$ $\displaystyle
0.050\pm 0.010,$ $\displaystyle\langle\xi^{7}\rangle$ $\displaystyle=$
$\displaystyle 0.017\pm 0.004.$ (41)
It should be noted that these moments are defined at the scale $\sim m_{c}$.
It should be also noted that the values of the moments (41) are in good
agreement with potential model estimation (see paper Braguta:2006wr ). For
instance, within Buchmuller-Tye potential model Buchmuller:1980su
$\langle\xi^{3}\rangle=0.18,\langle\xi^{5}\rangle=0.047,\langle\xi^{7}\rangle=0.016$;
within Cornell potential model Eichten:1978tg
$\langle\xi^{3}\rangle=0.16,\langle\xi^{5}\rangle=0.040,\langle\xi^{7}\rangle=0.013$.
Using values (41) one can find the matrix elements that control relativisitic
corrections to any process with $P$-wave charmonia in the initial or final
state. The relationships between these matrix elements and the moments are
given in equation (34). These relations are valid up to the higher order
relativistic corrections, which can be estimated as $\langle v^{2}\rangle$.
Taking into account this additional source of uncertainty one gets
$\displaystyle\langle v^{2}\rangle_{P}$ $\displaystyle=$
$\displaystyle\frac{5}{3}{\langle\xi^{3}\rangle_{P}}=0.30\pm 0.10,$
$\displaystyle\langle v^{4}\rangle_{P}$ $\displaystyle=$
$\displaystyle\frac{7}{3}{\langle\xi^{5}\rangle_{P}}=0.12\pm 0.04,$
$\displaystyle\langle v^{6}\rangle_{P}$ $\displaystyle=$
$\displaystyle\frac{9}{3}{\langle\xi^{7}\rangle_{P}}=0.051\pm 0.018.$ (42)
In the next sections results (41) will be used to build the model for the DAs
$\Phi(\xi)$ and $\Psi(\xi)$.
## V Model for charmonia distribution amplitudes.
To build the model of the function $\Phi(\xi,\mu\sim m_{c})$ we use Borel
version Shifman:1978bx ; Shifman:1978by of sum rules (36) but without
continuum contribution and power corrections
$\displaystyle f_{h_{c}}^{2}\langle\xi^{n+1}\rangle
e^{-{m_{h_{c}}^{2}}/{M^{2}}}=\frac{M^{2}}{4\pi^{2}}\int_{-1}^{1}d\xi~{}\xi^{n+2}~{}\frac{3}{4}(1-\xi^{2})\mbox{exp}\biggl{(}-\frac{4m_{c}^{2}}{M^{2}}\frac{1}{1-\xi^{2}}\biggr{)}.$
(43)
Evidently, within this approximation the function $\Phi(\xi,\mu\sim m_{c})$
can be written in the form
$\displaystyle\Phi(\xi,\mu\sim
m_{c})=c(\beta_{P})(1-\xi^{2})~{}\xi~{}\mbox{exp}\biggl{(}-\frac{\beta_{P}}{1-\xi^{2}}\biggr{)},$
(44)
where $c(\beta_{P})$ is a normalization constant and $\beta_{P}$ is some
constant. We propose function (44) as the model for DAs $\Phi(\xi,\mu\sim
m_{c})$. To fix the constant $\beta_{P}$ the value of the moment
$\langle\xi^{2}\rangle$ (41) will be used. Thus we get
$\beta_{P}=3.4^{+1.5}_{-0.9}$. The constant $c(\beta)$ can be determined from
normalization condition (6). The moments of the function (44) are
$\displaystyle\langle\xi^{3}\rangle$ $\displaystyle=$ $\displaystyle~{}0.18\pm
0.03,$ (45) $\displaystyle\langle\xi^{5}\rangle$ $\displaystyle=$
$\displaystyle~{}0.047\pm 0.014,$ $\displaystyle\langle\xi^{7}\rangle$
$\displaystyle=$ $\displaystyle 0.015\pm 0.006.$
Using the model for the function $\Phi(\xi,\mu\sim m_{c})$ and equation (32)
one can get the model of the DA $\Psi(\xi,\mu\sim m_{c})$
$\displaystyle\Psi(\xi,\mu\sim m_{c})=-\int_{-1}^{\xi}dt\Phi(t,\mu\sim
m_{c})=\frac{c(\beta_{P})}{2}(1-\xi^{2})^{2}~{}E_{3}\biggl{(}\frac{\beta_{P}}{1-\xi^{2}}\biggr{)},$
(46)
where the function $E_{3}(z)$ is the exponential integral function
$\displaystyle E_{3}(z)=\int_{1}^{\infty}dt\frac{e^{-zt}}{t^{3}}.$ (47)
To determine DAs (1)-(4) at a scale different from $m_{c}$ one should use
relations (29), (30) and than evolution equations for the corresponding DAs.
## VI Conclusion.
In this paper we have considered the leading twist distribution amplitudes
(DA) of $P$-wave nonrelativistic mesons. At the leading order approximation in
relative velocity of quark-antiquark pair inside the mesons these functions
can be expressed through one universal DA. We have derived the relations
between the DAs of $P$-wave nonrelativistic mesons and the universal DA.
As an example, we have considered the DAs of the $P$-wave charmonia mesons.
Within QCD sum rules we found the moments of the leading twist DA of $h_{c}$
meson, what allowed us to build the model of the universal DA for the $P$-wave
charmonia mesons.
In addition, we have found the relations between the moments and the
nonrelativistic QCD matrix elements that control relativistic corrections to
any amplitude involving $P$-wave charmonia. The calculation shows that
characteristic size of these corrections is $\sim 30\%$.
This work was partially supported by Russian Foundation of Basic Research
under grant 07-02-00417. The work of V. Braguta was partially supported by
CRDF grant Y3-P-11-05 and president grant MK-2996.2007.2. The work of A.
Luchinsky was partially supported by president grant MK-110.2008.2 and Russian
Science Support Foundation.
## References
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* (4) J. P. Ma and Z. G. Si, Phys. Lett. B 647, 419 (2007) [arXiv:hep-ph/0608221].
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|
arxiv-papers
| 2008-10-20T17:05:06 |
2024-09-04T02:48:58.354697
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V.V. Braguta, A.K. Likhoded, A.V. Luchinsky",
"submitter": "V Braguta",
"url": "https://arxiv.org/abs/0810.3607"
}
|
0810.3660
|
# Globular Cluster Systems in Nearby Dwarf Galaxies:
I. HST/ACS Observations and Dynamical Properties of Globular Clusters at Low
Environmental Density
Iskren Y. Georgiev1,2, Thomas H. Puzia3, Michael Hilker4, and Paul Goudfrooij2
1Argelander Institut für Astronomie der Universität Bonn, Auf dem Hügel 71,
D-53121 Bonn, Germany
2Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD
21218, USA
3Herzberg Institute for Astrophysics, 5071 Saanich Road, Victoria, BC V9E 2E7,
Canada
4European Southern Observatory, D-85748 Garching bei München, Germany E-mail:
iskren@astro.uni-bonn.de
(October 2008)
###### Abstract
We investigate the old globular cluster (GC) population of 68 faint
($M_{V}>-16$ mag) dwarf galaxies located in the halo regions of nearby ($\la
12$ Mpc) loose galaxy groups and in the field environment based on archival
HST/ACS images in F606W and F814W filters. The combined color distribution of
175 GC candidates peaks at $(V-I)=0.96\pm 0.07$ mag and the GC luminosity
function turnover for the entire sample is found at $M_{V,{\rm
TO}}\\!=\\!-7.6\pm 0.11$ mag, similar to the old metal-poor LMC GC population.
Our data reveal a tentative trend of $M_{V,{\rm TO}}$ becoming fainter from
late-type to early-type galaxies. The luminosity and color distributions of
GCs in dIrrs shows a lack of faint blue GCs. Our analysis reveals that this
might reflect a relatively younger GC system than typically found in luminous
early-type galaxies. If verified by spectroscopy this would suggest a later
formation epoch of the first metal-poor star clusters in dwarf galaxies. We
find several bright (massive) GCs which reside in the nuclear regions of their
host galaxies. These nuclear clusters have similar luminosities and structural
parameters as the peculiar Galactic clusters suspected of being the remnant
nuclei of accreted dwarf galaxies, such as M54 and $\omega$Cen. Except for
these nuclear clusters, the distribution of GCs in dIrrs in the half-light
radius vs. cluster mass plane is very similar to that of Galactic young halo
clusters, which suggests comparable formation and dynamical evolution
histories. A comparison with theoretical models of cluster disruption
indicates that GCs in low-mass galaxies evolve dynamically as self-gravitating
systems in a benign tidal environment.
###### keywords:
galaxies : dwarf – galaxies : irregular – galaxies : star clusters
††pagerange: Globular Cluster Systems in Nearby Dwarf Galaxies: I. HST/ACS
Observations and Dynamical Properties of Globular Clusters at Low
Environmental Density–LABEL:lastpage††pubyear: 2008
## 1 Introduction
Globular clusters (GCs) are observed in vast numbers in massive early-type
galaxies (e.g. Harris et al., 2006) and the integrated-light properties of GCs
were extensively studied in the last decade (e.g. Kissler-Patig et al., 1997;
Hilker et al., 1999; Kundu & Whitmore, 2001a, b; Larsen et al., 2001;
Goudfrooij et al., 2003; Puzia et al., 2002; Puzia et al., 2004; Harris et
al., 2006; Jordán et al., 2005; Jordán et al., 2007; Peng et al., 2006; Peng
et al., 2008) with the aim to understand how such populous GC systems were
assembled. These studies revealed the discovery of i) the bimodal
metallicity/color distribution of GCs (e.g. Ashman & Zepf, 1992; Gebhardt &
Kissler-Patig, 1999; Puzia et al., 1999; Kundu & Whitmore, 2001a, and
references therein) and ii) the presence of young and intermediate-age
globular clusters in merging, starburst, and quiescent galaxies (e.g. Whitmore
& Schweizer, 1995; Goudfrooij et al., 2001, 2007; Puzia et al., 2002;
Schweizer, 2002).
From the point of view of GC system assembly, multiple scenarios for galaxy
formation have been presented to explain these findings. The first scenario is
the hierarchical build-up of massive galaxies from pregalactic dwarf-sized gas
fragments Searle & Zinn (1978) in which the metal-poor GCs form in situ while
the metal-rich GCs originate from a second major star formation event Forbes
et al. (1997) or infalling gaseous fragments, the so-called mini-mergers at
high redshifts Beasley et al. (2002). The dissipative merger scenario (e.g.
Schweizer, 1987; Ashman & Zepf, 1992; Bekki et al., 2002) assumes that the
metal-poor GCs were formed early in “Searle-Zinn” fragments while metal-rich
(and younger) GCs formed during major merger events of galaxies with
comparable masses (Spiral–Spiral, Elliptical–Spiral, etc.). The
dissipationless merging and accretion scenario incorporates the classical
monolithic galaxy collapse in which the galaxies and their GCs were formed
after which the dwarf galaxies and their GCs were accreted by giant galaxies
Zinn (1993); Côté et al. (1998, 2002); Hilker et al. (1999); Lee et al.
(2007). For detailed discussions on the GC systems formation and assembly
scenarios and properties of GCs in various galaxy types we refer the reader to
Kissler-Patig (2000); van den Bergh (2000); Harris (2001, 2003); Brodie &
Strader (2006).
Dwarf galaxies play an important role in assessing the likelihoods of the
formation scenarios mentioned above. The prediction of the hierarchical growth
of massive galaxies through merging of many dwarf-sized fragments at early
times is backed by the steepening of the faint-end slope of the galaxy
luminosity function with redshift (e.g. Ryan et al., 2007; Khochfar et al.,
2007). In addition, the observed fraction of low-mass irregular galaxies
increases with redshift (e.g. Conselice et al., 2008), and Hubble Ultra Deep
field studies show that dwarf-sized irregular galaxies dominate at $z\ga 6$
(e.g. Stiavelli et al., 2004). Hence, the oldest stellar populations in nearby
dIrr galaxies may thus represent the probable surviving early building blocks
of massive galaxies.
Evidence for accretion events of dwarf galaxies and their dissolution in the
Galactic potential is manifested by a few tidal stellar streams observed in
the Milky Way, M31, and other nearby galaxies (e.g. Ibata et al., 2001;
Grillmair, 2006; Liu et al., 2006; Martinez-Delgado et al., 2008, and
references therein). A well-known example of a recent accretion event is the
Sagittarius dwarf galaxy, which is currently being added to the Milky Way halo
together with its globular clusters (e.g., Ibata et al., 1994). Many more such
minor accretion/merger events of dwarf sized galaxies are expected during the
hierarchical evolution of giant galaxies. The Pipino et al. (2007) galaxy
evolution and GC formation model, in which a low number of blue (metal-poor)
GCs in massive galaxies are predicted to form in the initial (”monolithic”)
collapse, contrasts with observations and suggests that subsequent accretion
of such metal-poor GCs from low-mass dwarfs is required (Côté et al., 1998,
2002). However, the observed large ratio between metal-poor GCs and metal-poor
field halo stars in massive galaxies Harris (2001); Harris & Harris (2002)
implies that, if such dwarfs were later accreted, they must have had large GC
specific frequencies $S_{N}$111$S_{N}$ is the number of GCs per unit galaxy
luminosity (Harris & van den Bergh, 1981). in order to keep the GC-to-field-
star ratio high. This is in line with the high specific frequencies observed
in low-mass early and late-type dwarfs in galaxy clusters Durrell et al.
(1996); Miller et al. (1998); Seth et al. (2004); Miller & Lotz (2007);
Georgiev et al. (2006); Peng et al. (2008) and in group/field environments
Olsen et al. (2004); Sharina et al. (2005); Georgiev et al. (2008); Puzia &
Sharina (2008).
In this series of papers we will present results from an analysis of old GCs
in 68 dwarf galaxies in nearby loose groups and in the field. The current
paper is organized as follows. In §2 we describe the target sample, discuss
the completeness and contamination, and define the globular cluster candidate
selection. In §3 the colors, luminosities, and structural parameters are
compared with old Local Group GCs, and a discussion of the dynamical state of
the globular clusters in our sample dwarfs. Section §4 summarizes our results.
## 2 Data
### 2.1 Observations
The current study is based on HST/ACS archival data from programs SNAP-9771
and GO-10235, conducted in HST cycles 12 and 13, respectively (PI: I.
Karachentsev), and archival HST/ACS data (GO-10210, PI: B. Tully) that were
described in Georgiev et al. (2008). A summary of the various datasets is
provided in Table LABEL:tab:sum.
Table 1: Summary of target sample. The table shows the number of morphological galaxy types observed in each program and the total sample. The bottom row shows the number of galaxies in which we found globular cluster candidates. Program ID | dIrr | dE | dSph | Sm
---|---|---|---|---
SNAP 9771 | 26 | 2 | 4 | 2
GO-10235 | 10 | 1 | 1 | 3
GO-10210 | 19 | – | – | –
$\Sigma_{\rm all}$ | 55 | 3 | 5 | 5
$\Sigma_{\rm w/GC}$ | 30 | 2 | 2 | 4
Non-dithered $2\times 600$ s F606W and $2\times 450$ s F814W exposures for
each galaxy were designed to reach the Tip of the Red Giant Branch (TRGB) at
$M_{I}=-4.05$ or $M_{V}=-2.75$ to $-1.45$ mag in the range [Z/H] $=-2.2$ to
$-0.7$ dex Da Costa & Armandroff (1990) and provide TRGB distances to the
sample dwarf galaxies as published in Karachentsev et al. (2006); Karachentsev
et al. (2007). All galaxies reside in the isolated outskirts of nearby ($\leq
6$ Mpc) galaxy groups (Sculptor, Maffei 1& 2, IC 342, M 81, CVn I cloud) with
the exception of 3 very isolated dwarfs within 12 Mpc.
The program SNAP-9771 observed 34 low-mass galaxies, of which 26 are dIrrs, 2
dEs, 4 dSphs and 2 late-type spiral dwarf galaxies. The program GO-10235
contains 15 targets of which 10 are dIrrs, 1 dE, 1 dSph and 3 late-type
spirals. All galaxies in our final sample have published TRGB distance
measurements Karachentsev et al. (2006); Karachentsev et al. (2007). A summary
of their general properties is given in Table 2. In column (1) the galaxy IDs
are listed, and in columns (2) and (3) their coordinates; columns (4) and (5)
contain the morphological classifications from LEDA222http://leda.univ-
lyon1.fr/ and NED333http://nedwww.ipac.caltech.edu/, (6) and (7) list the
distances and distance moduli, in (8) is the foreground galactic extinction
($E_{B-V}$), (9) and (10) give the absolute magnitudes and colors (measured in
this work, see Section 3.1), and in column (11) we provide their HI mass
(obtained from LEDA).
The additional archival HST/ACS data consists of 19 Magellanic-type dIrr
galaxies residing in nearby ($2-8$ Mpc) associations composed mainly of dwarfs
with similar luminosities as the target galaxies, with the only exception that
the previous sample was composed of dwarfs located in isolated associations of
only few dwarfs Tully et al. (2006), while the dwarfs in our new dataset are
in the halo regions of groups which contain massive dominant galaxies such as
M 83 and Cen A (Karachentsev et al., 2006; Karachentsev et al., 2007, see
Table 2). The HST/ACS F606W and F814W imaging of the previous data was also
designed to reach the tip of the red giant branch (TRGB) and measure TRGB
distances Tully et al. (2006).
In total, our sample consists of 55 dIrrs, 3 dEs, 5 dSphs, and 5 Sm galaxies
in the field environment. All dwarfs have apparent diameters smaller than the
HST/ACS field of view which provided us with a good sampling of their GCSs.
### 2.2 Data Reduction and Photometry
Image processing and photometry was performed in a manner identical to that in
our previous study described in Georgiev et al. (2008). In the following we
briefly summarize the basic steps. We retrieved archival HST/ACS images in
F606W and F814W filters which were processed with the ACS reduction pipeline
and the multidrizzle routine Koekemoer et al. (2002). To improve the object
detection and photometry we modeled and subtracted the underlying galaxy light
using a circular aperture median kernel of 41 pixel radius. This choice of
filter radius at 3 Mpc (the closest galaxy) corresponds to $\sim 30$ pc, which
is ten times the typical GC $r_{\rm h}$. This is large enough that after
median galaxy light subtraction, the structure of the GC candidates will be
preserved.
Object detection ($4\sigma$ above the background) and initial aperture
photometry (in 2,3,5 and 10 pixel radius) was performed with the IRAF444IRAF
is distributed by the National Optical Astronomy Observatories, which are
operated by the Association of Universities for Research in Astronomy, Inc.,
under cooperative agreement with the National Science Foundation.
DAOPHOT/DAOFIND and PHOT routines. Conversion from instrumental magnitudes to
the STMAG system and aperture corrections from 10 pixel to infinity were
performed using the ACS photometric calibration prescription by Sirianni et
al. (2005).
### 2.3 GC Candidate Selection
The first step in our GC candidate555In the following we refer to globular
cluster candidates as GCs. selection was based on the expected colors for old
($>\\!4$ Gyr) Simple Stellar Population (SSP) models Anders & Fritze-v.
Alvensleben (2003); Bruzual & Charlot (2003) and typical colors of old
Galactic GCs, namely objects in the color range $-0.4\leq$(F606W-F814W)$\leq
0.2$ (corresponding to $0.7<V-I<1.5$). Second, we have imposed a faint
magnitude cut-off at the TRGB ($M_{V}=-2.5$, $M_{I}=-4.05$; Da Costa &
Armandroff, 1990; Lee et al., 1993) converted to apparent magnitudes according
to the distance to each galaxy as derived by Tully et al. (2006) and
Karachentsev et al. (2006); Karachentsev et al. (2007). This is $\sim\\!5$ mag
fainter than the typical GC luminosity function turnover magnitude for metal-
poor Galactic globular clusters at $M_{V,\,{\rm TO}}=-7.66\pm 0.1$ mag (e.g.
Di Criscienzo et al., 2006).
Due to the deep ACS imaging and its high spatial resolution (1 ACS pixel being
equivalent to 1.2 pc at a mean distance to our sample dIrrs of $5$ Mpc), a
typical Local Group GC with half-light radius of $r_{\rm h}=3$ pc is resolved.
We used the imexam task within IRAF to confine the initial GC selection to
round objects (FWHM${}_{F606W}\simeq$ FWHMF814W) with an imexam ellipticity
$\epsilon\leq 0.15$ within a fixed $r=5$ pix aperture radius and a Moffat
index $\beta=2.5$, typical for stellar profiles. As we showed in Georgiev et
al. (2008), the upper imexam $\epsilon$ limit allows selection of GCs with
true ellipticity up to $\epsilon<0.4$. The final ellipticities and half-light
radii ($\epsilon,\ r_{\rm h}$) were measured with ishape (see Sect. 2.4).
The last GC selection step is to measure the differences between the aperture
photometry determined using 2, 3, and 5 pixel radii. These m${}_{2}-$m3 vs.
m${}_{2}-$m5 concentration indices separate well the unresolved foreground
stars from extended GCs and the majority of the contaminating background
galaxies during the automated GC selection.
We have detected 57 old GC candidates in 17 dIrrs, 6 GCs in 2 dSph, 25 GCs in
2 dEs and 25 GCs in 4 Sm dwarfs. Therefore, in combination with the 60 GCs
from Georgiev et al. (2008) we have in total 117 GCs in 30 dIrrs. The total
number of GCs in our sample contains: 119 in dIrrs, of which 64 are bGCs and
42 rGCs; in dEs/dSphs the number of GCs sums up to 31 with 21 bGCs and 10
rGCs; the low-mass late-type spirals contain 26 GCs in total, with 13 bGCs and
11 rGCs. The properties of all GC candidates are listed in Table 3 with the
cluster ID and its coordiantes in columns (2) and (3); cluster absolute
magnitude and foreground dereddened color $(V-I)_{0}$ in (3) and (4); cluster
half-light radius and ellipticity ($r_{\rm h},\ \epsilon$, respectively) in
(5) and (6) as derived in Sect. 2.4 and their projected distance from the
galactic center ($r_{\rm proj}$) and the normalized to the effective galaxy
radius projected distance ($r_{\rm proj}/r_{\rm eff}$) in (7) and (8),
respectively. Finding charts of all GCs are presented in the Appendix in
Figure 9.
The final GC photometry was derived from curve of growth analysis for each
individual object from images iteratively cleaned from contaminating sources
within the photometric apertures. A detailed description of this iterative
procedure is provided in Georgiev et al. (2008) and will not be repeated here.
Conversion from STMAG to Johnson/Cousins was performed using the
transformation and dereddening coefficients (for a G2 star) provided in
Sirianni et al. (2005). The Galactic foreground reddening $E_{B-V}$ toward
each galaxy was obtained from the Schlegel et al. (1998) maps.
### 2.4 Structural parameters
To derive the GC structural parameters (half-light radius, $r_{\rm h}$, and
ellipticities, $\epsilon$) we have used the ishape algorithm Larsen (1999),
which models the object’s surface brightness profile with analytical models
convolved with a subsampled point spread function (PSF). Since most clusters
are marginally resolved, a very good knowledge of the ACS PSF is required to
obtain reliable measurements of their sizes. Testing the influence of the PSF
library, we performed a comparison between the $r_{\rm h}$ measured using
TinyTim666http://www.stsci.edu/software/tinytim/tinytim.html
The TinyTim software package takes into account the field-dependent WFC
aberration, filter passband effects, charge diffusion variations, and varying
pixel area due to the significant field distortion in the ACS field of view
Krist & Hook (2004). model PSFs (TT-PSF) and the empirical, local PSF (lPSF),
built from stars in the image of each galaxy.
The ten times sub-sampled TT-PSF was convolved with the charge diffusion
kernel. The position-dependent lPSFs was built from images typically having
more than 20 isolated stars across the ACS field. For a few galaxies with a
very low point-source density, we use the lPSF library from ESO 223-09, which
has the maximum number of good PSF stars of all galaxies in our sample (69).
Both PSF types were used with ishape to model the cluster profiles for the all
available King model concentration parameters ($C=r_{\rm tidal}/r_{\rm
core}=5,15,30,100$). The final cluster parameters were adopted from the best
$\chi^{2}$ fit model for both PSFs. For the final cluster half-light radius we
have adopted the geometric mean from the ishape FWHMs measured along the semi-
major and semi-minor axis ($w_{y}$ and $w_{y}$), i.e. $r_{\rm h}=r_{{\rm
h},w_{y}}\sqrt[]{w_{x}/w_{y}}$.
Figure 1 shows a comparison between the $r_{\rm h}$ derived from the best-fit
TT-PSF and lPSF models. It is obvious that for every concentration parameter
the $r_{\rm h,lPSF}$ values are smaller than the $r_{\rm h,TT}$ values (see
$\Delta r$ values in Fig. 1). This shows that the TinyTim PSFs are sharper
than the empirical lPSFs which represent the actual imaging conditions (incl.
charge diffusion, telescope focus breathing, drizzle effects, etc.).
Figure 1: Comparison between $r_{\rm h}$ derived with TinyTim (TT) and
empirical PSFs build from the images (lPSF). From top to bottom are shown
relations for different King model concentration parameters and the best
$\chi^{2}$ fit models, respectively. The lower right panel shows the best
$\chi^{2}$ from our previous study Georgiev et al. (2008). The solid line
shows the 1:1 relation.
In this study we are compiling data from our previous study Georgiev et al.
(2008) in which the $r_{\rm h}$ were derived from the best $\chi^{2}$ TT-PSF
due to the lack of enough good PSF stars for modeling the lPSF in the field of
those extremely isolated dIrrs. Therefore, we convolved the GCs from that
study with the local PSF derived from the image of ESO 223-09. We point out
that variable telescope focus changes may introduce unknown systematics. In
the bottom right panel of Fig. 1 we show the $r_{\rm h}$ derived with the
TinyTim PSF versus the empirical PSF for the best $\chi^{2}$ model. As can be
seen, the values computed with TT-PSFs are $\sim 0.4$ pix larger than the
corresponding sizes computed with the empirical lPSFs.
A similar difference between the empirical lPSF and the TT-PSF of $\sim 0.5$
pix (the $r_{\rm h,TT}$ being larger) was also previously reported in the ACS
study of the Sombrero galaxy by Spitler et al. (2006) (their Fig.2), where a
fixed concentration index $C=30$ was assumed. In the following, we will use
the structural parameters of GCs obtained with the empirical lPSFs.
### 2.5 Completeness Tests
Figure 2: Left panels: Color-magnitude diagrams of the artificial clusters
with King profile concentrations $C=5,15,30$ and 100 from top to bottom,
respectively. The solid line rectangle indicates the color-magnitude region
used to generate artificial clusters and their subsequent selection. Right
panels: Magnitude concentration index of the artificial clusters as defined by
their 2,3 and 5 pixel aperture radius magnitudes. The rectangle defines the
final cluster selection. Figure 3: Completeness as a function of synthetic
cluster magnitude for $V$ and $I$ in the top and bottom panels, respectively.
To convert from instrumental STMAG to the Johnson/Cousins photometric system,
we have used the transformation relations tabulated in Sirianni et al. (2005).
To convert from apparent to absolute magnitudes, we have used the distance
modulus of $m-M=29.06$ mag to ESO 223-09 as derived from the TRGB measurement
by Karachentsev et al. (2007).
Artificial clusters were modeled with the mkcmppsf and mksynth routines in the
baolab package Larsen (1999). The procedure includes convolution of empirical
PSFs with analytic KING profiles for all concentrations ($C=5$, 15, 30 and
100). For the completeness tests we chose the galaxy ESO 223-09 which has the
highest surface brightness, foreground extinction ($E_{B-V}\\!=\\!0.260$) and
foreground star contamination in our sample (cf. Fig. 9). We point out that
this is the most conservative estimate of the photometric completeness among
our sample galaxies, i.e., all other sample galaxies have more complete GC
samples. To sample the parameter space typical for colors and luminosities of
old GCs ($t\geq 4$ Gyr) we generated synthetic clusters in the range $21<$
F606W $<26$ mag (STMAG) with the fainter limit matching the TRGB at the
distance of ESO 223-09 Karachentsev et al. (2007). The colors of the
artificial clusters were spread over the range $-0.4<F606W-F814W<0.25$
($0.5<V-I<1.5$) in fourteen color bins with 0.05 mag step. For each color bin
200 arificial clusters were randomly placed across the synthetic image. This
step was repeated 100 times thus providing 20 000 objects per color bin or in
total 280 000 artificial clusters for the completeness test for one King
profile concentration. The same procedure was applied to all four
concentration parameters provided within baolab.
The synthetic images consist of the modeled clusters on a constant (0 ADUs)
background, matching the mean background value of the original ACS images.
Every synthetic image, containing 200 objects, was then added to the original
ESO 223-09 image, thus preserving the image background level and noise
characteristics. The final images were then subjected through the same
routines for object detection, photometry, and GC selection as the ones used
to define the science sample.
In Figure 2 we show the color-magnitude diagrams (CMDs) and concentration
parameter plots of the retrieved artificial clusters for the four King models.
The rectangle in the left-column panels indicates the region in which the
artificial clusters were modeled and later selected. After applying the color-
magnitude and the imexam FWHM and $\epsilon$ cuts, the final cluster selection
was based on their concentration parameters as derived from the difference
between their magnitudes in 2, 3 and 5 pixel aperture radius. The right-column
panels of Figure 2 show the ${\rm m}_{2}-{\rm m}_{3}$ vs. ${\rm m}_{2}-{\rm
m}_{5}$ diagrams. The objects that survived all the selection criteria were
used to compute the completeness as a function of the cluster magnitude. In
Figure 3 we present the completeness functions in $V$ and $I$ for all King
models.
Figure 3 shows that for the case of a galaxy with high surface brightness,
strong foreground reddening and foreground star contamination (cf. Fig. 9),
the $90\%$ completness limit for extended sources is reached at
$M_{V}\simeq-4.5$ mag. As expected, the completeness is a function of the
cluster size, in the sense that more extended clusters suffer stronger
incompleteness as their detection and correct magnitude measurement are easily
affected by the variable galaxy background and bright foreground stars. We
point out that this is the least complete case in our sample and that all
other target galaxies have fainter completeness limits. For all our target
galaxies we sample more than 99% of the total GCLF, in terms of luminosity and
mass.
### 2.6 Background Contamination
Background contamination from bulges of compact galaxies at intermediate
redshifts, which resemble the colors and structural appearance of GCs need to
be taken into account. We have already assessed the expected contamination for
the ACS field of view and objects with identical magnitude and size
distributions as GCs in galaxies within 8 Mpc using the Hubble Ultra Deep
Field (HUDF)777http://heasarc.nasa.gov/W3Browse/all/hubbleudf.html Georgiev et
al. (2008). The expected number of contaminating background galaxies increases
significantly for $V_{0}\geq 25$ mag ($M_{V}\geq-4$), well beyond the
luminosity distribution of GCs. Nevertheless, down to this limit a
contamination of up to 2 objects per field is expected.
## 3 Analysis
### 3.1 Integrated Galaxy Magnitudes
Only six of the dwarf galaxies in our sample have total $V-$band magnitudes
available in the literature while the majority of them only have $B$
magnitudes. However, good knowledge of the $V$-band magnitudes is required to
estimate their GC specific frequencies for a robust comparison with previous
studies. Hence, we performed integrated-light photometry on the ACS images and
derived their total magnitudes.
Bright foreground stars and background galaxies were masked out and replaced
with the local background level and noise. The median smoothed image of each
galaxy was used to estimate the center of the galaxy. However, for most dwarfs
the derived centers were not representative of the visual center of the
extended galaxy light, but rather the region with the strongest starburst.
Thus, we adopted the geometric center of the isointensity contour at the
$10\sigma$ level above the background as the galaxy center. Since the galaxies
rarely extend beyond 1500 pixel radius, the estimate of the sky value
determined from the median value measured at the image corners, is
representative for the true photometric background. To measure the galaxy
magnitudes we used the ellipse task within IRAF. The initial values for galaxy
ellipticity and position angle were taken from NED. For dwarfs without
published values for those parameters, we estimated the center in interactive
mode with ellipse, i.e. to approximate the ellipticity and position angle of
the extended light (at the $10\sigma$ isointensity contour). We have measured
the total galaxy magnitudes within the ellipse with radius at the Holmberg
radius ($\mu_{B}=26.5$) quoted in NED.
Magnitudes were adopted from NED for two spiral galaxies (ESO 274-01 and NGC
247, see Table 2) which were extending beyond the ACS field of view as well as
for NGC 4605 which was off-center and for KKH 77 which was contaminated by a
very bright (and saturated) foreground star. These magnitudes from NED are
(deprojected) total magnitudes corrected for internal reddening assuming
$E_{B-V}=0.05$ mag. For IKN and VKN, two extremely low surface brightness
dwarfs ($M_{V}\approx-11.5$ and $-10.5$ mag, respectively), we could not
reliably determine their magnitudes due to an extremely bright foreground star
in the former and the very low surface brightness of the latter (close to the
level of background fluctuations and measurement errors). Their $V-$band
magnitudes were derived from their $B-$band magnitudes assuming an average
$B-V=0.45$ mag estimated from the rest of the dwarfs. Due to the fact that
UGCA 86 was centered in the middle of ACS chip 2, its Holmberg radius extended
beyond the ACS field by $\sim\\!1\arcmin$ and, therefore, we had to
extrapolate its magnitude with the gradient of the last 3 magnitude bins of
its curve of growth.
Finally, the derived magnitudes were compared with the published ones for six
dwarfs in our sample (DDO 52, ESO 269-58, IC 4662, NGC 5237, 4068, 4163) and
found consistent within the measurement errors. The magnitudes of all galaxies
are listed in Table 2.
Figure 4: Combined color-magnitude diagrams for GC candidates in our sample
galaxies split by the morphological type of the host. GCs in dIrr galaxies are
shown in the upper left panel; the other panels show the corresponding
distributions for dEs/dSphs, Sm, and of the combined sample. Filled and semi-
filled circles in the dIrr-panel indicate GC candidates from the current and
Georgiev et al. (2008) study. Open squares are old LMC GCs (data from
McLaughlin & van der Marel, 2005). The vertical dotted lines mark the color
separating blue and red GC candidates at $V\\!-\\!I=1.0$ mag. Thick solid and
dashed curves show Bruzual & Charlot (2003) SSP evolutionary tracks for two
clusters masses $M_{\rm cl}=3\cdot 10^{4}$ and $5\cdot 10^{4}\,M_{\odot}$ from
3 to 14 Gyr (left to right) and metallicities [Fe/H] $=-2.25$ and $-1.64$ dex,
respectively. Histograms illustrate the GC candidate color and luminosity
distributions. A reddening vector shows the effect of $E_{B-V}=0.2$ of
internal extinction.
### 3.2 Color Distributions
In Figure 4 we show the color-magnitude diagrams for GCs in dIrr, dE/dSph, and
Sm galaxies, and the composite sample, combining the results from this work
with our identically analyzed sample from Georgiev et al. (2008). The color
and luminosity distributions are illustrated as histograms in the top and
right sub-panels, where the curves indicate non-parametric Epanechnikov kernel
probability density estimates. We subdivide the samples in color into blue GCs
(bGCs) with $V\\!-\\!I\leq 1.0$ mag and red GCs (rGCs) with $V\\!-\\!I>1.0$
mag, which include the sub-sample of extremely red objects with colors
$V\\!-\\!I>1.4$ mag888The faintest clusters with $M_{V}\leq-6$ and
$(V\\!-\\!I)_{0}>1.0$ mag are likely background contaminants which passed our
selection criteria. Their nature will be confirmed by follow-up spectroscopy..
This division is motivated by the average location of the gap between the blue
and red color peaks of rich GCSs in massive early-type galaxies (e.g. Gebhardt
& Kissler-Patig, 1999).
An intriguing feature of Figure 4 is the lack of faint ($M_{V}\ga-6$) blue GCs
in our sample dIrr galaxies. Our artificial cluster tests show that this is
not a completeness effect as our 90% completeness limit is at $M_{V}=-4.5$ mag
(see Sect. 2.5). Finding one or more GCs in these faint dwarfs would increase
their already high specific frequencies even more (the specific frequencies
will be discussed in a subsequent paper of this series). On the other hand, we
do observe few clusters at $M_{V}\simeq-5.5$ to $-6.5$ mag and
$(V\\!-\\!I)<1.0$ mag in galaxies with dE/dSph and Sm morphology999We note
that all absolute magnitudes were calculated using the newly determined
distance moduli by Tully et al. (2006) and Karachentsev et al. (2006);
Karachentsev et al. (2007) which are fainter by $\sim\\!0.5$ mag on average
from the values listed in Karachentsev et al. (2004), hence, the GCs have
brighter absolute magnitudes..
One plausible explanation for the apparent lack of faint blue clusters is that
the metal-poor ($V-I<1.0$ mag) and low-mass GCs ($M_{V}>-6.5$ mag) are
actually younger (age $<4$ Gyr) than our selection criteria. If this is
spectroscopically confirmed would imply that GCs in dIrrs formed a bit later
than GCs in more massive early-type galaxies. To explore this effect we
investigate stellar evolution fading according to the SSP model of Bruzual &
Charlot (2003). Passive aging of a simple stellar population from 3 to 14 Gyr
reddens its $(V-I)_{0}$ color by $\sim 0.15$ mag and fades its $V-$band
luminosity by $\sim 1.5$ mag. This is illustrated in Figure 4 with
evolutionary tracks for two metallicities ([Fe/H] $=-2.25$ and $-1.64$ dex)
and two cluster masses ($M_{\rm cl}=3\cdot 10^{4}$ and $5\cdot
10^{4}\,M_{\odot}$) If some of the bluest GCs in our sample are indeed younger
clusters (with $t\la 4$ Gyr) they will end up on the faint-end of the
luminosity function at an age of 14 Gyr. However, those clusters would have to
have unusually low metallicities ([Fe/H] $\leq-2.0$), an interesting result
that calls for spectroscopic confirmation. Previous spectroscopic analyses of
GCs in other dwarf galaxies show that their blue colors are in general
consistent with old ages and low metallicities. However, some of these
clusters show spectroscopic intermediate ages ($\sim 4$ Gyr), in particular in
dIrrs Puzia & Sharina (2008). An alternative (though perhaps less likely)
explanation for the lack of old, metal-poor, low-mass clusters in dIrr
galaxies may be selective reddening of such objects. The reddening vector in
the CMDs of Figure 4 shows that a reddening of $E_{B-V}\\!=\\!0.2$ mag is
enough to dislocate intrinsically blue GCs to the red GC sub-sample. The age
and reddening effects should be tested with spectroscopic observations.
The probability density estimates, as shown in Figure 4, for all galaxy
subsamples give the highest probability values in the range
$(V-I)_{0,\rho}\approx 0.9-1.0$ mag and $M_{V,\rho}\approx-7.5$ to $-6.5$ mag.
Gaussian fits to the smoothed bGC luminosity function give peak values in the
range $M_{V,{\rm TO}}=-7.6$ to $-7.0$ mag with $\sigma_{\rm GCLF}=1.2-1.5$ mag
(see Sect. 3.3). The color distributions of GCs in our sample galaxies peak at
values typically found in other low-mass dwarfs Seth et al. (2004); Sharina et
al. (2005); Georgiev et al. (2006); Georgiev et al. (2008), and are very
similar to the canonical blue peak color of rich GCSs in massive early-type
galaxies (e.g. Peng et al., 2008). For comparison we also show ten of the
brightest old LMC GCs (McLaughlin & van der Marel, 2005). There are three more
LMC clusters that are fainter than $M_{V}=-6.5$ mag, however, without
available $V\\!-\\!I$ colors.
Figure 5: Luminosity distributions of selected blue and red GCs in the top and
bottom panels, respectively. Each sub-sample was split at $V-I\\!=\\!1.0$ mag
in bGCs and rGCs. Shaded histograms show the corresponding GC candidates
luminosity distribution, while open histograms are the total samples for a
given host morphology. In all panels, thick and dotted curves are non-
parametric Epanechnikov-kernel probability density estimates. The solid line
open histogram shows in the upper left panel the luminosity distribution of
old LMC GCs for comparison (data from McLaughlin & van der Marel, 2005).
### 3.3 Luminosity Functions
Figure 5 shows the luminosity functions of the blue and red GCs in the top and
bottom sub-panels, respectively, which were split at $V-I\\!=\\!1.0$ mag.
Thick curves are Epanechnikov kernel probability density estimates. We find
that the rGCs are biased toward fainter luminosities compared to the bGC sub-
sample. This indicates that these objects are either strongly affected by
background contamination or intrinsically fainter than their bGC counterparts.
Gaussian fits to the smoothed histogram distributions return $M_{V,{\rm
TO}}=-7.56\pm 0.02$ mag and $\sigma_{\rm GCLF}=1.23\pm 0.03$ for dIrr,
$M_{V,{\rm TO}}=-7.04\pm 0.02$ mag and $\sigma_{\rm GCLF}=1.15\pm 0.02$ for
dE/dSph, and $M_{V,{\rm TO}}=-7.30\pm 0.01$ mag and $\sigma_{\rm GCLF}=1.46\pm
0.02$ for Sm galaxies (note the different sample sizes when comparing sub-
populations in Fig. 5). Assuming a typical $M/L_{V}=1.8$ obtained for old
metal-poor Magellanic GCs McLaughlin & van der Marel (2005) the turnover
magnitude translates to a turnover mass $m_{\rm TO}\simeq 1.6\times 10^{5}$
M⊙, in excellent agreement with the results of Jordán et al. (2007).
The GC luminosity function turn over magnitude for dIrrs is slightly brighter
than those for dE/dSph and Sm galaxies, and it shows significantly broader
luminosity function peaks extending to fainter magnitudes. This may be due to
the interplay of different formation mechanisms and ages/metallicities or
perhaps due to contamination by background galaxies. In general, all
$M_{V,{\rm TO}}$ values are consistent with the luminosity function turn-over
magnitude for metal-poor Galactic GCs (Di Criscienzo et al., 2006), as well as
for GCs in early-type dwarfs (Sharina et al., 2005; Jordán et al., 2007;
Miller & Lotz, 2007), and virtually identical to the turn-over magnitude of
old LMC GCs at $M_{V}=-7.50\pm 0.16$ mag (data from McLaughlin & van der
Marel, 2005).
### 3.4 Nucleated Dwarf Irregular Galaxies
Another interesting feature in the CMDs of Figure 4 is the presence of a few
relatively bright GCs in dIrr and dE/dSph galaxies, which are similar in color
and magnitude to $\omega$ Cen and M 54. Those clusters are located in the
nuclear regions of their host galaxies. Such bright objects do not appear in
the Sm sub-sample. A dedicated study of the properties of these bright GCs
will be presented in a forthcoming paper of this series.
### 3.5 Structural Parameters
Since the clusters’ half-light radii, $r_{\rm h}$, and ellipticities,
$\epsilon$, are stable over many relaxation times (e.g. Spitzer & Thuan, 1972)
and, thus, contain information about the initial conditions and the dynamical
evolution of clusters over a Hubble time. In particular, the cluster half-
light radius is stable during $>\\!10$ relaxation times (i.e., $\sim$ 10 Gyr)
(e.g. Aarseth & Heggie, 1998), while the ellipticity decreases by a factor of
two within five relaxation times and reaches asymptotic values around 0.1 Fall
& Frenk (1985); Han & Ryden (1994); Meylan & Heggie (1997).
Figure 6: Luminosity $M_{V}$ versus half-light radius $r_{\rm h}$ for GCs in
dIrrs upper left panel, dE/dSph upper right panel, and Sm galaxies lower right
panel. Each sub-sample was split at $V-I\\!=\\!1.0$ mag in bGCs and rGCs. In
the upper left panel we show old GCs in the Magellanic Clouds (McLaughlin &
van der Marel, 2005). The dashed line indicates the upper envelope for the
distribution of Galactic GCs (Mackey & van den Bergh, 2005). The lower sub-
panels show the corresponding $r_{\rm h}$ distributions for the blue and red
GC candidates. The thick curves are Epanechnikov-kernel probability density
estimates. Note that in all our sub-samples the $r_{\rm h}$ distributions of
blue GCs appears more extended than that of the red GCs.
#### 3.5.1 Half-Light Radii
In Figure 6 we present the measurements of $r_{\rm h}$ for GCs in our sample
galaxies. The majority of GCs lies below the empirically established relation
$\log r_{\rm h}=0.25\times M_{V}+2.95$ (Mackey & van den Bergh, 2005), which
forms the upper envelope of Galactic GCs in the $M_{V}$ vs. $r_{\rm h}$ plane
(see lower left sub panels in Fig. 6). Some of these brightest GCs, that
reside in the nuclear regions of their hosts, tend to lie on or above that
envelope (towards larger $r_{h}$ at a given $M_{V}$), which seems typical for
nuclear star clusters (e.g. Böker et al., 2004; Haşegan et al., 2005). This
region is also occupied by the peculiar Galactic GCs $\omega$ Cen, NGC 2419,
NGC 2808, NGC 6441 and M 54, the nucleus of the Sagittarius dSph galaxy. For
an object with high S/N ($>50$) and good knowledge of the PSF, the theoretical
spatial resolution limit can be as good as $10\%$ of the PSF ($\sim 0.2$ pix
for ACS/PSF) Larsen (1999). Thus, at the median distance of the entire galaxy
sample of $\sim 5$ Mpc and taking into account the $r_{\rm h}$ measurement
error, the spatial resolution can be as good as $\sim 0.9$ pc ($\sim 0.8$
pix). Therefore, the majority of the clusters with high S/N are well resolved,
even for the distant most galaxy in our sample at $\sim 12$ Mpc.
The half-light radius distribution of the bGCs and rGCs is shown in the bottom
sub-panels of Figure 6. On average, bGCs appear more extended than rGCs in all
sub-samples, by about 9%, however with low statistical significance. On
average, the most compact bGC population is found in dE/dSph host galaxies
($r_{\rm h,med}=2.5$ pc), followed by bGCs in dIrrs ($r_{\rm h,med}=3.3$ pc)
and Sm galaxies ($r_{\rm h,med}=7.6$ pc), whose GC population is more
incomplete due to the restricted spatial coverage (cf. Sect. 3.1). We find the
highest value of the probability density estimate of the entire sample at
$r_{\rm h}\approx 2.9$ pc and a median $r_{\rm h,med}=3.2\pm 0.5$ pc. These
values are typical for Galactic GCs (Harris, 2001). The median value of the
old LMC GCs is $r_{\rm h,med}=5$ pc (based on measurements from McLaughlin &
van der Marel, 2005).
A comparison between the $r_{\rm h}$ distribution of the total bGC sample with
the sizes of different metal-poor, Galactic GC sub-populations shows that the
old halo (OH) GCs have comparable ($r_{\rm h,med}=3$ pc) and the young halo
(YH) GCs have larger median sizes ($r_{\rm h,med}=5.4$ pc) than blue GCs in
our sample dwarf galaxies. We note, however, that if all Galactic GCs are
considered, there is practically no difference in the mean $r_{\rm h}$ . In
the light of the accretion origin of the YH-GCs, their $r_{\rm h}$ might have
been influenced by the change of the host potential, which leads to an
increase of the cluster $r_{\rm t}$ at large Galactocentric distances, and
disk/bulge shocking causing mass loss, thus toward lower masses and larger
$r_{\rm h}$ (cf Fig. 8, Sect. 3.6). This is supported by the high orbital
energy (Etot, Zmax, eccentricity, velocity, angular momentum) found for YH-GCs
Lee et al. (2007) which also supports an accretion origin of those.
#### 3.5.2 Ellipticities
Figure 7: Ellipticity ($\epsilon=1-b/a$) distributions of GCs in dwarf
irregular galaxies. The top panel shows that bGCs in our sample dIrrs and in
LMC are similarly flattened. The even broader $\epsilon$ distribution for the
rGCs shows that the majority are likely background contaminants. The bottom
panel shows the $\epsilon$ distribution for the Galactic GC subpopulations: OH
= Old Halo; YH = Young Halo; BD = Bulge/Disk.
A difference between bGCs in dIrrs and those of the various Galactic GC sub-
populations is also found when clusters ellipticities are compared. In Figure
7 we show the ellipticity distribution of GCs in our sample dIrrs and in the
Magellanic Clouds for which we used data from Frenk & Fall (1982) and Kontizas
et al. (1989, 1990). The non-parametric kernel density estimate identifies
peaked distributions with values $\epsilon\simeq 0.1$ for both samples. This
is a markedly different ellipticity distributon than that of Galactic GCs
which is biased towards lower ellipticity values.
Fall & Frenk (1985) showed for self-gravitating clusters that during a period
$\sim 5t_{\rm rh}$ the ellipticity, $\epsilon$, decreases by a factor of two
and reaches an asymptotic value of $\epsilon\simeq 0.1$, corresponding to the
observed mean ellipticity of our bGCs in dIrr galaxies. Under the assumption
that the ages of the (bulk of the) GCs in our dIrr galaxies are similar to
those of Galactic GCs, this implies that our sample bGCs as well as the MC old
GCs evolved dynamically in isolation (i.e. mainly affected by internal
processes rather than external cluster dissolution processes).
The broad ellipticity distribution of the rGCs, which extends toward large
values, indicates that many or most of them are likely background
contaminants.
### 3.6 Dynamical State of Star Clusters in Dwarf Galaxies
The $r_{\rm h}$ versus cluster mass plane (Fig. 8) is often used to illustrate
cluster “survivability” that depends on the interplay between various external
and internal dissolution mechanisms (e.g. Fall & Rees, 1977; Gnedin &
Ostriker, 1997; Fall & Zhang, 2001; McLaughlin & Fall, 2008). In Figure 8 we
show the distribution of all sample GCs in this plane together with GCs in the
Galaxy and the Magellanic Clouds (MCs). To convert from luminosities to masses
we adopt a mean $M/L_{V}=1.8$ derived for the old MC clusters (McLaughlin &
van der Marel, 2005). The half-light radius estimates and cluster masses for
MC and Galactic clusters were taken from McLaughlin & van der Marel (2005),
where available, and from Harris (1996) for the remainder. It should be noted,
however, that if the GCs in our sample are on average younger, their
$M/L-$ratios would be smaller.
Figure 8: Half-light radius versus custer mass for GCs in low-mass galaxies.
The panels show from the top left panel clockwise: bGCs and rGCs (filled and
open circles) in dIrr galaxies (split at $V-I=1.0$ mag) and old Magellanic
Clouds GCs (open squares); GCs in dE/dSph galaxies, GCs in Sm galaxies; the
lower left panel shows the Galactic YH-GCs, which were split according to
their galactocentric distance being larger or smaller than 18 kpc (large or
small filled circles, respectively), and OH and BD GCs (open squares and
filled triangles), Sagittarius dwarf GCs (plus signs), and clusters with
unknown classification (asterisk). Lines indicate theoretical predictions for
cluster dissolution over a Hubble time due to tidal field effects for clusters
at 0.8 and 0.5 kpc from the center of the host (left and right vertical lines,
respectively), two-body relaxation processes as well as cluster re-expansion
after (dash-dotted lines), (for details see Sect. 3.6). With dotted lines are
also shown the cluster tidal radius $(r_{t,GC})$ at two assumed galactocentric
distances (0.5 and 0.8 kpc). Note that the larger error bars in the dE/dSph
panel are GCs in one of the most distant galaxies in our sample - the dE UGC
7369 at 11.6 Mpc.
In the following we discuss the influence of the various dynamical effects on
the evolution of our sample GCs. Dash-dotted lines show the cluster
evaporation limit due to two-body relaxation (Fall & Rees, 1977; Spitzer,
1987; Gnedin & Ostriker, 1997)
$r_{\rm h}=\left(\frac{\rm
12\,000{\rm[Myr]}}{20}\right)^{2/3}\left(\frac{0.138}{\sqrt{G}m_{\star}{\rm
ln}\left(\gamma\frac{M}{m_{\star}}\right)}\right)^{-2/3}M^{-1/3}$ (1)
for clusters that survived 12 Gyr after 20 initial relaxation times;
$m_{\star}=0.35M_{\odot}$ is the average stellar mass in a GC after 12 Gyr of
evolution and $\gamma=0.02$ is a correction constant taken from models which
simulate clusters with a specific mass spectrum (Giersz & Heggie, 1996). GCs
with $r_{\rm h}$ values lower values than this “survival” limit will have
dissolved after a Hubble time of dynamical evolution (e.g. Fall & Zhang,
2001).
The evolution of a star cluster in the Galactic tidal field was investigated
by Baumgardt & Makino (2003). We have used their equation 7 to calculate the
minimum mass of a cluster which can survive for 12 Gyr within a mean projected
distance of $d_{\rm proj}\approx 0.8$ kpc from the galaxy center, matching the
average $d_{\rm proj}$ of GCs in our sample. The result is shown as the left
vertical dashed line in Figure 8 and corresponds to a minimum mass of
$\sim\\!2.3\times 10^{4}M_{\odot}$ ($M_{V}=-5.5$ mag). Clusters below this
mass limit would have dissolved in the tidal field of the dwarf galaxy after a
Hubble time of evolution. The right vertical line indicates the minimum
cluster mass for $d_{\rm proj}=0.5$ kpc. It should be mentioned, however, that
effects due to a variable tidal field and dynamical friction were not taken
into account. Vesperini (2000) showed that low-mass galaxies efficiently
disrupt the majority ($\sim 90\%$) of their initial star-cluster population
due to dynamical friction.
An interesting aspect of cluster evolution is the re-expansion of star
clusters that can occur after the process of core collapse. The time evolution
of cluster mass and half-light radius is
$M(t)=M_{0}\left(\frac{t}{t_{0}}\right)^{-\nu}\ {\rm\ and}\ \ \ \ r_{\rm
h}(t)=r_{\rm h_{0}}\left(\frac{t}{t_{0}}\right)^{\frac{2+\nu}{3}}$ (2)
and was first obtained by Hénon (1965), where $M=M_{cl}/m_{\star}$ and $M_{0}$
and $r_{\rm h_{0}}$ are the cluster mass and half-light radius at the time of
core collapse $t_{0}$ and $\nu=0.01-0.1$ Baumgardt et al. (2002) is the
dynamical mass-loss factor. Combining both equations one obtains
$r_{\rm h}=r_{\rm h0}\left(\frac{M}{M_{0}}\right)^{-\frac{2+\nu}{3\nu}}$ (3)
Since core collapse occurs within several initial cluster relaxation times
$t_{0}=n_{\rm rh}t_{\rm rh,i}$ (see Gieles & Baumgardt, 2008) we can combine
Equations 1 and 3 and obtain
$r_{\rm h}=(t_{0})^{2/3}\left(\frac{0.138}{\sqrt{G}m_{\star}{\rm
ln}\left(\gamma\frac{M_{0}}{m_{\star}}\right)}\right)^{-2/3}M_{0}^{-1/3}\left(\frac{M}{M_{0}}\right)^{-\frac{2+\nu}{3\nu}}$
(4)
Consequently, if $M_{0}\\!=\\!M$ (i.e. no dynamical mass-loss after core
collapse) at $t_{0}=t=12$ Gyr ($\sim$Hubble time), and $n_{\rm rh}=20$ the
equation yields the limit labeled “$20t_{\rm rh}$” in Figure 8. A cluster that
starts its evolution below this relation will be prone to expansion with a
likely depletion of cluster stars prior to 20 relaxation times. In summary,
the distribution of GCs in all studied dwarfs shows that their dynamical
evolution was governed by internal processes (two-body relaxation, stellar
evolution). The absence of GCs at masses lower than the lower mass limit due
to disruption by the galactic tidal field within the mean projected radius of
the GCs in the sample galaxies ($\sim\\!3\\!\times 10^{4}M_{\odot}$, see Fig.
8) suggests that tidal disruption was important as well. With dotted lines are
in Figure 8 are shown the cluster tidal radius at two galactocentric
distances, 0.5 and 0.8 kpc for the dwarfs in our sample and at 8 and 18 kpc
for Galactic clusters. This sets up an upper limit to the cluster size. The
different distribution of rGCs with respect to that of bGCs in Fig. 8
indicates that most rGCs are background contaminants.
The comparison between our GC sample and Galactic GCs (bottom left panel in
Fig. 8) shows that bGCs, MC GCs, and Galactic YH GCs share very similar
distributions, which suggests that those objects have experienced similar
formation and/or dynamical evolution in the weak tidal fields of the dwarf and
the Milky Way halo regions. In contrast, the Galactic OH and BD clusters show
distributions that are significantly different from those of dwarf galaxy bGCs
and Galactic YH GCs. This may indicate that their dynamical evolution was
affected by the stronger inner Galactic tidal field and disk/bulge shocking.
## 4 Conclusions
We present the analysis of archival F606W and F814W HST/ACS data for 68 low-
mass faint ($M_{V}\\!>\\!-16$ mag) dwarf galaxies located in the halo regions
of nearby ($\la\\!12$ Mpc), loose galaxy groups. Most of the dwarf galaxies in
our sample are more than 2 mag fainter than the LMC ($M_{V}\\!=\\!-18.36$ mag)
and just as bright as the SMC ($M_{V}\\!=\\!-16.82$ mag). The morphological
makeup of our sample, summarized in Table LABEL:tab:sum, comprises 55 dIrrs, 3
dEs, 5 dSphs and 5 low-mass late-type dwarf spirals. Old GC candidates were
found in 30 dIrrs, 2 dEs, 2 dSphs and in 4 Sms. In total we found 175 GC
candidates and measure their colors and magnitudes which are consistent with
old and metal-poor stellar populations. The total sample contains 97 blue GCs
($0.7<(V-I)_{0}<1.0$ mag) GCs, 63 red GCs ($1.0<(V-I)_{0}<1.0$ mag) with the
rest being very red ($(V-I)_{0}>1.0$ mag) likely background contaminants.
The combined color distribution of GCs in dIrrs peaks at
$(V\\!-\\!I)\\!=\\!0.96\pm 0.07$ mag and the GC luminosity function turnover
is at $M_{V,TO}\\!=\\!-7.6$ mag, similar to the old LMC GCs. Gaussian fits to
the smoothed histogram distributions return $M_{V,{\rm TO}}\\!=\\!-7.56\pm
0.02$ mag and $\sigma\\!=\\!1.23\pm 0.03$ for dIrrs, $M_{V,{\rm
TO}}\\!=\\!-7.04\pm 0.02$ mag and $\sigma\\!=\\!1.15\pm 0.03$ for dE/dSph, and
$M_{V,{\rm TO}}\\!=\\!-7.30\pm 0.01$ mag and $\sigma\\!=\\!1.46\pm 0.02$ for
Sm galaxies. We thus find a tentative trend of $M_{V,{\rm TO}}$ becoming
fainter from late-type to early-type dwarf galaxies. Our artificial cluster
tests show that this trend is not due to incompleteness and may reflect
relatively younger GC systems in dIrrs by $\sim\\!2-5$ Gyr depending on the
metallicity. If confirmed, this would imply that GCs in dIrrs formed later
than blue GCs in massive galaxies. Thus we suggest that this result be
followed-up with spectroscopic observations.
The comparison of GC structural parameters with theoretical cluster disruption
models (including dynamical processes, such as stellar mass loss, relaxation,
tidal shocking indicates that the dynamical evolution of blue GCs in our
sample is consistent with the evolution of self-gravitating systems in a weak
tidal field. Their half-light radii and cluster masses evolve primarily due to
two-body relaxation. The $r_{\rm h}$ vs. cluster mass plane shows a similar
distribution between bGCs in our sample and Galactic Young Halo (YH) clusters,
which is indicative of a similar formation and dynamical evolution history.
Dynamical models of star clusters evolving in isolation show that they reach
an asymptotic ellipticity $\epsilon\approx 0.1$ after few cluster relaxation
times ($>5t_{\rm rh}$). Our analysis reveals that bGCs in our sample galaxies
on average more flattened ($\bar{\epsilon}=0.1$) than Galactic GCs, but have a
similar ellipticity distribution as GCs in the LMC. This suggests that old GCs
in low-mass galaxies are dynamically evolved stellar systems that spent most
of their evolution in benign tidal environment.
We briefly report on the discovery of several bright (massive and/or young)
GCs in the nuclear regions of some dIrr and dSph galaxies. These massive
nuclear clusters show similar structural parameters as the peculiar Galactic
clusters suspected of being the remnant nuclei of accreted dwarf galaxies,
such as M 54 and $\omega$Cen. If such accretion events happened early in the
assembly history of the Galaxy, any tidal streams from the host low-surface
brightness dIrr galaxies would likely have largely dissolved by now and thus
escape detection. We will present a more detailed analysis of the properties
of the nuclear clusters in a forthcoming paper in this series.
## Acknowledgments
IYG is grateful for the award of a STScI Graduate Research Fellowship, and
acknowledges support from the German Science Foundation through the grant DFG-
Projekt BO-779/32-1. THP acknowledges financial support from the National
Research Council of Canada in form of the Plaskett Research Fellowship. Based
on observations made with the NASA/ESA Hubble Space Telescope, and obtained
from the Hubble Multimission Archive, which is a collaboration between the
Space Telescope Science Institute (STScI/NASA), the Space Telescope European
Coordinating Facility (ST-ECF/ESA) and the Canadian Astronomy Data Centre
(CADC/NRC/CSA).
The authors are very thankful to the referee for the careful and thorough
reading whose suggestions improved the paper. We would also like to thank
Holger Baumgardt for the valuable discussions.
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Table 2: General properties of studied dwarf galaxies. ID | R.A. | Decl. | Morph. Type101010From LEDA/NED | D111111Distance and distance modulus from Karachentsev et al. (2006); Karachentsev et al. (2007) | $m-M^{b}$ | EB-V | $M_{V}$ | $(V-I)_{0}$ | $M_{HI}$
---|---|---|---|---|---|---|---|---|---
| (J2000.0) | (J2000.0) | T | Mpc | mag | mag | mag | mag | $10^{7}M_{\odot}$
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
— Cen A/M83 complex — | | | | | | | | |
Cen N | 13:48:09.2 | $-$47:33:54.0 | ? | dSph | 3.77 | 27.88 | 0.079 | $-$11.15 | 1.24 | –
ESO 059-01 | 07:31:19.3 | $-$68:11:10 | 10 | IB | 4.57 | 28.30 | 0.147 | $-$14.60 | 0.78 | 8.26
ESO 137-18 | 16:20:59.3 | $-$60:29:15 | 5.0 | SAsc | 6.4 | 29.03 | 0.243 | $-$17.21 | 0.79 | 34.14
ESO 215-09 | 10:57:30.2 | $-$48:10:44 | 10 | I | 5.25 | 28.60 | 0.221 | $-$14.08 | 0.80 | 64.21
ESO 223-09 | 15:01:08.5 | $-$48:17:33 | 9.7 | IAB | 6.49 | 29.06 | 0.260 | $-$16.47 | 0.88 | 63.89
ESO 269-58 | 13:10:32.9 | $-$46:59:27 | 9.4 | I | 3.8 | 27.90 | 0.106 | $-$15.78 | 0.98 | 2.31
ESO 269-66 | 13:13:09.2 | $-$44:53:24 | $-$5 | dE,N | 3.82 | 27.91 | 0.093 | $-$13.89 | 1.00 | –
ESO 274-01 | 15:14:13.5 | $-$46:48:45 | 6.6 | Scd | 3.09 | 27.45 | 0.257 | $-$17.47a | 1.03a | 20.18
ESO 320-14 | 11:37:53.4 | $-$39:13:14 | 10 | I | 6.08 | 28.92 | 0.142 | $-$13.67 | 0.80 | 2.25
ESO 381-18 | 12:44:42.7 | $-$35:58:00 | 9 | I | 5.32 | 28.63 | 0.063 | $-$13.39 | 0.72 | 2.71
ESO 381-20 | 12:46:00.4 | $-$33:50:17 | 9.8 | IBm | 5.44 | 28.68 | 0.065 | $-$14.80 | 0.63 | 15.71
ESO 384-16 | 13:57:01.6 | $-$35:20:02 | $-$5 | dSph/Im | 4.53 | 28.28 | 0.074 | $-$13.72 | 0.87 | –
ESO 443-09 | 12:54:53.6 | $-$28:20:27 | 10 | Im | 5.97 | 28.88 | 0.065 | $-$12.19 | 0.74 | 1.44
ESO 444-78 | 13:36:30.8 | $-$29:14:11 | 9.9 | Im | 5.25 | 28.60 | 0.053 | $-$13.48 | 0.86 | 2.06
HIPASS J1348-37 | 13:48:33.9 | $-$37:58:03 | 10 | I | 5.75 | 28.78 | 0.077 | $-$10.80 | 0.79 | 0.78
HIPASS J1351-47 | 13:51:22.0 | $-$47:00:00 | 10 | I | 5.73 | 28.79 | 0.145 | $-$11.55 | 0.68 | 2.69
IC 4247 | 13:26:44.4 | $-$30:21:45 | 1.6 | Sab | 4.97 | 28.48 | 0.062 | $-$14.69 | 0.66 | 3.45
IKN | 10:08:05.9 | $+$68:23:57 | $-$3 | dSph | 3.75 | 27.87 | 0.061 | $-$11.51121212Estimated from the total $B$ magnitudes assuming average $B-V=0.45$ and $V-I=0.7$ mag (see Sect.3.1). | 0.7c | –
KK 189 | 13:12:45.0 | $-$41:49:55 | $-$5 | dE | 4.42 | 28.23 | 0.114 | $-$11.99 | 0.92 | –
KK 196 | 13:21:47.1 | $-$45:03:48 | 9.8 | IBm | 3.98 | 28.00 | 0.084 | $-$10.72 | 0.71 | –
KK 197 | 13:22:01.8 | $-$42:32:08 | 10 | Im | 3.87 | 27.94 | 0.154 | $-$13.04 | 1.16 | 0.17
KKS 55 | 13:22:12.4 | $-$42:43:51 | $-$3 | dSph | 3.94 | 27.98 | 0.146 | $-$11.17 | 1.22 | –
KKS 57 | 13:41:38.1 | $+$42:34:55 | $-$3 | I | 3.93 | 27.97 | 0.091 | $-$10.73 | 1.08 | –
NGC 5237 | 13:37:38.9 | $-$42:50:51 | 1.4 | dSph/I | 3.4 | 27.66 | 0.096 | $-$15.45 | 0.91 | 3.10
— Sculptor group — | | | | | | | | |
ESO 349-31 | 00:08:13.3 | $-$34:34:42 | 10 | IB | 3.21 | 27.53 | 0.012 | $-$11.87 | 0.66 | 1.34
NGC 247 | 00:47:06.1 | $-$20:39:04 | 6.7 | SABd | 3.65 | 27.81 | 0.018 | $-$18.76a | 0.85a | 37.60
— Mafei 1 & 2 — | | | | | | | | |
KKH 6 | 01:34:51.6 | $+$52:05:30 | 10 | I | 3.73 | 27.86 | 0.351 | $-$12.66 | 0.80 | 1.34
— IC 342 group — | | | | | | | | |
KKH 37 | 06:47:46.9 | $+$80:07:26 | 10 | I | 3.39 | 27.65 | 0.076 | $-$12.07 | 0.80 | 0.48
UGCA 86 | 03:59:48.3 | $+$67:08:18.6 | 9.9 | Im | 2.96 | 27.36 | 0.942 | $-$16.13d | 0.80d | 48.25
UGCA 92 | 04:32:04.9 | $+$63:36:49.0 | 10 | Im | 3.01 | 27.39 | 0.792 | $-$14.71 | 0.51 | 7.62
— NGC 2903 group — | | | | | | | | |
D 564-08 | 09:19:30.0 | $+$21:36:11.7 | 10 | I | 8.67 | 29.69 | 0.029 | $-$12.76 | 1.00 | 1.93
D 565-06 | 09:19:29.4 | $+$21:36:12 | 10 | I | 9.08 | 29.79 | 0.039 | $-$12.88 | 0.95 | 0.54
— CVn I cloud — | | | | | | | | |
NGC 4068 | 12:04:02.4 | $+$52:35:19 | 9.9 | Im | 4.31 | 28.17 | 0.022 | $-$15.25 | 0.63 | 11.22
NGC 4163 | 12:12:08.9 | $+$36:10:10 | 9.9 | Im | 2.96 | 27.36 | 0.020 | $-$14.21 | 0.80 | 1.42
UGC 8215 | 13:08:03.6 | $+$46:49:41 | 9.9 | Im | 4.55 | 28.29 | 0.010 | $-$12.51 | 0.82 | 1.84
UGC 8638 | 13:39:19.4 | $+$24:46:33 | 9.9 | Im | 4.27 | 28.15 | 0.013 | $-$13.69 | 0.74 | 1.17
— Field — | | | | | | | | |
D 634-03 | 09:08:53.5 | $+$14:34:55 | 10 | I | 9.46 | 29.90 | 0.038 | $-$11.94 | 0.92 | 0.49
DDO 52 | 08:28:28.5 | $+$41:51:24 | 10 | I | 10.28 | 30.06 | 0.037 | $-$14.98 | 0.80 | 19.99
ESO 121-20 | 06:15:54.5 | $-$57:43:35 | 10 | I | 6.05 | 28.91 | 0.040 | $-$13.64 | 0.68 | 11.49
HIPASS J1247-77 | 12:47:32.6 | $-$77:35:01 | 10 | Im | 3.16 | 27.50 | 0.748 | $-$12.86 | 0.20 | 1.05
HS 117 | 10:21:25.2 | $+$71:06:58 | 10 | I | 3.96 | 27.99 | 0.115 | $-$11.31 | 0.91 | –
IC 4662 | 17:47:06.3 | $-$64:38:25 | 9.7 | IBm | 2.44 | 26.94 | 0.070 | $-$15.58 | 0.66 | 12.58
KK 182 | 13:05:02.9 | $-$40:04:58 | 10 | I | 5.78 | 28.81 | 0.101 | $-$13.10 | 0.63 | 4.45
KK 230 | 14:07:10.7 | $+$35:03:37 | 10 | I | 1.92 | 26.42 | 0.014 | $-$9.06 | 0.74 | 0.17
KK 246 | 20:03:57.4 | $-$31:40:54 | 10 | I | 7.83 | 29.47 | 0.296 | $-$13.77 | 0.83 | 11.92
KKH 77 | 12:14:11.3 | $+$66:04:54 | 10 | I | 5.42 | 28.67 | 0.019 | $-$14.58a | 0.7c | 4.67
NGC 4605 | 12:39:59.4 | $+$61:36:33 | 4.9 | SBc | 5.47 | 28.69 | 0.014 | $-$18.41a | 0.7c | 25.88
UGC 7369 | 12:19:38.8 | $+$29:52:59 | 7.6 | dE/dE,N? | 11.6 | 30.32 | 0.019 | $-$16.17 | 1.03 | –
VKN | 08:40:08.9 | $+$68:26:23 | $-$3 | dSph? | 3.4 | – | 0.035 | $-$10.52c | 0.7c | –
Table 3: Globular cluster candidate properties ID | RA,DEC (2000) | $M_{V}$ | $(V-I)_{0}$ | $r_{\rm h}$ | $\epsilon$ | rproj | r${}_{\rm proj}/{\rm r}_{\rm eff}$
---|---|---|---|---|---|---|---
| [hh:mm:ss],[dd:mm:ss] | [mag] | [mag] | [pc] | | [kpc] |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
dIrrs
E115-021-01 | 02:37:54.24 $-61:18:45.39$ | $-4.79\pm 0.05$ | $1.473\pm 0.052$ | $4.23\pm 0.26$ | 0.18 | 2.10 | 1.52
E154-023-01 | 02:56:49.17 $-54:33:16.51$ | $-4.78\pm 0.06$ | $1.045\pm 0.066$ | $2.27\pm 0.30$ | 0.13 | 2.01 | 0.75
E154-023-02 | 02:57:01.03 $-54:35:24.44$ | $-4.78\pm 0.06$ | $1.129\pm 0.062$ | $1.24\pm 0.25$ | 0.20 | 3.22 | 1.09
E154-023-03 | 02:57:01.06 $-54:35:11.54$ | $-4.40\pm 0.05$ | $1.277\pm 0.053$ | $2.18\pm 0.42$ | 0.14 | 3.07 | 1.02
IC1959-01 | 03:33:20.36 $-50:23:35.17$ | $-5.43\pm 0.05$ | $1.459\pm 0.051$ | $3.57\pm 0.32$ | 0.07 | 3.15 | 2.61
IC1959-02 | 03:33:14.24 $-50:25:40.16$ | $-6.19\pm 0.06$ | $1.028\pm 0.063$ | $6.65\pm 0.36$ | 0.37 | 1.51 | 1.42
IC1959-03 | 03:33:15.70 $-50:23:39.77$ | $-5.42\pm 0.05$ | $1.539\pm 0.051$ | $2.77\pm 0.35$ | 0.42 | 2.30 | 1.83
IC1959-04 | 03:33:12.43 $-50:24:52.60$ | $-9.83\pm 0.05$ | $0.968\pm 0.052$ | $2.92\pm 0.20$ | 0.08 | 0.03 | 0.19
IC1959-05 | 03:33:11.51 $-50:24:38.34$ | $-6.54\pm 0.06$ | $1.096\pm 0.062$ | $1.79\pm 0.23$ | 0.17 | 0.47 | 0.22
IC1959-06 | 03:33:07.70 $-50:23:17.16$ | $-7.60\pm 0.05$ | $0.893\pm 0.052$ | $1.91\pm 0.22$ | 0.08 | 3.08 | 2.39
IC1959-07 | 03:33:15.71 $-50:25:31.32$ | $-6.11\pm 0.06$ | $0.780\pm 0.074$ | $3.77\pm 0.28$ | 0.10 | 1.48 | 1.41
IC1959-08 | 03:33:09.50 $-50:25:45.00$ | $-4.66\pm 0.05$ | $0.990\pm 0.052$ | $1.26\pm 0.25$ | 0.14 | 1.77 | 1.56
IC1959-09 | 03:33:03.47 $-50:25:08.15$ | $-5.07\pm 0.06$ | $1.077\pm 0.070$ | $1.45\pm 0.26$ | 0.42 | 2.57 | 2.07
KK16-01 | 01:55:23.51 $+27:57:29.12$ | $-5.29\pm 0.06$ | $1.034\pm 0.062$ | $2.52\pm 0.30$ | 0.22 | 1.19 | 2.49
KK17-01 | 02:00:15.85 $+28:50:42.59$ | $-4.59\pm 0.05$ | $1.210\pm 0.058$ | $1.19\pm 0.24$ | 0.07 | 2.11 | 8.41
KK27-01 | 03:21:01.00 $-66:19:13.72$ | $-5.24\pm 0.06$ | $1.570\pm 0.067$ | $3.21\pm 0.19$ | 0.16 | 0.25 | 0.88
KK27-02 | 03:20:57.69 $-66:19:03.74$ | $-6.17\pm 0.06$ | $1.183\pm 0.071$ | $3.18\pm 0.19$ | 0.05 | 0.67 | 2.61
KK27-03 | 03:20:50.39 $-66:19:38.16$ | $-4.83\pm 0.05$ | $1.005\pm 0.053$ | $1.43\pm 0.18$ | 0.21 | 1.61 | 7.05
KK65-01 | 07:42:33.50 $+16:33:40.77$ | $-6.12\pm 0.06$ | $1.475\pm 0.070$ | $1.69\pm 0.48$ | 0.40 | 0.92 | 0.87
N1311-01 | 03:20:20.05 $-52:10:14.95$ | $-5.03\pm 0.05$ | $1.089\pm 0.055$ | $2.22\pm 0.29$ | 0.20 | 3.39 | 2.95
N1311-02 | 03:20:17.96 $-52:11:05.37$ | $-5.70\pm 0.05$ | $1.381\pm 0.051$ | $2.03\pm 0.25$ | 0.24 | 2.57 | 2.24
N1311-03 | 03:20:04.05 $-52:11:34.39$ | $-6.48\pm 0.06$ | $1.656\pm 0.064$ | $4.48\pm 0.31$ | 0.23 | 1.04 | 0.71
N1311-04 | 03:20:04.00 $-52:09:21.26$ | $-6.49\pm 0.06$ | $0.819\pm 0.064$ | $8.48\pm 0.21$ | 0.23 | 2.97 | 2.52
N1311-05 | 03:20:07.86 $-52:11:11.03$ | $-8.60\pm 0.05$ | $0.900\pm 0.052$ | $2.09\pm 0.20$ | 0.10 | 0.12 | 0.22
N1311-06 | 03:20:06.69 $-52:10:56.18$ | $-7.33\pm 0.06$ | $0.795\pm 0.067$ | $10.32\pm 0.18$ | 0.07 | 0.39 | 0.39
N784-01 | 02:01:18.09 $+28:48:42.92$ | $-7.54\pm 0.06$ | $0.956\pm 0.062$ | $2.93\pm 0.20$ | 0.15 | 2.02 | 0.98
N784-02 | 02:01:12.29 $+28:50:50.86$ | $-6.10\pm 0.06$ | $1.553\pm 0.064$ | $1.20\pm 0.24$ | 0.29 | 1.98 | 0.75
N784-03 | 02:01:16.42 $+28:49:52.81$ | $-6.65\pm 0.05$ | $0.866\pm 0.056$ | $4.54\pm 0.24$ | 0.12 | 0.29 | 0.25
N784-04 | 02:01:16.57 $+28:51:38.43$ | $-6.40\pm 0.06$ | $0.717\pm 0.076$ | $11.55\pm 0.25$ | 0.09 | 2.44 | 0.88
N784-07 | 02:01:18.31 $+28:49:44.61$ | $-5.72\pm 0.06$ | $1.202\pm 0.074$ | $2.60\pm 0.21$ | 0.10 | 0.62 | 0.37
N784-08 | 02:01:21.32 $+28:49:45.61$ | $-4.39\pm 0.06$ | $1.127\pm 0.078$ | $2.66\pm 0.34$ | 0.37 | 1.50 | 0.67
N784-09 | 02:01:18.54 $+28:52:05.14$ | $-6.62\pm 0.06$ | $0.752\pm 0.070$ | $12.85\pm 0.26$ | 0.08 | 3.15 | 1.18
U1281-01 | 01:49:38.95 $+32:35:09.73$ | $-7.41\pm 0.06$ | $1.240\pm 0.063$ | $6.14\pm 0.32$ | 0.07 | 2.34 | 1.53
U1281-02 | 01:49:32.54 $+32:35:25.26$ | $-7.59\pm 0.05$ | $0.959\pm 0.055$ | $2.01\pm 0.23$ | 0.08 | 0.38 | 0.12
U3755-01 | 07:13:51.95 $+10:31:42.11$ | $-9.20\pm 0.06$ | $0.773\pm 0.062$ | $1.62\pm 0.25$ | 0.27 | 0.92 | 0.96
U3755-02 | 07:13:50.61 $+10:30:40.01$ | $-7.28\pm 0.06$ | $0.986\pm 0.062$ | $2.16\pm 0.28$ | 0.09 | 1.42 | 0.85
U3755-03 | 07:13:50.40 $+10:31:48.33$ | $-6.60\pm 0.07$ | $0.928\pm 0.075$ | $4.31\pm 0.36$ | 0.25 | 1.27 | 1.26
U3755-04 | 07:13:50.12 $+10:32:15.02$ | $-8.75\pm 0.05$ | $1.003\pm 0.051$ | $1.73\pm 0.25$ | 0.08 | 2.21 | 1.95
U3755-05 | 07:13:52.78 $+10:30:42.62$ | $-6.66\pm 0.05$ | $1.038\pm 0.069$ | $6.81\pm 0.46$ | 0.09 | 1.41 | 0.71
U3755-06 | 07:13:52.13 $+10:31:23.98$ | $-8.24\pm 0.06$ | $0.870\pm 0.063$ | $5.97\pm 0.28$ | 0.05 | 0.40 | 0.48
U3755-07 | 07:13:52.06 $+10:31:12.28$ | $-7.35\pm 0.06$ | $0.835\pm 0.067$ | $7.75\pm 0.30$ | 0.16 | 0.33 | 0.18
U3755-08 | 07:13:51.43 $+10:30:57.23$ | $-8.27\pm 0.05$ | $0.885\pm 0.057$ | $6.96\pm 0.32$ | 0.20 | 0.72 | 0.29
U3755-09 | 07:13:50.24 $+10:31:10.73$ | $-6.64\pm 0.06$ | $1.194\pm 0.064$ | $8.54\pm 0.37$ | 0.01 | 0.72 | 0.66
U3974-01 | 07:41:59.49 $+16:49:00.92$ | $-7.21\pm 0.06$ | $0.840\pm 0.063$ | $6.60\pm 0.35$ | 0.16 | 2.77 | 0.86
U3974-02 | 07:41:58.16 $+16:48:16.12$ | $-8.77\pm 0.06$ | $0.959\pm 0.062$ | $2.23\pm 0.29$ | 0.11 | 1.13 | 0.44
U3974-03 | 07:41:58.08 $+16:47:50.52$ | $-7.86\pm 0.05$ | $0.929\pm 0.052$ | $3.34\pm 0.31$ | 0.14 | 1.23 | 0.47
U3974-04 | 07:41:54.87 $+16:47:54.72$ | $-8.28\pm 0.06$ | $0.979\pm 0.075$ | $2.73\pm 0.29$ | 0.09 | 0.91 | 0.18
U3974-05 | 07:41:54.57 $+16:48:33.71$ | $-8.72\pm 0.06$ | $1.321\pm 0.072$ | $10.43\pm 0.27$ | 0.17 | 1.38 | 0.30
U4115-01 | 07:57:03.79 $+14:22:41.01$ | $-7.62\pm 0.06$ | $0.952\pm 0.062$ | $3.30\pm 0.32$ | 0.08 | 1.93 | 0.91
U4115-02 | 07:56:54.05 $+14:23:40.30$ | $-5.73\pm 0.05$ | $1.189\pm 0.052$ | $4.20\pm 0.47$ | 0.24 | 4.28 | 3.07
U4115-03 | 07:56:59.20 $+14:21:53.09$ | $-6.45\pm 0.06$ | $1.221\pm 0.064$ | $4.50\pm 0.39$ | 0.14 | 3.68 | 2.28
U4115-04 | 07:57:01.16 $+14:24:22.25$ | $-4.95\pm 0.07$ | $1.251\pm 0.089$ | $3.26\pm 0.54$ | 0.36 | 2.22 | 1.76
U4115-05 | 07:57:03.97 $+14:22:26.30$ | $-4.86\pm 0.06$ | $1.342\pm 0.069$ | $0.52\pm 0.41$ | 0.77 | 2.45 | 1.25
U685-01 | 01:07:26.18 $+16:40:56.84$ | $-7.02\pm 0.06$ | $0.929\pm 0.062$ | $3.17\pm 0.16$ | 0.00 | 1.28 | 1.88
U685-02 | 01:07:26.77 $+16:40:30.71$ | $-4.67\pm 0.06$ | $1.730\pm 0.062$ | $1.06\pm 0.21$ | 0.29 | 1.66 | 2.39
U685-03 | 01:07:25.68 $+16:40:44.19$ | $-7.95\pm 0.06$ | $0.968\pm 0.068$ | $8.53\pm 0.16$ | 0.07 | 1.20 | 1.75
U685-04 | 01:07:23.60 $+16:41:21.88$ | $-8.64\pm 0.05$ | $0.896\pm 0.052$ | $1.80\pm 0.16$ | 0.12 | 0.56 | 0.88
U685-05 | 01:07:24.64 $+16:40:37.05$ | $-7.84\pm 0.05$ | $0.954\pm 0.053$ | $4.44\pm 0.18$ | 0.07 | 1.00 | 1.40
U685-06 | 01:07:22.24 $+16:41:15.14$ | $-8.36\pm 0.05$ | $0.893\pm 0.051$ | $1.66\pm 0.19$ | 0.10 | 0.22 | 0.36
U8760-01 | 13:50:50.73 $+38:01:48.27$ | $-4.80\pm 0.06$ | $1.275\pm 0.068$ | $5.29\pm 0.22$ | 0.26 | 0.60 | 0.89
D565-06-01 | 09:19:29.66 $+21:36:00.15$ | $-6.10\pm 0.07$ | $1.583\pm 0.094$ | $1.19\pm 0.41$ | 0.07 | 0.53 | 0.72
D634-03-01 | 09:08:53.72 $+14:34:55.87$ | $-7.08\pm 0.07$ | $1.039\pm 0.094$ | $5.93\pm 0.46$ | 0.13 | 0.30 | 0.08
DDO52-01 | 08:28:27.09 $+41:51:21.71$ | $-6.62\pm 0.07$ | $1.004\pm 0.099$ | $3.34\pm 0.42$ | 0.09 | 0.80 | 0.43
DDO52-02 | 08:28:32.70 $+41:52:26.84$ | $-7.05\pm 0.08$ | $0.958\pm 0.122$ | $6.78\pm 0.47$ | 0.03 | 3.90 | 2.04
E059-01-01 | 07:31:18.26 $-68:11:14.49$ | $-9.89\pm 0.06$ | $0.907\pm 0.077$ | $2.35\pm 0.15$ | 0.05 | 0.12 | 0.01
E121-20-01 | 06:15:50.13 $-57:43:27.48$ | $-6.01\pm 0.07$ | $1.335\pm 0.096$ | $2.70\pm 0.26$ | 0.09 | 1.11 | 2.56
E223-09-01 | 15:01:08.38 $-48:16:00.56$ | $-8.13\pm 0.06$ | $0.992\pm 0.078$ | $4.53\pm 0.23$ | 0.11 | 2.96 | 1.00
E223-09-02 | 15:01:10.40 $-48:16:00.95$ | $-7.87\pm 0.07$ | $0.982\pm 0.090$ | $3.37\pm 0.23$ | 0.02 | 3.00 | 1.01
E223-09-03 | 15:01:02.81 $-48:17:40.69$ | $-8.00\pm 0.06$ | $1.019\pm 0.078$ | $4.03\pm 0.25$ | 0.19 | 1.86 | 0.65
E223-09-04 | 15:01:04.59 $-48:17:29.36$ | $-7.76\pm 0.07$ | $0.853\pm 0.090$ | $5.77\pm 0.28$ | 0.02 | 1.30 | 0.46
E223-09-05 | 15:01:05.91 $-48:17:53.93$ | $-9.14\pm 0.06$ | $1.072\pm 0.077$ | $3.30\pm 0.23$ | 0.10 | 1.06 | 0.38
E223-09-06 | 15:01:09.85 $-48:17:33.95$ | $-9.72\pm 0.06$ | $0.921\pm 0.077$ | $3.51\pm 0.25$ | 0.21 | 0.36 | 0.11
E223-09-07 | 15:01:09.75 $-48:18:00.87$ | $-6.84\pm 0.07$ | $0.952\pm 0.094$ | $2.17\pm 0.23$ | 0.21 | 0.89 | 0.32
E223-09-08 | 15:01:17.37 $-48:17:54.92$ | $-6.70\pm 0.07$ | $0.915\pm 0.092$ | $2.30\pm 0.28$ | 0.28 | 2.79 | 0.95
E269-58-01 | 13:10:25.66 $-47:00:03.41$ | $-6.33\pm 0.07$ | $0.773\pm 0.092$ | $5.41\pm 0.33$ | 0.15 | 1.66 | 1.16
E269-58-02 | 13:10:29.41 $-46:58:19.36$ | $-7.97\pm 0.06$ | $0.820\pm 0.077$ | $2.57\pm 0.14$ | 0.10 | 1.54 | 1.17
E269-58-03 | 13:10:31.20 $-46:59:23.11$ | $-6.86\pm 0.06$ | $1.127\pm 0.078$ | $14.98\pm 0.13$ | 0.18 | 0.51 | 0.32
E269-58-04 | 13:10:32.08 $-47:00:57.67$ | $-7.43\pm 0.06$ | $0.923\pm 0.078$ | $2.98\pm 0.17$ | 0.04 | 1.65 | 1.20
E269-58-05 | 13:10:35.55 $-46:58:07.17$ | $-7.20\pm 0.06$ | $0.918\pm 0.078$ | $4.34\pm 0.13$ | 0.03 | 1.56 | 1.25
E269-58-06 | 13:10:35.88 $-46:59:32.51$ | $-6.62\pm 0.06$ | $1.104\pm 0.079$ | $18.56\pm 4.04$ | 0.08 | 0.39 | 0.39
E269-58-07 | 13:10:39.50 $-46:59:20.17$ | $-7.23\pm 0.06$ | $0.944\pm 0.078$ | $2.90\pm 0.15$ | 0.06 | 1.09 | 0.93
E269-58-08 | 13:10:41.34 $-46:59:00.47$ | $-6.69\pm 0.06$ | $0.941\pm 0.079$ | $7.79\pm 6.04$ | 0.11 | 1.52 | 1.26
E320-14-01 | 11:37:50.70 $-39:12:26.91$ | $-6.16\pm 0.07$ | $1.471\pm 0.092$ | $3.97\pm 0.29$ | 0.17 | 1.70 | 2.89
E349-031-01 | 00:08:15.16 $-34:36:35.17$ | $-5.15\pm 0.07$ | $1.293\pm 0.091$ | $0.69\pm 0.12$ | 0.24 | 1.81 | 4.21
E349-031-02 | 00:08:12.64 $-34:36:29.97$ | $-4.39\pm 0.07$ | $1.229\pm 0.101$ | $6.45\pm 0.18$ | 0.18 | 1.71 | 3.99
E381-20-01 | 12:46:03.21 $-33:49:16.77$ | $-5.39\pm 0.07$ | $1.344\pm 0.098$ | $4.13\pm 0.18$ | 0.15 | 1.78 | 1.87
E384-016-01 | 13:56:54.28 $-35:18:26.01$ | $-5.93\pm 0.07$ | $1.160\pm 0.090$ | $2.14\pm 0.15$ | 0.22 | 2.84 | 4.54
E384-016-02 | 13:56:59.90 $-35:19:37.17$ | $-5.29\pm 0.07$ | $1.271\pm 0.094$ | $1.41\pm 0.17$ | 0.17 | 0.67 | 1.01
KK197-01 | 13:21:59.81 $-42:32:06.51$ | $-5.69\pm 0.07$ | $1.050\pm 0.094$ | $1.95\pm 0.17$ | 0.01 | 0.46 | 1.68
KK197-02 | 13:22:02.04 $-42:32:08.14$ | $-9.83\pm 0.06$ | $0.932\pm 0.077$ | $2.95\pm 0.13$ | 0.11 | 0.00 | 0.00
KK197-03 | 13:22:02.53 $-42:32:13.82$ | $-7.26\pm 0.06$ | $0.925\pm 0.078$ | $2.56\pm 0.17$ | 0.07 | 0.15 | 0.53
KK246-01 | 20:03:57.47 $-31:40:55.93$ | $-6.72\pm 0.07$ | $0.864\pm 0.098$ | $4.46\pm 0.38$ | 0.18 | 0.19 | 0.34
KK246-02 | 20:03:57.07 $-31:40:58.93$ | $-8.30\pm 0.06$ | $0.960\pm 0.078$ | $4.24\pm 0.26$ | 0.06 | 0.16 | 0.42
KKH77-01 | 12:14:08.52 $+66:05:41.69$ | $-7.98\pm 0.06$ | $0.996\pm 0.077$ | $1.97\pm 0.23$ | 0.14 | 0.00 | 0.00
KKH77-02 | 12:14:22.72 $+66:05:38.58$ | $-5.41\pm 0.07$ | $1.074\pm 0.100$ | $1.77\pm 0.25$ | 0.11 | 2.27 | 3.01
KKH77-03 | 12:14:18.83 $+66:04:27.69$ | $-5.48\pm 0.07$ | $1.598\pm 0.096$ | $4.76\pm 0.18$ | 0.19 | 2.55 | 3.39
N4163-01 | 12:12:09.70 $+36:10:15.15$ | $-9.38\pm 0.06$ | $0.915\pm 0.077$ | $1.45\pm 0.10$ | 0.09 | 0.17 | 0.27
N4163-02 | 12:12:08.57 $+36:10:25.95$ | $-7.83\pm 0.06$ | $0.969\pm 0.077$ | $2.08\pm 0.10$ | 0.03 | 0.31 | 0.53
N5237-01 | 13:37:37.87 $-42:51:20.02$ | $-8.46\pm 0.06$ | $0.925\pm 0.077$ | $0.68\pm 0.16$ | 0.39 | 0.55 | 0.57
N5237-02 | 13:37:37.95 $-42:50:23.32$ | $-6.85\pm 0.06$ | $0.997\pm 0.078$ | $15.09\pm 0.18$ | 0.12 | 0.46 | 0.51
N5237-03 | 13:37:34.62 $-42:50:01.90$ | $-6.42\pm 0.06$ | $1.131\pm 0.078$ | $3.38\pm 0.11$ | 0.06 | 1.11 | 1.20
U8638-01 | 13:39:26.80 $+24:45:50.46$ | $-6.57\pm 0.06$ | $0.871\pm 0.078$ | $13.63\pm 0.88$ | 0.00 | 2.26 | 4.71
U8638-02 | 13:39:24.83 $+24:46:14.04$ | $-7.72\pm 0.06$ | $0.966\pm 0.077$ | $3.23\pm 0.17$ | 0.05 | 1.58 | 3.25
U8638-03 | 13:39:18.17 $+24:46:18.89$ | $-10.35\pm 0.06$ | $1.077\pm 0.077$ | $2.62\pm 0.14$ | 0.04 | 0.42 | 1.02
UA86-04 | 03:59:38.20 $+67:07:12.70$ | $-6.95\pm 0.07$ | $0.911\pm 0.092$ | $3.19\pm 0.20$ | 0.00 | 1.42 | 5.51
UA86-05 | 03:59:51.07 $+67:06:10.21$ | $-7.53\pm 0.07$ | $0.878\pm 0.089$ | $2.19\pm 0.11$ | 0.06 | 2.02 | 8.12
UA86-07 | 03:59:45.84 $+67:07:07.54$ | $-7.26\pm 0.07$ | $0.738\pm 0.091$ | $1.21\pm 0.16$ | 0.13 | 1.21 | 4.57
UA86-10 | 03:59:49.83 $+67:06:49.71$ | $-11.03\pm 0.06$ | $0.716\pm 0.077$ | $1.14\pm 0.10$ | 0.06 | 1.45 | 5.64
UA86-11 | 03:59:43.02 $+67:07:27.78$ | $-6.80\pm 0.07$ | $1.523\pm 0.089$ | $3.43\pm 0.14$ | 0.30 | 1.02 | 3.73
UA86-17 | 03:59:48.76 $+67:08:16.72$ | $-9.67\pm 0.06$ | $0.731\pm 0.077$ | $3.27\pm 0.13$ | 0.08 | 0.19 | 0.30
UA86-20 | 03:59:42.40 $+67:08:53.83$ | $-7.58\pm 0.07$ | $0.731\pm 0.089$ | $10.78\pm 0.14$ | 0.19 | 0.63 | 2.95
UA86-25 | 03:59:48.88 $+67:08:30.85$ | $-7.99\pm 0.06$ | $0.720\pm 0.078$ | $4.24\pm 0.10$ | 0.03 | 0.01 | 0.72
UA86-27 | 04:00:00.75 $+67:07:37.21$ | $-8.29\pm 0.07$ | $0.703\pm 0.088$ | $7.02\pm 0.18$ | 0.15 | 1.26 | 5.28
UA86-28 | 03:59:49.23 $+67:08:40.76$ | $-7.79\pm 0.07$ | $0.868\pm 0.088$ | $3.11\pm 0.15$ | 0.02 | 0.16 | 1.35
UA86-29 | 03:59:50.29 $+67:08:38.16$ | $-11.16\pm 0.06$ | $1.020\pm 0.077$ | $4.73\pm 0.14$ | 0.12 | 0.17 | 1.37
UA86-30 | 03:59:53.87 $+67:08:30.97$ | $-7.84\pm 0.07$ | $0.769\pm 0.088$ | $4.76\pm 0.24$ | 0.23 | 0.43 | 2.18
UA92-02 | 04:32:03.24 $+63:37:06.65$ | $-8.04\pm 0.06$ | $1.038\pm 0.077$ | $3.13\pm 0.17$ | 0.07 | 0.28 | 0.65
UA92-03 | 04:32:01.94 $+63:36:41.92$ | $-7.74\pm 0.06$ | $0.784\pm 0.078$ | $6.20\pm 0.12$ | 0.15 | 0.23 | 0.66
dSphs
IKN-01 | 10:08:07.14 $+68:23:36.65$ | $-6.65\pm 0.06$ | $0.911\pm 0.079$ | $6.62\pm 0.29$ | 0.13 | 0.40 | –
IKN-02 | 10:08:10.79 $+68:24:05.60$ | $-7.15\pm 0.06$ | $0.994\pm 0.078$ | $3.55\pm 0.13$ | 0.14 | 0.50 | –
IKN-03 | 10:08:05.26 $+68:24:33.78$ | $-6.76\pm 0.07$ | $1.085\pm 0.092$ | $14.81\pm 0.83$ | 0.13 | 0.66 | –
IKN-04 | 10:08:04.80 $+68:24:53.71$ | $-7.41\pm 0.06$ | $0.936\pm 0.077$ | $1.96\pm 0.16$ | 0.18 | 1.03 | –
IKN-05 | 10:08:05.52 $+68:24:57.99$ | $-8.47\pm 0.06$ | $0.906\pm 0.077$ | $2.89\pm 0.13$ | 0.12 | 1.10 | –
KKS55-01 | 13:22:12.41 $-42:45:11.76$ | $-7.36\pm 0.06$ | $0.907\pm 0.078$ | $4.51\pm 0.18$ | 0.11 | 1.48 | 4.75
dEs
E269-66-01 | 13:13:10.30 $-44:53:00.96$ | $-8.08\pm 0.06$ | $0.911\pm 0.078$ | $2.87\pm 0.15$ | 0.10 | 0.49 | 0.81
E269-66-03 | 13:13:08.84 $-44:53:22.59$ | $-9.99\pm 0.06$ | $0.926\pm 0.077$ | $2.50\pm 0.13$ | 0.13 | 0.00 | 0.00
E269-66-04 | 13:13:03.12 $-44:53:40.16$ | $-7.18\pm 0.06$ | $1.017\pm 0.078$ | $7.48\pm 0.13$ | 0.10 | 1.17 | 1.92
E269-66-05 | 13:13:11.79 $-44:53:09.79$ | $-7.18\pm 0.06$ | $0.873\pm 0.078$ | $4.77\pm 0.19$ | 0.16 | 0.63 | 1.03
U7369-01 | 12:19:40.78 $+29:52:04.71$ | $-6.90\pm 0.07$ | $0.899\pm 0.099$ | $2.77\pm 0.47$ | 0.17 | 3.43 | 2.39
U7369-02 | 12:19:41.50 $+29:52:45.68$ | $-5.83\pm 0.09$ | $0.846\pm 0.156$ | $2.89\pm 0.53$ | 0.07 | 2.18 | 1.53
U7369-03 | 12:19:39.93 $+29:52:37.20$ | $-7.03\pm 0.07$ | $1.027\pm 0.102$ | $2.91\pm 0.42$ | 0.03 | 1.54 | 1.07
U7369-04 | 12:19:37.86 $+29:52:06.76$ | $-5.75\pm 0.09$ | $0.956\pm 0.151$ | $6.59\pm 0.73$ | 0.19 | 3.03 | 2.11
U7369-05 | 12:19:37.32 $+29:52:08.36$ | $-4.95\pm 0.10$ | $1.284\pm 0.162$ | $2.87\pm 0.64$ | 0.18 | 3.04 | 2.12
U7369-06 | 12:19:39.91 $+29:52:52.40$ | $-6.65\pm 0.07$ | $0.996\pm 0.096$ | $3.16\pm 0.53$ | 0.08 | 0.97 | 0.68
U7369-09 | 12:19:38.94 $+29:53:00.92$ | $-6.62\pm 0.07$ | $1.305\pm 0.098$ | $3.42\pm 0.39$ | 0.41 | 0.19 | 0.14
U7369-10 | 12:19:38.70 $+29:52:59.48$ | $-12.08\pm 0.06$ | $0.824\pm 0.077$ | $2.31\pm 0.39$ | 0.16 | 0.00 | 0.01
U7369-11 | 12:19:38.88 $+29:53:05.55$ | $-6.11\pm 0.08$ | $0.901\pm 0.129$ | $2.23\pm 0.56$ | 0.28 | 0.37 | 0.26
U7369-12 | 12:19:39.01 $+29:53:08.45$ | $-5.89\pm 0.08$ | $1.152\pm 0.128$ | $0.55\pm 0.39$ | 0.29 | 0.55 | 0.39
U7369-13 | 12:19:40.37 $+29:53:29.93$ | $-6.84\pm 0.07$ | $0.895\pm 0.101$ | $1.56\pm 0.42$ | 0.18 | 2.10 | 1.47
U7369-14 | 12:19:37.83 $+29:52:57.41$ | $-6.46\pm 0.08$ | $1.069\pm 0.114$ | $3.48\pm 0.50$ | 0.20 | 0.65 | 0.45
U7369-15 | 12:19:37.45 $+29:52:52.53$ | $-7.43\pm 0.07$ | $0.929\pm 0.093$ | $1.13\pm 0.47$ | 0.05 | 0.99 | 0.69
U7369-16 | 12:19:38.51 $+29:53:09.36$ | $-5.66\pm 0.09$ | $1.204\pm 0.146$ | $2.29\pm 0.56$ | 0.16 | 0.57 | 0.40
U7369-17 | 12:19:37.81 $+29:53:00.54$ | $-8.22\pm 0.06$ | $0.895\pm 0.079$ | $1.33\pm 0.39$ | 0.05 | 0.65 | 0.45
U7369-18 | 12:19:36.97 $+29:52:58.99$ | $-7.05\pm 0.07$ | $1.081\pm 0.095$ | $1.14\pm 0.39$ | 0.04 | 1.27 | 0.88
U7369-19 | 12:19:38.27 $+29:53:26.03$ | $-7.68\pm 0.07$ | $0.917\pm 0.094$ | $1.86\pm 0.42$ | 0.15 | 1.52 | 1.07
U7369-20 | 12:19:39.43 $+29:53:46.89$ | $-6.27\pm 0.08$ | $1.007\pm 0.121$ | $1.31\pm 0.42$ | 0.17 | 2.72 | 1.90
U7369-21 | 12:19:37.42 $+29:53:16.35$ | $-7.18\pm 0.07$ | $0.910\pm 0.096$ | $1.34\pm 0.42$ | 0.13 | 1.33 | 0.93
U7369-22 | 12:19:37.68 $+29:53:27.93$ | $-6.46\pm 0.08$ | $0.940\pm 0.117$ | $1.54\pm 0.44$ | 0.13 | 1.77 | 1.23
U7369-23 | 12:19:36.34 $+29:53:12.58$ | $-6.68\pm 0.07$ | $0.848\pm 0.104$ | $1.35\pm 0.42$ | 0.20 | 1.88 | 1.31
Sms
E137-18-01 | 16:20:56.66 $-60:29:08.15$ | $-7.79\pm 0.06$ | $1.030\pm 0.078$ | $3.42\pm 0.21$ | 0.18 | 0.65 | 0.31
E137-18-02 | 16:21:05.09 $-60:27:50.06$ | $-8.16\pm 0.06$ | $0.772\pm 0.079$ | $3.20\pm 0.23$ | 0.10 | 2.96 | 1.40
E137-18-03 | 16:21:00.44 $-60:29:10.65$ | $-7.15\pm 0.07$ | $1.143\pm 0.090$ | $3.11\pm 0.21$ | 0.18 | 0.29 | 0.14
E137-18-04 | 16:21:00.42 $-60:29:43.48$ | $-7.78\pm 0.06$ | $0.900\pm 0.078$ | $3.97\pm 0.23$ | 0.04 | 0.91 | 0.43
E137-18-05 | 16:21:02.92 $-60:29:14.23$ | $-6.99\pm 0.07$ | $0.835\pm 0.092$ | $5.59\pm 0.49$ | 0.05 | 0.82 | 0.39
E137-18-06 | 16:21:00.98 $-60:30:04.40$ | $-6.98\pm 0.07$ | $0.929\pm 0.095$ | $8.10\pm 0.35$ | 0.00 | 1.57 | 0.74
E137-18-07 | 16:21:11.03 $-60:28:37.99$ | $-6.90\pm 0.07$ | $1.339\pm 0.092$ | $5.77\pm 1.30$ | 0.14 | 2.92 | 1.39
E274-01-01 | 15:14:15.65 $-46:47:31.48$ | $-8.56\pm 0.06$ | $1.004\pm 0.077$ | $0.45\pm 0.10$ | 0.41 | 0.73 | 0.39
E274-01-02 | 15:14:12.16 $-46:48:39.53$ | $-7.78\pm 0.06$ | $0.864\pm 0.077$ | $3.97\pm 0.17$ | 0.04 | 0.48 | 0.26
E274-01-03 | 15:14:15.32 $-46:48:09.76$ | $-7.88\pm 0.06$ | $1.093\pm 0.077$ | $2.15\pm 0.10$ | 0.09 | 0.18 | 0.10
E274-01-04 | 15:14:16.49 $-46:48:17.19$ | $-7.16\pm 0.06$ | $1.055\pm 0.078$ | $3.37\pm 0.10$ | 0.09 | 0.30 | 0.16
E274-01-06 | 15:14:19.07 $-46:48:00.91$ | $-7.26\pm 0.06$ | $1.051\pm 0.077$ | $4.92\pm 0.12$ | 0.11 | 0.75 | 0.40
E274-01-07 | 15:14:18.54 $-46:48:24.34$ | $-6.86\pm 0.06$ | $0.945\pm 0.078$ | $2.80\pm 0.10$ | 0.11 | 0.63 | 0.33
N247-01 | 00:47:09.72 $-20:37:40.25$ | $-7.42\pm 0.06$ | $0.859\pm 0.078$ | $16.02\pm 0.28$ | 0.05 | 1.75 | 0.31
N247-02 | 00:47:11.59 $-20:38:48.58$ | $-6.59\pm 0.06$ | $1.138\pm 0.080$ | $18.79\pm 1.01$ | 0.27 | 1.40 | 0.25
N4605-01 | 12:40:11.35 $+61:34:47.80$ | $-5.69\pm 0.07$ | $1.491\pm 0.090$ | $1.49\pm 0.18$ | 0.24 | 3.60 | 1.32
N4605-02 | 12:40:05.08 $+61:35:40.51$ | $-5.51\pm 0.07$ | $0.934\pm 0.101$ | $7.63\pm 0.40$ | 0.10 | 1.76 | 0.65
N4605-03 | 12:40:10.75 $+61:34:57.46$ | $-7.14\pm 0.06$ | $0.974\pm 0.079$ | $9.64\pm 0.40$ | 0.03 | 3.33 | 1.22
N4605-04 | 12:40:06.32 $+61:35:40.27$ | $-8.20\pm 0.06$ | $1.000\pm 0.077$ | $4.12\pm 0.20$ | 0.18 | 1.92 | 0.70
N4605-05 | 12:40:07.61 $+61:35:35.99$ | $-6.77\pm 0.06$ | $1.227\pm 0.079$ | $7.13\pm 0.21$ | 0.14 | 2.17 | 0.80
N4605-06 | 12:40:09.02 $+61:35:26.86$ | $-7.13\pm 0.06$ | $1.036\pm 0.079$ | $2.39\pm 0.22$ | 0.00 | 2.53 | 0.93
N4605-08 | 12:40:13.58 $+61:35:12.72$ | $-6.34\pm 0.07$ | $0.979\pm 0.091$ | $13.63\pm 0.29$ | 0.16 | 3.43 | 1.26
N4605-09 | 12:40:19.99 $+61:34:24.90$ | $-6.01\pm 0.07$ | $0.949\pm 0.098$ | $8.55\pm 0.18$ | 0.14 | 5.17 | 1.90
N4605-10 | 12:40:12.23 $+61:35:30.83$ | $-8.26\pm 0.06$ | $0.969\pm 0.077$ | $19.16\pm 0.25$ | 0.04 | 2.94 | 1.08
N4605-11 | 12:40:16.35 $+61:35:28.57$ | $-7.00\pm 0.06$ | $0.688\pm 0.080$ | $2.72\pm 0.22$ | 0.13 | 3.64 | 1.33
N4605-12 | 12:40:18.42 $+61:35:37.99$ | $-5.41\pm 0.08$ | $1.240\pm 0.113$ | $2.55\pm 0.25$ | 0.16 | 3.89 | 1.43
Figure 9: HST/ACS color composite images of the dwarf galaxies presented in
this study. With blue, red and magenta circles are shown the blue and red GC
candidates and likely background contaminants, respectively. For the blue
green and red channels we used $V,(V+I)/2$ and $I-$band HST/ACS images. (The
full version of this figure is available upon request).
|
arxiv-papers
| 2008-10-21T07:19:42 |
2024-09-04T02:48:58.361523
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Iskren Y. Georgiev (1,2), Thomas H. Puzia (3), Michael Hilker (4),\n Paul Goudfrooij (2) ((1) AIfA, Bonn, (2) STScI, Baltimore, (3) HIA, Victoria,\n (4) ESO, Garching)",
"submitter": "Iskren Georgiev",
"url": "https://arxiv.org/abs/0810.3660"
}
|
0810.3712
|
# Fossil remnants of reionization in the halo of the Milky Way
P. Madau11affiliation: Department of Astronomy & Astrophysics, University of
California, Santa Cruz, CA 95064, pmadau,diemand@ucolick.org. , M.
Kuhlen22affiliation: Institute for Advanced Study, Einstein Drive, Princeton,
NJ 08540, mqk@ias.edu. , J. Diemand11affiliation: Department of Astronomy &
Astrophysics, University of California, Santa Cruz, CA 95064,
pmadau,diemand@ucolick.org. , B. Moore33affiliation: Institute for Theoretical
Physics, University Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland,
dpotter,moore,stadel@physik.uzh.ch. , M. Zemp11affiliation: Department of
Astronomy & Astrophysics, University of California, Santa Cruz, CA 95064,
pmadau,diemand@ucolick.org. 44affiliation: Astronomy Department, University of
Michigan, Ann Arbor, MI, 48109, mzemp@umich.edu. , D. Potter33affiliation:
Institute for Theoretical Physics, University Zurich, Winterthurerstrasse 190,
8057 Zurich, Switzerland, dpotter,moore,stadel@physik.uzh.ch. , & J.
Stadel33affiliation: Institute for Theoretical Physics, University Zurich,
Winterthurerstrasse 190, 8057 Zurich, Switzerland,
dpotter,moore,stadel@physik.uzh.ch.
(ApJL, in press)
###### Abstract
Our recently completed one billion particle Via Lactea II simulation of a
Milky Way-sized dark matter halo resolves over 50,000 gravitationally bound
clumps orbiting today within the virialized region of the main host. About
2,300 of these subhalos have one or more “progenitors” with $M>10^{6}\,\,\rm
M_{\odot}$ at redshift $z=11$, i.e. massive enough for their gas to have
cooled via excitation of H2 and fragmented prior to the epoch of cosmic
reionization. We count 4,500 such progenitors: if these were able to convert a
fraction of their gas content into very metal-poor stars with a Salpeter
initial mass function (IMF), they would be shining today with a visual
magnitude $M_{V}=6.7$ per solar mass in stars. Assuming a universal baryon
fraction, we show that mean star formation efficiencies as low as 0.1% in
progenitors $\ll 10^{8}\,\,\rm M_{\odot}$ would overproduce the abundance of
the faint Galatic dwarf spheroidals observed by the Sloan Digital Sky Survey.
Star formation at first light must have occurred either with an IMF lacking
stars below $0.9\,\,\rm M_{\odot}$, or was intrinsically very inefficient in
small dark matter halos. If the latter, our results may be viewed as another
hint of a minimum scale in galaxy formation.
###### Subject headings:
cosmology: theory – dark matter – galaxies: dwarfs – halos
## 1\. Introduction
Galaxy halos are one of the crucial testing grounds for structure formation
scenarios, as they contain the imprints of past accretion events, from before
the epoch of reionization to the present. In the standard $\Lambda$CDM
concordance cosmology, objects like the halo of our Milky Way are assembled
via the hierarchical merging and accretion of many smaller progenitors.
Subunits collapse at high redshift, are dense, and have cuspy density
profiles, and when they merge into larger hosts they are able to resist tidal
disruption. Indeed, the Via Lactea Project, a suite of some of the largest
cosmological simulations of Galactic dark matter substructure, has shown that
galaxy halos today are teeming with surviving “subhalos” (Diemand et al. 2007,
2008). The predicted subhalo counts vastly exceed the number of known
satellites of the Milky Way, a “substructure problem” that has been the
subject of many recent studies. While a full characterization of this
discrepancy is hampered by luminosity bias in the observed satellite
luminosity function (LF) (Koposov et al. 2008), it is generally agreed that
cosmic reionization may offer a plausible solution to the apparent conflict
between the Galaxy’s relatively smooth stellar halo and the extremely clumpy
cold dark matter distribution. In this hypothesis, photoionization heating
after reionization breakthrough reduces the star forming ability of newly
forming halos that are not sufficiently massive to accrete intergalactic gas
(e.g. Bullock et al. 2000; Somerville 2002; Kravtsov et al. 2004; Ricotti &
Gnedin 2005; Moore et al. 2006; Strigari et al. 2007; Simon & Geha 2007; Madau
et al. 2008). The exact value of this mass threshold remains uncertain, as it
depends on poorly known quantities like the amplitude and spectrum of the
ionizing background and the ability of the system to self-shield against UV
radiation (e.g. Efstathiou 1992; Dijkstra et al. 2004).
In this Letter we take a different approach and focus instead on the sources
of reionization itself. We use our one billion particle Via Lactea II
simulation to constrain the character of star formation (initial mass function
and efficiency of gas conversion into stars) at first light, i.e. in
subgalactic halos prior to the epoch of reionization. The starting point of
our investigation is the finding that as many as 2,300 bound clumps that
survive today in the Via Lactea II halo have early progenitors that were
massive enough for their gas to cool via excitation of molecular hydrogen and
fragment before reionization breakthrough. If low-mass stars formed in the
process, their hosts would be shining today as faint Milky Way satellites.
## 2\. Via Lactea II
Our recently completed Via Lactea II simulation, one of the highest-precision
calculation of the assembly of the Galactic CDM halo to date (Diemand et al.
2008), offers the best opportunity for a systematic investigation of the
fossil records of reionization in the halo of the Milky Way. Via Lactea II
employs just over one billion $4,100\,\rm M_{\odot}$ particles to model the
formation of a $M_{200}=1.9\times 10^{12}\,\,\rm M_{\odot}$ Milky-Way size
halo and its substructure. It resolves 50,000 subhalos today within the host’s
$r_{200}=402$ kpc (the radius enclosing an average density 200 times the mean
matter value).
Figure 1.— Top: Projected dark matter density map of Via Lactea II at $z=11$.
The image covers 3.5 comoving Mpc across and shows (magenta) all progenitor
minihalos with $M>10^{6}\,\,\rm M_{\odot}$ that have a bound descendant today
within $r_{200}$. Bottom: the $z=0$ descendants of those early progenitors.
The image covers 800 kpc across.
The Wilkinson Microwave Anisotropy Probe (WMAP) 5-year data require the
universe to be fully reionized by redshift $z=11.0\pm 1.4$ (Dunkley et al.
2008). Assuming that the region around the Milky Way was reionized at about
the same epoch of the universe as a whole, we can trace the progenitors of
present-day surviving substructure back to a time prior to reionization
breakthrough, i.e. before star formation was quenched by photoheating. We use
the 6DFOF group finder described in Diemand et al. (2007) to identify peaks in
phase-space density at $z=11$ and $z=0$. The resulting groups contain between
16 and a few thousand particles linked together in the centers of halos and
subhalos. Note that the gravitationally bound region often extends beyond this
central group. We link a descendant $z=0$ group “A” to a high redshift
progenitor “B” if “A” contains more than 10 particles from “B” and more than
any other descendant of “B”. Thus a descendant can have more than one
progenitor, but a progenitor is linked to at most one descendant. Using the
halo centers provided by 6DFOF ensures that material from “B” does indeed
contribute to the central regions of “A”, i.e. that a star cluster that formed
in the core of a progenitor clump would end up today in the dwarf galaxy of a
descendant subhalo. We include in our analysis only progenitors that either
survive individually during the hierarchical clustering process or contribute
to a gravitationally-bound descendant at $z=0$. About 10,000 “first
generation” systems above $10^{6}\,\,\rm M_{\odot}$ are totally disrupted by
tidal forces: for half of them all of their particles lie today within
$r_{200}$ and contribute to the smooth stellar halo. Figure 1 shows an image
of all Via Lactea II progenitor halos above $10^{6}\,\,\rm M_{\odot}$ at
$z=11$ with a bound descendants within $r_{200}$ today, while Figure 2 shows
the cumulative mass functions of progenitors and parents. We track forward in
time 4,500 of these early minihalos into 2,300 descendants today.111Note that
the size of the Galaxy halo is not known very precisely, since the effects of
disk formation on the total mass distribution are still poorly constrained.
When comparing Via Lactea II to the Milky Way, this translates into an
uncertainty in the abundance of subhalos of about a factor of two (Klypin et
al. 2002; Dutton et al. 2007).
Figure 2.— Via Lactea II cumulative substructure mass function at different
epochs. Solid curve: All 4,500 progenitor halos with $M>10^{6}\,\,\rm
M_{\odot}$ at $z=11$ that contribute to a gravitationally bound descendant
(2,300 of them) today within $r_{200}$. Dot-dashed curve: same for the 2,900
$M>10^{6}\,\,\rm M_{\odot}$ progenitors at $z=13$. Also shown for comparison
is the mass function of all Via Lactea II subhalos at the present epoch
(dashed curve).
## 3\. Star formation at first light
Cosmological hydrodynamics simulations of structure formation in a
$\Lambda$CDM universe have found that in the early collapse of $\mathrel{\hbox
to0.0pt{\lower 3.0pt\hbox{$\mathchar 536\relax$}\hss}\raise
2.0pt\hbox{$\mathchar 318\relax$}}10^{6}\,\,\rm M_{\odot}$ systems, enough H2
is produced to cool the gas and allow star formation within a Hubble time
(e.g. Abel et al. 2002; Bromm et al. 2002). Figure 2 shows that a few hundred
progenitors are above the “atomic cooling” mass threshold of $\sim 3\times
10^{7}\,\,\rm M_{\odot}$ ($T_{\rm vir}\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{$\mathchar 536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar
318\relax$}}10^{4}\,$K), where gas can cool and fragment via excitation of
hydrogen Ly$\alpha$. Numerical studies also suggest that, while primordial
stars were likely very massive and formed in isolation at $z\mathrel{\hbox
to0.0pt{\lower 3.0pt\hbox{$\mathchar 536\relax$}\hss}\raise
2.0pt\hbox{$\mathchar 318\relax$}}20$ (e.g. O’Shea & Norman 2007), low-mass
second-generation stars can form as soon as a minimum pre-enrichment level of
$Z=10^{-5\pm 1}\,Z_{\odot}$ is reached (e.g. Schneider et al. 2002). The
existence of 0.8 $\,\rm M_{\odot}$ extremely iron-deficient stars in the halo
of the Milky Way (Christlieb et al. 2002; Frebel et al. 2005) indicates that
low-mass star formation was possible at very-low metallicities (e.g. Tumlinson
2007).
Figure 3.— The luminous fossil remnants of reionization. Solid curve: present-
day LF of all Via Lactea II subhalos that had one or more progenitor with
$M>10^{6}\,\,\rm M_{\odot}$ at $z=11$. At this epoch, a fraction $f_{*}$ of
the gas content of each progenitor is turned instantaneously into stars with a
Salpeter IMF and metallicity $Z=Z_{\odot}/200$. Dashed curve: same for
$M>5\times 10^{5}\,\,\rm M_{\odot}$. Dot-dashed curve: same for a Chabrier
(2003) IMF. Dotted curve: same for $Z=Z_{\odot}/5$. In all the above cases
visual magnitudes at age 13.7 Gyr have been calculated using Bruzual &
Charlot’s (2003) models (1994 Padova tracks).
In order to constrain the number of low-mass stars that lit up dark matter
halos before reionization, our default model makes a number of simplifying
assumptions: 1) all progenitor hosts above $M(z=11)=10^{6}\,\,\rm M_{\odot}$
have a gas content given by the universal baryon fraction, $M_{\rm
gas}=(\Omega_{b}/\Omega_{m})M=0.17\,M$; 2) at $z\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{$\mathchar 536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar
318\relax$}}11$ a fraction $f_{*}$ of this gas is turned into very metal-poor
stars with $Z=Z_{\odot}/200$ (the lowest metallicity allowed by Bruzual &
Charlot’s 2003 stellar population synthesis models) and a standard Salpeter
initial mass function (IMF) in the range $0.1<m_{*}<100\,\,\rm M_{\odot}$. The
star formation efficiency is independent of $M$; 3) star formation is
suppressed at later epochs in all progenitors and their descendants; 4)
primordial stellar systems are deeply embedded in progenitor minihalos and
remain largely unaffected by tidal stripping even if their hosts are not. The
complete tidal disruption of a host, however, also destroys its stellar
system.
We are now in the position to construct the present-day LF of the fossil
remnants of reionization in the halo of the Milky Way. According to Bruzual &
Charlot (2003), a stellar population undergoing an instantaneous burst will be
shining at age 13.7 Gyr with a visual magnitude per solar mass in stars of 6.7
mag (Salpeter IMF, $Z=Z_{\odot}/200$). The steep LF of Via Lactea II first-
generation halos predicted in this case is shown in Figure 3 for $\langle
f_{*}\rangle=0.03$. Luminous remnants are distributed between $M_{V}=-2$ and
$M_{V}=-10$, and there are more than two hundred relatively bright objects
with $M_{V}<-6$, i.e. as bright as Boötes. A mean star formation efficiency
three times smaller would shift this curve 1.2 mag to the left. The cutoff at
faint magnitudes simply reflects our assumption that clumps below
$M(z=11)=10^{6}\,\,\rm M_{\odot}$ never form stars: the effect of lowering
this mass threshold by a factor of two is also shown for comparison. Note that
implausibly increasing the metallicity of early stars to $Z=Z_{\odot}/5$ would
only shift the curve 0.3 mag to the left (Bruzual & Charlot 2003). Similarly,
a Chabrier (2003) IMF in the same mass range instead of Salpeter has only a
small (0.2 mag) brightening effect.
## 4\. Constraints from the SDSS
Over the last two years, the Sloan Digital Sky Survey (SDSS) has doubled the
number of known Milky Way dwarf spheroidals (dSphs) brighter than $M_{V}=-2$
(e.g. Belokurov et al. 2007). An accurate estimate of the total number of
dwarfs requires a correction to the observed LF that accounts for completeness
limits and that depends on the unknown intrinsic spatial distribution of
sources. Rather than producing an “unbiased” list of Galactic satellites as in
Tollerud et al. (2008), we apply here the completeness limit of the SDSS DR5
to our predicted fossil remnants in Via Lactea II (Fig. 3) to construct an
artificial DR5 sample of luminous first-generation substructure. As computed
by Koposov et al. (2008) (and fitted by Tollerud et al. 2008), the maximum
accessible distance beyond which an object of magnitude $M_{V}$ will go
undetected by the SDSS is
$r_{\rm max}=\left({3\over 4\pi f_{\rm
DR5}}\right)^{1/3}\,10^{(-0.6M_{V}-5.23)/3}\,{\rm Mpc},$ (1)
where $f_{\rm DR5}=0.194$ is the fraction of the sky covered by DR5.
Therefore, only subhalos brighter than $M_{V}=-6.6$ will be detected all the
way to $r_{200}$: first-generation remnants with $M_{V}=-4$ will be included
in the sample only if they are closer than 120 kpc. (The above detection limit
applies to objects with surface brightness $\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{$\mathchar 536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar
316\relax$}}30$ mag arcsec-2, i.e. we are implicitly assuming that our
luminous progenitors are less diffuse than this.) As a first order correction
we also assume that subhalos within $r_{\rm max}$ are detected with 100%
efficiency, and account for the partial sky coverage of SDSS by multiplying
the number of detectable subhalos by $f_{\rm DR5}$. The resulting cumulative
LF $N_{\rm DR5}(<M_{V})$ is plotted in Figure 4 for values of $\langle
f_{*}\rangle$ that decrease from 3% down to $3\times 10^{-4}$. In this range,
the faint-end of the predicted LF is dominated by the remnants of
$10^{6}-10^{7}\,\,\rm M_{\odot}$ progenitors. For an efficiency of 1%, SDSS
should have detected $\sim 150$ first-generation systems brighter than
$M_{V}=-2$, while only a dozen are actually observed. The known Milky Way
satellites are also plotted for comparison: these include the SDSS eleven
dwarfs with $M_{V}<-2$ (including Leo T which, at a distance of 417 kpc, lies
just outside $r_{200}$) as well as the eleven (times $f_{\rm DR5}$ to correct
for the partial sky coverage) classical (pre-SDSS) bright satellites. Clearly,
mean star formation efficiencies as low as 0.1% in progenitors below
$10^{7}\,\,\rm M_{\odot}$ would overproduce the abundance of ultra-faint
dSphs. The mean efficiency of star formation must actually scale with halo
mass in order to reproduce the observations with first-generation remnants.
This is because a constant efficiency translates into a steep LF that follows
the steep mass function of CDM field halos. By contrast, the 23 known
satellites of the Milky Way have a flat LF and shine with luminosities ranging
from about a thousand to a billion times solar. The dashed curve in Figure 4
shows a toy model with a mass-dependent efficiency, $\langle
f_{*}\rangle=(0.02,0.0025,0)$ for $M$ in the range $M=(>7\times
10^{7},3.5\times 10^{7}$-$7\times 10^{7},<3.5\times 10^{7}\,\,\rm M_{\odot})$.
Note how the data allow efficiencies of order 1% only above the atomic cooling
threshold.
Figure 4.— Via Lactea II first-generation ($z=11$) subhalos detectable today
by the SDSS DR5. All curves assume $M>10^{6}\,\,\rm M_{\odot}$, a Salpeter
IMF, and $Z=Z_{\odot}/200$. The data points are taken from the compilation of
Tollerud et al. (2008). Labelled solid curves: $\langle
f_{*}\rangle=0.03,0.01,0.003,0.001,0.0003$. Dashed curve: mass-dependent star
formation efficiency, $\langle f_{*}\rangle=(0.02,0.0025,0)$ for $M$ in the
range $(>7\times 10^{7},3.5\times 10^{7}$-$7\times 10^{7},<3.5\times
10^{7}\,\,\rm M_{\odot})$. Also plotted for comparison is the LF of the
remnants of $z=13$ halos with $\langle f_{*}\rangle=0.0005$ (short dot-dashed
curve at the bottom).
## 5\. Summary
In this work we have used results from the one billion particle Via Lactea II
simulation to study for the first time the fossil signatures of the pre-
reionization epoch in the Galactic halo and set constraints on the baryonic
building blocks of today’s galaxies. We have traced the progenitors of
present-day surviving substructure back to a time prior to reionization
breakthrough, i.e. before star formation was quenched by an external UV
background. We have then populated early-forming minihalos with very metal-
poor stars following a Salpeter IMF, and looked at the photometric properties
today of their 2,300 descendants within $r_{200}$. We have shown that even
star formation efficiencies as low as 0.1% in progenitor minihalos below
$10^{7}\,\,\rm M_{\odot}$ would overproduce the abundance of Galatic dSphs
observed by the SDSS. This value should be regarded as an upper limit since
luminous dSphs are known to have formed a fraction of their stars after
reionization (e.g. Orban et al. 2008). It is also meant to be taken as a mean
value, since an alternative possibility is to allow stars to form efficiently
in a small fraction of minihalos at some given mass and to suppress star
formation in the remainders, rather than reducing the efficiency across the
board. Note that the parameter $f_{*}$ is degenerate with the baryon fraction:
if the gas content of early virialized structure was smaller than the
universal value (as found, e.g., by O’Shea & Norman 2007), then our limits on
the star formation efficiency would be correspondingly higher.
We can now test a posteriori the main assumption underlying our work, i.e.
that early Via Lactea II progenitors were populated with ”pre-reionization”
stellar systems. In the simulated high-resolution volume, there are 17,700
$M>10^{6}\,\,\rm M_{\odot}$ subhalos at $z=11$, spread over a comoving volume
of 78 Mpc3 containing a total mass of $M_{\rm DM}=3.5\times 10^{12}\,\,\rm
M_{\odot}$. Stars distributed according to a Salpeter IMF produce during their
lifetime $f_{\gamma}\approx 4,000$ Lyman continuum photons per stellar proton,
of which only a fraction $f_{\rm esc}$ will escape into the intergalactic
medium (IGM). The total stellar mass in these early-forming progenitors is
$M_{*}=3.6\times 10^{10}\,\langle f_{*}\rangle\,\,\rm M_{\odot}$. Hydrogen
photoionization requires $(1+N_{\rm rec})$ photons above 13.6 eV per atom,
where $N_{\rm rec}$ is the number of radiative recombinations over a Hubble
time. The total mass of intergalactic gas that can be kept ionized is then
$M_{\rm ion}=f_{\gamma}M_{*}f_{\rm esc}/(1+N_{\rm rec})$. The condition
$M_{\rm ion}<(\Omega_{b}/\Omega_{m})\,M_{\rm DM}$ then implies $\langle
f_{*}\rangle<0.004\,[(1+N_{\rm rec})/f_{\rm esc}]$, where the factor in square
brackets is of order 10 or so. The low star formation efficiencies derived in
this work appear then to be consistent with the idea that the region around
the Milky Way was reionized at $z\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{$\mathchar 536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar
316\relax$}}11$ by external radiation, before stars in the Local Group formed
in sufficient numbers (Weinmann et al. 2007).
We conclude that star formation at first light must have occurred either with
an IMF lacking stars below $0.9\,\,\rm M_{\odot}$, or was intrinsically very
inefficient in small dark matter halos. If the former, this may be an
indication of an upward shift in the mass scale of the IMF at early
cosmological times owing, e.g., to the hotter cosmic microwave background
(Larson 2005). If the latter, our results may be viewed as another hint (see
Strigari et al. 2008) of a minimum scale in galaxy formation, below which
supernova feedback (Dekel & Silk 1986) and/or H2 photodissociation by a Lyman-
Werner background (Haiman et al. 1997) sharply suppress star formation.
We thank J. Bullock and Z. Haiman for providing useful comments on an earlier
draft. Support for this work was provided by NASA through grants HST-
AR-11268.01-A1 and NNX08AV68G (P.M.) and Hubble Fellowship grant HST-
HF-01194.01 (J.D.). M.K. gratefully acknowledges support from the William L.
Loughlin Fellowship at the Institute for Advanced Study.
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|
arxiv-papers
| 2008-10-21T00:23:32 |
2024-09-04T02:48:58.371606
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "P. Madau, M. Kuhlen, J. Diemand, B. Moore, M. Zemp, D. Potter, J.\n Stadel",
"submitter": "Piero Madau",
"url": "https://arxiv.org/abs/0810.3712"
}
|
0810.3760
|
# Measurement of the Solar Neutrino Flux with an Array of Neutron Detectors in
the Sudbury Neutrino Observatory
B. Jamieson (for the SNO collaboration) University of British Columbia,
Vancouver, BC V6T 1Z1, Canada
###### Abstract
The Sudbury Neutrino Observatory has measured the 8B solar neutrino flux using
an array of 3He proportional counters. Results obtained using a Markov-Chain
Monte-Carlo (MCMC) parameter estimation, integrating over a standard extended
likelihood, yield effective neutrino fluxes of:
$\phi_{nc}=5.54^{+0.33}_{-0.31}$(stat) ${}^{+0.36}_{-0.34}$(syst)$\times
10^{6}$ cm-2s-1, $\phi_{cc}=1.67^{+0.05}_{-0.04}$(stat)
${}^{+0.07}_{-0.08}$(syst)$\times 10^{6}$ cm-2s-1, and
$\phi_{es}=1.77^{+0.24}_{-0.21}$(stat) ${}^{+0.09}_{-0.10}$(syst)$\times
10^{6}$ cm-2s-1. These measurements are in agreement with previous solar
neutrino flux measurements, and with neutrino oscillation model results.
Including these flux measurements in a global analysis of solar and reactor
neutrino results yields an improved precision on the solar neutrino mixing
angle of $\theta~{}=~{}34.4^{+1.3}_{-1.2}$ degrees, and $\Delta
m^{2}=7.59^{+0.19}_{-0.21}eV^{2}$.
## I THE SNO DETECTOR
The Sudbury Neutrino Observatory (SNO) snonim is a low background neutrino
detector $\sim$2 km (6000 mwe overburden) underground in the Vale Inco
Creighton nickel mine in Sudbury, Canada. The detector consists of 1000 tonnes
of D2O in a 12m diameter acrylic vessel surrounded by an inner shield of 1700
tonnes of H2O. At the edge of the inner light water, a support structure
holding about 9500 photomultiplier tubes (PMTs) provides 54% coverage. An
additional outer shield of 5300 tonnes of light water surrounds the PMTs.
The SNO detector detects the neutrino reactions:
$\nu_{x}+e^{-}~{}\rightarrow~{}\nu_{x}+e^{-}$ (ES),
$\nu_{e}+d~{}\rightarrow~{}p+p+e^{-}$ (CC), and
$\nu_{x}+d~{}\rightarrow~{}p+n+\nu_{x}$ (NC). The SNO detector provides the
unique detection of the neutrons in the NC reaction by three different
methods, one for each phase of SNO running. The first phase detected gamma
rays from the triton production in the detector
($n+d~{}\rightarrow~{}t+\gamma+6.25~{}MeV$)snod2o . In SNO’s second phase NaCl
was added to the heavy water, and increased the neutron capture through
neutron capture on the Cl
($n+^{35}C~{}\rightarrow~{}^{36}Cl+\gamma+8.6~{}MeV$), where the 8.6 $MeV$ is
the sum of a cascade of gamma rayssnosalt .
For the final phase of SNO reported on here the neutron is detected when it is
captured in an array of 36 3He proportional counters (NCDs) via the reaction
$n+^{3}He~{}\rightarrow~{}p+t+0.76~{}MeV$ncdnim ncdprl . The NCD phase
measurement separates the NC and CC signal detection which significantly
reduces the CC spectrum contamination by the 6.25 $MeV$ neutron captures on
deuterium. The NCD phase is more complex however since $\sim$10% of the
Cherenkov light is blocked by the array, the radioactivity of the counters
adds a non-negligible background, and the signal rate of $\sim$1000
neutrons/year is fairly low.
## II NCD DETECTOR ENERGY SPECTRUM CALIBRATION
The neutron energy spectrum is measured with calibration data from a 24NaCl
brine that produces neutrons by the gamma capture on deuterium
($\gamma+d~{}\rightarrow~{}p+n$). The spectrum is characterized by a peak at
0.76 $MeV$ with features at 0.57 $MeV$ and 0.19 $MeV$ where either only the
proton or triton are seen in the proportional counter.
The 24NaCl brine is the calibration source most like the neutrons produced
from solar neutrinos since it can be uniformly distributed in the D2O, and it
also provides a measurement of the neutron detection efficiency ($0.211\pm
0.007$). The mixing of the brine can be seen by looking at the light output
from different parts of the detector, and only data from after the brine was
uniformly mixed was used. In addition, the detection efficiency from the MCNP
MCNP Monte Carlo code yielded an efficiency of 0.210(3). Finally a time-
series based analysis using neutron bursts from a 252Cf source confirmed these
neutron efficiency measurements.
A simulation of the NCD detector was used to model the energy spectrum from
background alphas from U, Th, and Po in the nickel of the NCD walls. The model
included effects of the energy loss, multiple scattering, electron-ion pair
generation, electron drift and diffusion, electron multiple scattering, ion
mobility, electron avalanche, space charge, signal generation, and a detailed
propagation through the electronics. The Monte Carlo simulation was tuned for
the surface to bulk alpha ratio, energy scale, energy resolution, alpha depth,
and contributions from different parts of the NCD using the alphas above 2
$MeV$. The simulation was found to reproduce the pulse width and energy
spectrum very well, including the effects of alphas from the NCD anode wires.
Instrumental events in the NCD detector were easily separated from ionization
events using an amplitude versus energy cut. Six of the 36 NCD strings with
high instrumental rates were removed from the analysis. Two probability
distribution functions for the instrumental backgrounds were included in the
signal-extraction to fit for an unconstrained number of instrumental events.
## III BLIND ANALYSIS
Three blindfolds were implemented on the NCD phase measurement. One month of
the data was open for analysts to tune cuts. A hidden fraction of neutrons
that follow muons were added to the data, and an unknown fraction of candidate
events were omitted. Detailed internal documentation was reviewed by topic
committees before the box was opened to reveal the true solar neutrino flux
measurement.
The box was opened on May 2, 2008, and the results are presented as found
after correcting two inconsistencies. The three separate signal-extraction
codes had to correct pilot errors on the inputs to the final fit, which
resulted in no change in the central values, and made the final uncertainties
reported agree. An incorrect algorithm in fitting the peak value of the ES
posterior distribution was replaced.
## IV NEUTRON BACKGROUNDS
The neutron backgrounds were measured for the D2O radioactivity, atmospheric
neutrinos, 16N neutrons, NCD counter neutrons from the bulk of the counters,
from hot-spots on the counters, and from the NCD cables. In addition,
backgrounds from the acrylic-vessel, reactor neutrinos, and other sources were
included in the background estimates. The neutron backgrounds are summarized
in the following Table, and were included in the signal extraction broadening
the uncertainties in the measured solar neutrino fluxes.
Table 1: Table of neutron backgrounds in the PMT and NCD data. Source | PMT neutrons | NCD neutrons
---|---|---
D2O radioactivity | 7.6$\pm$1.2 | 28.7$\pm$4.7
Atmospheric $\nu$, 16N | 24.7$\pm$4.6 | 13.6$\pm$2.7
Other backgrounds | 0.7$\pm$0.1 | 2.3$\pm$0.3
NCD bulk PD, 17,18O($\alpha$,n) | 4.6${}^{+2.1}_{-1.6}$ | 27.6${}^{+12.9}_{-10.3}$
NCD hot-spots | 17.7$\pm$1.8 | 64.4$\pm$6.4
NCD cables | 1.1$\pm$1.0 | 8.0$\pm$5.2
External-source neutrons | 20.6$\pm$10.4 | 40.9$\pm$20.6
Total | 77${}^{+12}_{-10}$ | 185${}^{+25}_{-22}$
## V SIGNAL EXTRACTION METHODS
Parameter estimation, and estimation of the uncertainties on all fit
parameters (both fluxes and nuisance parameters for systematics) is done with
a Metropolis algorithm Markov-Chain Monte Carlo (MCMC)mcmc .
In SNO’s previous signal extractions, the negative log-likelihood (NLL)
function was simply minimized with respect to all parameters to get the best-
fit value, and the curvature of -$\log(L)$ at the minimum was used to
determine the uncertainties. The floating systematics approach also uses a
minimization, although with additional nuisance parameters added to account
for systematic uncertainties.
Minimizing the NLL is very challenging for the 27 flux parameters
($\phi_{nc}$,$\phi_{cc1...13}$,$\phi_{es1...13}$) and 35 systematic parameters
in the fit. The systematic parameters include PMT reconstruction uncertainties
estimated from calibration data pmtcalib , both PMT and NCD efficiencies, NCD
Monte Carlo and NCD instrumental uncertainties. In addition because the
likelihood function can be a bit choppy near the minimum, traditional
minimizers such as MINUIT run into trouble and often will not converge in
reasonable periods of time.
The MCMC method gets around this problem by interpreting NLL as the negative
log of a joint probability distribution for all of the free parameters. We
then integrate over all nuisance parameters to determine the distributions for
the fluxes. The origins of this procedure go back to Bayesian probability
theory, and in fact our approach could be considered to be a Bayesian analysis
with uniform priors assumed for the fluxes.
The advantages of the MCMC method are twofold. First, it converges much faster
than a 50+ parameter MINUIT minimization. Rather than minimizing over
parameters, we in fact integrate over nuisance parameters. Second, since we
integrate over nuisance parameters with the MCMC instead of trying to find a
best-fit point, we are insensitive to and in fact average over choppiness in
the NLL that would interfere with finding a minimum. Both the speed of
convergence and the insensitivity to numerical noise in the NLL means that the
MCMC method is better suited to handling large numbers of nuisance parameters.
## VI RESULTS
The final corrected solar neutrino fluxes above a 6 $MeV$ Kinetic Energy
threshold from the unblinded NCD data are:
$\phi_{nc}~{}=~{}5.54^{+0.48}_{-0.46}\times 10^{6}$ cm-2s-1,
$\phi_{cc}~{}=~{}1.67^{+0.08}_{-0.09}\times 10^{6}$ cm-2s-1, and
$\phi_{es}~{}=~{}1.77^{+0.26}_{-0.23}\times 10^{6}$ cm-2s-1. The correlation
between $\phi_{cc}$ and $\phi_{nc}$ was only -0.19 in the NCD phase fit
including all systematic uncertainties. The $\phi_{es}$ is a 2.2 sigma lower
than the Super Kamiokande measurement, but the full set of fluexes has a
probability of 32.8% of being consistent with the six other flux measurements
from all of the SNO phases.
The NCD energy fit, unconstrained PMT energy fit, PMT angle to the Sun, and
radial position fits are shown in FIG. 1. It can be seen in the PMT energy fit
that the number of neutrons in the PMT is considerably less than either the
SNO D2O or salt phases, and provides the better CC separation than in those
phases.
Figure 1: These figures show the NCD energy spectrum fit (top left), the PMT
energy unconstrained fit (top right), the PMT angle to the sun fit (bottom
left), and the PMT radial position fit (bottom right).
Including the NCD phase flux measurements in a global oscillation analysis
results in an improved precision on the solar neutrino mixing angle of
$\theta~{}=~{}34.4^{+1.3}_{-1.2}$ degrees, and
$\Delta~{}m^{2}=7.59^{+0.19}_{-0.21}eV^{2}$.
###### Acknowledgements.
SNO gratefully acknowledges NSERC, Industry Canada, NRC, Northern Ontario
Heritage Fund, Vale INCO, Atomic Energy of Canada, Ontario Power Generation,
High Performance Computing Virtual Laboratory, Canada Foundation for
Innovation, Canada Research Chairs, Westgrid, US Department of Energy, NERSC
PDSF, UK STFC, and Portugal FCT.
## References
* (1) SNO Collaboration, Nucl. Instrum. and Methods Phys. Res., Sect. A 449, (2000) 172.
* (2) SNO Collaboration, Phys. Rev. Lett. 87, (2001) 071301. SNO Collaboration, Phys. Rev. Lett. 89, (2002) 011301. SNO Collaboration, Phys. Rev. Lett. 89, (2002) 011302.
* (3) SNO Collaboration, Phys. Rev. Lett. 92, (2004) 181301. Phys. Rev. C72, (2005) 055502. SNO Collaboration, Phys. Rev. C75, (2007) 045502.
* (4) J.F. Amsbaugh _et al._ , Nucl. Instrum. and Methods Phys. Res., Sect. A 579, (2007) 1054.
* (5) SNO Collaboration, Phys. Rev. Lett. 101, (2008) 111301.
* (6) Refer to the MCNP website for details:
http://mcnp-green.lanl.gov/publication/mcnp_publications.html
* (7) Metropolis _et al._ , Journ. Chem. Phys. 21,(1953) 1087. W.K. Hastings, Biometrika 57, (1970) 97.
* (8) M.R. Dragowsky _et al._ , Nucl. Instrum. and Methods Phys. Res., Sect. A 481, (2002) 284. I. Blevis _et al._ , Nucl. Instrum. and Methods Phys. Res., Sect. A 517, (2004) 139\.
|
arxiv-papers
| 2008-10-21T07:49:17 |
2024-09-04T02:48:58.379496
|
{
"license": "Public Domain",
"authors": "Blair Jamieson (for the SNO Collaboration)",
"submitter": "Blair Jamieson",
"url": "https://arxiv.org/abs/0810.3760"
}
|
0810.3789
|
# Bosonic Spectral Function in HTSC Cuprates:
Part I - Experimental Evidence for Strong Electron-Phonon Interaction
1E. G. Maksimov, 2,3M. L. Kulić, 4O. V. Dolgov 1 Lebedev Physical Institute,
119991 Moscow, Russia
2Goethe-Universität Frankfurt, Theoretische Physik, 60054 Frankfurt/Main,
Germany
3Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie,
12489 Berlin,Germany
4Max-Planck-Institut für Festkörperphysik,70569 Stuttgart, Germany
###### Abstract
In Part I we discuss accumulating experimental evidence related to the
structure and origin of the bosonic spectral function $\alpha^{2}F(\omega)$ in
high-temperature superconducting (HTSC) cuprates near optimal doping. Some
global properties of $\alpha^{2}F(\omega)$, such as number and positions of
peaks, are extracted by combining optics, neutron scattering, ARPES and
tunnelling measurements. These methods give convincing evidence for strong
electron-phonon interaction (EPI) with $1<\lambda\lesssim 3$ in cuprates near
optimal doping. Here we clarify how these results are in favor of the
Eliashberg-like theory for HTSC cuprates near optimal doping.We argue that the
neglect of EPI in some previous studies of HTSC was based on a number of
deceptive prejudices related to the strength of EPI, on some physical
misconceptions and misleading interpretation of experimental results.
In Part II we discuss some theoretical ingredients which are necessary to
explain the experimental results related to pairing mechanism in optimally
doped cuprates. These comprise the Migdal-Eliashberg theory for EPI in
strongly correlated systems which give rise to the forward scattering peak.
The latter is due to the combined effects of the weakly screened Madelung
interaction in the ionic-metallic structure of layered cuprates and many body
effects of strong correlations. While EPI is responsible for the strength of
pairing the residual Coulomb interaction (by including spin fluctuations)
triggers the d-wave pairing.
## I Introduction
In spite of an unprecedentedly intensive experimental and theoretical study
after the discovery of high-temperature superconductivity (HTSC) in cuprates
there is, even twenty-two years after, no consensus on the pairing mechanism
in these materials. At present there are two important experimental facts
which are not under dispute: (1) the critical temperature $T_{c}$ in cuprates
is high, with the maximum $T_{c}^{\max}\sim 160$ $K$; (2) the pairing in
cuprates is d-wave like, i.e.
$\Delta(\mathbf{k},\omega)\approx\Delta_{s}(k,\omega)+\Delta_{d}(\omega)(\cos
k_{x}-\cos k_{y})$ with $\Delta_{s}<0.1\Delta_{d}$. On the contrary there is a
dispute concerning the scattering mechanism which governs normal state
properties and pairing in cuprates. To this end, we stress that in the HTSC
cuprates, a number of properties can be satisfactorily explained by assuming
that the quasi-particle dynamics is governed by some electron-boson scattering
and in the superconducting state bosonic quasi-particles are gluing electrons
in Cooper pairs. Which bosonic quasi-particles are dominating in the cuprates
is the subject which will be discussed in this work. It is known that the
electron-boson (phonon) scattering is well described by the Migdal-Eliashberg
theory if the adiabatic parameter $A_{B}\equiv$ $\lambda(\omega_{B}/W_{b})$
fulfills the condition $A_{B}\ll 1$, where $\lambda$ is the electron-boson
coupling constant, $\omega_{B}$ is the characteristic bosonic energy and
$W_{b}$ is the electronic band width. The important characteristic of the
electron-boson scattering is the Eliashberg spectral function
$\alpha^{2}F(\mathbf{k},\mathbf{k}^{\prime},\omega)$ (or its average
$\alpha^{2}F(\omega)$) which characterizes scattering of quasi-particle from
$\mathbf{k}$ to $\mathbf{k}^{\prime}$ by exchanging bosonic energy $\omega$.
Therefore, in systems with electron-boson scattering the knowledge of this
function is of crucial importance. There are at least two approaches differing
in assumed gluing bosons. The first one is based on the electron-phonon
interaction (EPI) Maksimov-Review , Kulic-Review , Alexandrov , Gunnarsson-
review-2008 , Falter where mediating bosons are phonons and where the the
average spectral function $\alpha^{2}F(\omega)$ is similar to the phonon
density of states $F_{ph}(\omega)$. Note, $\alpha^{2}F(\omega)$ is not the
product of two functions although sometimes one defines the function
$\alpha^{2}(\omega)=\alpha^{2}F(\omega)/F(\omega)$ which should approximate
the energy dependence of the strength of the EPI coupling. There are numerous
experimental evidence in cuprates which support the dominance of the EPI
scattering mechanism with a rather large coupling constant
$1<\lambda^{ep}\lesssim 3$ and which will be discussed in detail below. In the
EPI approach $\alpha^{2}F(\omega)$ is extracted from tunnelling measurements
in conjunction with IR optical measurements. We stress again that the Migdal-
Eliashberg theory is well justified framework for EPI since in most
superconductors the condition $A_{ph}\ll 1$ is fulfilled. The HTSC are on the
borderline and it is a natural question - under which condition can high Tc be
realized in the non-adiabatic limit $A_{ph}\sim 1$? The second approach Pines
assumes that EPI is too weak to be responsible for high $T_{c}$ in cuprates
and it is based on a phenomenological model for spin-fluctuation interaction
($SFI$) as the dominating scattering mechanism, i.e. it is a non-phononic
mechanism. In this approach the spectral function is proportional to the
imaginary part of the spin susceptibility
$\mathop{\mathrm{I}m}\chi(\mathbf{k}-\mathbf{k}^{\prime},\omega)$, i.e.
$\alpha^{2}F(\mathbf{k},\mathbf{k}^{\prime},\omega)\sim\mathop{\mathrm{I}m}\chi(\mathbf{k}-\mathbf{k}^{\prime},\omega)$.
NMR spectroscopy and magnetic neutron scattering give that in HTSC cuprates
$\chi(\mathbf{q},\omega)$ is peaked at the antiferromagnetic wave vector
$Q=(\pi/a,\pi/a)$ and this property is favorable for d-wave pairing. The $SFI$
theory roots basically on the strong electronic repulsion on Cu atoms, which
is usually studied by the Hubbard model or its (more popular) derivative the
t-J model. Regarding the possibility to explain high Tc solely by strong
correlations, as it is reviewed in Patrick-Lee , we stress two facts. First,
at present there is no viable theory which can justify these (non-phononic)
mechanisms of pairing. Second, the central question in this approach is - do
models based on the Hubbard Hamiltonian show up superconductivity at
sufficiently high critical temperatures ($T_{c}\sim 100$ $K$) ? A number of
numerical studies of these models offer a negative answer. For instance, the
sign-free variational Monte Carlo algorithm in the 2D repulsive ($U>0$)
Hubbard model gives no evidence for HTSC, neither the BCS- nor Berezinskii-
Kosterlitz-Thouless (BKT)-like Imada-MC . At the same time, similar
calculations show that there is a strong tendency to superconductivity in the
attractive ($U<0$) Hubbard model for the same strength of $U$, i.e. at finite
temperature in the 2D model with $U<0$ the BKT superconducting transition is
favored. Concerning HTSC in the $t-J$ model, various numerical calculations
such as Monte Carlo calculations of the Drude spectral weight Scalapino-Drude-
weight and high temperature expansion Pryadko have shown that there is no
superconductivity at temperatures characteristic for cuprates and if it exists
$T_{c}$ must be rather low - few Kelvins. These numerical results tell us that
the lack of high $T_{c}$ (even in $2D$ BKT phase) in the repulsive ($U>0$)
single-band Hubbard model and in the $t-J$ model is not only due to
thermodynamical $2D$-fluctuations (which at finite T suppress and destroy
superconducting phase coherence in large systems) but it is mostly due to an
inherent ineffectiveness of strong correlations to produce solely high $T_{c}$
in cuprates. These numerical results certainly mean that the simple single-
band Hubbard, as well as its derivative the t-J model, are insufficient to
explain the pairing mechanism in cuprates and some other ingredients must be
included. Having in mind these facts there is no room at present for any kind
of celebration of the victory of non-phononic mechanisms of pairing as some
prefer to do.
Since $EPI$ is as a rule strong in oxides, then it is plausible that it should
be accounted for in cuprates at least in the normal metallic state. As it will
be argued later on, the experimental support for the importance of EPI comes
from optics, tunnelling, and recent ARPES measurements Shen-review . It is
worth mentioning that recent ARPES activity was a strong impetus for renewed
experimental and theoretical studies of EPI in cuprates. However, in spite of
accumulating experimental evidence for importance of EPI with
$\lambda^{ep}>1$, there are occasionally reports which doubt its importance in
cuprates. This is the case with recent interpretation of some optical
measurements in terms of SFI only Hwang-Timusk-1 and with LDA band structure
calculations Bohnen-Cohen , Giuistino , where both claim that EPI is
negligibly small, i.e. $\lambda^{ep}<0.3$.
The paper is organized as follows. There are two parts - in Part I we will
mainly discuss experimental results in cuprates near optimal doping and
minimal theoretical explanations which are related to the spectral function
$\alpha^{2}F(\omega)$ as well to the transport spectral function
$\alpha_{tr}^{2}F(\omega)$ and how these are related to EPI in cuprates. In
this work we consider mainly those direct one-particle and two-particles
probes of low energy quasi-particle excitations (by including gap and
pseudogap) and scattering rates which give informations on the structure of
the spectral functions $\alpha^{2}F(\mathbf{k},\mathbf{k}^{\prime},\omega)$
and $\alpha_{tr}^{2}F(\omega)$ in systems near optimal doping. These are
angle-resolved photoemission ($ARPES$), various arts of tunnelling
spectroscopies such as superconductor/insulator/ normal metal ($SIN$)
junctions and break junctions, scanning-tunnelling microscope spectroscopy
($STM$), infrared ($IR$) and Raman optics, inelastic neutron scattering
($INS$) etc. We shall argue that these direct probes give evidence for a
rather strong EPI in cuprates as dominating scattering mechanism of quasi-
particles. Some other experiments on EPI are also discussed in order to
complete the arguments for the importance of EPI in cuprates. The detailed
contents of Part I is the following. In Section II we discuss some prejudices
related to $EPI$ and the Fermi-liquid behavior of HTSC cuprates. We argue that
any non-phononic mechanism of pairing should have very large bare critical
temperature $T_{c0}\gg T_{c}$ in the presence of the large EPI coupling
constant, $\lambda^{ep}\geq 1$, if its spectral function is weakly momentum
dependent, i.e. if
$\alpha^{2}F(\mathbf{k},\mathbf{k}^{\prime},\omega)\approx\alpha^{2}F(\omega)$.
The fact that EPI is large in the normal state of cuprates and the condition
that it must conform with d-wave pairing implies inevitably that EPI in
cuprates must be strongly momentum dependent. In Section III we discuss direct
and indirect experimental evidence for the importance of EPI and for the
weakness of SFI in cuprates. These are:
(A) Magnetic neutron scattering measurements - These measurements provide
dynamic spin susceptibility $\chi(\mathbf{q},\omega)$ which is in the $SFI$
phenomenological approach Pines related to the Eliashberg spectral function,
i.e.
$\alpha^{2}F_{sf}(\mathbf{k},\mathbf{k}^{\prime},\omega)\sim\mathop{\mathrm{I}m}\chi(\mathbf{q}=\mathbf{k}-\mathbf{k}^{\prime},\omega)$.
We stress that such an approach can be qualitatively justified only in the
weak coupling limit, $g_{sf}\ll W_{b}$, where $W_{b}$ is the band width. Here
we discuss experimental results which give evidence for strong rearrangement
(with respect to $\omega$) of $\mathop{\mathrm{I}m}\chi(\mathbf{Q},\omega)$ by
doping toward the optimal doped HTSC Bourges . It turns out that in the
optimally doped cuprates with $T_{c}=92.5$ $K$
$\mathop{\mathrm{I}m}\chi(\mathbf{Q},\omega)$ is drastically suppressed
compared to that in slightly underdoped ones with $T_{c}=91$ $K$, and this is
strong evidence for the smallness of the SFI coupling constant.
(B) Optical conductivity measurements \- From these measurements one can
extract the transport relaxation rate $\gamma_{tr}(\omega)$ and indirectly an
approximative shape of the transport spectral function
$\alpha^{2}F_{tr}(\omega)$. In that respect we discuss: (i) the misleading
concept concerning the relation between the optical relaxation rate
$\gamma_{tr}(\omega)$ and the quasi-particle relaxation rate $\gamma(\omega)$.
This (misleading) concept has been appearing repeatedly in the last twenty
years despite the fact that this controversy is resolved many years ago Allen
, Dolgov-Shulga , Shulga , Maksimov-Review , Kulic-Review , Kulic-AIP ; (ii)
some methods of extraction of the optical spectral function
$\alpha_{tr}^{2}F(\omega)$ from optical reflectivity measurements. It turns
out that the width and the shape of the extracted $\alpha_{tr}^{2}F(\omega)$
favor EPI; (iii) the restricted sum-rule for the optical weight as a function
of temperature which can also be explained by strong $EPI$ Maks-Karakoz-1 ,
Maks-Karakoz-2 ; (iv) good agreement with experiments of the temperature
dependence of the resistivity $\rho(T)$ calculated with the extracted
$\alpha_{tr}^{2}F(\omega)$. Recent femtosecond time-resolved optical
spectroscopy in $La_{2-x}Sr_{x}CuO_{4}$ gives additional evidence for
importance of EPI and ineffectiveness of SFI Kusar-2008 .
(C) ARPES measurements and EPI \- From these measurements one can extract the
self-energy $\Sigma(\mathbf{k},\omega)$ from which one can extract some
properties of $\alpha^{2}F(\mathbf{k},\mathbf{k}^{\prime},\omega)$. Here we
discuss the following items: (i) appearance of the nodal and anti-nodal kinks
in optimally and slightly underdoped cuprates, as well as the structure of the
ARPES self-energy ($\Sigma(\mathbf{k},\omega)$) and its isotope dependence,
which are all due to EPI; (ii) appearance of different slopes of
$\Sigma(\mathbf{k},\omega)$ at low ($\omega\ll\omega_{ph}$) and high energies
($\omega\gg\omega_{ph}$) which can be explained with strong EPI; (iii)
formation of small polarons in the undoped HTSC which is due to strong EPI -
this gives rise to phonon side bands which are clearly seen in ARPES of
undoped HTSC.
(D) Tunnelling spectroscopy - It is well known that this method is of an
immense importance in obtaining the spectral function $\alpha^{2}F(\omega)$
from tunnelling conductance. In this part we discuss the following items: (i)
extraction of the Eliashberg spectral function $\alpha^{2}F(\omega)$ with
$\lambda=2-4$ from the tunnelling conductance of break-junctions Tunneling-
Vedeneev -Ponomarev-Tunnel which gives that the maxima of
$\alpha^{2}F(\omega)$ coincide with the maxima in the phonon density of
states; (ii) the presence of the dip in dI/dV in STM which shows the
pronounced oxygen isotope effect and important role of these phonons: (iii)
the presence of fine and doping independent structure in I(V) characteristics
due to phonon emission by the Josephson current in layered HTSC cuprates with
intrinsic Josephson junctions.
(E) Phonon neutron scattering measurements \- From these experiments one can
extract the phonon density of state $F_{ph}(\omega)$ and strengths of the
quasi-particle coupling with various phonon modes. Here we argue, that the
large softening and broadening of the half-breathing Cu-O bond-stretching
phonon, of apical oxygen phonons and of oxygen $B_{1g}$ buckling phonons (in
LSCO, BISCO,YBCO) cannot be explained by LDA. It is curious that the magnitude
of softening can be partially obtained by LDA but the calculated widths of
some important modes are an order of magnitude smaller than the neutron
scattering data show. This remarkable fact implies the inadequacy of LDA in
strongly correlated systems and a more sophisticated many body theory for EPI
is needed. This problem will be discussed in more details in Part II MaKuDoAk
. In Section IV brief summary of the Part I is given. Since we are dealing
with electron-boson scattering in cuprate near optimal doping, then in Section
V \- Appendix we introduce the reader briefly into the Migdal-Eliashberg
theory for superconductors (and normal metals) where the quasi-particle
spectral function $\alpha^{2}F(\mathbf{k},\mathbf{k}^{\prime},\omega)$ and the
transport spectral function $\alpha_{tr}^{2}F(\omega)$ are defined.
Finally, at the end of the day one poses a question - do the experimental
results of the above enumerated spectroscopic methods allow a building of a
satisfactory and physically reasonable microscopic theory for basic scattering
and pairing mechanism in cuprates? The posed question is very modest compared
with a much stringent request for the theory of everything \- which would be
able to explain all properties of HTSC materials. Such an ambitious project is
not realized even in those low-temperature conventional superconductors where
it is definitely proved that the pairing is due to EPI and many properties are
well accounted for by the Migdal-Eliashberg theory. Let us mention only two
examples. First, the experimental value for the coherence peak in the
microwave response $\sigma_{s}(T<T_{c})$ at $17$ $GHz$ in $Nb$ is much higher
than the theoretical value obtained by the Migdal-Eliashberg theory
Marsiglio-1994 . So to say, the theory explains the coherence peak at $17$
$GHz$ in $Nb$ qualitatively but not quantitatively. However, the measurements
at higher frequency $\sim 60$ $GHz$ are in agreement with the Migdal-
Eliashberg theory Klein-1994 . Second, the experimental boron (B) isotope
effect in $MgB_{2}$ (with $T_{c}\approx 40$ $K$) is much smaller than the
theoretical value, i.e. $\alpha_{B}^{\exp}\approx 0.3<\alpha_{B}^{th}=0.5$,
although the pairing is due solely by EPI for boron vibrations MgB2-isotop .
Since the theory of everything is impossible in the complex materials such as
HTSC cuprates in Part I and II we shall not discuss those phenomena which need
much more microscopic details and/or more sophisticated many-body theory.
These are selected by chance: (i) peculiarities of the coherence peak in the
microwave response $\sigma(T)$ in HTSC cuprates, which is peaked at $T$ much
smaller than $T_{c}$, contrary to the case of LTSC where it occurs near
$T_{c}$; (ii) $T_{c}$ dependence on the number of $CuO_{2}$ in the unit cell;
(iii) temperature dependence of the Hall coefficient; (iv) distribution of
states in the vortex core, etc. However, in a separate paper - Part II
MaKuDoAk we shall discuss some minimal theoretical concepts which can explain
at least qualitatively and semi-quantitatively results related to the above
enumerated spectroscopic methods. Due to the presence of strong correlations
and quasi-2D electronic structure some of these concepts go beyond the LDA
approach. In our opinion at this stage of the HTSC physics some important
ingredients of the future theory are already recognized. These are: (1) very
peculiar quasi-2D ionic-metallic structure with a rather weak screening along
the c-axis, which is a prerequisite for strong EPI; (2) strong Coulomb
interaction and correlations which are responsible for strong magnetism in
undoped cuprates and for important renormalizations of EPI. Since both
ingredients belong to the class of strong coupling problems, at present there
is no quantitative theory and therefore we must rely on approximative and
model theories. Even these approaches allow us qualitative (and semi-
quantitative) explanations of some important properties which are due to the
interplay of EPI and strong correlations. The latter two cause the appearance
of momentum dependent EPI - peaked at small transfer momenta Kulic-Review .
Based on such an approach we are able to explain (understand) at least
qualitatively some very puzzling experimental results, for instance: (a) why
is d-wave pairing realized in the presence of strong EPI? (b) why is the
transport coupling constant ($\lambda_{tr}$) smaller than the pairing one
$\lambda$, i.e. $\lambda_{tr}\lesssim\lambda/3$? (c) Why is the mean-field
(one-body) LDA approach unable to give reliable values for the EPI coupling
constant in cuprates and how many-body effects help; (d) why is d-wave pairing
robust in presence of non-magnetic impurities and defects? (e) why the ARPES
nodal and antinodal kinks are differently renormalized in the superconducting
states?
In real materials there are numerous experimental evidence for nanoscale
inhomogeneities in HTSC oxides . For instance recent STM experiments show
rather large gap dispersion at least on the surface of BISCO crystals Davis
giving rise for a pronounced inhomogeneity of the superconducting order
parameter, i.e. $\Delta(\mathbf{k},\mathbf{R})$ where $\mathbf{k}$ is the
relative momentum of the Cooper pair and $\mathbf{R}$ is the center of mass of
Cooper pairs. One possible reason for the inhomogeneity of
$\Delta(\mathbf{k},\mathbf{R})$ and disorder on the atomic scale can be due to
extremely high doping level of $\sim(10-20)$ $\%$ in HTSC cuprates which is
many orders of magnitude larger than in standard semiconductors ($10^{21}$ vs
$10^{15}$ carrier concentration). There are some claims that high $T_{c}$ is
exclusively due to these inhomogeneities (of an extrinsic or intrinsic origin)
which may increase EPI Phillips , while other try to explain high $T_{c}$
within the inhomogeneous Hubbard or t-J model. In Part II MaKuDoAk we argue
that the concept of an increase of Tc by inhomogeneity is ill-defined, since
the increase of $T_{c}$ is defined with respect to the average value
$\bar{T}_{c}$. However, the latter quantity is experimentally not well defined
and an alleged increase of $T_{c}$ by the material inhomogeneity cannot be
tested at all.
## II EPI vs non-phononic mechanisms - facts vs prejudices
Concerning the high $T_{c}$ in cuprates, one of the central questions is -
which interaction(s) is(are) responsible for strong quasi-particle scattering
in the normal state and for the superconducting pairing? In the last twenty-
two years, the scientific community was overwhelmed by all kinds of
(im)possible proposed pairing mechanisms, most of which are hardly verifiable
in any material, if at all. This trend is still continuing nowadays (although
with smaller slope), in spite of the fact that the accumulated experimental
results eliminate all but few.
A. Fermi vs non-Fermi liquid in cuprates
After discovery of HTSC in cuprates there was a large amount of evidence on
strong scattering of quasi-particles which contradicts the canonical (popular
but narrow) definition of the Fermi liquid, thus giving rise to numerous
proposals of the so called non-Fermi liquids, such as Luttinger liquid, RVB
theory, marginal Fermi liquid, etc. In our opinion there is no need for these
radical approaches in explaining basic physics in cuprates at least in
optimally, slightly underdoped and overdoped metallic and superconducting HTSC
cuprates. This subject will be discussed more in Part II and here we give some
clarifications related to the dilemma of Fermi vs non-Fermi liquid. The
definition of the canonical Fermi liquid (based on the Landau work) in
interacting Fermi systems comprises the following properties: (1) there are
quasi-particles with charge $q=\pm e$, spin $s=1/2$ and low-laying energy
excitations $\xi_{\mathbf{k}}(=\epsilon_{\mathbf{k}}-\mu)$ which are much
larger than their inverse life-times, i.e. $\xi_{\mathbf{k}}\gg
1/\tau_{\mathbf{k}}\sim\xi_{\mathbf{k}}^{2}/W_{b}$. Since the level width
$\Gamma=2/\tau_{\mathbf{k}}$ of the quasi-particle is negligibly small, this
means that the excited states of the Fermi liquid are placed in one-to-one
correspondence with the excited states of the free Fermi gas; (2) at $T=0$ $K$
there is an energy level with the Fermi surface at which
$\xi_{\mathbf{k}_{F}}=0$ and the Fermi quasi-particle distribution function
$n_{F}(\xi_{\mathbf{k}})$ has finite jump; (3) the number of quasi-particles
under the Fermi surface is equal to the total number of conduction particles
(we omit here other valence and core electrons) - the Luttinger theorem; (4)
the interaction between quasi-particles are characterized with a few (Landau)
parameters which describe low-temperature thermodynamics and transport
properties. Having this definition in mind one can say that if fermionic
quasi-particles interact with some bosonic excitation (for instance with
phonons) and if the coupling is sufficiently strong, then the former are not
described by the canonical Fermi liquid since at energies and temperatures of
the order of the Debay temperature $k_{B}\Theta_{D}(\equiv\hbar\omega_{D})$
(more precisely $\sim\Theta_{D}/5$), i.e. for $\xi_{\mathbf{k}}\sim\Theta_{D}$
one has $\tau_{\mathbf{k}}^{-1}\gtrsim\xi_{\mathbf{k}}$ and the quasi-particle
picture (in the sense of the Landau definition) is broken down. In that
respect an electron-boson system can be classified as a non-canonical Fermi
liquid for sufficiently strong electron-boson coupling. It is nowadays well
known that for instance Al, Zn are weak coupling systems since for
$\xi_{\mathbf{k}}\sim\Theta_{D}$ one has
$\tau_{\mathbf{k}}^{-1}\ll\xi_{\mathbf{k}}$ and they are well described by the
Landau theory. However, the electron-phonon system is satisfactory described
by the Migdal-Eliashberg theory and the Boltzmann theory, where thermodynamic
and transport properties depend on the spectral function
$\alpha^{2}F_{sf}(\mathbf{k},\mathbf{k}^{\prime},\omega)$ and its higher
momenta. Since in HTSC cuprates the electron-boson (phonon) coupling is rather
strong and $T_{c}$ is large, i.e. of the order of characteristic boson
energies ($\omega_{B}$), $T_{c}\sim\omega_{B}/4$, then it is natural that in
the normal state (at $T>$ $T_{c}$) we deal inevitably with a strong
interacting non-canonical Fermi liquid which is at least qualitatively and
semi-quantitatively described by the Migdal-Eliashberg theory. In order to
justify this statement we shall in the following elucidate some properties in
more details by studying optical, ARPES, tunnelling and other experiments.
B. Prejudice on the limitation of the strength of EPI
In spite of reach experimental evidence in favor of strong EPI in HTSC oxides
there was a disproportion in the research activity (especially theoretical) in
the past, since the investigation of the SFI mechanism of pairing prevailed in
the literature. This retrograde trend was partly due to an incorrect statement
in Cohen on the possible upper limit of Tc in the phonon mechanism of
pairing. Since in the past we have discussed this problem thoroughly in
numerous papers - for the recent one see Maksimov-Dolgov-2007 , we shall
outline here the main issue and results only.
It is well known that in an electron-ion crystal, besides the attractive EPI,
there is also repulsive Coulomb interaction. In case of an isotropic and
homogeneous system with weak quasi-particle interaction, the effective
potential $V_{eff}(\mathbf{k},\omega)$ in the leading approximation looks like
as for two external charges ($e$) embedded in the medium with the total
longitudinal dielectric function $\varepsilon_{tot}(\mathbf{k},\omega)$
($\mathbf{k}$ is the momentum and $\omega$ is the frequency) Kirzhnitz , i.e.
$V_{eff}(\mathbf{k},\omega)=\frac{V_{ext}(\mathbf{k})}{\varepsilon_{tot}(\mathbf{k},\omega)}=\frac{4\pi
e^{2}}{k^{2}\varepsilon_{tot}(\mathbf{k},\omega)}.$ (1)
In case of strong interaction between quasi-particles, the state of embedded
quasi-particles changes significantly due to interaction with other quasi-
particles, giving rise to $V_{eff}(\mathbf{k},\omega)\neq 4\pi
e^{2}/k^{2}\varepsilon_{tot}(\mathbf{k},\omega)$. In that case $V_{eff}$
depends on other (than $\varepsilon_{tot}(\mathbf{k},\omega)$) response
functions. However, in the case when Eq.(1) holds, i. e. when the weak-
coupling limit is realized, $T_{c}$ is given by
$T_{c}\approx\bar{\omega}\exp(-1/(\lambda^{ep}-\mu^{\ast})$ Allen-Mitrovic ,
Kirzhnitz $)$. Here, $\lambda^{ep}$ is the EPI coupling constant,
$\bar{\omega}$ is an average phonon frequency and $\mu^{\ast}$ is the Coulomb
pseudo-potential, $\mu^{\ast}=\mu/(1+\mu\ln E_{F}/\bar{\omega})$ ($E_{F}$ is
the Fermi energy). The couplings $\lambda^{ep}$ and $\mu$ are expressed by
$\varepsilon_{tot}(\mathbf{k},\omega=0)$
$\mu-\lambda^{ep}=\langle N(0)V_{eff}(\mathbf{k},\omega=0)\rangle$
$=N(0)\int_{0}^{2k_{F}}\frac{kdk}{2k_{F}^{2}}\frac{4\pi
e^{2}}{k^{2}\varepsilon_{tot}(\mathbf{k},\omega=0)},$ (2)
where $N(0)$ is the density of states at the Fermi surface and $k_{F}$ is the
Fermi momentum - see more in Maksimov-Review . In Cohen it was claimed that
lattice stability of the system with respect to the charge density wave
formation implies the condition $\varepsilon_{tot}(\mathbf{k},\omega=0)>1$ for
all $\mathbf{k}$. If this was correct then from Eq.(2) it follows that
$\mu>\lambda^{ep}$, which limits the maximal value of Tc to the value
$T_{c}^{\max}\approx E_{F}\exp(-4-3/\lambda^{ep})$. In typical metals
$E_{F}<(1-10)$ $eV$ and if one accepts the statement in Cohen , i.e. that
$\lambda^{ep}\leq\mu\leq 0.5$, one obtains $T_{c}\sim(1-10)$ $K$. The latter
result, if it would be true, means that EPI is ineffective in producing not
only high-Tc superconductivity but also low-temperature superconductivity
(LTS). However, this result is apparently in conflict first of all with
experimental results in LTSC, where in numerous systems $\mu\leq\lambda^{ep}$
and $\lambda^{ep}>1$. For instance, $\lambda^{ep}\approx 2.6$ is realized in
$PbBi$ alloy which is definitely much higher than $\mu(<1)$.
Moreover, the basic theory tells us that $\varepsilon_{tot}(\mathbf{k}\neq
0,\omega)$ is not the response function Kirzhnitz . Namely, if a small
external potential $\delta V_{ext}(\mathbf{k},\omega)$ is applied to the
system it induces screening by charges of the medium and the total potential
is given by $\delta V_{tot}(\mathbf{k},\omega)=\delta
V_{ext}(\mathbf{k},\omega)/\varepsilon_{tot}(\mathbf{k},\omega)$ which means
that $1/\varepsilon_{tot}(\mathbf{k},\omega)$ is the response function. The
latter obeys the Kramers-Kronig dispersion relation which implies the
following stability condition: $1/\varepsilon_{tot}(\mathbf{k},\omega=0)<1$
for $\mathbf{k}\neq 0$, i.e. either $\varepsilon_{tot}(\mathbf{k}\neq
0,\omega=0)>1$ or $\varepsilon_{tot}(\mathbf{k}\neq 0,\omega=0)<0$. This
important theorem invalidates the above restriction on the maximal value of Tc
in the EPI mechanism. We stress that the condition
$\varepsilon_{tot}(\mathbf{k}\neq 0,\omega=0)<0$ is not in conflict with the
lattice instability. For instance, in inhomogeneous systems such as crystal,
the total longitudinal dielectric function is matrix in the space of
reciprocal lattice vectors ($\mathbf{Q}$), i.e.
$\hat{\varepsilon}_{tot}(\mathbf{k+Q},\mathbf{k+Q}^{\prime},\omega)$, and
$\varepsilon_{tot}(\mathbf{k},\omega)$ is defined by
$\varepsilon_{tot}^{-1}(\mathbf{k},\omega)=\hat{\varepsilon}_{tot}^{-1}(\mathbf{k+0},\mathbf{k+0},\omega)$.
It is well known that in dense metallic systems with one ion per cell (such as
metallic hydrogen ) and with the electronic dielectric function
$\varepsilon_{el}(\mathbf{k},0)$, one has DKM
$\varepsilon_{tot}(\mathbf{k},0)=\frac{\varepsilon_{el}(\mathbf{k},0)}{1-1/\varepsilon_{el}(\mathbf{k},0)G_{ep}(\mathbf{k})}.$
(3)
At the same time the frequency of the longitudinal phonon
$\omega_{l}(\mathbf{k})$ is given by
$\omega_{l}^{2}(\mathbf{k})=\frac{\Omega_{p}^{2}}{\varepsilon_{el}(\mathbf{k},0)}[1-\varepsilon_{el}(\mathbf{k},0)G_{ep}(\mathbf{k})],$
(4)
where $\Omega_{p}^{2}$ is the ionic plasma frequency, $G_{ep}$ is the local
(electric) field correction - see Ref. DKM . The real condition for lattice
stability requires that $\omega_{l}^{2}(\mathbf{k})>0$, which implies that for
$\varepsilon_{el}(\mathbf{k},0)>0$ one has
$\varepsilon_{el}(\mathbf{k},0)G_{ep}(\mathbf{k})<1$. The latter condition
gives automatically $\varepsilon_{tot}(\mathbf{k},0)<0$. Furthermore, the
calculations DKM show that in the metallic hydrogen crystal,
$\varepsilon_{tot}(\mathbf{k},0)<0$ for all $\mathbf{k\neq 0}$. Moreover, the
analyzes of crystals with more ions per unit cell DKM gives that
$\varepsilon_{tot}(\mathbf{k\neq 0},0)<0$ is more a rule than an exception \-
see Fig. 1. The physical reason for $\varepsilon_{tot}(\mathbf{k\neq 0},0)<0$
are local field effects described above by $G_{ep}(\mathbf{k})$. Whenever the
local electric field $\mathbf{E}_{loc}$ acting on electrons (and ions) is
different from the average electric field $\mathbf{E}$, i.e.
$\mathbf{E}_{loc}\neq\mathbf{E}$, there are corrections to
$\varepsilon_{tot}(\mathbf{k},0)$ (and to $\varepsilon_{e}(\mathbf{k},0)$)
which may lead to $\varepsilon_{tot}(\mathbf{k},0)<0$.
Figure 1: Inverse total static dielectric function
$\varepsilon^{-1}(\mathbf{p})$ for normal metals (K, Al, Pb and metallic H) in
$\mathbf{p}=(1,0,0)$ direction. $\mathbf{G}$ is the reciprocal lattice vector.
The above analysis tells us that in real crystals
$\varepsilon_{tot}(\mathbf{k},0)$ can be negative in the large portion of the
Brillouin zone giving rise to $\lambda^{ep}-\mu>0$ in Eq.(2). This means that
the dielectric function $\varepsilon_{tot}$ does not limit $T_{c}$ in the
phonon mechanism of pairing. This result does not mean that there is no limit
on Tc at all - see more in Maksimov-Dolgov-2007 and references therein. We
mention in advance that the local field effects play important role in HTSC
oxides, due to their layered structure with very unusual ionic-metallic
binding, thus giving rise to large $EPI$ \- see more in the subsequent
sections. It is pertinent to note that one of the author of Cohen recognizes
the possibility $\varepsilon_{tot}(\mathbf{k},0)<0$ and in Cohen2 even makes
interesting proposals for compounds with large EPI and $T_{c}>100$ $K$, while
the other author Anderson2 still ignores rigors of scientific arguments and
negates importance of EPI in HTSC cuprates.
In conclusion we point out that there are no theoretical and experimental
arguments for ignoring EPI in HTSC cuprates. To this end it is necessary to
answer several important questions which are related to experimental findings
in HTSC cuprates (oxides): (1) if EPI is responsible for pairing in HTSC
cuprates and if superconductivity is of $d-wave$ type, how are these two facts
compatible? (2) why is the transport EPI coupling constant $\lambda_{tr}$
(entering resistivity) much smaller than the pairing EPI coupling constant
$\lambda^{ep}(>1)$ (entering Tc), i.e. why one has $\lambda_{tr}(\approx
0.4-1.2)\ll\lambda^{ep}(\sim 2-4)$? (3) is high Tc possible for a moderate EPI
coupling constant, let say for $\lambda^{ep}\leq 1$, and under which
conditions? (4) if EPI is ineffective for pairing in HTSC oxides, inspite of
$\lambda^{ep}>1$, why it is so?
C. Is a non-phononic pairing realized in HTSC?
Regarding EPI one can pose a question - whether it contributes significantly
to d-wave pairing in cuprates? Surprisingly, despite numerous experiments in
favor of EPI, a number of researchers still believe that EPI is irrelevant for
pairing Pines . This belief is mainly based: (i) on the, previously discussed,
incorrect lattice stability criterion, which implies small EPI; (ii) on the
well established experimental fact that d-wave pairing is realized in cuprates
Tsui-Kirtley , which is believed to be incompatible with EPI. Having in mind
that EPI in HTSC is strong with $1<\lambda^{ep}<3$ (see below), we assume for
the moment that the leading pairing mechanism in cuprates, which gives d-wave
pairing, is due to some non-phononic mechanism, like the exitonic one, with
the high energy gluing boson ($\Omega_{nph}\gg\omega_{ph}$) and with the bare
critical temperature $T_{c0}$ and look for the effect of EPI. If EPI is
isotropic, like in most LTSC materials, then it would be very detrimental for
d-wave pairing - the pair breaking effect. In the case of dominating isotropic
EPI in the normal state and the exitonic-like pairing, then near $T_{c}$ the
linearized Eliashberg equations have an approximative form for weak non–phonon
interaction (with the characteristic frequency $\Omega_{nph}$)
$Z(\omega_{n})\Delta(\mathbf{k},\omega_{n})=\pi
T_{c}\sum_{n^{\prime}}^{\Omega_{{}_{nph}}}\sum_{\mathbf{q}}V_{nph}(\mathbf{k},\mathbf{q},n,n_{{}^{\prime}})\frac{\Delta(\mathbf{q},\omega_{n^{\prime}})}{\left|\omega_{n^{\prime}}\right|}$
$Z(\omega_{n})\approx 1+\Gamma_{ep}/\omega_{n}.$ (5)
For pure d-wave pairing one has
$V_{nph}(\mathbf{k},\mathbf{q},n,n_{{}^{\prime}})=V_{nph}\Theta(\Omega_{nph}-\left|\omega_{n}\right|)\Theta(\Omega_{nph}-\left|\omega_{n^{\prime}}\right|)\times(\cos
k_{x}-\cos k_{y})(\cos q_{x}-\cos q_{y})$ and
$\Delta(\mathbf{k},\omega_{n})=\Delta_{d}\Theta(\Omega_{nph}-\left|\omega_{n}\right|)(\cos
k_{x}-\cos k_{y})$ which gives the equation for Tc \- see Maksimov-Review
$\ln\frac{T_{c}}{T_{c0}}=\Psi(\frac{1}{2})-\Psi(\frac{1}{2}+\frac{\Gamma_{ep}}{2\pi
T_{c}}).$ (6)
Here $\Psi$ is the di-gamma function. At temperatures near $T_{c}$ one has
$\Gamma_{ep}\approx 2\pi\lambda_{ep}T_{c}$ and the solution of Eq. (6) is
approximately $T_{c}\approx T_{c0}\exp\\{-\lambda^{ep}\\}$, which means that
for $T_{c}^{\max}\sim 160$ $K$ and $\lambda^{ep}>1$ the bare $T_{c0}$ due to
the non-phononic interaction must be very large, i.e. $T_{c0}>500$ $K$.
Concerning other non-phononic mechanisms, such as the SFI one, the effect of
the isotropic EPI in the framework of Eliashberg equations was studied
numerically in Licht . The latter is based on Eqs.(42-44) in Appendix A. with
kernels
$\lambda_{\mathbf{kp}}^{Z}(i\nu_{n})=\lambda_{\mathbf{kp}}^{sf}(i\nu_{n})+\lambda_{\mathbf{kp}}^{ep}(i\nu_{n})$
(7)
$\lambda_{\mathbf{kp}}^{\Delta}(i\nu_{n})=\lambda_{\mathbf{kp}}^{ep}(i\nu_{n})-\lambda_{\mathbf{kp}}^{sf}(i\nu_{n}),$
(8)
where $\lambda_{\mathbf{kp}}^{sf}(i\nu_{n})$ is taken in the FLEX
approximation Scalapino-Review . The calculations Licht confirm the very
detrimental effect of the isotropic EPI on the d-wave pairing due to SFI. For
the bare SFI $T_{c0}\sim 100$ $K$ and $\lambda^{ep}>1$ the calculations give
very small $T_{c}\ll 100$ $K$. These results tell us that a more realistic
pairing interaction must be operative in cuprates and that EPI is strongly
momentum dependent Kulic1 . Only in that case is strong EPI conform with
d-wave pairing, either as its main cause or as a supporter of a non-phononic
mechanism. In Part II we shall argue that the strongly momentum dependent EPI
is the main player in cuprates providing the strength of the pairing
mechanism, while the residual Coulomb interaction and SF, although weaker,
trigger it to d-wave pairing.
## III Experimental evidence for strong EPI and weak SFI
In the following we discuss some important experiments which give evidence for
strong EPI in cuprates. Before doing it; we shall discuss some magnetic
neutron scattering measurements in cuprates whose results are against the SFI
mechanism of pairing. The experimental results related to the pronounced
imaginary part of the susceptibility $Im\chi(\mathbf{k},k_{z},\omega)$ at the
AF wave vector $\mathbf{k}=\mathbf{Q}=(\pi,\pi)$ were interpreted in a number
of papers as a support for the SFI mechanism for pairing Pines . We briefly
explain inadequacy of such an interpretation.
### III.1 Magnetic neutron scattering and the spin fluctuation spectral
function
A. SFI affects $T_{c}$ very little
Before discussing experimental results in cuprates on the imaginary part of
the spin susceptibility $Im\chi(\mathbf{k},\omega)$ we point out that in
theories based on spin fluctuations the effective pairing potential
$V_{sf}(\mathbf{k},\omega)$, which is repulsive, is assumed in the form Pines
$V_{sf}(\mathbf{q},\omega+i0^{+})=g_{sf}^{2}\int_{-\infty}^{\infty}\frac{d\nu}{\pi}\frac{Im\chi(\mathbf{q},\nu+i0^{+})}{\nu-\omega}.$
(9)
This form of $V_{sf}$ can be theoretically justified in the weak coupling
limit ($U\ll W_{b}$) only. This mechanism of pairing could be effective in
cuprates only if the spin susceptibility (spectral function)
Im$\chi(\mathbf{q},\omega)$ is strongly peaked at the AF wave vector
$\mathbf{Q}=(\pi/a,\pi/a)$. What is the experimental situation? The
breakthrough came from magnetic neutron scattering experiments on
$YBa_{2}Cu_{3}O_{6+x}$ by Bourges group Bourges . They showed that
$\mathop{\mathrm{I}m}\chi^{(odd)}(\mathbf{q},\omega)$ (the odd part of the
spin susceptibility in the bilayer system) at
$\mathbf{q}=\mathbf{Q}=(\pi,\pi)$ is strongly dependent on the hole doping as
it is shown in Fig. 2.
Figure 2: Magnetic spectral function $Im\chi^{(-)}(\mathbf{k},\omega)$ in
$YBa_{2}Cu_{3}O_{6+x}$. (Top) In the normal state at $T=100$ $K$ and at
$Q=(\pi,\pi)$. $100$ counts in the vertical scale corresponds to
$\chi_{max}^{(-)}\approx 350\mu_{B}^{2}/eV$. (Bottom) In the superconducting
state at $T=5$ $K$ and at $Q=(\pi,\pi)$. From Ref. Bourges .
By varying doping there is a huge rearrangement of
$\mathop{\mathrm{I}m}\chi^{(odd)}(\mathbf{Q},\omega)$ in the frequency
interval which is important for superconducting pairing, let say $5$
$meV<\omega<60$ $meV$ as it is seen in the last two curve in Fig. 2(top). At
the same time there is only a small variation of the corresponding critical
temperature $T_{c}$! For instance, in the underdoped $YBa_{2}Cu_{3}O_{6.92}$
crystal Im$\chi^{(odd)}(\mathbf{Q},\omega)$, and
$S(\mathbf{Q})=N(\mu)g_{sf}^{2}\int_{0}^{60}d\omega\mathop{\mathrm{I}m}\chi^{(odd)}(\mathbf{Q},\omega)\sim\lambda^{sf}\cdot\left\langle\omega\right\rangle$,
is much larger than that in the near optimally doped $YBa_{2}Cu_{3}O_{6.97}$,
i.e. $S_{6.92}(\mathbf{Q})\gg S_{6.97}(\mathbf{Q})$, although the difference
in the corresponding critical temperatures $T_{c}$ is very small, i.e.
$T_{c}^{(6.92)}=91$ $K$ (in $YBa_{2}Cu_{3}O_{6.92}$) and $T_{c}^{(6.97)}=92.5$
$K$ (in $YBa_{2}Cu_{3}O_{6.97}$). This pronounced rearrangement and decrease
of Im$\chi^{(odd)}(\mathbf{Q},\omega)$ by doping, but a negligible change in
$T_{c}$ in YBCO is clearly seen in Fig. 2(top), which is strong evidence
against the $SFI$ mechanism of pairing. These results in fact mean that the
$SFI$ coupling constant $\lambda^{sf}(\sim g_{sf}^{2})$ is small, i.e.
$\lambda_{sf}^{(\exp)}\ll 1$, and the $SFI$ pairing mechanism is ineffective
in cuprates. We stress that in the phenomenological theory of the SFI pairing
Pines , an unrealistically large coupling $g_{sf}>0.7$ $eV$ was assumed which
gives $\lambda^{sf}\sim 2$. The latter value cannot be justified neither
experimentally nor theoretically. Let us add that the anti-correlation between
the decrease of $\mathop{\mathrm{I}m}\chi(\mathbf{Q},\omega)$ and increase of
$T_{c}$ by increasing doping toward the optimal value is also present in the
NMR spectral function $I_{\mathbf{Q}}=\lim_{\omega\rightarrow
0}\mathop{\mathrm{I}m}\chi(\mathbf{Q},\omega)/\omega$ which determines the
longitudinal relaxation rate $1/T_{1}$ \- see Kulic-Review . This result
additionally disfavors the SFI model of pairing Pines , i.e. the strength of
pairing interaction is little affected by SFI. As we shall discuss below the
role of SFI together with the stronger direct Coulomb interaction is to
trigger d-wave pairing.
A less direct argument for smallness of the SFI coupling constant, i.e.
$g_{sf}\leq 0.2$ $eV$ and $\lambda^{sf}\sim 0.2-0.3$ comes from other
experiments related to the magnetic resonance peak in the superconducting
state, and this will be discussed next.
B. Ineffectiveness of the magnetic resonance peak
In the superconducting state of optimally doped YBCO and BISCO,
$\mathop{\mathrm{I}m}\chi(\mathbf{Q},\omega)$ is significantly suppressed at
low frequencies except near the resonance energy $\omega_{res}\approx 41$
$meV$ where a pronounced narrow peak appears - the magnetic resonance peak. We
stress that there is no magnetic resonance peak in LSCO sand consequently one
can question the importance of the resonance peak in the scattering processes.
The relative intensity of this peak (compared to the total one) is small, i.e.
$I_{0}\sim(1-5)\%$ \- see Fig 2 (bottom). In underdoped cuprates this peak is
present also in the normal state as it is seen in Fig 2 (top). After the
discovery of the resonance peak there were attempts to relate it: (i) to the
origin of the superconducting condensation energy and (ii) to the kink in the
energy dispersion or the peak-dimp structure in the ARPES spectral function.
In order that the property (i) holds it is necessary that the peak intensity
$I_{0}$ is small Kivelson . $I_{0}$ is obtained by equating the condensation
energy $E_{con}$ with the change of the magnetic energy in the superconducting
state, i.e. $\delta E_{mag}\approx 4I_{0}E_{mag}$, where
$E_{con}\approx N(0)\Delta^{2}/2$ $E_{mag}=J\iint\frac{d\omega
d^{2}k}{(2\pi)^{3}}(1-\cos k_{x}-\cos k_{y})S(\mathbf{k},\omega).$ (10)
By taking $2\Delta\approx 4T_{c}$ and the realistic value $N(0)\sim
1/(10J)\sim 1$ $states/eV\cdot spin$, one obtains $I_{0}\sim
10^{-1}(T_{c}/J)^{2}\sim 10^{-3}$. However, such a small intensity cannot be
responsible for the anomalies in ARPES and optical spectra since it gives rise
to small coupling constant $\lambda^{res}$ for the interaction of holes with
the resonance peak, i.e.
$\lambda^{res}\approx(2I_{0}N(0)g_{sf}^{2}/\omega_{res})\ll 1$. Such a small
coupling does not affect superconductivity. Moreover, by studying the width of
the resonance peak one can extract the SFI coupling constant $g_{sf}$. Thus,
the magnetic resonance disappears in the normal state of the optimally doped
YBCO, which can be qualitatively understood by assuming that its broadening
scales with the resonance energy $\omega_{res}$, i.e.
$\gamma^{res}<\omega_{res}$, where the line-width is given by
$\gamma^{res}=4\pi(N(0)g_{sf})^{2}\omega_{res}$ Kivelson . This limits to
$g_{sf}<0.2$ $eV$. We stress that the obtained $g_{sf}$ is much smaller than
the one assumed in the phenomenological spin-fluctuation theory Pines where
$g_{sf}\sim 0.6-0.7$ $eV$, but much larger than in Kivelson (where
$g_{sf}<0.02$ $eV$). The smallness of $g_{sf}$ comes out also from the
analysis of the antiferromagnetic state in underdoped metals of LSCO and YBCO
Kulic-Kulic , where the small magnetic moment $\mu(<0.1$ $\mu_{B})$ points to
an itinerant antiferromagnetism with small coupling constant $g_{sf}\leq 0.2$
$eV$. The conclusion is that the magnetic resonance in the optimally doped
YBCO is a consequence of the onset of superconductivity and not its cause.
There is also a principal reason against the pairing due to the resonance peak
at least in optimally doped cuprates. Since its intensity near $T_{c}$ is
vanishingly small, though not affecting pairing at the second order phase
transition at $T_{c}$, then if it would be the origin for superconductivity
the phase transition at Tc would be first order, contrary to experiments.
Recent ARPES experiments give evidence that the magnetic resonance cannot be
related to the kinks in ARPES spectra Lanzara , Valla \- see the discussion
below.
We shall argue below that despite its smallness, spin fluctuations can,
together with other contributions of the residual Coulomb interaction, trigger
d-wave pairing, while the strength of pairing is due to EPI which is peaked at
small transfer momenta - see more below and in Kulic-Review , Kulic-AIP .
### III.2 Optical conductivity and EPI
Optical spectroscopy gives information on optical conductivity
$\sigma(\omega)$ and on two-particle excitations, from which one can
indirectly extract the transport spectral function $\alpha_{tr}^{2}F(\omega)$.
Since this method probes bulk sample (on the skin depth), contrary to ARPES
and tunnelling methods which probe tiny regions ($10-15$ Å) near the sample
surface, this method is very indispensable. However, $\sigma(\omega)$ is not a
directly measured quantity but it is derived from the reflectivity
$R(\omega)=\left|(\sqrt{\varepsilon_{ii}(\omega)}-1)/(\sqrt{\varepsilon_{ii}(\omega)}+1)\right|^{2}$
with the transversal dielectric tensor
$\varepsilon_{ii}(\omega)=\varepsilon_{ii,\infty}+\varepsilon_{ii,latt}+4\pi
i\sigma_{ii}(\omega)/\omega$. Here, $\varepsilon_{ii,\infty}$ is the high
frequency dielectric function, $\varepsilon_{ii,latt}$ describes the
contribution of the lattice vibrations and $\sigma_{ii}(\omega)$ describes the
optical (dynamical) conductivity of conduction carriers. $R(\omega)$ was
usually measured in the limited frequency interval
$\omega_{\min}<\omega<\omega_{\max}$. Therefore, some physical modelling for
$R(\omega)$ is needed in order to guess it outside this range - see more in
reviews Maksimov-Review , Kulic-Review . This was the reason for numerous
inadequate interpretations of optic measurements in cuprates, as well as the
misconceptions and misinterpretations that will be uncover below. An
illustrative example for this claim is large dispersion in the reported value
of $\omega_{pl}$ \- from $0.06$ to $25$ $eV$, i.e. almost three orders of
magnitude - see discussion in Bozovic-Plasma . This tells us also that in some
periods science suffers from a lack of rigorousness and objectiveness.
However, it turns out that $IR$ measurements of $R(\omega)$ in conjunction
with elipsometric measurements of $\varepsilon_{ii}(\omega)$ at high
frequencies allows reliable determination of $\sigma(\omega)$.
1\. Transport and quasiparticle relaxation rates
The widespread misconception in studying the quasi-particle scattering in
cuprates was an ad hoc assumption that the transport relaxation rate
$\gamma_{tr}(\omega)$ is equal to the quasi-particle relaxation rate
$\gamma(\omega)$, in spite of the well known fact that
$\gamma_{tr}(\omega)\neq\gamma(\omega)$ Allen . This incorrect assumption led
to the abandoning of EPI as relevant scattering mechanism in cuprates.
Although we have discussed this problem several times before, we want to do it
again, since the correct understanding of the scattering mechanism in cuprates
will take us forward in understanding of the pairing mechanism.
The dynamical conductivity $\sigma(\omega)$ consists of two parts, i.e.
$\sigma(\omega)=\sigma^{inter}(\omega)+\sigma^{intra}(\omega)$ where
$\sigma^{inter}(\omega)$ describes interband transitions which contribute at
higher frequencies, while $\sigma^{intra}(\omega)$ is due to intraband
transitions which are relevant at low frequencies $\omega<1$ $eV$. (In $IR$
measurements the frequency is usually given in $cm^{-1}$, where the following
conversion holds: $1cm^{-1}=29.98$ $GHz=0.123985$ $meV=1.44$ $K$.) The
experimental data for $\sigma(\omega)=\sigma_{1}+i\sigma_{2}$ in cuprates are
usually processed by the generalized (extended) Drude formula Allen ,
Schlesinger , Dolgov-Shulga , Shulga
$\sigma(\omega)=\frac{\omega_{p}^{2}}{4\pi}\frac{1}{\gamma_{tr}(\omega)-i\omega
m_{tr}(\omega)/m_{\infty}}\equiv\frac{1}{\tilde{\omega}_{tr}(\omega)},$ (11)
which is a useful representation for systems with single band electron-boson
scattering which is justified in HTSC cuprates - see the discussion below.
(The usefulness of introducing the optic relaxation
$\tilde{\omega}_{tr}(\omega)$ will be discussed in Appendix B.) Here, $i=a,b$
enumerates the plane axis, $\omega_{p}$, $\gamma_{tr}(\omega,T)$ and
$m_{op}(\omega)$ are the electronic plasma frequency, the transport (optical)
scattering rate and the optical mass, respectively. Very frequently, the
quantity
$\gamma_{tr}^{\ast}(\omega,T)=\gamma_{tr}(\omega,T)(m_{\infty}/m_{tr}(\omega))=\mathop{\mathrm{I}m}\sigma(\omega)/\omega\mathop{\mathrm{R}e}\sigma(\omega)$
Schlesinger , which is determined from the half-width of the Drude-like
expression for $\sigma(\omega)$, was analyzed since it is independent of
$\omega_{p}^{2}$. In the weak coupling limit $\lambda^{ep}<1$, the formula for
conductivity given in Eqs. (64-67) can be written in the form of Eq.(11) where
$\gamma_{tr}$ reads Dolgov-Shulga -Shulga
$\gamma_{tr}(\omega,T)=\pi\sum_{l}\int_{0}^{\infty}d\nu\alpha_{tr,l}^{2}F_{l}(\nu)[2(1+2n_{B}(\nu))$
$-2\frac{\nu}{\omega}-\frac{\omega+\nu}{\omega}n_{B}(\omega+\nu)+\frac{\omega-\nu}{\omega}n_{B}(\omega-\nu)].$
(12)
Here $n_{B}(\omega)$ is the Bose distribution function. (For the explicit form
of the transport mass $m_{tr}(\omega)$ see Allen , Dolgov-Shulga , Shulga ,
Maksimov-Review , Kulic-Review .) In the presence of impurity scattering one
should add $\gamma_{tr}^{imp}$ to $\gamma_{tr}$. It turns out that Eq.(12)
holds within a few percents also for large $\lambda^{ep}(>1)$. Note, that
$\alpha_{tr,l}^{2}F_{l}(\nu)\neq\alpha_{l}^{2}F_{l}(\nu)$ and the index $l$
enumerates all scattering bosons - phonons, spin fluctuations, etc. For
comparison, we give the quasi-particle scattering rate $\gamma(\omega,T)$
$\gamma(\omega,T)=2\pi\int\limits_{0}^{\infty}d\nu\alpha^{2}F(\nu)\\{2n_{B}(\nu)$
$+n_{F}(\nu+\omega)+n_{F}(\nu-\omega)\\}+\gamma^{imp},$ (13)
where $n_{F}$ is the Fermi distribution function. By comparing Eq.(13) and
Eq.(12), it is seen that $\gamma_{tr}$ and $\gamma$ are different quantities,
$\gamma_{tr}\neq\gamma$, i.e. the former describes the relaxation of Bose
particles (electron-hole pairs) while the latter one the relaxation of Fermi
particles. This difference persists also at $T=0$ $K$ where one has (due to
simplicity we omit in the following summation over $l$ ) Allen
$\gamma_{tr}(\omega)=\frac{2\pi}{\omega}\int_{0}^{\omega}d\nu(\omega-\nu)\alpha_{tr}^{2}(\nu)F(\nu)$
(14)
and
$\gamma(\omega)=2\pi\int_{0}^{\omega}d\nu\alpha^{2}(\nu)F(\nu).$ (15)
In the case of EPI, the above equations give that $\gamma^{ep}(\omega)=const$
for $\omega>\omega_{ph}^{\max}$ while $\gamma_{tr}^{ep}(\omega)$ (as well as
$\gamma_{tr}^{\ast}$) is monotonic growing for $\omega>\omega_{ph}^{\max}$,
where $\omega_{ph}^{\max}$ is the maximal phonon frequency. This is clearly
seen by comparing $\gamma(\omega,T)$, $\gamma_{tr}(\omega,T)$ and
$\gamma_{tr}^{\ast}$ which are calculated for the EPI spectral function
$\alpha_{ep}^{2}(\omega)F_{ph}(\omega)$ extracted from tunnelling experiments
in YBCO (with $\omega_{ph}^{\max}\sim 80$ $meV$) Tunneling-Vedeneev \- see
Fig. 3.
Figure 3: (a) Scattering rates $\gamma(\omega,T)$, $\gamma_{tr}(\omega,T)$ and
$\gamma_{tr}^{\ast}$ \- from top to bottom, for the Eliashberg function in
(b). From Dolgov-Shulga . (b)Eliashberg spectral function
$\alpha_{ep}^{2}(\omega)F_{ph}(\omega)$ obtained from tunnelling experiments
on break junctions Tunneling-Vedeneev . Inset shows $\gamma_{tr}^{\ast}$ with
(full line) and without (dashed line) interband transitions protectMaksimov-
Review .
The results shown in Fig. 3 clearly demonstrate the physical difference
between two scattering rates $\gamma^{ep}$ and $\gamma_{tr}^{ep}$. It is also
seen that $\gamma_{tr}^{\ast}(\omega,T)$ is more a linear function of $\omega$
than $\gamma_{tr}(\omega,T)$. From these calculations one concludes that the
quasi-linearity of $\gamma_{tr}(\omega,T)$ (and $\gamma_{tr}^{\ast}$) is not
in contradiction with the EPI scattering mechanism but it is in fact a natural
consequence of EPI. We stress that such behavior of $\gamma^{ep}$ and
$\gamma_{tr}^{ep}$, shown in Fig. 3, is in fact not exceptional for HTSC
cuprates but it is generic for many metallic systems, for instance 3D metallic
oxides, low temperature superconductors such as $Al$, $Pb$, etc. - see more in
Maksimov-Review , Kulic-Review .
Let us discuss briefly the experimental results for $R(\omega)$ and
$\gamma_{tr}^{\ast}(\omega,T)$ and compare these with theoretical predictions
obtained by using a single band model and
$\alpha_{ep}^{2}(\omega)F_{ph}(\omega)$ from tunnelling data with the EPI
coupling $\lambda=2$ Tunneling-Vedeneev . In the case of YBCO the agreement
between measured and calculated $R(\omega)$ is very good up to frequencies
$\omega<6000$ $cm^{-1}$ which confirms the importance of EPI in scattering
processes. For higher frequencies, where a mead infrared peak appears, it is
necessary to account for interband transitions Maksimov-Review . In optimally
doped $Bi_{2}Sr_{2}CaCu_{2}O_{6}$ Romero92 the experimental results for
$\gamma_{tr}^{\ast}(\omega,T)$ are explained theoretically by assuming that
the EPI spectral function $\alpha_{ep}^{2}(\omega)F(\omega)\sim
F_{ph}(\omega)$, where $F_{ph}(\omega)$ is the phononic DOS in BISCO while
$\alpha_{ep}^{2}(\omega)\sim\omega^{1.6}$, $\lambda=1.9$ and
$\gamma_{im}\approx 320$ $cm^{-1}$ \- see Fig. 4(a). The agreement is rather
good. At the same time the fit of $\gamma_{tr}^{\ast}(\omega,T)$ by the
marginal Fermi liquid fails as it is evident in Fig. 4(b).
Figure 4: (Top) Experimental transport scattering rate $\gamma_{tr}^{\ast}$
(solid lines) for BISCO and the theoretical curve by using Eq. (64) and
transport mass $m_{tr}^{\ast}$ with $\alpha^{2}F(\omega)$ described in text
(dashed lines). (Bottom) Comparison with the marginal Fermi liquid theory -
dashed lines. From Maksimov-Review
Now we will comment on the so called pronounced linear behavior of
$\gamma_{tr}(\omega,T)$ (and $\gamma_{tr}^{\ast}(\omega,T)$) which served in
the past for numerous inadequate conclusions. We stress that the measured
quantity is reflectivity $R(\omega)$ and derived ones are $\sigma(\omega)$,
$\gamma_{tr}(\omega,T)$ and $m_{tr}(\omega)$, which are very sensitive to the
value of the dielectric constant $\varepsilon_{\infty}$. This is clearly
demonstrated in Fig. 5 for Bi2212 where it is seen that
$\gamma_{tr}(\omega,T)$ (and $\gamma_{tr}^{\ast}(\omega,T)$) for
$\varepsilon_{\infty}=1$ is linear up to much higher $\omega$ than in the case
$\varepsilon_{\infty}>1$.
Figure 5: Dependence of $\gamma_{tr}^{\ast}(\omega,T)$ on
$\varepsilon_{\infty}$ in $Bi_{2}Sr_{2}CaCu_{2}O_{8}$ on different
temperatures for $\varepsilon_{\infty}=4$ (solid lines) and
$\varepsilon_{\infty}=1$ (dashed lines). From Kaufmann .
In some experiments Puchkov , Timusk-old $\gamma_{tr}(\omega,T)$ (and
$\gamma_{tr}^{\ast}(\omega,T)$) is linear up to very high $\omega$ which means
that the ion background and interband transitions (contained in
$\varepsilon_{\infty}$) are not properly taken into account since it is
assumed too small $\varepsilon_{\infty}$. The recent elipsometric measurements
on YBCO BorisMPI give the reliable value for $\varepsilon_{\infty}\approx
4-6$. The latter gives rise to a much less spectacular linearity in the
relaxation rates than it was the case immediately after the discovery of HTSC
cuprates.
Furthermore, we would like to comment on two points concerning $\sigma$,
$\gamma_{tr}$, $\gamma$ and their interrelations. First, the parametrization
of $\sigma(\omega)$ with the generalized Drude formula in Eq.(11) and its
relation to the transport scattering rate $\gamma_{tr}(\omega,T)$ and the
transport mass $m_{tr}(\omega,T)$ is useful if we deal with electron-boson
scattering in a single band problem. In Shulga it is shown that
$\sigma(\omega)$ of a two-band model with only elastic impurity scattering can
be represented by the generalized (extended) Drude formula with $\omega$ and
$T$ dependence of effective parameters $\gamma_{tr}^{eff}(\omega,T)$,
$m_{tr}^{eff}(\omega,T)$ despite the fact that the inelastic electron-boson
scattering is absent. To this end we stress that the single-band approach is
fully justified for a number of HTSC cuprates such as LSCO, BISCO etc. Second,
at the beginning we said that $\gamma_{tr}(\omega,T)$ and $\gamma(\omega,T)$
are physically different quantities and it holds
$\gamma_{tr}(\omega,T)\neq\gamma(\omega,T)$. In order to give the physical
picture and qualitative explanation we assume that
$\alpha_{tr}^{2}F(\nu)\approx\alpha^{2}F(\nu)$. In that case the renormalized
frequencies, the quasi-particle one
$\tilde{\omega}(\omega)=Z(\omega)\omega=\omega-\Sigma(\omega)$ (for the
definition of $Z(\omega)$ see Appendix A.) and the transport one
$\tilde{\omega}_{tr}(\omega)$ \- defined above, are related and at $T=0$, they
are given by Allen , Shulga
$\tilde{\omega}_{tr}(\omega)=\frac{1}{\omega}\int_{0}^{\omega}d\omega^{\prime}2\tilde{\omega}(\omega^{\prime}).$
(16)
It gives the relation between $\gamma_{tr}(\omega)$ and $\gamma(\omega)$,
$m_{tr}(\omega)$ and $m^{\ast}(\omega)$ respectively
$\gamma_{tr}(\omega)=\frac{1}{\omega}\int_{0}^{\omega}d\omega^{\prime}\gamma(\omega^{\prime})$
(17) $\omega
m_{tr}(\omega)=\frac{1}{\omega}\int_{0}^{\omega}d\omega^{\prime}2\omega^{\prime}m^{\ast}(\omega^{\prime}).$
(18)
The physical meaning of Eq.(16) is the following: in optical measurements one
photon with the energy $\omega$ is absorbed and two excited particles
(electron and hole) are created above and below the Fermi surface. If the
electron has energy $\omega^{\prime}$ and the hole $\omega-\omega^{\prime}$,
then they relax as quasi-particles with the renormalized $\tilde{\omega}$.
Since $\omega^{\prime}$ takes values $0<\omega^{\prime}<\omega$ then the
optical relaxation $\tilde{\omega}_{tr}(\omega)$ is the energy-averaged
$\tilde{\omega}(\omega)$ according to Eq.(16). The factor 2 is due to the two
quasi-particles, electron+hole. At finite $T$, the generalization reads Allen
, Shulga
$\tilde{\omega}_{tr}(\omega)=\frac{1}{\omega}\int_{0}^{\infty}d\omega^{\prime}[1-n_{F}(\omega^{\prime})-n_{F}(\omega-\omega^{\prime})]2\tilde{\omega}(\omega^{\prime}).$
(19)
2\. Inversion of the optical data and $\alpha_{tr}^{2}(\omega)F(\omega)$
In principle, the transport spectral function
$\alpha_{tr}^{2}(\omega)F(\omega)$ can be precisely extracted from
$\sigma(\omega)$, i.e. $\gamma_{tr}(\omega)$, only at $T=0$ $K$, which follows
from Eq.( 14)
$\alpha_{tr}^{2}(\omega)F(\omega)=\frac{1}{2\pi}\frac{\partial^{2}}{\partial\omega^{2}}(\omega\gamma_{tr}(\omega)$
$=\frac{\omega_{p}^{2}}{8\pi^{2}}\frac{\partial^{2}}{\partial\omega^{2}}[\omega
Re\frac{1}{\sigma(\omega)}]\mid_{T=0}.$ (20)
However, real measurements are performed at finite $T$ (and also at $T>T_{c}$)
and the inversion procedure is in principle an ill-posed problem since
$\alpha_{tr}^{2}(\omega)F(\omega)$ is the deconvolution of the inhomogeneous
Fredholm integral equation of the first kind with the temperature dependent
Kernel $K_{2}(\omega,\nu,T)$ in Eq.(12). An ill-posed mathematical problem,
like this one, is very sensitive to input since experimental data contain less
information than one needs. This can cause the fine structure of
$\alpha_{tr}^{2}(\omega)F(\omega)$ gets blurred in the extraction procedures
and it can be temperature dependent even when the true
$\alpha_{tr}^{2}(\omega)F(\omega)$ is $T$ independent. In the context of HTSC
cuprates, this problem was first studied in Dolgov-Shulga , Shulga with the
following results: (1) the extracted shape of
$\alpha_{tr}^{2}(\omega)F(\omega)$ in $YBa_{2}Cu_{3}O_{7-x}$ is not unique and
it is temperature dependent, i.e. at higher $T>T_{c}$ the peak structure is
smeared and only a single peak (slightly shifted to higher $\omega$) is
present. For instance, the experimental data of $R(\omega)$ in YBCO were
reproduced by two different spectral functions
$\alpha_{tr}^{2}(\omega)F(\omega)$, one with single peak and the other with
three peaks structure as it is shown in Fig. 6 The similar situation is
realized in optimally doped BISCO as it is seen in Fig. 7. It is important to
stress that the width of the extracted $\alpha_{tr}^{2}(\omega)F(\omega)$ in
both compounds coincide with the width of the phonon density of states
$F_{ph}(\omega)$ Dolgov-Shulga , Shulga , Kaufmann ; (2) the upper energy
bound for $\alpha_{tr}^{2}(\omega)F(\omega)$ can be extracted with certainty
and it coincides approximately with the maximal phonon frequency in cuprates
$\omega_{ph}^{\max}\lesssim 80$ $meV$ as it is seen in Figs. 6-7.
Figure 6: Experimental (solid lines) and calculated (dashed lines) of
$R(\omega)$ in optimally doped YBCO Schutzmann at T=100, 200, 300 K (from top
to bottom). Inset: the two reconstructed $\alpha_{tr}^{2}(\omega)F(\omega)$ at
T=100 K. The phonon density of states $F(\omega)$ \- dotted line. From Dolgov-
Shulga
Figure 7: Experimental (solid line) and calculated (dashed line) of
$R(\omega)$ in optimally doped BISCO Kamaras at T=100 K. Inset: the
reconstructed $\alpha_{tr}^{2}(\omega)F(\omega)$ \- solid line. The phonon
density of states $F(\omega)$ \- dotted line. From Dolgov-Shulga
These results undoubtedly demonstrate the importance of EPI in cuprates
Dolgov-Shulga , Shulga , Maksimov-Review . We point out that the width of
$\alpha_{tr}^{2}(\omega)F(\omega)$ which is extracted from the optical
measurements Dolgov-Shulga , Shulga , Maksimov-Review coincides with the
width of the quasi-particle spectral function $\alpha^{2}(\omega)F(\omega)$
obtained in tunneling and ARPES spectra (which we shall discuss below), i.e.
both functions are spread over the energy interval
$0<\omega<\omega_{ph}^{\max}(\lesssim 80$ $meV)$. Since in cuprates this
interval coincides with the width in the phononic density of states
$F(\omega)$ and since the maxima of $\alpha^{2}(\omega)F(\omega)$ and
$F(\omega)$ almost coincide, this is further strong evidence for the dominance
of EPI.
To this end, we would like to comment on two important points. First, in some
reports Carbotte ,Hwang-Timusk-1 , Hwang-Timusk-2 it was assumed that
$\alpha_{tr}^{2}(\omega)F(\omega)$ of cuprates can be extracted also in the
superconducting state by using Eq. (20). However, Eq. (20) holds exclusively
in the normal state (at T=0) since $\sigma(\omega)$ can be described by the
generalized (extended) Drude formula in Eq. (11) only in the normal state.
Such an approach apparently does not hold in the superconducting state since
the dynamical conductivity depends not only on the electron-boson scattering
but also on coherence factors and on the momentum and energy dependent order
parameter $\Delta(\mathbf{k},\omega)$. In such a case it is unjustified to
extract $\alpha_{tr}^{2}(\omega)F(\omega)$ from Eq. (20). Second, if
$R(\omega)$ (and $\sigma(\omega)$) in cuprates are due to some other bosonic
scattering which is pronounced up to much higher energies
$\omega_{c}\gg\omega_{ph}^{\max}$, this should be seen in the extracted
spectral function $\alpha_{tr}^{2}(\omega)F(\omega)$. Such an assumption is
made, for instance, in the phenomenological spin-fluctuation approach Pines
where it is assumed that
$\alpha^{2}(\omega)F(\omega)=g_{sf}^{2}$Im$\chi(\omega)$ where
Im$\chi(\omega)$ is extended up to the large energy cutoff $\omega_{c}\approx
400$ $meV$. This assumption is apparently in conflict with the above
theoretical and experimental analysis which shows that solely EPI can describe
$R(\omega)$ very well and that the contribution from higher energies
$\omega\gg\omega_{ph}^{\max}$ must be small and therefore irrelevant for
pairing Dolgov-Shulga , Shulga , Kaufmann . This is also confirmed by
tunnelling measurements - see below.
Despite the experimentally established facts for the importance of EPI, some
reports appeared recently claiming that SFI dominates and that
$\alpha_{tr}^{2}(\omega)F(\omega)\approx
g_{sf}^{2}\mathop{\mathrm{I}m}\chi(\omega)$ where
$\mathop{\mathrm{I}m}\chi(\omega)=\int d^{2}k\chi(\mathbf{k},\omega)$ Hwang-
Timusk-1 , Hwang-Timusk-2 . This claim is based on reanalyzing some old IR
measurements Hwang-Timusk-1 , Hwang-Timusk-2 . The transport spectral function
$\alpha_{tr}^{2}(\omega)F(\omega)$ is extracted in Hwang-Timusk-1 by using
the maximum entropy method in solving the Fredholm equation. However, in order
to exclude negative values in the extracted $\alpha_{tr}^{2}(\omega)F(\omega)$
they imply a biased condition that $\alpha_{tr}^{2}(\omega)F(\omega)$ has a
rather large tail at large energies - up to 400 meV. Then it is not surprising
at all that their extracted $\alpha_{tr}^{2}(\omega)F(\omega)$ at large
$\omega$ resembles qualitatively $\mathop{\mathrm{I}m}\chi(\omega)$ obtained
by the magnetic neutron scattering on $La_{2-x}Sr_{x}CuO_{4}$ Vignolle . In
other words one obtains as output what is assumed in the input. It turns out
that even such a biased assumption in Hwang-Timusk-1 by extracting
$\alpha_{tr}^{2}(\omega)F(\omega)$ does not reproduce the experimental curve
$\mathop{\mathrm{I}m}\chi(\omega)$ Vignolle in some important respects. (1)
The relative heights of the two peaks in the extracted spectral function
$\alpha_{tr}^{2}(\omega)F(\omega)$ at lower temperatures are opposite to that
in $\mathop{\mathrm{I}m}\chi(\omega)$ Vignolle \- see Fig. 1 in Hwang-
Timusk-1 . (2) The strong temperature dependence of the extracted
$\alpha_{tr}^{2}(\omega)F(\omega)$, found in Hwang-Timusk-1 , Hwang-Timusk-2
is in fact not the intrinsic property of the spectral function but it is due
to the high sensitivity of the extraction procedure on temperature. As we
already explained before, this is due to the ill-posed problem of solving the
Fredholm integral equation of the first kind with strong $T$-dependent kernel.
(3) The extracted spectral weight $\alpha_{tr}^{2}(\omega)F(\omega)$ in Hwang-
Timusk-1 has much smaller values at larger frequencies ($\omega>100$ $meV$)
than it is the case for the measured Im$\chi(\omega)$, i.e. $(I(\omega>100$
$meV)/I(\omega_{\max}))\ll$Im$\chi(\omega>100$ $meV)/$Im$\chi(\omega_{\max})$
\- see Fig. 1 in Hwang-Timusk-1 . In spite of the fact that the main weight of
the extracted $\alpha_{tr}^{2}(\omega)F(\omega)$ Hwang-Timusk-1 lies in the
range of phononic frequencies, $0<\omega<\omega_{\max}^{ph}$ it is not in
agreement with that obtained in tunnelling and ARPES measurements. (4) To this
end it is suspicious that the transport coupling constant $\lambda_{tr}$
extracted in Hwang-Timusk-1 is so large, i.e. $\lambda_{tr}>3$ contrary to
the previous findings that $\lambda_{tr}<1.5$ Dolgov-Shulga , Shulga ,
Kaufmann . Since in HTSC one has $\lambda>\lambda_{tr}$ this would probably
give $\lambda\approx 6$ that is not confirmed by other experiments. It is
necessary to stress that the estimated $\lambda_{tr}$ depends strongly on the
value of plasma frequency, i.e. on $\omega_{pl}^{2}$ \- see Fig. 12 below, and
it might be that for the latter the larger value is assumed in Hwang-Timusk-1
. (5) The interpretation of $\alpha_{tr}^{2}(\omega)F(\omega)$ in LSCO and
BISCO solely in terms of Im$\chi(\omega)$ is in contradiction with the
magnetic neutron scattering in the optimally doped and slightly underdoped
YBCO Bourges \- that was discussed above, where in the former
Im$\chi(\mathbf{Q},\omega)$ is small in the normal state - it is even below
the experimental noise. This means that if the assumption that
$\alpha_{tr}^{2}(\omega)F(\omega)\approx g_{sf}^{2}$Im$\chi(\omega)$ were
correct then the contribution to Im$\chi(\omega)$ from the momenta $0<k<Q$
would be dominant and very detrimental for d-wave superconductivity and Tc
would be rather low. Since the results of magnetic neutron scattering in YBCO
Bourges are very convincing and trustful, then the conclusion is that the SFI
coupling constant $\lambda^{sf}(\sim g_{sf}^{2})$ must be small, i.e.
$g_{sf}<0.2$ $eV$ and $\lambda^{sf}<0.2$. The latter is in accordance with
other independent estimates (discussed also above) of $\lambda^{sf}(\ll 2)$.
Finally, we point out that very similar (to cuprates) properties, of
$\sigma(\omega)$, $R(\omega)$ (and $\rho(T)$ and electronic Raman spectra)
were observed in 3D isotropic metallic oxides $La_{0.5}Sr_{0.5}CoO_{3}$ and
$Ca_{0.5}Sr_{0.5}RuO_{3}$ which are non-superconducting Bozovic and in
$Ba_{1-x}K_{x}BiO_{3}$ which superconducts at $T_{c}\simeq 30$ $K$ at $x=0.4$.
This means that in all of them, the scattering mechanism might be of similar
origin. Since in these compounds there are no signs of antiferromagnetic
fluctuations (which are present in cuprates), then EPI plays important role.
3\. Restricted optical sum-rule
The restricted optical sum-rule was studied intensively in HTSC cuprates. It
shows peculiarities not present in low-temperature superconductors. It turns
out that the restricted spectral weight $W(\Omega_{c},T)$ is strongly
temperature dependent in the normal and superconducting state, that was
interpreted either to be due to EPI Maks-Karakoz-1 , Maks-Karakoz-2 or to
some non-phononic mechanisms Hirsch . In the following we demonstrate that the
temperature dependence of $W(\Omega_{c},T)=W(0)-\beta T^{2}$ in the normal
state can be explained in a natural way by the $T$-dependence of the EPI
transport relaxation rate $\gamma_{tr}^{ep}(\omega,T)$ Maks-Karakoz-1 , Maks-
Karakoz-2 . Since the problem of the restricted sum-rule attracted much
interest, it will be considered here in some details. In fact there are two
kinds of sum rules related to $\sigma(\omega)$. The first one is the total sum
rule which in the normal state reads
$\int_{0}^{\infty}\sigma_{1}^{N}(\omega)d\omega=\frac{\omega_{pl}^{2}}{8}=\frac{\pi
ne^{2}}{2m},$ (21)
while in the superconducting state it is given by the Tinkham-Ferrell-Glover
(TFG) sum-rule
$\int_{0}^{\infty}\sigma_{1}^{S}(\omega)d\omega=\frac{c^{2}}{8\lambda_{L}^{2}}+\int_{+0}^{\infty}\sigma_{1}^{S}(\omega)d\omega=\frac{\omega_{pl}^{2}}{8}.$
(22)
Here,$\ n$ \- the total electron density, $e$ \- the electron charge, $m$ \-
the bare electron mass and $\lambda_{L}$ \- the London penetration depth. The
first (singular) term $c^{2}/8\lambda_{L}^{2}$ is due to the superconducting
condensate which contributes
$\sigma_{1,cond}^{S}(\omega)=(c^{2}/4\lambda_{L}^{2})\delta(\omega)$. The
total sum rule represents the fundamental property of matter - the
conservation of the electron number, and to calculate it one should use the
total Hamiltonian $\hat{H}_{tot}=\hat{T}_{e}+\hat{H}_{int}$ where all
electrons, bands and their interactions $\hat{H}_{int}$ (Coulomb, EPI, with
impurities, etc.) are accounted for. Here, $T_{e}$ is the kinetic energy of
bare electrons
$\hat{T}_{e}=\sum_{\sigma}\int
d^{3}x\hat{\psi}_{\sigma}^{\dagger}(x)\frac{\mathbf{\hat{p}}^{2}}{2m}\hat{\psi}_{\sigma}(x)=\sum_{\mathbf{p},\sigma}\frac{\mathbf{p}^{2}}{2m_{e}}\hat{c}_{\mathbf{p}\sigma}^{\dagger}\hat{c}_{\mathbf{p}\sigma}.$
(23)
The partial sum rule is related to the energetics in the conduction (valence)
band which is described by the Hamiltonian of the valence (band) electrons
$\hat{H}_{v}=\sum_{\mathbf{p},\sigma}\epsilon_{\mathbf{p}}\hat{c}_{v,\mathbf{p}\sigma}^{\dagger}\hat{c}_{v,\mathbf{p}\sigma}+\hat{V}_{v,Coul}.$
(24)
It contains the band-energy with the dispersion $\epsilon_{\mathbf{p}}$ and
the effective Coulomb interaction of the valence electrons $\hat{V}_{v,Coul}$.
In this case the partial sum-rule in the normal state reads Maldague (for
general form of $\epsilon_{\mathbf{p}}$)
$\int_{0}^{\infty}\sigma_{1,v}^{N}(\omega)d\omega=\frac{\pi
e^{2}}{2V}\sum_{\mathbf{p}}\frac{\langle\hat{n}_{v,\mathbf{p}}\rangle_{H_{v}}}{m_{\mathbf{p}}}$
(25)
where the number operator
$\hat{n}_{v,\mathbf{p}}=\sum_{\sigma}\hat{c}_{\mathbf{p}\sigma}^{\dagger}\hat{c}_{\mathbf{p}\sigma}$;
$1/m_{\mathbf{p}}=\partial^{2}\epsilon_{\mathbf{p}}/\partial p_{x}^{2}$ is the
reciprocal mass and $V$ is volume . In practice measurements are performed up
to finite frequency and the integration over $\omega$ goes up to some cutoff
frequency $\Omega_{c}$ (of the order of the band plasma frequency). In this
case the restricted sum-rule has the form
$W(\Omega_{c},T)=\int_{0}^{\Omega_{c}}\sigma_{1,v}^{N}(\omega)d\omega$
$=\frac{\pi}{2}\left[K^{d}+\Pi(0)\right]-\int_{0}^{\Omega_{c}}\frac{Im\Pi(\omega)}{\omega}d\omega.$
(26)
where $K^{d}$ is the diamagnetic Kernel and $\Pi(\omega)$ is the paramagnetic
(current-current) response function - see more in Maks-Karakoz-1 , Maks-
Karakoz-2 . In the case when the interband gap $E_{g}$ is the largest scale in
the problem, i.e. when $W_{b}<\Omega_{c}<E_{g}$, in this region one has
approximately Im$\Pi(\omega)\approx 0$ and the limit
$\Omega_{c}\rightarrow\infty$ in Eq.(26) is justified. In that case one has
$\Pi(0)\approx\int_{0}^{\Omega_{c}}($Im$\Pi(\omega)/\omega)d\omega$ which
gives the approximate formula for $W(\Omega_{c},T)$
$W(\Omega_{c},T)=\int_{0}^{\Omega_{c}}\sigma_{1,v}^{N}(\omega)d\omega\approx\frac{\pi}{2}K^{d}$
$=e^{2}\pi\sum_{\mathbf{p}}\frac{\partial^{2}\epsilon_{\mathbf{p}}}{\partial\mathbf{p}^{2}}n_{\mathbf{p}},$
(27)
where $\epsilon_{\mathbf{p}}$ is the band-energy and
$n_{\mathbf{p}}=\left\langle\hat{n}_{v,\mathbf{p}}\right\rangle$ is the quas-
iparticle distribution function in the interacting system. Note that the right
hand side of Eq.(27) does not depend on the cutoff energy $\Omega_{c}$. So one
should be careful not to interpret blindly the experimental result in cuprates
by this formula and for that reason the best way is to calculate
$W(\Omega_{c},T)$ by using the exact result in Eq.(26) which apparently
depends on $\Omega_{c}$. However, Eq.(27) is useful for appropriately chosen
$\Omega_{c}$, since it allows us to get semi-quantitative and qualitative
results. In most papers related to the restricted sum-rule in HTSC, it was
assumed, due to simplicity, the tight-binding model with nearest neighbors
(n.n.) with the energy $\epsilon_{\mathbf{p}}=-2t(\cos p_{x}a+\cos p_{y}a)$
and $1/m_{\mathbf{p}}=-2ta^{2}\cos p_{x}a$. It is straightforward to show that
in this case one has
$W(\Omega_{c},T)=\int_{0}^{\Omega_{c}}\sigma_{1,v}^{N}(\omega)d\omega$
$\approx\frac{\pi e^{2}a^{2}}{2V}\langle-T_{v}\rangle,$ (28)
where $\langle
T_{v}\rangle_{H_{v}}=\sum_{\mathbf{p}}\epsilon_{\mathbf{p}}\langle
n_{v}\rangle_{H_{v}}$ is the averaged kinetic energy of the band electrons and
$\omega_{pl,v}$ is the (band) plasma frequency. In this approximation
$W(\Omega_{c},T)$ is a direct measure of the averaged kinetic energy. In the
superconducting state the partial sum-rule reads
$W_{s}(\Omega_{c},T)=\frac{c^{2}}{8\lambda_{L}^{2}}+\int_{+0}^{\Omega_{c}}\sigma_{1,v}^{S}(\omega)d\omega$
$=\frac{\pi e^{2}a^{2}}{2}\langle-T_{v}\rangle_{s}.$ (29)
In order to introduce the reader to the complexity of the problem of
T-dependence of $W(\Omega_{c},T)$, let us consider the electronic system in
the normal state and in absence of quasi-particle interaction. In that case
one has $n_{\mathbf{p}}=f_{\mathbf{p}}$ ($f_{\mathbf{p}}$ is the Fermi
distribution function) and $W_{n}(\Omega_{c},T)$ increases with the decrease
temperature, i.e. $W_{n}(\Omega_{c},T)=W_{n}(0)-\beta_{b}T^{2}$ where
$\beta_{b}\sim 1/W_{b}$. To this end, let us mention in advance that the
experimental value $\beta_{\exp}$ is much larger than $\beta_{b}$, i.e.
$\beta_{\exp}\gg\beta_{b}$ thus telling us that the simple Sommerfeld-like
smearing of $f_{\mathbf{p}}$ by the temperature effects cannot explain the
T-dependence of $W(\Omega_{c},T)$ as it was put forward in some papers. We
stress that the smearing of $f_{\mathbf{p}}$ by temperature lowers the
spectral weight compared to that at $T=0$ $K$, i.e.
$W_{n}(\Omega_{c},T)<W_{n}(\Omega_{c},0)$. In that respect it is not
surprising at all that there is a lowering of $W_{s}(\Omega_{c},T)$ in the BCS
superconducting state, $W_{s}^{BCS}(\Omega_{c},T=0)<W_{n}(\Omega_{c},0)$ since
$f_{\mathbf{p}}$ is smeared due to the appearance of the superconducting gap,
$2f_{\mathbf{p}}=1-(\xi_{\mathbf{p}}/E_{\mathbf{p}})th(E_{\mathbf{p}}/2T)$,
$E_{\mathbf{p}}=\sqrt{\xi_{\mathbf{p}}^{2}+\Delta^{2}}$,
$\xi_{\mathbf{p}}=\epsilon_{\mathbf{p}}-\mu$, and the maximal decrease of
$W_{s}(\Omega_{c},T)$ is at $T=0$.
Let us enumerate and analyze the main experimental results in cuprates. 1. In
the normal state ($T>T_{c}$) of most cuprates, one has
$W_{n}(\Omega_{c},T)=W_{n}(0)-\beta_{ex}T^{2}$ with
$\beta_{\exp}\gg\beta_{b}$, i.e. $W_{n}(\Omega_{c},T)$ is increasing by
decreasing $T$, even at T below the opening of the pseudogap. The change of
$W_{n}(\Omega_{c},T)$ from room temperature down to $T_{c}$ is no more than
$5$ $\%$. 2. In the superconducting state ($T<T_{c}$) of some underdoped and
optimally doped Bi-2212 compounds Molegraaf , Carbone (and underdoped Bi-2212
films Santander-2003 ) there is an effective increase of $W_{s}(\Omega_{c},T)$
with respect to that in the normal state, i.e.
$W_{s}(\Omega_{c},T)>W_{n}(\Omega_{c},T)$ for $T<T_{c}$. This is non-BCS
behavior shown in Fig. 8.
Figure 8: Measured spectral weight $W_{s}(\Omega_{c},T)$ for
$\omega_{c}\approx 1.25eV$ in two underdoped $Bi2212$ (with $T_{c}=88$ $K$ and
$T_{c}=66$ $K$). From Molegraaf .
In some optimally doped and in most overdoped cuprates, there is decreasing of
$W_{s}(\Omega_{c},T)$ at $T<T_{c}$ which is the BCS-like behavior Deutscher-
optics as it is seen in Fig. 9
Figure 9: (Top) Spectral weight $W_{n}(\Omega_{c},T)$ of the overdoped
$Bi2212$ for $\Omega_{c}=1eV$. Closed symbols - normal state. Open symbols \-
superconducting state. (Bottom) Change of the kinetic energy $\Delta
E_{kin}=E_{kin,S}-E_{kin,N}$ in $meV$ per Cu site vs the charge $p$ per Cu
with respect to the optimal value $p_{opt}$. From Deutscher-optics .
We stress that the non-BCS behavior of $W_{s}(\Omega_{c},T)$ for underdoped
and optimally doped systems was obtained by assuming that
$\Omega_{c}\approx(1-1.2)$ $eV$. However, in Ref. BorisMPI these results have
been questioned and the conventional BCS-like behavior was observed
($W_{s}(\Omega_{c},T)<W_{n}(\Omega_{c},T)$) in the optimally doped YBCO and
slightly underdoped Bi-2212 by using larger cutoff energy $\Omega_{c}=1.5$
$eV$. Although the results obtained in BorisMPI looks very trustfully, it is
fair to say that the issue of the reduced spectral weight in the
superconducting state of cuprates is still unsettled and under dispute. In
overdoped Bi-2212 films, the BCS-like behavior
$W_{s}(\Omega_{c},T)<W_{n}(\Omega_{c},T)$ was observed, while in LSCO it was
found that $W_{s}(\Omega_{c},T)\approx const$, i.e.
$W_{s}(\Omega_{c},T<T_{c})\approx W_{n}(\Omega_{c},T_{c})$.
How to explain the strong temperature dependence of $W(\Omega_{c},T)$ in the
normal and superconducting state? In Maks-Karakoz-1 , Maks-Karakoz-2 it was
shown that the EPI relaxation $\gamma^{ep}(T)$ plays the main role in the
$T$-dependence of $W(\Omega_{c},T)$. The main theoretical results of Maks-
Karakoz-1 , Maks-Karakoz-2 are the following. (1) The calculations based on
the exact formula in Eq.(27) give that for $\Omega_{c}\gg\Omega_{D}$, the
difference in spectral weights of the normal and superconducting state is
small, i.e. $W_{n}(\Omega_{c},T)\approx W_{s}(\Omega_{c},T)$ (in the following
we call it $W$) since
$W_{n}(\Omega_{c},T)-W_{s}(\Omega_{c},T)\sim\Delta^{2}/\Omega_{c}^{2}$. In the
case of large $\Omega_{c}$ based on the approximate formula Eq.(27), one
obtains
$W(\Omega_{c},T)\approx\frac{\omega_{pl}^{2}}{8}\left[1-\frac{\gamma(T)}{W_{b}}-\frac{\pi^{2}}{2}\frac{T^{2}}{W_{b}^{2}}\right].$
(30)
In the case of EPI, one has $\gamma=\gamma^{ep}(T)+\gamma^{imp}$ where
$\gamma^{ep}(T)=\int_{0}^{\infty}dz\alpha^{2}(z)F(z)\coth(z/2T)$. It turns out
that for $\alpha^{2}(\omega)F(\omega)$ shown in Fig. 3, one obtains: (i)
$\gamma^{ep}(T)\sim T^{2}$ in the temperature interval $100$ $K<T<200$ $K$ as
it is seEn in Fig. 10, Maks-Karakoz-1 , Maks-Karakoz-2 ; (ii) the second term
in Eq.(30) is much larger than the last one (the Sommerfeld-like term). For
the EPI coupling constant $\lambda_{tr}^{ep}=1.5$ one obtains rather good
agreement with experiments. At lower temperatures, $\gamma^{ep}(T)$ deviates
from the $T^{2}$ behavior and the deviation depends on the structure of the
spectrum in $\alpha^{2}(\omega)F(\omega)$. It is seen in Fig. 10 that for a
softer Einstein spectrum (with $\Omega_{E}=200$ $K$), $W(\Omega_{c},T)$ lies
above the curve with the $T^{2}$ asymptotic, while the one with a harder
phononic spectrum (with $\Omega_{E}=400$ $K$) lies below it.
Figure 10: Spectral weight $W(\Omega_{c},T)$ for Einstein phonons with
$\Omega_{E}=200$ $K$ (full triangles) and $\Omega_{E}=400$ $K$ (open circles,
left axis). Dashed lines is $T^{2}$ asymptotic. From Maks-Karakoz-2 .
This result means that different behavior of $W(\Omega_{c},T)$ in the
superconducting state of cuprates for different doping might be simply related
to different contributions of low and high frequency phonons. We stress that
such a behavior of $W(\Omega_{c},T)$ was observed in experiments Molegraaf ,
Carbone , BorisMPI and the above analysis tells us that the theory based on
EPI explains in a consistent way the strange temperature behavior of
$W(\Omega_{c},T)$ above and below $T_{c}$ and that there is no need to invoke
exotic scattering mechanisms.
4\. Resistivity $\rho(T)$
The temperature dependence of the in-plane resistivity $\rho_{ab}(T)$ in
cuprates is a direct consequence of the quasi-$2D$ motion of quasi-particles
and of the inelastic scattering which they experience. At present, there is no
consensus on the origin of the linear temperature dependence of the in-plane
resistivity $\rho_{ab}(T)$ in the normal state and there is rather widespread
believe that it can not be due to EPI. The inadequacy of this belief was
already demonstrated by analyzing the dynamic conductivity $\sigma(\omega)$
which is successfully explained by EPI. Since $\rho(T)=1/\sigma(\omega=0)$
$\rho(T)=\frac{4\pi}{\omega_{p}^{2}}\gamma_{tr}(T)+\rho_{imp}$ (31)
$\gamma_{tr}(T)=\frac{\pi}{T}\int_{0}^{\infty}d\omega\frac{\omega}{\sin^{2}(\omega/2T)}\alpha_{tr}^{2}(\omega)F(\omega).$
(32)
It is quite natural that in some temperature region, $\rho(T)$ in cuprates can
be explained by EPI as it is shown in Fig. 11. It turns out that
$\gamma_{tr}(T)\sim T$ for $T>\alpha\Theta_{D}$, $\alpha<1$ depending on the
shape of $\alpha_{tr}^{2}(\omega)F(\omega)$. In case of the Debye spectrum, it
is realized for $T>\Theta_{D}/5$ i.e.
$\rho(T)\simeq
8\pi^{2}\lambda_{tr}^{ep}\frac{k_{B}T}{\hbar\omega_{p}^{2}}=\rho^{\prime}T.$
(33)
Figure 11: (a) Calculated resistivity $\rho(T)$ for the EPI spectral function
$\alpha_{tr}^{2}(\omega)F(\omega)$ in KMS . (b) Measured resistivity in a(x)-
and b(y)-crystal direction of YBCO Friedman and calculated Bloch-Grüneisen
curve for $\lambda^{ep}=1$, Allen-kinky .
There is an experimental constraint on $\lambda_{tr}$, i.e.
$\lambda_{tr}\approx
0.25\omega_{pl}^{2}(eV)\rho^{\prime}(\mu\Omega\mathrm{cm}/K),$ (34)
which imposes a limit on it. For instance, for $\omega_{pl}\approx(2-3)$ $eV$
Bozovic and $\rho^{\prime}\approx 0.6$ in the oriented YBCO films and
$\rho^{\prime}\approx 0.3-0.4$ in single crystals of BSCO, one obtains
$\lambda_{tr}\approx 0.4-1.2$. In case of YBCO single crystals, there is a
pronounced anisotropy in $\rho_{a,b}(T)$ Friedman which gives
$\rho_{x}^{\prime}(T)=0.6\mu\Omega\mathrm{cm}/K$ and
$\rho_{y}^{\prime}(T)=0.25\mu\Omega\mathrm{cm}/K$. According to Eq.(34), one
obtains $\lambda_{tr}(\omega_{pl})$ which is shown in Fig.12, where the plasma
frequency $\omega_{pl}$ which enters Eqs.(31-33) can be calculated by LDA and
also extracted from the width ($\sim$ $\omega_{pl}^{\ast})$ of the Drude peak
at small frequencies, where
$\omega_{pl}=\sqrt{\varepsilon_{\infty}}\omega_{pl}^{\ast}$.
Figure 12: Transport EPI spectral function coupling constant in YBCO as a
function of plasma frequency $\omega_{p}$ as derived from the experimental
slope of resistivity $\rho^{\prime}(T)$ in Eq.(34). $\lambda_{x}$ for
$\rho_{x}^{\prime}(T)=0.6\mu\Omega\mathrm{cm}/K$ and $\lambda_{y}$ for
$\rho_{y}^{\prime}(T)=0.25\mu\Omega\mathrm{cm}/K$ Friedman . Squares are LDA
values Mazin-Dolgov .
We shall argue below that from tunnelling experiments Tunneling-Vedeneev
-Tsuda one obtains in the framework of the Eliashberg theory that the EPI
coupling constant is large $\lambda^{ep}\approx 2-3$ which implies that
$\lambda_{tr}\sim(\lambda/3)$, i.e. EPI is reduced in transport properties due
to some reasons that shall be discussed in Part II. Such a large reduction of
$\lambda_{tr}$ cannot be obtained within the LDA band structure calculations
which means that $\lambda^{ep}$ and $\lambda_{tr}$ contain renormalization
which do not enter in the $LDA$ theory. In Part II we shall argue that the
strong suppression of $\lambda_{tr}$ may have its origin in strong electronic
correlations and the long-range Madelung energy Kulic1 , Kulic2 .
4\. Femtosecond time-resolved optical spectroscopy
The femtosecond time-resolved optical spectroscopy (FTROS) has been developed
intensively in the last couple of years and applied successfully to HTSC
cuprates. In this method a femtosecond ($1fs=10^{-15}\sec$) laser pump excites
in materials electron-hole pairs via interband transitions. These hot carriers
release their energy via electron-electron (with the relaxation time
$\tau_{ee}$) and electron-phonon scattering reaching states near the Fermi
energy within $10-100$ $fs$ \- see Mihailovic-Kabanov . The typical energy
density of the laser pump pulses with the wavelength $\lambda\approx 810$ nm
($1.5$ $eV$) was around $F\sim 1\mu J/cm^{2}$ (the excitation fluenc $F$)
which produces approximately $3\times 10^{10}$ carriers per puls (by assuming
that each photon produces $\hbar\omega/\Delta$ carriers, $\Delta$ is the
superconducting gap). By measuring photoinduced changes of the reflectivity in
time, i.e. $\Delta R(t)/R_{0}$, one can extract information on the further
relaxation dynamics of the low-laying electronic excitations. Since $\Delta
R(t)$ relax to equilibrium the fit with exponential functions is used
$\frac{\Delta
R(t)}{R_{0}}=f(t)\left[Ae^{-\frac{t}{\tau_{A}}}+Be^{-\frac{t}{\tau_{B}}}+...\right],$
(35)
where $f(t)=H(t)[1-\exp\\{-t/\tau_{ee}\\}]$ ($H(t)$ is the Heavyside function)
describes the finite rise-time. The parameters $A$, $B$ depends on the fluenc
$F$. This method was used in studying the superconucting phase of
$La_{2-x}Sr_{x}CuO_{4}$, with $x=0.1$, $0.15$ and $T_{c}=30$ $K$ and $38$ $K$
respectively Kusar-2008 . In that case the signal $A\neq 0$ for $T$$<$$T_{c}$
and $A=0$ for $T>T_{c}$, while the signal $B$ was present also at $T>T_{c}$.
It turns out that the signal $A$ is related to the quasi-particle
recombination across the superconducting gap $\Delta(T)$ and has a relaxation
time of the order $\tau_{A}>10$ $ps$ at $T=4.5$ $K$. At the so called
threshold fluenc ($F_{T}=4.2\pm 1.7$ $\mu J/cm^{2}$ for $x=0.1$ and
$F_{T}=5.8\pm 2.3$ $\mu J/cm^{2}$ for $x=0.15$) occurs the vaporization
(destroying) of the superconducting phase, where the parameter $A$ saturates.
This vaporization process takes place at the time scala $\tau_{r}\approx 0.8$
$ps$. The external fluenc is distributed in the sample over the excitation
volume which is proportional to the optical penetration depth
$\lambda_{op}$($\approx 150$ $nm$ at $\lambda\approx 810$ $nm$) of the pump.
The energy densities stored in the excitation volume at the vaporization
threshold for $x=0.1$ and $x=0.15$ are $U_{p}=F_{T}/\lambda_{op}=2.0\pm 0.8$
$K/Cu$ and $2.6\pm 1.0$ $K/Cu$, respectively. The important fact is that
$U_{p}$ is much larger than the superconducting condensation energy which is
$U_{cond}\approx 0.12$ $K/Cu$ for $x=0.1$ and $U_{cond}\approx 0.3$ $K/Cu$ for
$x=0.15$, i.e. $U_{p}\gg U_{cond}$. This means that the energy difference
$U_{p}-U_{cond}$ must be stored elsewhere on the time scale $\tau_{r}$. The
only present reservoir which can absorb the difference in energy are the
bosonic baths of phonons and spin fluctuations. The energy required to heat
the spin reservoir from $T=4.5K$ to $T_{c}$ is
$U_{sf}=\int_{T}^{T_{c}}C_{sf}(T)dT$. The measured $C_{sf}(T)$ in
$La_{2}CuO_{4}$ Kusar-2008 gives very small value $U_{sf}\approx 0.01$ $K$.
In the case of the phonon reservoir on obtains
$U_{ph}=\int_{T}^{T_{c}}C_{ph}(T)dT=9$ $K/Cu$ for $x=0.1$ and $28$ $K/Cu$ for
$x=0.15$. Since $U_{sf}\ll U_{p}-U_{cond}$ the spin reservoir cannot absorb
the rest energy $U_{p}-U_{cond}$. The situation is opposite with phonons since
$U_{ph}\gg U_{p}-U_{cond}$ and phonon can absorb the rest energy in the
excitation volume. The complete vaporization dynamics can be described in the
framework of the Rothwarf-Taylor model which describes approaching of
electrons and phonons to quasi-equilibrium on the time scale of 1 ps Kabanov-
PRL . We shall not go into details but only summarize, that only phonon-
mediated vaporization is consistent with the experiments, thus ruling out
spin-mediated quasi-particle recombination and pairing in HTSC cuprates. This
is additional proof for the ineffectivness of the SFI scattering in cuprates.
In conclusion, optics and resistivity measurements in normal state of cuprates
are much more in favor of EPI than against it. However, some intriguing
questions still remain to be answered: (i) what are the values of
$\lambda_{tr}$ and $\omega_{pl}$; (ii) what is the reason that
$\lambda_{tr}\ll\lambda$ is realized in cuprates; (iii) what is the role of
Coulomb scattering in $\sigma(\omega)$ and $\rho(T)$. Later on we shall argue
that ARPES measurements in cuprates give evidence for a contribution of
Coulomb scattering at higher frequencies, where
$\gamma(\omega)\approx\gamma_{0}+\lambda_{c}\omega$ for
$\omega>\omega_{\max}^{ph}$ with $\lambda_{c}\approx 0.4$. So, despite the
fact that EPI is suppressed in transport properties it can be sufficiently
strong in the self-energy in some frequency and temperature range.
### III.3 ARPES and the EPI self-energy
ARPES is nowadays a leading spectroscopy method in the solid state physics
Shen-review . It provides direct information to the one-electron removal
spectrum in a complex many system. The method involves shining light (photons)
with energies between $5-1000$ $eV$ on the sample and by detecting momentum
($\mathbf{k}$) - and energy($\omega$)-distribution of the outgoing electrons.
The resolution of ARPES has been significantly increased in the last decade
with the energy resolution of $\Delta E\approx 1-2$ $meV$ (for photon energies
$\sim 20$ $eV$) and angular resolution of $\Delta\theta\lesssim
0.2{{}^{\circ}}$. The ARPES method is surface sensitive technique, since the
average escape depth ($l_{esc}$) of the outgoing electrons is of the order of
$l_{esc}\sim 10$ Å, depending on the energy of incoming photons. Therefore,
very good surfaces are needed in order that the results be representative for
bulk samples. The most reliable studies were done on the bilayer
$Bi_{2}Sr_{2}CaCu_{2}O_{8}$ ($Bi2212$) and its single layer counterpart
$Bi_{2}Sr_{2}CuO_{6}$ ($Bi2201$), since these materials contain weakly coupled
$BiO$ planes with the longest inter-plane separation in the cuprates. This
results in a natural cleavage plane making these materials superior to others
in ARPES experiments. After a drastic improvement of sample quality in other
families of HTSC materials, the ARPES technique has became a central method in
theoretical considerations. Potentially, it gives valuable information on the
quasi-particle Green’s function, i.e. on the quasi-particle spectrum and life-
time effects. The ARPES can indirectly give information on the momentum and
energy dependence of the pairing potential. Furthermore, the electronic
spectrum of the (above mentioned) cuprates is highly quasi-2D which allows an
unambiguous determination of the initial state momentum from the measured
final state momentum, since the component parallel to the surface is conserved
in photoemission. In this case, the ARPES probes (under some favorable
conditions) directly the single particle spectral function
$A(\mathbf{k},\omega)$. In the following we discuss only those ARPES
experiments which give evidence for the importance of the EPI in cuprates -
see more in Shen-review .
The photoemission measures a nonlinear response function of the electron
system, and under some conditions it is analyzed in the so-called three-step
model, where the total photoemission intensity
$I_{tot}(\mathbf{k},\omega)\approx I\cdot I_{2}\cdot I_{3}$ is the product of
three independent terms: (1) $I$ \- describes optical excitation of the
electron in the bulk; (2) $I_{2}$ \- describes the scattering probability of
the travelling electrons; (2) $I_{3}$ \- the transmission probability through
the surface potential barrier. The central quantity in the three-step model is
$I(\mathbf{k},\omega)$ and it turns out that it can be written in the form
(for $\mathbf{k=k}_{\parallel}$) Shen-review $I(\mathbf{k},\omega)\simeq
I_{0}(\mathbf{k},\upsilon)f(\omega)A(\mathbf{k},\omega)$ with
$I_{0}(\mathbf{k},\upsilon)\sim\mid\langle\psi_{f}\mid\mathbf{pA\mid}\psi_{i}\rangle\mid^{2}$
and the quasi-particle spectral function
$A(\mathbf{k},\omega)=-$Im$G(\mathbf{k},\omega)/\pi$
$A(\mathbf{k},\omega)=-\frac{1}{\pi}\frac{Im\Sigma(\mathbf{k},\omega)}{[\omega-\xi(\mathbf{k})-\mathop{\mathrm{R}e}\Sigma(\mathbf{k},\omega)]^{2}+Im\Sigma^{2}(\mathbf{k},\omega)}.$
(36)
Here, $\langle\psi_{f}\mid\mathbf{pA\mid}\psi_{i}\rangle$ is the dipole matrix
element which depends on $\mathbf{k}$, polarization and energy $\upsilon$ of
the incoming photons. The knowledge of the matrix element is of a great
importance and its calculation from first principles was done carefully in
Bansil . $f(\omega)$ is the Fermi function, $G$ and
$\Sigma=\mathop{\mathrm{R}e}\Sigma+i\mathop{\mathrm{I}m}\Sigma$ are the quasi-
particle Green’s function and the self-energy, respectively.
We summarize and comment here on some important ARPES results which were
obtained recently and which confirm the existence of the Fermi surface and
importance of EPI in quasi-particle scattering Shen-review .
ARPES in the normal state
($N1$) There is a well defined Fermi surface in the metallic state with the
topology predicted by the LDA. However, the bands are narrower than LDA
predicts which points to a strong quasi-particle renormalization. ($N2$) The
spectral lines are broad with
$\mid$Im$\Sigma(\mathbf{k},\omega)\mid\sim\omega$ (or $\sim T$ for $T>\omega$)
which tells us that the quasi-particle liquid is a non-canonical Fermi liquid.
($N3$) There is a bilayer band splitting in $Bi2212$ (at least in the over-
doped state). The previous experiments did not show this splitting and served
for various speculations on some exotic non-Fermi liquid scenarios. ($N4$) At
temperatures $T_{c}<T<T^{\ast}$ and in the under-doped cuprates there is a
d-wave like pseudogap $\Delta_{pg}(\mathbf{k})\sim\Delta_{pg,0}(\cos
k_{x}-\cos k_{y})$ in the quasi-particle spectrum where $\Delta_{pg,0}$
increases by lowering doping. We stress that the pseudogap phenomenon is not
well understood at present and we shall discuss this problem in Part II. Its
origin can be due to a precursor superconductivity or due to a competing
order, such as spin- or charge-density wave or something similar. ($N5$) The
ARPES self-energy gives clear evidence that EPI interaction is rather strong.
For instance, at $T>T_{c}$ there are kinks in the quasi-particle dispersion
$\omega(\xi_{\mathbf{k}})$ in the nodal direction (along the $(0,0)-(\pi,\pi)$
line) at the characteristic phonon energy $\omega_{ph}^{(70)}\sim(60-70)$
$meV$ Lanzara , see Fig. 13, and near the anti-nodal point $(\pi,0)$ at $40$
$meV$ Cuk \- see Fig. 13.
Figure 13: (Top) Quasi-particle dispersion of $Bi2212$, $Bi2201$ and $LSCO$
along the nodal direction, plotted vs the momentum $k$ for $(a)-(c)$ different
doping, and $(d)-(e)$ different $T$; black arrows indicate the kink energy;
the red arrow indicates the energy of the $q=(\pi,0)$ oxygen stretching phonon
mode; inset of $(e)$\- T-dependent $\Sigma^{\prime}$ for optimally doped
$Bi2212$; $(f)$ \- doping dependence of the effective coupling constant
$\lambda^{\prime}$ along $(0,0)-(\pi,\pi)$ for the different HTSC oxides. From
Ref. Lanzara . ( Bottom) Quasi-particle dispersion $E(k)$ in the normal state
(a1, b1, c), at 107 K and 115 K, along various directions $\phi$ around the
anti-nodal point. The kink at $E=40meV$ is shown by the horizontal arrow. (a2
and b2) is $E(k)$ in the superconducting state at 10 K with the shifted kink
to $70meV$. (d) kink positions as a function of $\phi$ in the anti-nodal
region. From Ref. Cuk .
That these kinks exist also above $T_{c}$ excludes the scenario with the
magnetic resonance peak in $Im\chi_{s}(\mathbf{Q},\omega)$. Since the magnetic
neutron scattering give small SFI coupling constant $\lambda^{sf}<0.3$. The
kinks cannot be due to SFI as we already discussed above. ($N6$) The position
of the nodal kink is practically doping independent which points towards
phonons as the scattering (gluing) boson. ($N7$) The quasi-particles (holes)
couple practically to the whole spectrum of phonons since at least three group
of phonons were extracted from the ARPES effective self-energy in
$La_{2-x}Sr_{x}CuO_{4}$ Zhou-PRL \- Fig. 14.
Figure 14: (a) Effective real self-energy for the non-superconducting
$La_{2-x}Sr_{x}CuO_{4}$, $x=0.03$. Extracted
$\alpha_{eff}^{2}(\omega)F(\omega)$ is in the inset. (b) Top: the total MDC
width - open circles. Bottom: the EPI contribution shows saturation, impurity
contribution - dotted black line. The residual part is growing
$\sim\omega^{1.3}$. From Zhou-PRL .
This result is in a qualitative agreement with numerous tunnelling
measurements Tunneling-Vedeneev -Tsuda which apparently demonstrate that the
broad spectrum of phonons couples with holes without preferring any particular
phonons - see discussion below. ($N8$) Recent ARPES measurements in B2212
Valla show very different slope $d\omega/d\xi_{\mathbf{k}}$ of the quasi-
particle energy $\omega(\xi_{\mathbf{k}})$ at very small
$\mid\xi_{\mathbf{k}}\mid\ll\omega_{ph}$ and large energies
$\mid\xi_{\mathbf{k}}\mid\gg\omega_{ph}$ \- see Fig. 15. The theoretical
analysis Kulic-Dolgov-lambda of these results gives the total coupling
constant $\lambda>3$, the EPI one $\lambda^{ep}>2$ while the Coulomb
scattering (SFI is a part of it) is $\lambda^{c}\approx 1$ Kulic-Dolgov-lambda
\- see Fig. 15. To this end let us mention some confusion related to the value
of the EPI coupling constant extracted from ARPES. Namely, in Shen-review ,
Shen-Cuk-review , Lanzara-isotope the EPI self-energy was obtained by
subtracting the high energy slope of the quasi-particle spectrum
$\omega(\xi_{k})$ at $\omega\sim 0.3$ $eV$. The latter is apparently due to
the Coulomb interaction. Although the position of the low-energy kink is not
affected by this procedure (if $\omega_{ph}^{\max}\ll\omega_{c}$), the above
(subtraction) procedure gives in fact an effective EPI self-energy
$\Sigma_{eff}^{ep}(\mathbf{k},\omega)$ and the coupling constant
$\lambda_{z,eff}^{ep}(\mathbf{k})$ only. The latter is smaller than the real
EPI coupling constant $\lambda_{z}^{ep}(\mathbf{k})$. The total self-energy is
$\Sigma(\mathbf{k},\omega)=\Sigma^{ep}(\mathbf{k},\omega)+\Sigma^{c}(\mathbf{k},\omega)$
where $\Sigma^{c}$ is the contribution due to the Coulomb interaction. At very
low energies $\omega\ll\omega_{c}$ one has usually
$\Sigma^{c}(\mathbf{k},\omega)=-\lambda_{z}^{c}(\mathbf{k})\omega$, where
$\omega_{c}(\sim 1$ $eV)$ is the characteristic Coulomb energies and
$\lambda_{z}^{c}$ the Coulomb coupling constant. The quasi-particle spectrum
$\omega(\mathbf{k})$ is determined from the condition
$\omega-\xi(\mathbf{k})-\mathop{\mathrm{R}e}[\Sigma^{ep}(\mathbf{k},\omega)+\Sigma^{c}(\mathbf{k},\omega)]=0,$
(37)
where $\xi(\mathbf{k})$ is the bare band structure energy. At low energies
$\omega<\omega_{ph}^{\max}\ll\omega_{c}$, Eq.(37) can be rewritten in the form
$\omega-\xi^{ren}(\mathbf{k})-\mathop{\mathrm{R}e}\Sigma_{eff}^{ep}(\mathbf{k},\omega)=0,$
(38)
$\xi^{ren}(\mathbf{k})=[1+\lambda_{z}^{c}(\mathbf{k})]^{-1}\xi(\mathbf{k})$
(39)
$Re\Sigma_{eff}^{ep}(\mathbf{k},\omega)=\frac{Re\Sigma_{eff}^{ep}(\mathbf{k},\omega)}{1+\lambda_{z}^{c}(\mathbf{k})}.$
(40)
Since at very low energies $\omega\ll\omega_{ph}^{\max}$, one has
$\mathop{\mathrm{R}e}\Sigma^{ep}(\mathbf{k},\omega)=-\lambda_{z}^{ep}(\mathbf{k})\omega$
and
$\mathop{\mathrm{R}e}\Sigma_{eff}^{ep}(\mathbf{k},\omega)=-\lambda_{z,eff}^{ep}(\mathbf{k})\omega$,
then the real coupling constant is related to the effective one by
$\lambda_{z}^{ep}(\mathbf{k})=[1+\lambda_{z}^{c}(\mathbf{k})]\lambda_{z,eff}^{ep}(\mathbf{k})>\lambda_{z,eff}^{ep}(\mathbf{k}).$
At higher energies $\omega_{ph}^{\max}<\omega<\omega_{c}$, which are less
important for pairing, the EPI effects are suppressed and
$\Sigma^{ep}(\mathbf{k},\omega)$ stops growing, one has
$\mathop{\mathrm{R}e}\Sigma(\mathbf{k},\omega)\approx\mathop{\mathrm{R}e}\Sigma^{ep}(\mathbf{k},\omega)-\lambda_{z}^{c}(\mathbf{k})\omega$.
The measured $\mathop{\mathrm{R}e}\Sigma^{\exp}(\mathbf{k},\omega)$ at $T=10$
$K$ near and slightly away from the nodal point in the optimally doped Bi2212
with $T_{c}=91$ $K$ Valla-2006 is shown in Fig. 15.
Figure 15: Fig.4b from Valla-2006 : $Re\Sigma(\omega)$ measured in Bi2212
(thin line) and model $Re\Sigma(\omega)$ (bold line) obtained in Valla-2006 .
The three thin lines ($\lambda_{1},\lambda_{2},\lambda_{3}$) are the slopes of
$Re\Sigma(\omega)$ in different energy regions - see the text.
It is seen that $Re\Sigma^{\exp}(\mathbf{k},\omega)$ has two kinks \- the
first one at low energy $\omega_{1}\approx\omega_{ph}^{high}\approx 50-70$
$meV$ which is most probably of the phononic origin Shen-review , Shen-Cuk-
review , Lanzara-isotope , while the second kink at higher energy
$\omega_{2}\approx\omega_{c}\approx 350$ $meV$ is probably due to the Coulomb
interaction. However, the important results in Ref. Valla-2006 is that the
slopes of $\mathop{\mathrm{R}e}\Sigma^{\exp}(\mathbf{k},\omega)$ at low
($\omega<\omega_{ph}^{high}$) and high energies
($\omega_{ph}^{high}<\omega<\omega_{c}$) are different. The low-energy and
high-energy slope _near the nodal point_ are depicted and shown in Fig. 15
schematically (thin lines). From Fig. 15 it is obvious that EPI prevails at
low energies $\omega<\omega_{ph}^{high}$. More precisely digitalization of
$\mathop{\mathrm{R}e}\Sigma^{\exp}(\mathbf{k},\omega)$ in the interval
$\omega_{ph}^{high}<\omega<0.4$ $eV$ gives the Coulomb coupling
$\lambda_{z}^{c}\approx 1.1$ while the same procedure at $20$
$meV\approx\omega_{ph}^{low}<\omega<\omega_{ph}^{high}\approx 50-70meV$ gives
the total coupling constant
$(\lambda_{2}\equiv)\lambda_{z}=\lambda_{z}^{ep}+\lambda_{z}^{c}\approx 3.2$
and the EPI coupling constant
$\lambda_{z}^{ep}(\equiv\lambda_{z,high}^{ep})\approx
2.1>2\lambda_{z,eff}^{ep}(\mathbf{k})$, i.e. the EPI coupling is at least
twice larger than in the previous analysis of ARPES results. This estimation
tells us that at (and near) the nodal point, the EPI interaction dominates in
the quasi-particle scattering at low energies since $\lambda_{z}^{ep}(\approx
2.1)\approx 2\lambda_{z}^{c}>2\lambda_{z}^{sf}$, while at large energies
(compared to $\omega_{ph}$), the Coulomb interaction with
$\lambda_{z}^{c}\approx 1.1$ dominates. We point out that EPI near the anti-
nodal point can be even larger than in the nodal point, mostly due to the
higher density of states near the anti-nodal point. ($N8$) Recent ARPES
spectra in the optimally doped $Bi2212$ near the nodal and anti-nodal point
Lanzara-isotope show a pronounced isotope effect in
$\mathop{\mathrm{R}e}\Sigma^{\exp}(\mathbf{k},\omega)$, thus pointing to the
important role of EPI - see more in the part related to the isotope effect.
The isotope effect in $\mathop{\mathrm{R}e}\Sigma(\mathbf{k},\omega)$ can be
well described in the framework of the Migdal-Eliashberg theory for EPI Ma-Ku-
Do as it will be discussed in Part II. ($N9$) ARPES experiments on
$Ca_{2}CuO_{2}Cl_{2}$ give strong evidence for the formation of small polarons
in undoped cuprates which can be only due to phonons and strong EPI, while by
doping quasi-particles appear and there are no small polarons Shen-polarons .
Thus in Shen-polarons , it a broad peak (around $-0.8$ $eV$) is observed at
the top of the band ($\mathbf{k}=(\pi/2,\pi/2)$) with the dispersion similar
to that predicted by the $t-J$ model - see Fig. 16.
Figure 16: (Left) The ARPES spectrum of undoped $Ca_{2}CuO_{2}Cl_{2}$ at
$\mathbf{k}=(\pi/2,\pi/2)$. Gaussian shape - solid line, Lorentzian shape -
dashed line. (Right) Dispersion of the polaronic band - A and of the quasi-
particle band - B along the nodal direction. Horizontal lines are the chemical
potentials for a large number of samples. From Shen-polarons
However, the peak in Fig. 16 is of Gaussian shape and can be described only by
coupling to bosons, i.e. this peak is a boson side band \- see more in
Gunnarsson-review-2008 and references therein. The theory based on the t-J
model (in the antiferromagnetic state of the undoped compound) by including
coupling to several (half-breathing, apical oxygen, low-lying) phonons, which
is given in Gunnar-Nagaosa-Ciuchi , explains successfully this broad peak of
the boson side band by the formation of small polarons due to the EPI coupling
($\lambda^{ep}\approx 1.2$). Note that this $\lambda^{ep}$ is for the polaron
at the bottom of the band while in the case whwn the Fermi surface exists this
coupling is even larger Gunnar-Nagaosa-Ciuchi . In Gunnar-Nagaosa-Ciuchi it
was stressed that even when the electron-magnon interaction is stronger than
the EPI one, the polarons are formed due to EPI. The latter involves
excitation of many phonons at the lattice site (where the hole is seating),
while it is possible to excite only one magnon at the given site.
($N10$) Recent soft x-ray ARPES measurements on the electron doped HTSC
$Nd_{1.85}Ce_{0.15}CuO_{4}$ Tsunekawa , and $Sm_{(2-x)}Ce_{x}CuO_{4}$
($x=0.1,$ $0.15,$ $0.18$), $Nd_{1.85}Ce_{0.15}CuO_{4}$,
$Eu_{1.85}Ce_{0.15}CuO_{4}$ Eisaki show kink at energies $50-70$ $meV$ in the
quasi-particle dispersion relation along both nodal and antinodal directions
as it is shown in Fig. 17.
Figure 17: NCCO electron-doped: (a) $Im\Sigma(\omega)$ measured in the nodal
point. Curves are offsets by 50 meV for clarity. The change of the slope in
the last bottom curve is at the phonon energy. (b) $Im\Sigma(\omega)$ for the
antinodal direction with $30$ $meV$ offset. (c) Experimental phonon dispersion
of the bond stretching modes. (d) Estimated $\lambda_{eff}^{ep}$ from
$Im\Sigma(\omega)$. From Eisaki .
It is seen from this figure that the effective EPI coupling constant
$\lambda_{eff}^{ep}(<\lambda^{ep})$ is isotropic and
$\lambda_{eff}^{ep}\approx 0.8-1$. The kink in the electron-doped cuprates is
due solely to EPI and in that respect the situation is similar to the one in
the hole-doped cuparates.
ARPES results in the superconducting state
($S1$) There is an anisotropic superconducting gap in most HTSC compounds
Shen-review , which is predominately d-wave like, i.e.
$\Delta_{sc}(\mathbf{k})\sim\Delta_{0}(\cos k_{x}-\cos k_{y})$ with
$2\Delta_{0}/T_{c}\approx 5-6$. ($S2$) The kink at $(60-70)$ $meV$ in the
quasi-particle energy around the nodal point is not-shifted in the
superconducting state while the antinodal kink at $\omega_{ph}^{(40)}\sim 40$
$meV$ is shifted in the superconducting state by $\Delta_{0}(=(25-30)meV)$,
i.e. $\omega_{ph}^{(40)}\rightarrow\omega_{ph}^{(40)}+\Delta_{0}=(65-70)meV$
Shen-review . To remind the reader, in the standard Eliashberg theory the kink
in the normal state at $\omega=\omega_{ph}$ should be shifted in the
superconducting state to $\omega_{ph}+\Delta_{0}$ at any point at the Fermi
surface. This puzzling result might be a smoking gun result since it makes a
constraint on the quasi-particle interaction in cuprates. Until now there is
only one plausible explanation Kulic-Dolgov-shift of this shift-non-shift
puzzle which is based on an assumption of the forward scattering peak (FSP) in
EPI - see more in Part II. The FSP in EPI means that electrons scatter into a
narrow region around the starting point in the k-space, so that at the most
part of the Fermi surface, there is weak (or no) mixing of states with
different signs of the order parameter $\Delta(\mathbf{k})$. ($S3$) The recent
ARPES spectra Chen-Shen-4-layered on an undoped single crystalline 4-layered
cuprate with apical fluorine (F), $Ba_{2}Ca_{3}Cu_{4}O_{8}F_{2}$ (F0234) gives
strong evidence against SFI - see Fig. 18.
Figure 18: Crystal structure of
$Ba_{2}Ca_{3}Cu_{4}O_{8}(O_{\delta}F_{1-\delta})_{2}$. There are four
$CuO_{2}$ layers in a unit cell with the outer having apical F atoms. From
Chen-Shen-4-layered .
Namely, F0234 is not a Mott insulator - as expected from valence charge
counting which puts $Cu$ valence as $2^{+}$, but it is a superconductor with
$T_{c}=60$ $K$. Moreover, the ARPES data Chen-Shen-4-layered reveal at least
two metallic Fermi-surface sheets with corresponding volumes equally below and
above half-filling - see Fig. 19.
Figure 19: (a) Fermi surface (FS) contours from two samples of F0234. $n$ \-
electron-like; $h$ \- hole-like. Bold arrow is $(\pi,\pi)$ scattering vector.
Angle $\theta$ defines the horizontal axis in (b). (b) Leading gap edge along
k-space angle from the two FS contours. From Chen-Shen-4-layered .
One of the Fermi-surfaces is due to the electron-like ($N$) band (with $20\pm
6$% electron-doping) and the other one due to the hole-like ($P$) band (with
$20\pm 8$ $\%$ hole-doping) and their split along the nodal direction is
significant and cannot be explained by the LDA (or DFA) method Shen-
Anders-4layer . This electron and hole self-doping of inner and outer layers
is in an appreciable contrast to other multilayered cuprates where there is
only hole self-doping. For instance, in HgBa2CanCun+1O2n+2 ($n=2,3$) and
(Cu,C)Ba2CanCun+1O3n+2 ($n=2,3,4$), the inner CuO2 layers are less hole-doped
than outer layers. It turns out, unexpectedly, that the superconducting gap on
the $N$-band Fermi-surface is significantly larger than on the $P$-one, i.e.
their ratio is anomalous $(\Delta_{N}/\Delta_{P})\approx 2$ and $\Delta_{N}$
is an order of magnitude larger than in the electron-doped cuprate
$Nd_{2-x}Ce_{x}CuO_{4}$. Furthermore, the $N$-band Fermi-surface is rather far
from the antinodal point at ($\pi/2,0$). This is an extremely important result
which means that the antiferromagnetic spin fluctuations with the AF wave-
vector $\mathbf{Q}=(\pi/2,\pi/2)$, as well as the van Hove singularity, are
not important for the pairing in the $N$-band. To remind the reader, the SFI
scenario assumes that the pairing is due to spin fluctuations with the wave-
vector $\mathbf{Q}$ which connects two anti-nodal points which are near the
van Hove singularity at the hole-surface (at ($\pi/2,0$) and ($0,\pi/2$))
giving rise to large density of states. This is apparently not the case for
the $N$-band Fermi-surface - see Fig. 19. The ARPES data give further that
there is a kink at $\sim 85$ $meV$ in the quasi-particle dispersion of both
bands, while the kink in the $N$-band is stronger than that in the $P$-band.
This result, together with the anomalous ratio $(\Delta_{N}/\Delta_{P})\approx
2$, strongly disfavors SFI as a pairing mechanism.
($S4$) Despite the presence of significant elastic scattering in optimally
doped Bi-2212, there are dramatic sharpening of the spectral function near the
anti-nodal point $(\pi,0)$ at $T<T_{c}$ Zhu . This can be explained by
assuming that the small q-scattering (the forward scattering peak) dominates
in the elastic impurity scattering KuOudo , Kulic-Dolgov-imp . As a result,
one finds that the impurity scattering rate in the superconducting state is
almost zero, i.e.
$\gamma_{imp}(\mathbf{k},\omega)=\gamma_{n}(\mathbf{k},\omega)+\gamma_{a}(\mathbf{k},\omega)=0$
for $\mid\omega\mid<\Delta_{0}$ for any kind of pairing (s- p- d-wave etc.)
since the normal ($\gamma_{n}$) and the anomalous ($\gamma_{a}$) scattering
rates compensate each other - the collapse of the elastic scattering rate.
This result is a consequence of the Anderson-like theorem for unconventional
superconductors which is due to the dominance of the small q-scattering KuOudo
-Kulic-Dolgov-imp . In such a case d-wave pairing is weakly unaffected by
impurities - there is small reduction in $T_{c}$ Kulic-Dolgov-imp , Kee-Tc .
The physics behind this result is rather simple. The small q-scattering
(forward scattering) means that electrons scatter into a small region in the
k-space, so that at the most part of the Fermi surface there is no mixing of
states with different signs of the order parameter $\Delta(\mathbf{k})$, and
the detrimental effect of impurities is reduced. For states near the nodal
points, there is mixing but since $\Delta(\mathbf{k})$ is small in this
region, there is only small reduction in $T_{c}$ Licht . This result points to
the importance of strong correlations in the renormalization of the impurity
scattering - see discussion in Part II.
In conclusion, in order to explain ARPES results in cuprates it is necessary
to take into account: (1) EPI interaction since it dominates in the
quasiparticle scattering in the energy region responsible for pairing; (2)
effects of elastic nonmagnetic impurities with FSP; (3) the Coulomb
interaction which dominates at higher energies $\omega>\omega_{ph}$. In this
respect, the presence of ARPES kinks and the knee-like shape of the spectral
width are serious constraints for the pairing theory.
### III.4 Tunnelling spectroscopy and spectral function $\alpha^{2}F(\omega)$
By measuring current-voltage $I-V$ characteristics in $NIS$ (normal metal-
insulator-superconductor) tunnelling junctions with large tunnelling barrier -
see more below, one obtains from tunnelling conductance $G_{NS}(V)=dI/dV$ the
so called tunnelling density of states in superconductors $N_{T}(\omega)$.
Moreover, by measuring of $G_{NS}(V)$ at voltages $eV>\Delta$ it is possible
to determine the Eliashberg spectral function $\alpha^{2}F(\omega)$ and
finally to confirm (definitely) the phonon mechanism of pairing in $LTSC$
materials, except heavy fermions. Four tunnelling techniques were used in the
study of $HTSC$ cuprates: $\mathbf{(1)}$ vacuum tunneling by using the $STM$
technique - scanning tunnelling microscope; $\mathbf{(2)}$ point-contact
tunnelling; $\mathbf{(3)}$ break-junction tunnelling; $\mathbf{(4)}$ planar-
junction tunnelling. Each of these techniques has some advantages although in
principle the most potential one is the STM technique Kirtley . It should be
stressed that there are still difficulties in understanding tunnelling
experiments in $HTSC$ cuprates because of non-ideal tunnelling behavior of
contacts Kirtley . Since tunnelling measurements probe a surface region of the
order of superconducting coherence length $\xi_{0}$, then this kind of
measurements in $HTSC$ materials with small coherence length $\xi_{0}$
($\xi_{ab}\sim 20$ Å in the $a-b$ plane and $\xi_{c}\sim 1-3$ Å along the
$c-axis$) depends strongly on the surface quality and sample preparation.
Nowadays, many of these material problems in $HTSC$ cuprates are understood
and as a result consistent picture of tunnelling features is starting to
emerge.
From tunnelling experiments one obtains the energy gap in the superconducting
state. Since we have already discussed this problem in Kulic-Review , we will
only briefly mention some important result, that in most cases, $G_{NS}(V)$
has V-shape in all families of HTSC hole and electron doped cuprates. This is
due to $d-wave$ pairing with the gapless spectrum, which is definitely
confirmed in the interference experiments on hole and electron doped cuprates
Tsui-Kirtley . Some experiments give the U-shape of $G_{NS}(V)$ which
resembles s-wave pairing. This controversy is explained to be the property of
the tunneling matrix element which filters out states with the maximal gap.
Here we are interested in the electron-boson spectral function
$\alpha^{2}F(\omega)$ in HTSC cuprates which can be extracted by using
tunnelling spectroscopy. We inform the reader in advance, that the shape and
the energy width of $\alpha^{2}F(\omega)$, which are extracted from the second
derivative $d^{2}I/dV^{2}$ at voltages above the superconducting gap, in most
HTSC cuprates resembles the phonon density of states $F(\omega)$. This result
is strong evidence for the importance of $EPI$ in the pairing potential of
$HTSC$ cuprates. For instance, plenty of break-junctions made from $Bi2212$
single crystals Tunneling-Vedeneev show that the negative peaks in
$d^{2}I/dV^{2}$coincide with the peaks in the generalized phonon density of
states $F_{ph}(\omega)$ measured by neutron scattering - see Fig. 20.
Figure 20: (a) Second derivative of $I(V)$ for a $Bi2212$ break junction in
various magnetic fields. The structure of minima in $d^{2}I/dV^{2}$ can be
compared with the phonon density of states $F(\omega)$; (b) the spectral
functions $\alpha^{2}F(\omega)$ in various magnetic fields. From Tunneling-
Vedeneev
The tunnelling spectra in Bi2212 break junctions Tunneling-Vedeneev , which
are shown in Fig. 20, indicates that the spectral function
$\alpha^{2}F(\omega)$ is unchanged in magnetic field which disfavors SFI since
in the latter case, this function should be sensitive to the magnetic field.
The reported broadening of the peaks in $\alpha^{2}F(\omega)$ are partly due
to the gapless spectrum of $d-wave$ pairing in $HTSC$ cuprates. Additionally,
the tunnelling density of states $N_{T}(\omega)$ at very low $T$ show a
pronounced gap structure and it was found that $2\bar{\Delta}/T_{c}=6.2-6.5$,
where $T_{c}=74-85$ $K$ and $\bar{\Delta}$ is some average value of the gap.
In order to obtain $\alpha^{2}F(\omega)$ the inverse procedure was$<$used by
assuming $s-wave$ superconductivity and the effective Coulomb parameter
$\mu^{\ast}\approx 0.1$ Tunneling-Vedeneev . The obtained
$\alpha^{2}F(\omega)$ gives large $EPI$ coupling constant $\lambda^{ep}\approx
2.3$. Although this analysis Tunneling-Vedeneev was done in terms of $s-wave$
pairing, it mimics qualitatively the case of $d-wave$ pairing, since one
expects that $d-wave$ pairing does not change significantly the global
structure of $d^{2}I/dV^{2}$ at $eV>\Delta$ albeit introducing a broadening in
it - see the physical meaning in Appendix A. We point out that the results
obtained in Tunneling-Vedeneev were reproducible on more than 30 junctions.
In that respect very important results on slightly overdoped $Bi2212-GaAs$ and
on $Bi2212-Au$ planar tunnelling junctions are obtained in Tun2 \- see Fig.
21.
Figure 21: The spectral functions $\alpha^{2}F(\omega)$ and the calculated
density of states at $0K$ (upper solid line) obtained from the conductance
measurements the $Bi(2212)-Au$ planar junctions. From Tun2
These results show very similar features to those obtained in Tunneling-
Vedeneev on break-junctions. It is worth mentioning that several groups Tun3
, Tun4 , Gonnelli have obtained similar results for the shape of the spectral
function $\alpha^{2}F(\omega)$ from the $I-V$ measurements on various $HTSC$
cuprates - see the comparison in Fig. 22. The latter results leave no much
doubts about the importance of the $EPI$ in pairing mechanism of $HTSC$
cuprates.
Figure 22: The spectral functions $\alpha^{2}F(\omega)$ from measurements of
various groups: Vedeneev et al. Tunneling-Vedeneev , Gonnelli et al.Gonnelli ,
Miyakawa et al. Tun3 , Shimada et al. Tun2 . The generalized density of states
GPDS for Bi2212 is at the bottom. From Tun2 .
In that respect tunnelling measurements on slightly overdoped
$Bi_{2}Sr_{2}CaCu_{2}O_{8}$ Tun2 , Tsuda are impressive, since the Eliashberg
spectral function $\alpha^{2}F(\omega)$ was extracted from the measurements of
$d^{2}I/dV^{2}$ and by solving the inverse problem. The extracted
$\alpha^{2}F(\omega)$ has several peaks in broad energy region up to $80$
$meV$ as it is seen in Fig. 21\- 22, which coincide rather well with the peaks
in the phonon density of states $F_{ph}(\omega)$. In Tsuda numerous peaks,
from $P1-P13$, in $\alpha^{2}F(\omega)$ are discerned as shown in Fig.24,
which correspond to various groups of phonon modes - laying in (and around)
these peaks. Moreover, in Tun2 , Tsuda are extracted the coupling constants
for these modes and their contribution ($\Delta T_{c}$) to $T_{c}$ as it is
seen in Fig. 23. Note, due to the nonlinearity of the problem, the sum of
$\Delta T_{c}$ is not equal to $T_{c}$.
Figure 23: Table I - Phonon frequency $\omega$, EPI coupling constant
$\lambda_{i}$ of the peaks $P1-P13$ and contribution $\Delta T_{c}$ to $T_{c}$
of each peak in $\alpha^{2}F(\omega)$-shown in Fig.24, obtained from the
tunnelling conductance of $Bi_{2}Sr_{2}CaCu_{2}O_{8}$. $\Delta T_{c}$ is the
decrease in $T_{c}$ when the peak in $\alpha^{2}F(\omega)$ is eliminated. From
Tsuda .
The next remarkable result is that the extracted $EPI$ coupling constant is
very large,i.e. $\lambda^{ep}(=2\int
d\omega\alpha^{2}F(\omega)/\omega)=\sum_{i}\lambda_{i}\approx 3.5$ \- see Fig.
23. It is obvious from Figs. (23-24) that almost all phonon modes contribute
to $\lambda^{ep}$and $T_{c}$, which means that on the average, each particular
phonon mode is not too strongly coupled to electrons thus keeping the lattice
stable. None of the modes has too large $\lambda_{i}<1.3$, thus keeping the
lattice stability.
Figure 24: The spectral functions $\alpha^{2}F(\omega)$ from the tunnelling
conductance of $Bi_{2}Sr_{2}CaCu_{2}O_{8}$ for the positive and the negative
bias voltages, and the averaged one Tun2 . The averaged one is divided into
$13$ components. The origin of the ordinate is $2,1,0$ and $-0.5$ from the top
down. From Tsuda , Tun2
For a better understanding of the the EPI coupling in these systems we show in
Fig. 25 the total and partial density of phononic states.
Figure 25: The phonon density of states $F(\omega)$ (PDOS) of
$Bi_{2}Sr_{2}CaCu_{2}O_{8}$ compared with the generalized density of states
(GPDOS) Renker . Atomic vibrations: O1 - O in the $CuO_{2}$ plane; O2 - apical
O; O3 - O in the BiO plane. From Tun2
In Fig. 23 it is seen that lower frequency modes from $P1-P3$, corresponding
to $Cu,Sr$ and $Ca$ vibrations, are rather strongly coupled to electrons (with
$\lambda_{\kappa}\sim 1$) and give appreciable contributions to $T_{c}$. It is
also seen in Fig. 23 that the coupling constants $\lambda_{i}$ of the high-
energy phonons ($P9-P13$ with $\omega\geq 70$ $meV$) have $\lambda_{i}\ll 1$
and give moderate contribution to $T_{c}$ \- around 10 %. These results
confirm the importance of modes which cause the change of the Madelung energy
in the ionic-metallic structure of HTSC cuprates, the idea also conveyed in
Maksimov-Review , Kulic-Review \- see more in Part II. If definitely
confirmed, these results are in accordance with the moderate oxygen isotope
effect in cuprates near the optimal doping. We stress that each peak $P1-P13$
in $\alpha^{2}F(\omega)$ corresponds to many modes. In order to get filling on
the structure of vibrations possibly strongly involved in pairing, we show in
Figs. 26-27 the structure of these vibrations at special points in the
Brillouin zone. It is seen in Fig. 26 that the low-frequency phonons $P1-P2$
are dominated by Cu, Sr, Ca vibrations.
Figure 26: Atomic polarization vectors and their frequencies (in $meV$) at
special points in the Brillouin zone. The larger closed circles in the lattice
are O-ions. $\Gamma-X$ is along the Cu-O-Cu direction. Arrows indicate
displacements. The modes in square and round brackets are the transverse and
longitudinal optical modes respectively. (Top) - modes of the P1 peak.
(Bottom) - modes of the P2 peak. From Tun2 , Tsuda .
Figure 27: Atomic polarization vectors and their frequencies (in $meV$) at
special points in the Brillouin zone. The larger closed circles in the lattice
are O-ions.$\Gamma-X$ is along the Cu-O-Cu direction. Arrows indicate
displacements. The modes in square and round brackets are the transverse and
longitudinal optical modes, respectively.(Top) - modes of the P3 peak.
(Bottom) - modes of the P4 peak. From Tun2 , Tsuda .
It is seen in Fig. 23 that $P3$ modes are stronger coupled to electrons than
$P4$ ones, although the density of state for the $P4$ modes is larger. The
reason for such an anomalous behavior might be due to symmetries of
corresponding phonons as it is shown in Fig. 27. Namely to the $P3$ peak
contribute axial vibrations of O(1) in the Cu2 plane which are odd under
inversion, while in the $P4$ peak these modes are even. The in-plane modes of
Ca and O(1) are present in $P3$ which are in-phase and out-of-phase modes,
while in $P4$ they are all out-of-phase modes. For more information on other
modes $P5-P13$ see Tsuda . We stress that the Eliashberg equations based on
the extracted $\alpha^{2}F(\omega)$ of the slightly overdoped
$Bi_{2}Sr_{2}CaCu_{2}O_{8}$ with the ratio $(2\Delta/T_{c})\approx 6.5$
describe rather well numerous optical, transport and thermodynamic properties
Tsuda . However, in underdoped systems with $(2\Delta/T_{c})\approx 10$, where
the pseudogap phenomena are pronounced, there are serious disagreements
between experiments and the Eliashberg-like theory.
Similar conclusion, regarding the properties of the $EPI$ spectral function
$\alpha^{2}F(\omega)$ in $HTSC$ cuprates, comes out from tunneling
measurements on the Andreev ($Z\ll 1,$ low barrier)- and Giaver ($Z\gg 1$,
high barrier) -type junctions in $La_{2-x}Sr_{x}CuO_{4}$ and $YBCO$ compounds
Deutscher , where the extracted $\alpha^{2}F(\omega)$ is in accordance with
the phonon density of states $F_{ph}(\omega)$ see Fig. 28.
Figure 28: (a) $d^{2}I/dV^{2}$ of a Giaver-like contact in $La$
${}_{2-x}Sr_{x}CuO_{4}$ \- note the large structure below $50meV$; (b)
$d^{2}I/dV^{2}$ of an Andreev- and Giaver-like contact compared to the peaks
in the phonon density of states. From Deutscher
Note that the BTK parameter $Z$ is related to the transmission and reflection
coefficients for the normal metal $(1+Z^{2})^{-1}$ and $Z^{2}(1+Z^{2})^{-1}$
respectively.
Although most of the peaks in $\alpha^{2}F(\omega)$ in $HTSC$ cuprates
coincide with the peaks in the phonon density of states it is legitimate to
put the question - can the magnetic resonance in the superconducting state
give contribution to the $\alpha^{2}F(\omega)$? In that respect very important
inelastic neutron scattering measurements of the magnetic resonance as a
function of doping Keimer-pss give that the resonance energy $E_{r}$ scales
with $T_{c}$, i.e. $E_{r}=(5-6)T_{c}$ as shown in Fig.29.
Figure 29: Doping dependence of the energy $E_{r}$ of the magnetic resonance
peak at $\pi,\pi$ in YBCO and Bi2212 measured at low temperatures by inelastic
neutron scattering. From Keimer-pss
This means that if one of the peaks in $\alpha^{2}F(\omega)$ is due to the
magnetic resonance at $\omega=E_{r}$, then it shifts strongly with doping as
it is observed in Keimer-pss . This is contrary to phonon peaks (energies)
whose positions are doping independent. To this end, recent tunnelling
experiments on Bi2212 Ponomarev-Tunnel show clear doping independence of
$\alpha^{2}F(\omega)$ as it is seen in Fig. 30. This remarkable result is an
additional and strong evidence in favor of EPI and against the SFI mechanism
of pairing in HTSC cuprates.
Figure 30: Second derivative of $I(V)$ for a $Bi2212$ tunnelling junctions for
various doping: UD-underdoped; OD-optimally doped; OVD-overdoped system. The
structure of minima in $d^{2}I/dV^{2}$ can be compared with the phonon density
of states $F(\omega)$. The full and vertical lines mark the positions of the
magnetic resonance energy $E_{r}\approx 5.4T_{c}$ for various doping taken
from Fig.29. Red tiny arrows mark positions of the magnetic resonance $E_{r}$
in various doped systems. Dotted vertical lines mark various phonon modes.
From Ponomarev-Tunnel
It is interesting that in the vacuum tunneling $STM$ measurements Fischer-
Renner-RM the fine structure in $d^{2}I/dV^{2}$ at $eV>\Delta$ was not seen
below $T_{c}$, while the pseudogap structure is observed at temperatures near
and above $T_{c}$. This result could mean that the $STM$ tunnelling is likely
dominated by the nontrivial structure of the tunnelling matrix element (along
the $c$-axis), which is derived from the band structure calculations AJLM .
However, recent $STM$ experiments on $Bi2212$ Davis give important
information on possible nature of the bosonic mode which couples with
electrons. In Davis the local conductance $dI/dV(\mathbf{r},E)$ is measured
where it is found that $d^{2}I/dV^{2}(\mathbf{r},E)$ has peak at
$E(\mathbf{r})=\Delta(\mathbf{r})$ $+\Omega(\mathbf{r})$ where
$dI/dV(\mathbf{r},E)$ has the maximal slope - see Fig. 31(a).
Figure 31: (a) Typical conductance $dI/dV(\mathbf{r},E)$. The ubiquitous
feature at $eV>\Delta$(gap) with maximal slopes, which give peaks in
$d^{2}I/dV^{2}(\mathbf{r},E)$ are indicated by arrows. (b) The histograms of
all values of $\Omega(\mathbf{r})$ for samples with $O^{1}6$ \- right curve
and with $O^{1}8$ \- left curve. From Davis
It turns out that the average phonon energy $\bar{\Omega}$ depends on the
oxygen mass, i.e. $\bar{\Omega}\sim M_{O}^{-1/2}$, with $\bar{\Omega}_{16}=52$
$meV$ and $\bar{\Omega}_{18}\approx 48$ $meV$ \- as it is seen in Fig. 31(b).
This result is a convincing evidence that phonons are strongly involved in the
quasi-particle scattering. A possible explanation is put forward in Davis by
assuming that this isotope effect is due to the B1g phonon which interacts
with anti-nodal quasi-particles.
In our opinion the important message of tunnelling experiments in $HTSC$
cuprates (by including $Ba_{1-x}K_{x}BiO_{3}$ too Huang , Jensen ) is that
there is strong evidence for the importance of EPI and that no particular
phonon mode can be singled out in the spectral function $\alpha^{2}F(\omega)$
as being the only one which dominates in pairing mechanism. This important
result means that the high $T_{c}$ is not attributable to a particular phonon
mode in the $EPI$ mechanism, i.e. all phonon modes contribute to
$\lambda^{ep}$. Having in mind that the phonon spectrum in $HTSC$ cuprates is
very broad (up to $80$ $meV$ ), then the large $EPI$ constant
($\lambda^{ep}\gtrsim 2$) obtained in tunnelling experiments is not surprising
at all.
### III.5 Phonon spectra and EPI
Although experiments, such as inelastic neutron and Raman scattering, related
to phonon spectra and their renormalization by $EPI$ do not give directly
$\alpha^{2}F(\omega)$, as the tunnelling and optic spectra do, they
nevertheless give useful information on the strength of EPI for some
particular phonons. We stress in advance that the interpretation of
experimental results in terms of EPI for weakly interacting electrons might be
risky since in strongly correlated systems, such as HTSC cuprates, the phonon
renormalization due to EPI is rather different from that in weakly correlated
metals - see Gunnarsson-Rosch-epi . Since these questions are thoroughly
studied in the excellent review Gunnarsson-Rosch-epi \- see also Part II, we
shall briefly enumerate the main points: (1) In strongly correlated systems,
the EPI coupling for a number of phononic modes can be significantly larger
than the LDA and Hartree-Fock methods predict. This is due to many-body
effects Gunnarsson-Rosch-epi . (2) In strongly correlated systems the quasi-
particle charge susceptibility, which enters the phonon self-energy
$\Pi\mathbf{(q},\omega)(\sim\chi_{c}\mathbf{(q},\omega))$, is much more
suppressed than in weakly correlated metals and these effects are out of the
LDA possibilities Kulic-Review , Khaliullin , Gunnarsson-Rosch-epi . This is
one of the reasons that the analysis of experiments on phonon renormalizations
in the framework of LDA underestimates the EPI coupling constant significantly
\- on all these questions, see more details in Part II.
#### III.5.1 The phonon Raman scattering
The phonon Raman scattering gives also evidence for appreciable EPI in
cuprates Cardona1 , Cardona2 , Hadjiev . We enumerate some of them - see more
in Kulic-Review and References therein. (i) There is a pronounced asymmetric
line-shape (of the Fano resonance) in the metallic state. For instance, in
$YBa_{2}Cu_{3}O_{7}$ two Raman modes at $115$ $cm^{-1}$ (Ba dominated mode)
and at $340$ $cm^{-1}$ (O dominated mode in the CuO2 planes) show pronounced
asymmetry which is absent in $YBa_{2}Cu_{3}O_{6}$. This result points to
appreciable interaction of Raman active phonons with continuum states (quasi-
particles) Cardona1 , Cardona2 ,. (ii) The phonon frequencies for some
$A_{1g}$ and $B_{1g}$ are strongly renormalized in the superconducting state,
between $(6-10)$ $\%$, pointing again to large EPI Hadjiev \- see also in
Kulic-Review , Kulic-AIP . To this point we mention that the electronic Raman
scattering in cuprates show a remarkable correlation between the Raman cross-
section $\tilde{S}_{\exp}(\omega)$ and the optical conductivity in the a-b
plane $\sigma(\omega)$, i.e. $\tilde{S}_{\exp}(\omega)\sim\sigma(\omega)$
Kulic-Review . It was argued above that EPI with the very broad spectral
function $\alpha^{2}F(\omega)$ explains in a natural way the $\omega$ and $T$
dependence of $\sigma(\omega)$. This means that the electronic Raman spectra
in cuprates can be explained by EPI in conjunction with strong correlations.
This conclusion is supported by calculations of the Raman cross-section
Rashkeev which take into account EPI with $\alpha^{2}F(\omega)$ extracted
from tunnelling measurements in $YBa_{2}Cu_{3}O_{6+x}$ and
$Bi_{2}Sr_{2}CaCu_{2}O_{8+x}$ Kulic-Review , Tunneling-Vedeneev -Tsuda . Quite
similar properties (to cuprates) of the electronic Raman scattering, as well
as of $\sigma(\omega)$, $R(\omega)$ and $\rho(T)$, were observed in
experiments Bozovic on isotropic 3D metallic oxides $La_{0.5}Sr_{0.5}CoO_{3}$
and $Ca_{0.5}Sr_{0.5}RuO_{3}$ where there are no signs of antiferromagnetic
fluctuations. This means that low-dimensionality and antiferromagnetic spin
fluctuations cannot be a prerequisite for anomalous scattering of quasi-
particles and EPI must be inevitably taken into account since it is present in
all these compounds.
#### III.5.2 Neutron scattering, phonon spectra and EPI
The softening of numerous phonon modes has been observed in the normal state
of cuprates giving important evidence for pronounced EPI. There are several
important reviews on this subject Pintschovius and here we discuss briefly
two important examples which demonstrate in an impressive way the inefficiency
of the LDA band structure calculations to treat quantitatively and
qualitatively EPI in HTSC cuprates. Namely, by doping the Cu-O bond-stretching
phonon mode shows a substantial softening at $\mathbf{q}_{hb}=(0.5,0,0)$ \-
called the half-breathing phonon, and a large broadening by 5 meV at 15%
doping. While the softening can be partly described by the LDA method Bohnen-
Heid-2003 , it predicts an order of magnitude smaller broadening than the
experimental one. The reason for this failure lies in strong correlations,
which are not included in the LDA method as explained in Kulic-Review ,
Gunnarsson-review-2008 , Khaliullin , Gunnarsson-Rosch-epi . They give rise to
an increase of the EPI coupling and to strong suppression of the charge
fluctuations which enter the phonon self-energy via the charge susceptibility
- see below and in Part II. The neutron scattering in
$La_{1.85}Sr_{0.15}CuO_{4}$ gives evidence for large (30%) softening of the
O${}_{Z}^{Z}$ with $\Lambda_{1}$ symmetry with the energy $\omega\approx 60$
$meV$, which is theoretically predicted in Falter-O-phonon , and for the large
line-width about 17 meV which also suggest strong EPI.
As it is discussed in the Introduction, there are recently several
calculations of the EPI coupling constant $\lambda^{ep}$ in the framework of
DFT (or LDA), where very small $\lambda^{ep}\approx 0.3$ was obtained Bohnen-
Cohen , Giuistino . However, the LDA method is inadequate for strongly
correlated systems as it does not correctly take into account exchange-
correlations and many-body effects, and therefore overestimates the screening
in cuprates. If DFT is able to describe EPI correctly, it must also be able to
calculate phonon renormalization, such as softening and broadening of the
spectrum. In fact DFT completely fails to describe this renormalization for
some important phonon modes and therefore fails to describe the effect of EPI
on the electronic spectrum. The critique of LDA (DFT) results in HTSC cuprates
is done in Kulic-Review and recently strongly argued in Gunnarsson-
review-2008 , ReSaGuDe by its disagreement with neutron scattering
measurements as it is shown in Fig. 32.
Figure 32: Comparison of DFT calculations with experimental results:(a) in
$La_{1.85}Sr_{0.15}CuO_{4}$; (b) in $Y_{B}a_{2}Cu_{3}O_{7}$. (c) Phonon line
widths in $La_{1.85}Sr_{0.15}CuO_{4}$. DFT calculations Bohnen-Cohen gives
much smaller width than experiments Phonon-Exp . From ReSaGuDe .
The point is that DFT (LDA) can reproduce the phonon softening in
$La_{1.85}Sr_{0.15}CuO_{4}$ and $Y_{B}a_{2}Cu_{3}O_{7}$ rather good at low
momenta $\mathbf{q}=(h,0,0)$ but predicts smooth softening at higher
$\mathbf{q}$, while experiments show pronounced features for $h=0.3$. At the
same time DFT predicts an order of magnitude smaller line width than
experiments Phonon-Exp .
In Part II we shall discuss some theoretical approaches related to EPI
renormalization of phonons in strongly correlated systems. Here, we point out
two results. First, there is an appreciable difference in the phonon
renormalization in strongly and weakly correlated systems. Namely, the change
of phonon frequencies in the presence of conduction electrons is proportional
to the coupling constant $\left|g_{\mathbf{q}}\right|$ and charge
susceptibility $\chi_{c}$, i.e.
$\delta\omega(\mathbf{q})\sim\left|g_{\mathbf{q}}\right|^{2}\mathop{\mathrm{R}e}\chi_{c}$,
while the line width is given by
$\Gamma_{\omega(\mathbf{q})}\sim\left|g_{\mathbf{q}}\right|^{2}\left|\mathop{\mathrm{I}m}\chi_{c}\right|$.
It turns out that in strongly correlated systems doped with hole concentration
$\delta\ll 1$ the charge fluctuations are suppressed in which case the
following sum-rule holds Gunnarsson-review-2008 , Khaliullin
$\frac{1}{\pi N}\sum_{\mathbf{q}\neq
0}\int_{-\infty}^{\infty}\left|\mathop{\mathrm{I}m}\chi_{c}(\mathbf{q})\right|d\omega=2\delta(1-\delta)N,$
while in the LDA method one has
$\frac{1}{\pi N}\sum_{\mathbf{q}\neq
0}\int_{-\infty}^{\infty}\left|\mathop{\mathrm{I}m}\chi_{c}(\mathbf{q})\right|^{LDA}d\omega=(1-\delta)N.$
This means that for low doping $\delta\ll 1$ (note $n=1-\delta$), one has
$\left|\mathop{\mathrm{I}m}\chi_{c}\right|\ll\left|\mathop{\mathrm{I}m}\chi_{c}^{LDA}\right|$
and LDA strongly underestimates the coupling constant, i.e.
$\left|g_{\mathbf{q}}\right|_{LDA}\ll\left|g_{\mathbf{q}}\right|$. We stress
that there is no such strong suppression in the quasi-particle self-energy
Gunnarsson-review-2008 .
Second, the theory gives that the coupling constant
$\left|g_{\mathbf{q}}\right|$ in HTSC cuprates can be significantly larger
than LDA predicts, which is due to some many-body effects not present in the
latter Gunnarsson-review-2008 , Khaliullin . It can be shown that for some
phonon modes one has $\left|g_{\mathbf{q}}\right|^{2}\gg$
$\left|g_{\mathbf{q}}\right|_{LDA}^{2}$. For instance, for the half-breathing
mode, one has $\left|g_{\mathbf{q}}\right|^{2}\approx
3\left|g_{\mathbf{q}}\right|_{LDA}^{2}$ that is first calculated in Khaliullin
. These results point to inadequacy of LDA in calculations of EPI effects in
HTSC cuprates.
### III.6 Isotope effect for various doping
The isotope effect $\alpha_{T_{C}}$ in the critical temperature $T_{c}$ was
one of the very important proof for the EPI pairing in low-temperature
superconductors (LTSC). As a curiosity the isotope effect in $LTSC$ systems
was measured almost exclusively in monoatomic systems and in few polyatomic
systems: the hydrogen isotope effect in $PdH$, the $Mo$ and $Se$ isotope shift
of $T_{c}$ in $Mo_{6}Se_{8}$, and the isotope effect in $Nb_{3}Sn$ and
$MgB_{2}$. We point out that very small ($\alpha_{T_{C}}\approx 0$ in $Zr$ and
$Ru$) and even negative (in $PdH$) isotope effect in some polyatomic systems
of $LTSC$ materials are compatible with the $EPI$ pairing mechanism but in the
presence of substantial Coulomb interaction or lattice anharmonicity. The
isotope effect $\alpha_{T_{C}}$cannot be considered as the smoking gun since
it is sensitive to numerous influences. For instance, in $MgB_{2}$ it is with
certainty proved that the pairing is due to EPI and strongly dominated by the
boron vibrations, but the boron isotope effect is significantly reduced, i.e.
$\alpha_{T_{C}}\approx 0.3$. It is still unexplained. The situation in HTSC
cuprates is much more complicated because they contain many-atoms in unit
cell. Additionally, the situation is complicated with the presence of
intrinsic and extrinsic inhomogeneities which can mask real effects. On the
other hand new techniques such as ARPES, STM, $\mu SR$ allow studies of the
isotope effects in quasi-particle self-energies, i.e. $\alpha_{\Sigma}$, which
will be discussed below.
#### III.6.1 Isotope effect $\alpha_{T_{C}}$ in $T_{c}$
This problem will be discussed only briefly since more extensive discussion
can be found in Kulic-Review . It is well known that in the pure EPI pairing
mechanism, the total isotope coefficient $\alpha$ is given by
$\alpha_{T_{C}}=\sum_{i,p}\alpha_{i}^{(p)}=-\sum_{i,p}\frac{d\ln T_{c}}{d\ln
M_{i}^{(p)}},$ (41)
where $M_{i}^{(p)}$ is the mass of the i-th element in the p-th
crystallographic position. Note that in the case when the screened Coulomb
interaction is negligible, i.e. $\mu_{c}^{\ast}=0$, one has
$\alpha_{T_{C}}=1/2$. From this formula one can deduce that the relative
change of $T_{c}$, $\delta T_{c}/T_{c}$, for heavier elements is rather small
- for instance it is 0.02 for ${}^{135}Ba\rightarrow^{138}Ba$, 0.03 for
${}^{63}Cu\rightarrow^{65}Cu$ and 0.07 for ${}^{138}La\rightarrow^{139}La$.
This means that measurements of $\alpha_{i}$ for heavier elements are at/or
beyond the ability of present day experimental techniques. Therefore most
isotope effect measurements were done by substituting light atoms ${}^{16}O$
by ${}^{18}O$ only. It turns out that in most optimally doped HTSC cuprates
$\alpha_{O}$ is small. For instance $\alpha_{O}\approx 0.02-0.05$ in
$YBa_{2}Cu_{3}O_{7}$ with $T_{c,\max}\approx 91$ $K$, but it is appreciable in
$La_{1.85}Sr_{0.15}CuO_{4}$ with $T_{c,\max}\approx 35$ $K$ where$\
\alpha_{O}\approx 0.1-0.2$. In $Bi_{2}Sr_{2}CaCu_{2}O_{8}$ with
$T_{c,\max}\approx 76$ $K$ one has $\alpha_{O}\approx 0.03-0.05$ while
$\alpha_{O}\approx 0.03$ and even negative ($-0.013$) in
$Bi_{2}Sr_{2}Ca_{2}Cu_{2}O_{10}$ with $T_{c,\max}\approx 110$ $K$. The
experiments on $Tl_{2}Ca_{n-1}BaCu_{n}O_{2n+4}$ ($n=2,3$) with
$T_{c,\max}\approx 121$ $K$ are still unreliable and $\alpha_{O}$ is unknown:
In the electron-doped $(Nd_{1-x}Ce_{x})_{2}CuO_{4}$ with $T_{c,\max}\approx
24$ $K$ one has $\alpha_{O}<0.05$ while in the underdoped materials
$\alpha_{O}$ increases. The largest $\alpha_{O}$ is obtained even in the
optimally doped compounds like in systems with substitution, such as
$La_{1.85}Sr_{0.15}Cu_{1-x}M_{x}O_{4}$, $M=Fe,Co$, where $\alpha_{O}\approx
1.3$ for $x\approx 0.4$ $\%$. In $La_{2-x}M_{x}CuO_{4}$ there is a $Cu$
isotope effect which is of the order of the oxygen one, i.e.
$\alpha_{Cu}\approx\alpha_{O}$ giving $\alpha_{Cu}+\alpha_{O}\approx
0.25-0.35$ for optimally doped systems ($x=0.15$). In case when $x=0.125$ with
$T_{c}\ll T_{c,\max}$ one has$\ \alpha_{Cu}\approx 0.8-1$ with
$\alpha_{Cu}+\alpha_{O}\approx 1.8$ Franck . The appreciation of copper
isotope effect in $La_{2-x}M_{x}CuO_{4}$ tells us that vibrations other than
oxygen ions are important in giving high Tc. In that sense one should have in
mind the tunnelling experiments discussed above, which tell us that all
phonons contribute to the Eliashberg pairing function
$\alpha^{2}F(\mathbf{k},\omega)$ and according to these results the oxygen
modes give moderate contribution to $T_{c}$ Tsuda . Having these facts in
mind, then the small oxygen isotope effect $\alpha_{T_{c}}^{(O)}$ in optimally
doped cuprates, if it is intrinsic property, does not exclude the EPI
mechanism of pairing.
#### III.6.2 Isotope effect $\alpha_{\Sigma}$ in the self-energy
The fine structure of the quasi-particle self-energy
$\Sigma(\mathbf{k},\omega)$, such as kinks, can be resolved in ARPES
measurements and in some respect in STM. It turns out that there is an isotope
effect in the self-energy in the optimally doped $Bi2212$ samples Lanzara-
isotope , Douglas-isotop , Iwasawa-isotop . In the first paper on this subject
Lanzara-isotope , there is a red shift $\delta\omega_{k,70}\sim-(10-15)$ $meV$
of the nodal kink at $\omega_{k,70}\simeq 70$ $meV$ for the
${}^{16}O\rightarrow^{18}O$ substitution. This isotope shift of the self-
energy $\delta\Sigma=\Sigma_{16}-\Sigma_{18}\sim 10$ $meV$ is more pronounced
at large energies $\omega=100-300$ $meV$ . However, there is a dispute on the
latter result which is not confirmed experimentally Douglas-isotop , Iwasawa-
isotop . The isotope effect in $\mathop{\mathrm{R}e}\Sigma(\mathbf{k},\omega)$
Douglas-isotop , Iwasawa-isotop can be well described in the framework of the
Migdal-Eliashberg theory for EPI Ma-Ku-Do which is in accordance with the
recent ARPES measurements with low-energy photons $\sim 7$ $eV$ Iwasawa-
isotop-2 . The latter allowed very good precision in measuring the isotope
effect in the nodal point of Bi2212 with $T_{c}^{16}=92.1$ $K$ and
$T_{c}^{18}=91.1$ $K$ Iwasawa-isotop-2 . They observed shift in the maximum of
Re$\Sigma(\mathbf{k}_{N},\omega)$ \- at $\omega_{k,70}\approx 70$ $meV$ which
corresponds to the half-breathing or breathing phonon, by
$\delta\omega_{k,70}\approx 3.4\pm 0.5$ $meV$ as shown in Fig. 33.
Figure 33: (a) Effective $Re\Sigma$ for five samples for $O^{16}$ (blue) and
$O^{18}$ (red) along the nodal direction. (b) Effective $Im\Sigma$ determined
from MDC full widths. An impurity term is subtracted at $\omega=0$. From
Iwasawa-isotop-2 .
By analyzing the shift in Im$\Sigma(\mathbf{k}_{N},\omega)$ \- shown in Fig.
33, one finds similar result $\delta\omega_{k,70}\approx 3.2\pm 0.6$ $meV$.
The similar shift was obtained in STM measurements Davis which is shown in
Fig. 31(b) and can have its origin in different phonons. We would like to
stress two points: (i) in compounds with $T_{c}\sim 100$ $K$ the isotope
effect in $T_{c}$ is moderate, i.e. $\alpha_{T_{c}}^{(O)}\lesssim 0.1$,
Iwasawa-isotop-2 . If we consider this value to be intrinsic then it is not in
conflict with the tunnelling experiments in Tsuda which give evidence that
vibrations of heavier ions contribute significantly to $T_{c}$ \- see the
above discussion on the tunnelling spectroscopy; (ii) the extracted value in
Iwasawa-isotop-2 of the effective EPI coupling constant $\lambda_{eff}^{ep}$
$\sim 0.6$, which is smaller than the real $\lambda^{ep}$ \- see above the
discussion on ARPES, is significantly larger than the LDA theory predicts
$\lambda_{LDA}^{ep}<0.3$ Bohnen-Cohen , Giuistino . This again points that the
LDA method does not pick up the many-body effects due to strong correlations-
see Part II.
## IV Summary of Part I
The analysis of experimental data in HTSC cuprates which are related to
optics, tunnelling and ARPES measurements near the optimal doping give
evidence for the large electron-phonon interaction with the coupling constant
$1<\lambda^{ep}<3.5$. The analysis is done in the framework of the Migdal-
Eliashberg theory which is a reliable approach for systems near the optimal
doping. The spectral function averaged over the Fermi surface
$\alpha^{2}F(\omega)$ is extracted from various tunnelling measurements and it
contains peaks at the same positions as the phonon density of states. The
energy width of $\alpha^{2}F(\omega)$ is the same as the width of the phonon
density of states $F(\omega)$. This is an unambiguous proof for the important
role which EPI plays in the pairing mechanism of cuprates. The optical IR
reflectivity data provide additional support for this finding since the
transport spectral function has the width and global properties similar to
$F(\omega)$. These findings are additionally and strongly supported by ARPES
measurements on BISCO compounds. The ARPES kinks in the quasi-particle self-
energy can be explained exclusively by EPI and there is no much room for spin
fluctuations (SFI) effects. The weakness of SFI is unambiguously proved in
magnetic neutron scattering on YBCO where the imaginary part of the
susceptibility is drastically reduced in the low energy region by going from
slightly underdoped toward optimally doped systems, while $T_{c}$ is
practically unchanged. This implies that the SFI coupling constant is limited
to the value $\lambda^{sf}\lesssim 0.3$. All these results do not leave doubts
on the significance of EPI and weakness of SFI. The obtained total EPI
coupling constant is rather large, i.e. $1<\lambda^{ep}<3.5$, while the
transport coupling constant is $\lambda_{tr}\sim\lambda/3$. The different
renormalization of the quasi-particle and transport self-energies by the
Coulomb interaction and strong correlations points to dominance of the small-
momentum scattering in EPI. This will be discussed in Part II.
Inelastic neutron scattering measurements in cuprates show that the broadening
of phonon lines is by an order of magnitude larger than the LDA (DFA) method
predicts. Since the phonon line-widths depend on the EPI coupling and the
charge susceptibility it is evident that calculations of both quantities are
beyond the range of applicability of LDA. As a consequence, LDA overestimates
electronic screening and thus underestimates the EPI coupling. This means that
LDA is suitable only for weakly correlated systems, while many-body effects
due to strong correlations are not contained in this mean-field type theory.
In spite of the very promising and encouraging results about the dominance of
EPI in cuprates the theory is still confronted with the task of obtaining
sufficiently large coupling constant in the d-channel in order that EPI
conforms with d-wave pairing. At present we do not have such a detailed
microscopic theory although some concepts, such as the the dominant EPI
scattering at small transfer momenta, are understood at least qualitatively.
These set of problems and questions will be discussed in Part II.
Acknowledgement. We are thankful to Vitalii Lazarevich Ginzburg for his
permanent interest in our work and for support in many respects over many
years. One of us (M. L. K.) is thankful to Karl Bennemann for inspiring
discussions on many subjects related to physics of HTSC cuprates. We thank
Godfrey Akpojotor for careful reading of the manuscript. M. L. K. is thankful
to the Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie,
Berlin for the hospitality and financial support during his stay where part of
this work has been done.
## V Appendix - Spectral functions
### V.1 Spectral functions
$\alpha^{2}F(\mathbf{k},\mathbf{k}^{\prime},\omega)$ and $\alpha^{2}F(\omega)$
Before discussing experimental results in cuprate superconductors which are
related to various spectral functions, we introduce the reader briefly into
the subject. The quasiparticle-bosonic (Eliashberg) spectral function
$\alpha^{2}F(\mathbf{k},\mathbf{k}^{\prime},\omega)$ and its Fermi surface
average
$\alpha^{2}F(\omega)=\left\langle\left\langle\alpha^{2}F(\mathbf{k},\mathbf{k}^{\prime},\omega)\right\rangle\right\rangle_{\mathbf{k},\mathbf{k}^{\prime}}$
enter the quasi-particle self-energy $\Sigma(\mathbf{k},\omega)$, while the
transport spectral function $\alpha^{2}F_{tr}(\omega)$ enters the transport
self-energy $\Sigma_{tr}(\mathbf{k},\omega)$ and dynamical conductivity
$\sigma(\omega)$. Since the Migdal-Eliashberg theory for EPI is well defined
we define the spectral functions for this case and the generalization to other
electron-boson interaction is straightforward. In the superconducting state
the Matsubara Green’s function $\hat{G}(\mathbf{k},\omega_{n})$ and
$\hat{\Sigma}(\mathbf{k},\omega_{n})$ are $2\times 2$ matrices with the
diagonal elements $G_{11}\equiv
G(\mathbf{k},\omega_{n}),G_{22},\Sigma_{11}\equiv\Sigma(\mathbf{k},\omega_{n}),\Sigma_{22}$
and off-diagonal elements $G_{12}\equiv
F(\mathbf{k},\omega_{n}),G_{21},\Sigma_{12}\equiv\Phi(\mathbf{k},\omega_{n}),\Sigma_{21}$
which describe superconducting pairing. By defining
$i\omega_{n}\left[1-Z(\mathbf{k},\omega_{n})\right]=\left[\Sigma(\mathbf{k},\omega_{n})-\Sigma(\mathbf{k},-\omega_{n})\right]/2$
and
$\chi(\mathbf{k},\omega_{n})=\left[\Sigma(\mathbf{k},\omega_{n})+\Sigma(\mathbf{k},-\omega_{n})\right]/2$,
the Eliashberg functions for EPI in the presence of the Coulomb interaction
(in the pairing channel) read Allen-Mitrovic , Maksimov-Eliashberg ,
Marsiglio-Carbotte-Book
$Z(\mathbf{k},\omega_{n})=1+\frac{T}{N}\sum_{\mathbf{p},m}\frac{\lambda_{\mathbf{kp}}^{Z}(\omega_{n}-\omega_{m})\omega_{m}}{N(\mu)\omega_{n}}\frac{Z(\mathbf{p},\omega_{m})}{D(\mathbf{p},\omega_{m})},$
(42)
$\chi(\mathbf{k},\omega_{n})=-\frac{T}{N}\sum_{\mathbf{p},m}\frac{\lambda_{\mathbf{kp}}^{Z}(\omega_{n}-\omega_{m})}{N(\mu)}\frac{\epsilon(\mathbf{p})-\mu+\chi(\mathbf{p},\omega_{m})}{D(\mathbf{p},\omega_{m})},$
(43)
$\Phi(\mathbf{k},\omega_{n})=\frac{T}{N}\sum_{\mathbf{p},m}\left[\frac{\lambda_{\mathbf{kp}}^{\Delta}(\omega_{n}-\omega_{m})}{N(\mu)}-V_{\mathbf{kp}}\right]\frac{\Phi(\mathbf{p},\omega_{m})}{D(\mathbf{p},\omega_{m})},$
(44)
where $N(\mu)$ is the density of states at the Fermi surface, $\omega_{n}=\pi
T(2n+1)$, $\Phi(\mathbf{k},\omega_{n})\equiv
Z(\mathbf{k},\omega_{n})\Delta(\mathbf{k},\omega_{n})$ and
$D=\omega_{m}^{2}Z^{2}+\left(\epsilon-\mu+\chi\right)^{2}+\Phi^{2}$. (For
studies of optical properties - see below, it is useful to introduce the
renormalized frequency $i\tilde{\omega}_{n}(i\omega_{n})(\equiv
i\omega_{n}Z(\omega_{n}))=\omega_{n}-\Sigma(\omega_{n})$ (or its analytical
continuation $\tilde{\omega}(\omega)=Z(\omega)\omega=\omega-\Sigma(\omega)$).
These equations are supplemented with the electron number equation $n(\mu)$
($\mu$ is the chemical potential)
$n(\mu)=\frac{2T}{N}\sum_{\mathbf{p},m}G(\mathbf{p},\omega_{m})e^{i\omega_{m}0^{+}}$
$=1-\frac{2T}{N}\sum_{\mathbf{p},m}\frac{\epsilon(\mathbf{p})-\mu+\chi(\mathbf{p},\omega_{m})}{D(\mathbf{p},\omega_{m})}.$
(45)
Note that in the case of EPI one has
$\lambda_{\mathbf{kp}}^{\Delta}(\nu_{n})=\lambda_{\mathbf{kp}}^{Z}(\nu_{n})(\equiv\lambda_{\mathbf{kp}}(\nu_{n}))$
(with $\nu_{n}=\pi Tn$) where $\lambda_{\mathbf{kp}}(\nu_{n})$ is defined by
$\lambda_{\mathbf{kp}}(\nu_{n})=2\int_{0}^{\infty}\frac{\nu\alpha_{\mathbf{kp}}^{2}F(\nu)d\nu}{\nu^{2}+\nu_{n}^{2}}$
(46)
$\alpha_{\mathbf{kp}}^{2}F(\nu)=N(\mu)\sum_{\kappa}\left|g_{\kappa,\mathbf{kp}}^{ren}\right|^{2}B_{\kappa}(\mathbf{k}-\mathbf{p,}\nu)$
(47)
where $B_{\kappa}(\mathbf{k}-\mathbf{p;}\nu)$ is the phonon spectral function
of the $\kappa$-th phonon mode related to the phonon propagator
$D_{\kappa}(\mathbf{q,}i\nu_{n})=-\int_{0}^{\infty}\frac{\nu}{\nu^{2}+\nu_{n}^{2}}B_{\kappa}(\mathbf{q,}\nu)d\nu.$
(48)
The renormalized coupling constant $g_{\kappa,\mathbf{kp}}^{ren}(\approx
g_{\kappa,\mathbf{kp}}^{0}\varepsilon^{-1}\gamma)$ comprises the screening
effect due to long-range Coulomb interaction ($\sim\varepsilon^{-1}$ \- the
inverse electronic dielectric function) and short-range strong correlations
($\sim\gamma$ \- the vertex function) - see more in Part II. Usually in the
case of low-temperature superconductors (LTS) with s-wave pairing the
anisotropy is rather small (or in the presence of impurities it is averaged
out) which allows an averaging of the Eliashberg equations Allen-Mitrovic ,
Maksimov-Eliashberg , Marsiglio-Carbotte-Book
$Z(\omega_{n})=1+\frac{\pi
T}{\omega_{n}}\sum_{m}\frac{\lambda(\omega_{n}-\omega_{m})\omega_{m}}{\sqrt{\omega_{m}^{2}+\Delta^{2}(\omega_{m})}},$
(49) $Z(\omega_{n})\Delta(\omega_{n})=\pi
T\sum_{m}[\lambda(\omega_{n}-\omega_{m})$
$-\mu(\omega_{c})\theta(\omega_{c}-\left|\omega_{m}\right|)]\frac{\Delta(\omega_{m})}{\sqrt{\omega_{m}^{2}+\Delta^{2}(\omega_{m})}},$
(50)
$\lambda(\omega_{n}-\omega_{m})=\int_{0}^{\infty}d\nu\frac{2\nu\alpha^{2}F(\nu)}{\nu^{2}+(\omega_{n}-\omega_{m})^{2}}.$
(51)
Here
$\alpha^{2}F(\omega)=\left\langle\left\langle\alpha^{2}F(\mathbf{k},\mathbf{k}^{\prime},\omega)\right\rangle\right\rangle_{\mathbf{k},\mathbf{k}^{\prime}}$where
$\left\langle\left\langle...\right\rangle\right\rangle_{\mathbf{k},\mathbf{k}^{\prime}}$
is the average over the Fermi surface. The above equations can be written on
the real axis by the analytical continuation
$i\omega_{m}\rightarrow\omega+i\delta$ where the gap function is complex i.e.
$\Delta(\omega)=\Delta_{R}(\omega)+i\Delta_{I}(\omega)$. The solution for
$\Delta(\omega)$ allows the calculation of the current-voltage characteristic
$I(V)$ and tunnelling conductance $G_{NS}(V)=dI_{NS}/dV$ in the
superconducting state of the $NIS$ tunneling junction where $I_{NS}(V)$ is
given by
$I_{NS}(V)=2e\sum_{\mathbf{k},\mathbf{p}}\mid
T_{\mathbf{k},\mathbf{p}}\mid^{2}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}$
$A_{N}(\mathbf{k},\omega)A_{S}(\mathbf{p},\omega+eV)[f(\omega)-f(\omega+eV)].$
(52)
Here, $A_{N,S}(\mathbf{k},\omega)=-2ImG_{N,S}(\mathbf{k},\omega)$ are the
spectral functions of the normal metal and superconductor, respectively and
$f(\omega)$ is the Fermi distribution function. Since the angular and energy
dependence of the tunnelling matrix elements $\mid
T_{\mathbf{k},\mathbf{p}}\mid^{2}$ is practically unimportant for $s-wave$
superconductors, then in that case the relative conductance
$\sigma_{NS}(V)\equiv G_{NS}(V)/G_{NN}(V)$ is proportional to the tunnelling
density of states $N_{T}(\omega)=\int
A_{S}(\mathbf{k},\omega)d^{3}k/(2\pi)^{3}$, i.e. $\sigma_{NS}(\omega)\approx
N_{T}(\omega)$ where
$N_{T}(\omega)=Re\left\\{\frac{\omega+i\tilde{\gamma}(\omega)}{\sqrt{(\omega+i\tilde{\gamma}(\omega))^{2}-\tilde{Z}^{2}(\omega)\Delta(\omega)^{2}}}\right\\}.$
(53)
Here, $\tilde{Z}(\omega)=Z(\omega)/ReZ(\omega)$,
$\tilde{\gamma}(\omega)=\gamma(\omega)/ReZ(\omega)$,
$Z(\omega)=ReZ(\omega)+i\gamma(\omega)/\omega$ and the quasi-particle
scattering rate in the superconducting state
$\gamma_{s}(\omega,T)=-2Im\Sigma(\omega,T)$ is given by
$\gamma_{s}(\omega,T)=2\pi\int\limits_{0}^{\infty}d\nu\alpha^{2}F(\nu)N_{s}(\nu+\omega)\\{2n_{B}(\nu)$
$+n_{F}(\nu+\omega)+n_{F}(\nu-\omega)\\}+\gamma^{imp},$ (54)
where
$N_{s}(\omega)=\mathop{\mathrm{R}e}\\{\omega/(\omega^{2}-\Delta^{2})^{1/2}$ is
the quasi-particle density of states in the superconducting state,
$n_{B,F}(\nu)$ are Bose and Fermi distribution function respectively. Since
the structure of phononic spectrum is contained in $\alpha^{2}F(\omega)$, it
is reflected on $\Delta(\omega)$ for $\omega>\Delta_{0}$ (the real gap
obtained from $\Delta_{0}=Re\Delta(\omega=\Delta_{0})$) which gives the
structure in $G_{S}(V)$ at $V=\Delta_{0}+\omega_{ph}$. On the contrary one can
extract the spectral function $\alpha^{2}F(\omega)$ from $G_{NS}(V)$ by the
inversion procedure proposed by McMillan and Rowell McMillan-Rowell , Kulic-
Review . It turns out that in low-temperature superconductors, negative peaks
of $d^{2}I/dV^{2}$ at $eV_{i}=\Delta+\omega_{ph,i}$, correspond to the peak
positions of $\alpha^{2}F(\omega)$ and $F(\omega)$. However, we would like to
point out that in HTSC cuprates the gap function is unconventional and very
anisotropic, i.e. $\Delta(\mathbf{k},i\omega_{n})\sim\cos k_{x}a-\cos k_{y}a$.
Since in this case the extraction of
$\alpha^{2}F(\mathbf{k},\mathbf{k}^{\prime},\omega)$ is extremely difficult
and at present rather unrealistic task, then an ”average”
$\alpha^{2}F(\omega)$ is extracted from the experimental curve $G_{S}(V)$.
There is belief that it gives relevant information on the real spectral
function such as the energy width of the bosonic spectrum
($0<\omega<\omega_{\max}$) and positions and distributions of peaks due to
bosons. It turns out that even such an approximate procedure gives valuable
information in HTSC cuprates - see discussion in Section III D.
Note that in the case of spin-fluctuation interaction (the SFI model) one
should make difference between $\lambda_{\mathbf{kp}}^{Z}(i\nu_{n})$ and
$\lambda_{\mathbf{kp}}^{\Delta}(i\nu_{n})$ since they differ by sign i.e.
$\lambda_{\mathbf{kp}}^{Z}(i\nu_{n})=-\lambda_{\mathbf{kp}}^{\Delta}(i\nu_{n})>0$
since SFI is repulsive in the pairing-channel - see Eqs. (7-8).
#### V.1.1 Inversion of tunnelling data
Phonon features in the conductance $\sigma_{NS}(V)$ at
$eV=\Delta_{0}+\omega_{ph}$ makes the tunnelling spectroscopy a powerful
method in obtaining the Eliashberg spectral function $\alpha^{2}F(\omega)$.
Two methods were used in the past for extracting $\alpha^{2}F(\omega)$.
The first method is based on solving the inverse problem of the nonlinear
Eliashberg equations. Namely, by measuring $\sigma_{NS}(V)$, one obtains the
tunnelling density of states $N_{T}(\omega)(\sim\sigma_{NS}(\omega))$ and by
the inversion procedure one gets $\alpha^{2}F(\omega)$ McMillan-Rowell . In
reality the method is based on the iteration procedure - the McMillan-Rowell
($MR$) inversion, where in the first step an initial
$\alpha^{2}F_{ini}(\omega)$, $\mu_{ini}^{\ast}$ and $\Delta_{ini}(\omega)$ are
inserted into Eliashberg equations (for instance
$\Delta_{ini}(\omega)=\Delta_{0}$ for $\omega<\omega_{0}$ and
$\Delta_{ini}(\omega)=0$ for $\omega>\omega_{0}$) and then $\sigma_{ini}(V)$
is calculated. In the second step the functional derivative
$\delta\sigma(\omega)/\delta\alpha^{2}F(\omega)$ ($\omega\equiv eV$) is found
in the presence of a small change of $\alpha^{2}F_{ini}(\omega)$ and then the
iterated solution
$\alpha^{2}F_{\mathbf{(1)}}(\omega)=\alpha^{2}F_{ini}(\omega)+$
$\delta\alpha^{2}F(\omega)$ is obtained, where the correction
$\delta\alpha^{2}F(\omega)$ is given by
$\delta\alpha^{2}F(\omega)=\int
d\nu[\frac{\delta\sigma_{ini}(V)}{\delta\alpha^{2}F(\nu)}]^{-1}[\sigma_{exp}(\nu)-\sigma_{ini}(\nu)].$
(55)
The procedure is iterated until $\alpha^{2}F_{(n)}(\omega)$ and
$\mu_{(n)}^{\ast}$ converge to $\alpha^{2}F(\omega)$ and $\mu^{\ast}$which
reproduce the experimentally obtained conductance $\sigma_{NS}^{\exp}(V)$. In
such a way the obtained $\alpha^{2}F(\omega)$ for $Pb$ resembles the phonon
density of states $F_{Pb}(\omega)$, that is obtained from neutron scattering
measurements. Note that the method depends explicitly on $\mu^{\ast}$ but on
the contary it requires only data on $\sigma_{NS}(V)$ up to the voltage
$V_{\max}=\omega_{ph}^{\max}+\Delta_{0}$ where $\omega_{ph}^{\max}$ is the
maximum phonon energy ($\alpha^{2}F(\omega)=0$ for
$\omega>\omega_{ph}^{\max}$) and $\Delta_{0}$ is the zero-temperature
superconducting gap. One pragmatical feature for the interpretation of
tunnelling spectra (and for obtaining the spectral pairing function
$\alpha^{2}F(\omega)$) in $LTS$ and $HTSC$ cuprates is that the negative peaks
of $d^{2}I/dV^{2}$ are at the peak positions of $\alpha^{2}F(\omega)$ and
$F(\omega)$. This feature will be discussed later on in relation with
experimental situation in cuprates.
The second method has been invented in GDS-method and it is based on the
combination of the Eliashberg equations and dispersion relations for the
Greens functions - we call it GDS method. First, the tunnelling density of
states is extracted from the tunneling conductance in a more rigorous way
Ivanshenko
$N_{T}(V)=\frac{\sigma_{NS}(V)}{\sigma_{NN}(V)}-\frac{1}{\sigma^{\ast}(V)}\int_{0}^{V}du$
$\times\frac{d\sigma^{\ast}(u)}{du}\left[N_{T}(V-u)-N_{T}(V)\right]$ (56)
where $\sigma^{\ast}(V)=\exp\\{-\beta V\\}\sigma_{NN}(V)$ and the constant
$\beta$ is obtained from $\sigma_{NN}(V)$ at large biases - see GDS-method .
$N_{T}(V)$ under the integral can be replaced by the BCS density of states.
Since the second method is used in extracting $\alpha^{2}F(\omega)$ in a
number of LTSC as well as in HTSC cuprates - see below, we describe it briefly
for the case of isotropic EPI at T=0 K. In that case the Eliashberg equations
are Allen-Mitrovic , Maksimov-Eliashberg , Marsiglio-Carbotte-Book , GDS-
method
$Z(\omega)\Delta(\omega)=\int_{\Delta_{0}}^{\infty}d\omega^{\prime}\mathop{\mathrm{R}e}\left[\frac{\Delta(\omega^{\prime})}{\left[\omega^{\prime
2}-\Delta^{2}(\omega^{\prime})\right]^{1/2}}\right]$
$\times\left[K_{+}(\omega,\omega^{\prime})\right]-\mu^{\ast}\theta(\omega_{c}-\omega)$
(57)
$Z(\omega)=\frac{1}{\omega}\int_{\Delta_{0}}^{\infty}d\omega^{\prime}\mathop{\mathrm{R}e}\left[\frac{\omega^{\prime}}{\left[\omega^{\prime
2}-\Delta^{2}(\omega^{\prime})\right]^{1/2}}\right]K_{-}(\omega,\omega^{\prime})$
(58)
where
$K_{\pm}(\omega,\omega^{\prime})=\int_{\Delta_{0}}^{\omega_{ph}^{\max}}d\nu\alpha^{2}F(\nu)(\frac{1}{\omega^{\prime}+\omega+\nu+i0^{+}}$
$\pm\frac{1}{\omega^{\prime}-\omega+\nu-i0^{+}}).$ (59)
Here $\mu^{\ast}$ is the Coulomb pseudopotential, the cutoff $\omega_{c}$ is
approximately $(5-10)$ $\omega_{ph}^{\max}$, $\Delta_{0}=\Delta(\Delta_{0})$
is the energy gap. Now by using the dispersion relation for the matrix Greens
functions $\hat{G}(\mathbf{k},\omega_{n})$ one obtains GDS-method
$\mathop{\mathrm{I}m}S(\omega)=\frac{2\omega}{\pi}\int_{\Delta_{0}}^{\infty}d\omega^{\prime}\frac{N_{T}(\omega^{\prime})-N_{BCS}(\omega^{\prime})}{\omega^{2}-\omega^{\prime
2}}$ (60)
where $S(\omega)=\omega/\left[\omega^{2}-\Delta^{2}(\omega)\right]^{1/2}$.
From Eqs. (57-58) one obtains
$\int_{0}^{\omega-\Delta_{0}}d\nu\alpha^{2}F(\omega-\nu)\mathop{\mathrm{R}e}\left\\{\Delta(\nu)\left[\nu^{2}-\Delta^{2}(\nu)\right]^{1/2}\right\\}$
$=\frac{\mathop{\mathrm{R}e}\Delta(\omega)}{\omega}\int_{0}^{\omega-\Delta_{0}}d\nu\alpha^{2}F(\nu)N_{T}(\omega-\nu)+\frac{\mathop{\mathrm{I}m}\Delta(\omega)}{\pi}$
$+\frac{\mathop{\mathrm{I}m}\Delta(\omega)}{\pi}\int_{0}^{\infty}d\omega^{\prime}N_{T}(\omega^{\prime})\int_{0}^{\omega_{ph}^{\max}}d\nu\frac{2\alpha^{2}F(\nu)}{(\omega^{\prime}+\nu)^{2}-\omega^{2}}.$
(61)
Based on Eqs. (56-61) one obtains the scheme for extracting
$\alpha^{2}F(\omega)$
$\sigma_{NS}(V),\sigma_{NN}(V)\rightarrow N_{T}(V)$
$\rightarrow\mathop{\mathrm{I}m}S(\omega)\rightarrow\Delta(\omega)\rightarrow\alpha^{2}F(\omega).$
The advantage in this method is that the explicit knowledge of $\mu^{\ast}$ is
not required GDS-method . However, the integral equation for
$\alpha^{2}F(\omega)$ is linear Fredholm equation of the first kind which is
ill-defined - see the duscussion in Section II.B.2.
#### V.1.2 Phonon effects in $N_{T}(\omega)$
We briefly discuss the physical origin for the phonon effects in
$N_{T}(\omega)$ by considering a model with only one peak, at $\omega_{0}$, in
the phonon density of states $F(\omega)$ by assuming for simplicity
$\mu^{\ast}=0$ and neglecting the weak structure in $N_{T}(\omega)$ at
$n\omega_{0}+\Delta_{0}$, which is due to the nonlinear structure of the
Eliashberg equations SSW .
Figure 34: (a) Model phonon density of states $F(\omega)$ with the peak at
$\omega_{0}$. (b) The real (solid) $\Delta_{R}$ and imaginary part
$\Delta_{I}$ (dashed) of the gap $\Delta(\omega)$. (c) The normalized
tunnelling density of states $N_{T}(\omega)/N(0)$ (solid) compared with the
BCS density of states (dashed). From SSW .
In Fig.34 it is seen that the real gap $\Delta_{R}(\omega)$ reaches a maximum
at $\omega_{0}+\Delta_{0}$ then decreases, becomes negative and zero, while
$\Delta_{I}(\omega)$ is peaked slightly beyond $\omega_{0}+\Delta_{0}$ that is
the consequence of the effective electron-electron interaction via phonons.
It follows that for $\omega<\omega_{0}+\Delta_{0}$ most phonons have higher
energies than the energy $\omega$ of electronic charge fluctuations and there
is over-screening of this charge by ions giving rise to attraction. For
$\omega\approx\omega_{0}+\Delta_{0}$ charge fluctuations are in resonance with
ion vibrations giving rise to the peak in $\Delta_{R}(\omega)$. For
$\omega_{0}+\Delta_{0}<\omega$ the ions move out of phase with respect to
charge fluctuations giving rise to repulsion and negative
$\Delta_{R}(\omega)$. This is shown in Fig. 34(b). The structure in
$\Delta(\omega)$ is reflected on $N_{T}(\omega)$ as shown in Fig. 34(c) which
can be reconstructed from the approximate formula for $N_{T}(\omega)$ expanded
in powers of $\Delta/\omega$
$\frac{N_{T}(\omega)}{N(0)}\approx
1+\frac{1}{2}\left[\left(\frac{\Delta_{R}(\omega)}{\omega}\right)^{2}-\left(\frac{\Delta_{I}(\omega)}{\omega}\right)^{2}\right].$
As $\Delta_{R}(\omega)$ increases above $\Delta_{0}$ this gives
$N_{T}(\omega)>N_{BCS}(\omega)$, while for
$\omega\gtrsim\omega_{0}+\Delta_{0}$ the real value $\Delta_{R}(\omega)$
decreases while $\Delta_{I}(\omega)$ rises and $N_{T}(\omega)$ decreases
giving rise for $N_{T}(\omega)<N_{BCS}(\omega)$.
### V.2 Transport spectral function $\alpha_{tr}^{2}F(\omega)$
The spectral function $\alpha_{tr}^{2}F(\omega)$ enters the dynamical
conductivity $\sigma_{ij}(\omega)$ ($i,j=a,b,c$ axis in $HTS$ systems) which
generally speaking is a tensor quantity given by the following formula
$\sigma_{ij}(\omega)=-\frac{e^{2}}{\omega}\int\frac{d^{4}q}{(2\pi)^{4}}\gamma_{i}(q,k+q)$
$\times G(k+q)\Gamma_{j}(q,k+q)G(q),$ (62)
where $q=(\mathbf{q},\nu)$ and $k=(\mathbf{k}=0,\omega)$ and the bare current
vertex $\gamma_{i}(q,k+q;\mathbf{k}=0)$ is related to the Fermi velocity
$v_{F,i}$, i.e. $\gamma_{i}(q,k+q;\mathbf{k}=0)=v_{F,i}.$ The vertex function
$\Gamma_{j}(q,k+q)$ takes into account the renormalization due to all
scattering processes responsible for finite conductivity SchrieffLTS . In the
following we study only the in-plane conductivity at $\mathbf{k}=0$. The
latter case is realized due to the long penetration depth in HTSC cuprates and
the skin depth in the normal state are very large. In the $EPI$ theory,
$\Gamma_{j}(q,k+q)\equiv\Gamma_{j}(\mathbf{q},i\omega_{n},i\omega_{n}+i\omega_{m})$
is a solution of an approximative integral equation written in the symbolic
form KMS
$\Gamma_{j}=v_{j}+V_{eff}GG\Gamma_{j}$ (63)
where the effective potential $V_{eff}$ (due to EPI) is given by
$V_{eff}=\sum_{\kappa}\mid g_{\kappa}^{ren}\mid^{2}D_{\kappa}$, where
$D_{\kappa}$ is the phonon Green’s function. In such a case the Kubo theory
predicts $\sigma_{ii}^{intra}(\omega)$ ($i=x,y,z$)
$\sigma_{ii}(\omega)=\frac{\omega_{p,ii}^{2}}{4i\pi\omega}\\{\int_{-\omega}^{0}d\nu
th(\frac{\omega+\nu}{2T})S^{-1}(\omega,\nu)$
$+\int_{0}^{\infty}d\nu[th(\frac{\omega+\nu}{2T})-th(\frac{\nu}{2T})]S^{-1}(\omega,\nu)\\},$
(64)
where
$S(\omega,\nu)=\omega+\Sigma_{tr}^{\ast}(\omega+\nu)-\Sigma_{tr}(\nu)+i\gamma_{tr}^{imp}$,
and $\gamma_{tr}^{imp}$ is the impurity contribution. In the following we omit
the tensor index $ii$ in $\sigma_{ii}(\omega)$. In the presence of several
bosonic scattering processes the transport self-energy
$\Sigma_{tr}(\omega)=Re\Sigma_{tr}(\omega)+iIm\Sigma_{tr}(\omega)$ is given by
$\Sigma_{tr}(\omega)=-\sum_{l}\int_{0}^{\infty}d\nu\alpha_{tr,l}^{2}F_{l}(\nu)[K_{1}(\omega,\nu)+iK_{2}(\omega,\nu)],$
(65) $K_{1}(\omega,\nu)=Re[\Psi(\frac{1}{2}+i\frac{\omega+\nu}{2\pi
T})-\Psi(\frac{1}{2}+i\frac{\omega-\nu}{2\pi T})],$ (66)
$K_{2}(\omega,\nu=\frac{\pi}{2}[2cth(\frac{\nu}{2T})-th(\frac{\omega+\nu}{2T})+th(\frac{\omega-\nu}{2T})]$
(67)
Here $\alpha_{tr,l}^{2}F_{l}(\nu)$ is the transport spectral function which
measures the strength of the $l$-th (bosonic) scattering process and $\Psi$ is
the digamma function. The index $l$ enumerates EPI, charge and spin-
fluctuation scattering processes. Like in the case of EPI, the transport
bosonic spectral function $\alpha_{tr,l}^{2}F(\Omega)$ is given explicitly by
$\alpha_{tr,l}^{2}F(\omega)=\frac{1}{N^{2}(\mu)}\int\frac{dS_{\mathbf{k}}}{v_{F,\mathbf{k}}}\int\frac{dS_{\mathbf{k}^{\prime}}}{v_{F,\mathbf{k}^{\prime}}}\times$
$\left[1-\frac{v_{F,\mathbf{k}}^{i}v_{F,\mathbf{k}}^{i}}{(v_{F,\mathbf{k}}^{i})^{2}}\right]\alpha_{\mathbf{kk}^{\prime},l}^{2}F(\omega).$
(68)
We stress that in the phenomenological SFI theory Pines one assumes
$\alpha_{\mathbf{kk}^{\prime}}^{2}F(\omega)\approx
N(\mu)g_{sf}^{2}$Im$\chi(\mathbf{k}-\mathbf{p},\omega)$, which, as we have
repeated several times, can be justified only for small $g_{sf}$, i.e.
$g_{sf}\ll W_{b}$ (the band width).
In case of weak coupling ($\lambda<1$), $\sigma(\omega)$ can be written in the
generalized (extended) Drude form as discussed in Section III.B.
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|
arxiv-papers
| 2008-10-21T10:25:59 |
2024-09-04T02:48:58.387341
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. G. Maksimov, M. L. Kulic, O. V. Dolgov",
"submitter": "Evgenii Maksimov",
"url": "https://arxiv.org/abs/0810.3789"
}
|
0810.3971
|
# A QSO plus host system lensed into a 6” Einstein ring by a low redshift
galaxy
Kajal K Ghosh11affiliationmark: USRA/NSSTC/MSFC/NASA, 320 Sparkman Drive,
Huntsville, AL 35805, USA kajal.k.ghosh@nasa.gov D.
Narasimha22affiliationmark: Tata Institute of Fundamental Research, Mumbai
400 005, India dna@tifr.res.in
###### Abstract
We report the serendipitous discovery of an “Einstein Ring” in the optical
band from the Sloan Digital Sky Survey (SDSS) data and associated four images
of a background source. The lens galaxy appears to be a nearby dwarf spheroid
at a redshift of 0.0375$\pm$0.002. The lensed quasar is at a redshift of
0.6842$\pm$0.0014 and its multiple images are distributed almost 360o around
the lens nearly along a ring of radius $\sim$6.”0. Single component lens
models require a mass of the galaxy of almost 1012 M⊙ within 6”.0 from the
lens center. With the available data we are unable to determine the exact
positions, orientations and fluxes of the quasar and the galaxy, though there
appears evidence for a double or multiple merging image of the quasar. We have
also detected strong radio and X-ray emissions from this system. It is
indicative that this ring system may be embedded in a group or cluster of
galaxies. This unique ring, by virtue of the closeness of the lens galaxy,
offers possible probe to some of the key issues like mass-to-light ratio of
intrinsically faint galaxies, existence of large scale magnetic fields in
elliptical galaxies etc.
Gravitational lensing: quasar - galaxies : ellipticals – Cosmology:
observations, cosmological parameters, dark matter - SDSS J091949.16+342304.0
## 1 Introduction
Galaxy formation has remained an open issue in astronomy and cosmology.
Several multiwavelength deep surveys have been carried out to measure the
evolution of the mass-to-light ratio and to constrain the process(es) of
galaxy formation and evolution, but they have their inherent limitations.
Gravitational lensing offers an unbiased technique to directly measure the
mass of galaxies as well as constraints on the mass distribution (Kochanek &
Narayan 1992; Warren et al. 1996; Kochanek et al. 1999; Kochanek, Keeton &
McLoed, 2001; Myers et al. 2003; Warren & Dye 2003; Bolton et al. 2004;
Cabanac et al. 2007; Gavazzi et al. 2008 and references therin) Tight
constraints on these measurements can be made by observing nearby lens systems
having multiple images as well as perfect Einstein rings because these systems
will not suffer from well-known ellipticity-shear degeneracy (cf. Keeton,
Kochanek, Seljak, 1997). In addition, a detailed observation of the galaxies
in the vicinity as well as the shape of the luminous mass in the main galaxy
is feasible for very nearby galaxies. Consequently, a strong constraint on the
ratio of the mass–to–light ratio of the lens galaxy is, in principle, possible
from such a system. For the formation of a perfect (360o) “Einstein Ring”, the
lens and the source have to be aligned such that an extended structure in the
source straddles over at least three of the vertexes of the tangential lens
caustic. In that process the lens and an inner image of the source will appear
almost as a single object close to the center of the ring. To date, not many
Einstein rings are known that have large circumferences (Belokurov et al. 2007
and references therein), though recently they have discovered a $\sim$300o
Einstein ring system. Here we report the discovery of an almost perfect
optical “Einstein Ring” (SDSS J091949.16+342304.0), a quasar at a redshift of
0.684 lensed by a galaxy at a very low redshift of 0.0375 forming a ring of
nearly 6” radius and associated four images. Observations and data analysis
are described in §2. Computations of models and interpretations of results are
presented in §3. §4 presents significance of this powerful lens system and
discussion and conclusions are presented in §5.
## 2 Observations, data analysis and results
Fig. 1 show the SDSS composite image in the field of SDSS J091949.16+342304.0.
Close-up view of the unique Einstein ring system of SDSS J091949.16+342304.0
is shown in Fig. 2. It can be seen from this figure that there appears to be
two objects at the center. This is further supported from the results of the
image-projection profile of the central region. Nature of these two objects
has been determined from their spectra, which are shown in Fig. 3. The SDSS
spectrum of the quasar at the center (SDSS J091949.16+342304.0) shows that it
is at a redshift of 0.6842$\pm$0.0014. This spectrum was de-reddened using the
Galactic extinction curve, (Schlegel et al. 1998), then the wavelength scale
was transformed from the observed to the source frame. This is shown in black
and the SDSS composite quasar spectrum is shown in magenta color (vanden Berk
et al. 2001). We adjusted the relative extinction between these two spectra to
match their red-wings of Mg II (2800 Å) line and the fluxes at 5100 Å. It can
be seen from these two spectra that the absorption features between 5100-5300
Å region is present only in SDSS J091949.16+342304.0 and not in the composite
spectrum. In order to check the validity of these lines, the FeII emission
lines were subtracted from the observed spectrum of SDSS J091949.16+342304.0.
We obtained the Fe II model spectrum in the optical (3530 – 7570 Å) band from
Anabela C. Goncalves. First, we determined the required broadening by
comparing the full width at half intensity maximum (FWHM) of the iron lines in
the observed and the template spectra. Then by varying the scaling factor we
created a large number of template spectra, which were subtracted from the
observed spectrum to determine the residual spectra. Standard deviations at
the continuum around 4600 Å region of the residual spectra were computed.
Finally, the residual spectrum with the least value of the standard deviation
was subtracted from the observed spectrum (Veron-Cetty, Veron & Goncalves
2001; Vestergaard & Wilkes 2001). The spectrum of SDSS J091949.16+342304.0
without iron emission lines clearly displays the presence of Ca II lines with
high significances. These lines are clearly absent in the SDSS composite
quasar spectrum, which can be seen from Fig. 3.
Next, we subtracted the SDSS composite quasar spectrum from the observed
spectrum of SDSS J091949.16+342304.0 and the residual spectrum, at the
observer’s frame, is shown at the upper plot of Fig. 4. Clearly, two
absorption lines around 9000 Å (8863 and 8988 Å) and one around 5040 Å are
present. Other prominent absorption and emission fetures present in this
spectrum are the artifacts. All these three absorption lines (H$\beta$ and Ca
II) are consistent with the redshift of 0.0375$\pm$0.002. Thus, we identified
these absorption features as redshifted (0.0375$\pm$0.002) H$\beta$ and Ca II
absorption lines (8498, 8542 and 8662 Å), which are marked on this figure. We
could not identify the first Ca II triplet line (8498 Å), which will be
redshifted at 8817 Å as this line is located at the red-wing of the redshifted
(0.6842$\pm$0.0014) Mg triplet lines (Mg1, Mg2 and Mgb around 5175Å) of the
host galaxy of the quasar. In addition, we could not identify the H$\alpha$
line from the foreground galaxy, because it is located on the blue-side of the
quasar’s H$\delta$ emission line. To identify the nature of the foreground
galaxy we compared the residual spectrum with the SDSS spectra of different
types of galaxies, which are at redshifts between 0.037 and 0.038 and are
fainter than 18.2 mag in the SDSS r-band. The limit of 18.2 mag comes from the
SDSS photometric results of the central region of Fig. 2. In this figure we
have marked two objects with a circle and an ellipse, whose measured
brightnesses are 19.6$\pm$0.2 and 18.4$\pm$0.1 mag in the SDSS r-band,
respectively. Finally, from the results of the comparison of spectra we find
that the residual spectrum is similar to that of the dwarf elliptical (dE)
galaxy, which is shown in the lower plot of Fig. 4. In addition, the flux and
the photometric magnitude of the elliptical object of Fig. 2 is consistent
with that of the dE galaxies at similar redshift. Furthermore, we found from
the SDSS database that there are at least a few quasars at redshifts between
0.68 and 0.69 (SDSS J154127.26+405720.2, SDSS J155900.8+062412.0, etc.) whose
radio and optical spectral properties are similar to that of SDSS
J091949.16+342304.0. Photometric magnitudes of these quasars are between 19.5
and 20.0 mag. In addition, there are many quasars at redshifts between 0.68
and 0.69, whose optical spectral properties are similar to that of SDSS
J091949.16+342304.0 and are fainter than 21 mag. All these photometric and
spectroscopic results indicate that the central region of Figs. 1 and 2
contains a nearby dE galaxy and a quasar. It is important to mention here that
these results should be taken cautiously until future high spatial resolution,
peferably with the Hubble Space Telescope, images confirm the positions,
orientations and brightnesses of these objects.
In Fig. 2 we have marked two objects with “A” and “B”. It appears that this
object “A” could be composed of multiple images. We obtained its optical
spectrum on March 06, 2007 using the Low Resolution Spectrograph at the
Nasmyth B focus of the Telescopio Nazionale Galileo with 2000 s exposure (TNG
is a 3.58 m optical/infrared telescope located in the Island of San Miguel de
La Palma). This observation was carried out in the Long Slit Spectroscopy mode
(LR-R Grism #3) with a camera, which is equipped with a 2048 x 2048 Loral
thinned and back-illuminated CCD. This spectrum is shown in Fig. 5 in red
color (middle plot), without corrections for the atmospheric absorptions,
which are marked with “T” and their presence have affected the strengths of
some emission lines. For comparison, the SDSS spectrum of the quasar is shown
in black. Five redshifted emission lines of [OII 3727 Å], [NeIII 3869 Å],
H$\gamma$ (4340 Å), H$\beta$ (4860 Å) and [OIII 5007 Å] that are common
between these two spectra, have been marked with the vertical dashed lines.
While the spectrum of object “A” was obtained, the position of the slit of the
spectrograph was aligned in such a way that the object “B” was on the slit. We
extracted the spectrum of object “B” and the highly smoothed spectrum is also
shown in Fig. 5 (bottom plot in green color). Four redshifted emission lines
of [OII 3727 Å], [NeIII 3869 Å], [OIII 4959Å) and [OIII 5007 Å] are labeled in
this figure and their redshifted wavelengths are same as those of object “A”
and the SDSS quasar. These results indicate that the central quasar and
objects “A” and “B” are at the same redshift (0.6842$\pm$0.0014). However,
these results have to be confirmed with high signal-to-noise ratio spectra.
We also obtained the near-infrared spectra of the quasar and the object “A” on
March 07, 2007, using the Near Infrared Camera Spectrometer at the TNG with a
HgCdTe Hawaii 1024x1024 array detector for 500 s exposure. These spectra are
shown in Fig. 6 with the upper one being the quasar spectrum (black) and the
lower one is for the object “A” (magenta). A broad emission feature around
1.105 $\mu$ is present in both the spectra and we identify these features as
redshifted H$\alpha$ emission line with z=0.684. Again, these results suggest
that the quasar and object “A” are at the same redshift.
We searched the 2MASS database for the counterparts of the central objects of
the ring system. A central extended-blob was detected only in the J-band with
few more objects within a circle of 30” radius. Positions of these objects
coincide with the bright optical counterparts present in the SDSS images. The
central objects were also detected in the VLA/FIRST and NVSS surveys ( 1.4
GHz) with peak-flux densities at 2.15$\pm$0.13 and 2.3$\pm$0.45 mJy,
respectively. We make the reasonable assumption that the radio emission comes
mainly from the quasar and negligible contribution from the dwarf spheroid
lens galaxy or other nearby galaxies. Then, using the VLA/FIRST flux density,
the SDSS i–band magnitude and eqn. (3) of Shen et al. (2006), we find that
this quasar is a radio-loud object with radio-loudness index $\sim$1.3. Fig. 7
shows the ROSAT/PSPC image of the ring system, which has been adaptively
smoothed. It can be clearly seen from this figure that bright X-ray emission
is present in and around the ring system. This is indicative of X-ray emission
from the quasar, its images and from the surroundings, which may contain X-ray
emitting group of galaxies or similar objects. This image was located on the
outer part of the ROSAT/PSPC detector. The PSF of the PSPC detector varies too
much across the field of view and in the outer parts it shows strong non-
symmetric features. This did not permit us to deconvolve this image in a
meaningful way (private communication with Frank Haberl). Future, high spatial
resolution X-ray observations will reveal the details of this ring system. In
summary, all the observed results suggest that we have detected a system,
which consists of a foreground dE galaxy at a redshift of 0.0375$\pm$0.002
and, at least, three quasars (central quasar and objects “A” and “B”) at the
redshifts of 0.6842$\pm$0.0014.
## 3 Physical triple quasars or lens$?$
Many physical binary quasars have been detected (Djorgovski et al. 1987;
Meylan et al. 1990; Hennawi et al. 2006; Myers et al. 2006). However,
presently, it is not clear whether these are physical binary quasars or
gravitational lens systems (Kochanek et al. 1999; Mortlock et al. 1999). To
date, no physical triple quasars have been unambiguously detected (Djorgovski
et al. 2007; Sochting et al. 2008). All the known triple quasars associated
with a galaxy or a group of galaxies or cluster of galaxies are gravitational
lens systems (Kochanek & Narayan 1992; Warren et al. 1996; Kochanek et al.
1999; Kochanek, Keeton & McLoed, 2001; Myers et al. 2003; Warren & Dye 2003;
Bolton et al. 2004; Cabanac et al. 2007; Gavazzi et al. 2008 and references
therin) Thus, it is strongly indicative that SDSS J091949.16+342304.0 system
(Figs. 1 and 2) is a gravitational lens system. In this system we do not see
the source quasar, which is most likely fainter than 21 mag. At the center we
see a lensing galaxy (dE galaxy) and an image of this quasar. Objects “A” and
“B” are the other two images of the quasar. Image “A” is most likely composed
of two or more images.
Figure 1: SDSS composite image around the Einstein ring system, SDSS
J091949.16+342304.0. North is up and east to the left. The size of the image
around 82”.0$\times$82”.0. Figure 2: Same as Fig. 1, but for small size
(25”.0$\times$25”.0). The ring is clearly visible. A small circle and the
ellipse show the positions of the source and the lensing galaxy. “A” and “B”
are the two images of the source. It appears that the image “A” may consists
of multiple images. Figure 3: The Galactic extinction corrected rest-frame
observed spectrum of SDSS J091949.16+342304.0, which is at the redshift of
0.6842$\pm$0.0014 (black). For comparison, SDSS composite quasar spectrum, at
a redshift of 0.5, is shown in magenta color. From the comparison of the two
spectra, it can be seen that the absorption lines are absent in the SDSS
composite quasar spectrum. The vertical dashed lines are drawn at the position
of three absorption lines, which correspond to H$\beta$ and two Ca II lines
(8542 and 8662 Å) of the foreground galaxy, which is at a redshift of
0.0375$\pm$0.002. Figure 4: The upper plot shows the residual spectrum
between SDSS J091949.16+342304.0 and the SDSS composite quasar spectrum. The
lower plot shows the SDSS composite spectrum of dwarf elliptical galaxy at a
redshift of 0.037, which we generated using the spectra from the SDSS
database. H$\beta$ and two Ca II lines (8542 and 8662 Å) are marked with
vertical dashed lines. The residual spectrum has been shifted up for clarity.
Figure 5: Observed spectra of objects “A” and “B”, in red and green colors,
respectively. at the bottom. The atmospheric telluric bands are marked
with“T”. For comparison, the spectrum of the quasar, SDSS J091949.16+342304.0,
is shown in black at the top. Four emission lines, which are at the same
positions between the three spectra are marked with vretical dashed lines. In
addition, the H$\gamma$ line is marked between object “A” and the quasar. This
shows that the quasar and objects “A” and “B” are at the same redshift. Figure
6: Observed near infrared spectra of the quasar, SDSS J091949.16+342304.0
(black), and its image, “A” (magenta). The broad emission lines around 11000 Å
are the redshifted H$\alpha$ line. Figure 7: Adaptively smoothed ROSAT/PSPC
image at the position of SDSS J091949.16+342304.0. It can be seen that the
emission is extended at the arcminute scale.
## 4 Models
The system consists of a quasar and its host at redshift of 0.6842$\pm$0.0014
lensed by a dwarf spheroid galaxy at a redshift of 0.038 forming mutiple
images of the quasar and an almost perfect 360o ring of radius $\sim$6.”0. We
can identify atleast three distinct images of the quasar, though their
positions are not robust. Due to poor seeing, we cannot resolve the images
fully, but the configuration is similar to the well-studied systems B1422+231
and Q1938+666 (cf. Narasimha & Patnaik, 1993). We believe that the brightest
image, “A”, is probably a double, which should be resolvable under improved
signal and a seeing of better than 1 arcsec. The position of an image at
arcsecond separation from the centre of the lens galaxy is still very
uncertain and its flux cannot be determined accurately with the present set of
data since the galaxy at a redshift of 0.038 is much brighter than any of the
images.
The lens is at a very low redshift and hence its surface mass density is not
much greater than the critical value for multiple imaging. Consequently, the
central region of the lens galaxy turns out to be important in determining the
characteristics of the inner image. The profile of the Einstein ring of radius
6”.0 is determined by the large scale shear produced by the main galaxy or any
galaxy groups. Consequently, a good map of the image configuration could
render this system a valuable probe of the dynamical mass distribution in
elliptical galaxies and at larger scales.
### 4.1 Model1:Model independent minimal limts for lens mass:
A single component spherical mass at a redshift of 0.0375 (distance of the
order of 160 Mpc) acting as gravitational lens for a background source of
redshift 0.68 and producing three images at approximately 6 arcsecond from the
lens centre should have a minimum mass given by
$M_{min}={{D_{eff}c^{2}\theta_{E}^{2}}\over{4G}}$
where $D_{eff}$ = ${{D_{S}D_{L}}\over D_{LS}}$, the combination of distances
to the source DS, to the lens DL and the distance from the source to the lens
DLS is essentially the distance to the lens in this case. c=velocity of light,
G=Gravitational constant and $\theta_{E}$ is the Einstein radius, which is
about 5 arcseconds. Consequently, the minimum mass in the absence of large
scale shear due to the outer regions of the galaxy or additional galaxies is
of the order of 9$\times$ 1011 M⊙. However, the ellipticity of the lens can
reduce the minimum surface mass required for multiple image formation
(Subramanian & Cowling, 1986). At present we cannot determine the ellipticity
of the lens or importance of external shear, without an observational
information about the orientationalong which the double images in A merge or
their substructures. Consequently, we have constructed a model for a
spheroidal lens with ellipticity to match the approximate position of the
images “A” and “B” and the apparent direction along which two subimages appear
to be merging in image “A” (Narasimha, Subramanian & Chitre, 1982). In Figure
8, a single lens produces the essentials of the model, though due to
limitations of data the model is just indicative. For instance, the high
eccentricity of 0.9 used to get the curvature of the ring near the images will
change with better quality data on the images and then, the mass of the lens
could increase. In this model, the central cusp like mass profile is simulated
through a 200 pc bulge of mass 1010 solar mass and a truncated King profile
has core radius of 1.2 kpc, eccentricity of 0.9 and mass of 6$\times$ 1011
M$\odot$ as expected from the simplistic considerations. For the observed
luminosity of the order of 109 L$\odot$, the Mass to Light ratio is
substantial. Such a model cannot be completely ruled out, specially in view of
the massive low surface brightness galaxies observed (Impey et al, 1988;
Bothun et al, 1997), but from the observed optical flux distribution, the lens
galaxy is unlikely to be of that type.
Figure 8: Image configuration due to the lensing action of a spheroidal mass
distribution. The solid lines are the critical curves in the source plane
delineating regions of image multiplicities while the long dashed lines are
the corresponding caustics in the image plane. The tangnetial magnification of
images tend to infinity for a pair of images when the source lies on the
diamond shaped critical curve and they will appear to be merging. The shape
and position of this critical line is dependent on the large scale shear of
the lens. The outer oval shaped long dashed curve is its counterpart in the
image plane. The radial magnification tends to infinity for image pairs when
the source lies on the oval shaped solid line and the radially merging images
lie on the inner dashed caustic. The dotted oval shaped curve is the
spheroidal lens. The source (a small circle just inside the diamond shaped
critical curve) has an extended arc like image consisting of two merging
images, additional external image as well as an inner image apprx. 1 arcsecond
from the lens centre. A fifth very dim image is not shown.
However, a configuration with three images forming almost an extended arc and
another image close to the centre of the main lensing galaxy can be a natural
result when shear due to many galaxies en route to a background source
qualitatively modify the lensing action due to a foreground galaxy. Formation
of multiple images and arc–like features due to shear dominated gravitational
lens action is discussed by Narasimha (1993) and Narasimha & Chitre (1993),
who studied the effects of the central mass and the large scale shear when the
lens can barely form multiple images. The essential features of such a
configuration can be studied by taking the gravitational lens action of a
spherical mass along with a constant shear. This will reduce the mass of the
main lensing galaxy, and the ellipticity of the model is a natural consequence
of the shear due to surrounding mass distribution. But we cannot get any idea
of the shear upto when we isolate the central faint image and resolve the
double image A. However, we should add that such configurations are likely to
be detectable in surveys extending to fainter levels because at arcsecond
scales the gravitational pull due to multiple galaxies at cosmological
distances en route to a distant source become important.
In our system, there is indeed evidence for the existence of at least three
groups of galaxies along three directions around the observed foreground
galaxy. We believe that their combined action is likely to be responsible for
the observed multiple images. When two lenses of equal strength are located at
two vertices of a triangle and additional lenses are distributed along the
third direction, the midpoint of the first two lenses and surrounding regions
have nearly constant shear due to the comined action of the lenses. This
scenario is not as unlikely as it might appear: The central region of a weak
Group of galaxies could produce this configuration or meeting points of
filaments in large scale structures have similar morphology. Even by pure
conincidence, sometimes large number of galaxies can simulate this effect
since the bending angle drops very slowly as the inverse of distance while the
number of lenses that can act at a point, which is proportional to the area of
the region, increases as the square of the distance to the line of sight to
the background source. If indeed this is the scanario, it will affect both
inferences of cosmological matter power density based on multiply–imaged large
separation lenses as well as estimation of masses of the individual lenses.
## 5 Significance of SDSS J091949.16+342304.0 as a diagnostic tool to probe
mass inhomogeneities at small and large scales:
We have only preliminary data for this powerful lens system and consequently,
the model given in the previous section should be treated as illustrative. But
the shear dominated lens system and its diagnostic power should not be
overlooked, while analysing images from deep surveys. We could speculate some
of the results expected from a detailed multifrequency study of this system.
### 5.1 Large scale mass distribution in the main lens:
The object appearing as the main lens is a dwarf spheroid of R magnitude 18 at
a distace of apprx. 160 Mpc and hence abs R mag of -18.1. It would be very
difficult to have reliable direct observation and analysis of such an object
in a lens system if it were, say, at a redshift of 0.3 because the lens will
be typically 5 mag fainter and the images of the background source will
dominate at almost al wavelengths. However, for the present system we have the
lens five times brighter than the background images in optical bands and
extends over a few arcseconds. At a distance of 160 Mpc, 1 arcsecond
corresponds to about 800 pc and hence, with deeper optical and infrared
images, we can obtain high dynamical range images of the galaxy to determine
the scale length at which the flux from the lens decreases. A good model of
the lens system can independetly give an idea of how the gravitational mass
drops off, if we have good radio images of the system showing the details of
the curvature of the Einstein Ring at subarcsecond scales or we can map the
images at radio to determine the shape of the merging images.
At this distance, galaxy clusters of even moderate masses can be studied in
X-ray and the temperature of the intracluster gas can be estimated. If we have
good X–ray map showing the massive objects in the field and their temperature,
we can determine the mass distribution at 100s of kpc corresponding to
arcminute scale or rule out the possibility of any cluster of galaxies
associated with the main lens. If indeed we are able to detect any warm X–ray
corona at a few tens of arcseconds which can satisfactorily explain the
observed image configuration, it might open up the possibility of dwarf–like
galaxies being centres of massive dark elliptical halos.
### 5.2 Small scale mass distribution:
At present we have limited information about the inneer image, though we can
determine its separation from the lens centre of 0.85 arcsecond. However,
further details should await better quality data. If we have accurate position
of this image and magnification with respect to image “B”, it could provide
some constraints on the cusp like mass distribution at the centre of the dwarf
galaxy as well as possible presence of a bulge component at a kiloparsec
scale.
Ideally, we expect image “A” to be a double and from an inspection of the
available images we feel they are separated by 0.”5-0.”7. If radio, optical
and X-ray observations confirm this hypothesis and their relative flux ratios
are consistent, we should expect a smooth mass distribution at scales of 1 kpc
in the lens. But a million solar mass globular cluster at 160 Mpc has an
Einstein radius of 6 milliarcsec. Consequently, any possible mass
inhomogeneity of this scale, along with the main lens can introduce
microimages separated by a few tens of milliarcsecond and their signature can
be seen in the high-reoslution radio image. Certainly, a high sensitivity
radio image of the Einstein Ring should show the signatures of such mass
inhomogeneity, mainly due to the proximity of the lens. We feel that this
system, due to its proximity, radio loudness and the Einstein Ring of large
radius is an ideal candidate to probe inhomogeneities at tens of millions of
solar mass at 100 parsec scale.
### 5.3 Role of other massive galaxies in the field:
It is not clear if many other galaxies within an arcminute are part a Group or
Cluster of galaxies or other normal field galaxies at various redshifts which
happen to be near the line of sight to the Einstein Ring. The radius of the
Einstein Ring is not small, and we do expect a few tens of galaxies in the
field within 103 arcsec2 area. Consequently, statistics does not help us
differentiate between the possibilities in the absence of redshift or X-ray
flux measurements. But, even if the galaxies are chance coincidence, their
role in the formation of the Einstein Ring is important. The bending angle
drops off as 1/b where b is the impact parameter of the photon path with
respect to any such galaxy. The number of lenses in the plane of the sky will
increase as the square of the impact parameter, if the lenses are
homogeneously distributed. Consequently, if there is an extra concentration of
mass along some direction even at tens of arcsecond away, it can have
noticeable efect on the image formation, which might dominate over the
gravitational effects of the main dwarf galaxy lens. This might not have been
the case for a typical giant elliptical galaxy at a redshift of 0.3 acting as
the lens, producing an arcsecond scale ring mainly due to the central mass of
the galaxy and shear due to its large scale mass. For the present lens
configuration, even if there is no extra concentration along a specific
direction, we can still have a fairly wide region where an almost constant
shear due to many of these galaxies act. This appears to be likely because
there are about a dozen or more galaxies at ten to 30 arcsecond away. We do
not have redshift or other details of these gaalxies, but from an inspection
of the position and brightness of these galaxies, the Einstein Ring appears to
be almost at the centroid of a triangle formed by these galaxies. If indeed
this is a chance coincidence and we notice it only due to the proximity of the
main lens, the possibility of many of the multiple image sysetms reported in
the literature being artefact of the specific large scale galaxy distribution
along their line of sight should be taken into account while constructing
their model as well as estimating the amplitude of matter power fluctuation
based on the statistic of image sepration in strong lens systems.
## 6 Discussion and conclusions
We have discovered a gravitational lens system consisting of possible four
images of a quasar and an almost perfect Einstein Ring of radius nearly 6”.
The quasar has a redshift of 0.684 and the main lens appears to be a 18.1
magnitude galaxy at a redshift of 0.0375. Since the lens is at a very low
redshift and the source is radio-loud and X–ray luminous, the system provides
a powerful tool to study mass distribution within lens galaxy, specially in
the central regions. A detailed observation and analysis of the system could
provide many direct tests or cinfirmations in lensing as well as strcuture of
galaxies: e.g. the ring morphology as well as the imaging of the field of
galaxies around could give an indication of the importance of Groups of
Galaxies at even very low redshifts as powerful lenses, possible chance
conincidences producing many of the eye–catching lens configurations, mass to
light ratio of lens galaxy as a function of radial distance, possible
existence of large scale magetic fields in the elliptical lens galaxy are some
of the important problems that can be addressed with this system simply
because of its proximity, scale and being loud in radio, optical as well as
X–ray.
Though we have only preliminary data for this system, it could have some
important implications to cosmolgy:
1. 1.
If a single galaxy is the main lens, it should have a mass to light ratio of
upwards of 500 M⊙/L⊙. This will possibly be a new result for a dwarf spheroid
of similar luminosity.
2. 2.
If the lens consists of a Group of galaxies, the possibility of some of the
weak groups of galaxies having high surface mass density to produce multiple
images and extended arcs even at a very low redshift of 0.0375 should be
considered while estimating the matter power density from a surveylike SDSS,
even though those groups may not be conspicous in optical luminosity. In this
context, the lensing due to filaments along favourable directions should be
taken into account while estimating cosmological parameters, for instance,
from cosmic shear data.
3. 3.
If the lens shear is due to chance location of many galaxies along the line of
sight to a distant background source, this fact should be taken into account
while using gravitational lens systems of large angular separation to estimate
the mass of very massive objects. This will have far reaching implications
when gravitational lens is used to calibrate, for instance, masses of
galaxy–clusters and hence, the $\sigma_{8}$ parameter for the amplitude of
matter power is derived.
Our sincere thanks to the referee for valuable comments and suggestions that
helped to improve the paper. We thank Carlos M Gutierrez de la Cruz and Martin
Lopez-Corredoira who obtained the optical and near-infrared spectra of the
quasar and its images, presented in this paper, during their observations. In
this paper, we have extensively used data from the Sloan Digital Sky Survey
(SDSS). Funding for the SDSS has been provided by the Alfred P. Sloan
Foundation, the Participating Institutions, the National Aeronautics and Space
Administration, the National Science Foundation, the U. S. Department of
Energy, the Japanese Monbuka- gakusho, and the Max Planck Society. The SDSS
website is http://www.sdss.org/. This research has made use of the NASA/IPAC
Extragalactic Database (NED) which is operated by the Jet Propulsion
Laboratory, California Institute of Technology, under contract with NASA; of
data products from the Two Micron All Sky Survey, VLA/FIRST, NVSS and
ROSAT/PSPC.
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|
arxiv-papers
| 2008-10-22T05:34:35 |
2024-09-04T02:48:58.403607
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kajal K Ghosh, D. Narasimha",
"submitter": "Delampady Narasimha",
"url": "https://arxiv.org/abs/0810.3971"
}
|
0810.4326
|
# TOPICS IN MITIGATING RADAR BIAS
Demetrios Serakos1, John E. Gray2 and Hazim Youssef1
Naval Surface Warfare Center
1Warfare Systems Department
2Electromagnetic and Sensor Systems Department
Dahlgren, VA 22448
###### Abstract
In this paper, we investigate two topics related to mitigating the effect of
radar bias in ballistic missile tracking applications. We determine the
absolute bias between two radars in polar coordinates when their relative bias
is given in rectangular coordinates. Using this result, we then obtain the
optimized steady-state filter to handle the random bias.
## 1
Introduction
There are several facets to the problem of tracking ballistic missiles with
radar that require enhanced error correction to effectively track threats. In
this paper, we obtain the exact form of the bias error for the coordinate
transformation problem. This result is useful in Ballistic Missile Defense
bistatic applications where one sensor is used for launching an interceptor,
while another is used to track the threat. Thus, the problem of translation
between internal sensor coordinate frames to a common frame (that is, used by
all sensors) is important. The coordinate transformation problem from
Cartesian to spherical coordinates introduces a bias that, if accounted for,
can be corrected in the design of a filter. This problem occurs when one has
multiple launch platforms, because each local track must be formatted for a
common reference frame. When bias correction is accomplished correctly, one
can improve tracking performance of the filter and increase the likelihood
that an interceptor can successfully engage a threat.
## 2 An Optimized Method of Obtaining Absolute Bias
Although relative bias calculation can be used to provide correct association
of tracks from two sensors, the calculation of the absolute bias is required
to correct the track state and is needed for track fusion and for producing a
Single Integrated Air Picture. Methods for obtaining the relative bias between
two radars tracking the same ballistic missile are presented in Levedahl [2]
and Brown, Weisman and Brock [3]. The methods presented in these reports have
to do with maximizing a likelihood function. The relative biases obtained in
these papers are determined in rectangular coordinates. In this paper, the
absolute bias for the two sensors is calculated from the relative bias by
solving a minimization problem. The problem is set up to minimize the weighted
sum of the two absolute biases while viewing the given relative bias as a
constraint.
A point in $3$-dimensional space in both rectangular and spherical
coordinates111Denote yaw (azimuth) by $\psi$, pitch (elevation) by $\theta$.
$\phi$ is normally reserved for roll; however, roll is not used here. is
denoted by:
$\overrightarrow{p}=\left[\begin{array}[]{c}x\\\ y\\\
z\end{array}\right];\text{
and}\qquad\overrightarrow{\pi}=\left[\begin{array}[]{c}r\\\ \psi\\\
\theta\end{array}\right]\text{, respectively.}$ (1)
The transformations between the coordinates are
$\overrightarrow{p}=f(\overrightarrow{\pi})$ and
$\overrightarrow{\pi}=f^{-1}(\overrightarrow{p})$, which are given by:
$f(\overrightarrow{\pi})=\left[\begin{array}[]{c}r\cos\theta\cos\psi\\\
r\cos\theta\sin\psi\\\ r\sin\theta\end{array}\right];\qquad\qquad
f^{-1}(\overrightarrow{p})=\left[\begin{array}[]{c}\sqrt{x^{2}+y^{2}+z^{2}}\\\
\arctan\left(y/x\right)\\\
\arctan\left(z/\sqrt{x^{2}+y^{2}}\right)\end{array}\right]\text{ .}$ (2)
We need the following definitions:
$P_{1}$ | Target position as seen by sensor 1
---|---
$P_{2}$ | Target position as seen by sensor 2
$P_{T}$ | True target position (unknown)
$B_{1}$ | Sensor 1 bias
$B_{2}$ | Sensor 2 bias
$B_{R}$ | Relative bias
$P_{1TO2}$ | Sensor 2 position from sensor 1
$P_{1,ENU(1)}=(x_{1},y_{1},z_{1})_{ENU(1)}^{\prime}$,
$P_{2,ENU(2)}=(x_{2},y_{2},z_{2})_{ENU(2)}^{\prime}$. $B_{1,ENU(1)}=(\Delta
x_{1},\Delta y_{1},\Delta z_{1})_{ENU(1)}^{\prime}$, $B_{2,ENU(2)}=(\Delta
x_{2},\Delta y_{2},\Delta z_{2})_{ENU(2)}^{\prime}$. (ENU denotes the East
North Up coordinate system.) Thus, we have in the sensor coordinates
$P_{T,ENU(1)}=P_{1,ENU(1)}+B_{1,ENU(1)}$ (3)
$P_{T,ENU(2)}=P_{2,ENU(2)}+B_{2,ENU(2)}\text{ .}$ (4)
If we use an ENU coordinate system located at sensor 1, (4) becomes
$P_{T,ENU(1)}=P_{1TO2,ENU(1)}+P_{2,ENU(1)}+B_{2,ENU(1)}$ (5)
where $P_{1TO2,ENU(1)}$ is the position vector from the first sensor to the
second sensor in $ENU(1)$. The relative bias in $ENU(1)$ is
$B_{R,ENU(1)}=B_{2,ENU(1)}-B_{1,ENU(1)}$
$=(P_{T,ENU(1)}-P_{1TO2,ENU(1)}-P_{2,ENU(1)})-(P_{T,ENU(1)}-P_{1,ENU(1)})$
$=P_{1,ENU(1)}-P_{1TO2,ENU(1)}-P_{2,ENU(1)}\text{ .}$ (6)
We consider the coordinate transformations to allow us to go from $ENU$ to
radar-face coordinates for a particular sensor. Each sensor has its own face
and ENU coordinate systems. The face coordinate system (denoted FACE) of a
sensor is related to the ENU coordinate system of a sensor by the following
transformation:
$T_{ENU(i)2FACE(i)}=\
\left[\begin{array}[]{ccc}\cos\theta_{i}&0&\sin\theta_{i}\\\ 0&1&0\\\
-\sin\theta_{i}&0&\cos\theta_{i}\end{array}\right]\left[\begin{array}[]{ccc}\cos\psi_{i}&\sin\psi_{i}&0\\\
-\sin\psi_{i}&\cos\psi_{i}&0\\\ 0&0&1\end{array}\right]$
$=\left[\begin{array}[]{ccc}\cos\theta_{i}\cos\psi_{i}&\cos\theta_{i}\sin\psi_{i}&\sin\theta_{i}\\\
-\sin\psi_{i}&\cos\psi_{i}&0\\\
-\sin\theta_{i}\cos\psi_{i}&-\sin\theta_{i}\sin\psi_{i}&\cos\theta_{i}\end{array}\right]\allowbreak$
(7)
where $i=1,2$. We also have that
$T_{FACE(i)2ENU(i)}=\left[\begin{array}[]{ccc}\cos\theta_{i}\cos\psi_{i}&-\sin\psi_{i}&-\sin\theta_{i}\cos\psi_{i}\\\
\cos\theta_{i}\sin\psi_{i}&\cos\psi_{i}&-\sin\theta_{i}\sin\psi_{i}\\\
\sin\theta_{i}&0&\cos\theta_{i}\end{array}\right]\text{ ,}\allowbreak$ (8)
which is the transpose of (7). We can also have the matrix
$T_{ENU(i)2FACE(j)}$, which is
$T_{ENU(i)2FACE(j)}=\left[\begin{array}[]{ccc}\cos\theta_{i,j}\cos\psi_{i,j}&\cos\theta_{i,j}\sin\psi_{i,j}&\sin\theta_{i,j}\\\
-\sin\psi_{i,j}&\cos\psi_{i,j}&0\\\
-\sin\theta_{i,j}\cos\psi_{i,j}&-\sin\theta_{i,j}\sin\psi_{i,j}&\cos\theta_{i,j}\end{array}\right]\text{
.}$ (9)
The absolute (as opposed to relative) bias can be expressed in the face
coordinates:
$B_{i,FACE(i)}=\Delta r\cdot\overrightarrow{u_{r}}+\Delta
c_{A}\cdot\overrightarrow{u_{cA}}+\Delta c_{B}\cdot\overrightarrow{u_{cB}}$
(10)
where $\overrightarrow{u_{r}}$ is the unit vector in the range coordinate and
$\overrightarrow{u_{cA}},$ $\overrightarrow{u_{cB}}$ are the two cross range
coordinate unit vectors. Substituting $p_{T}\Delta\psi=\Delta c_{A}$ and
$p_{T}\Delta\theta=\Delta c_{B}$ where $p_{Ti}=\left\|P_{T}(i)\right\|$ (see
note222True position is not available. When applying this method measured
position is used for this calculation instead.), the distance from sensor to
the target, we get
$B_{i,FACE(i)}=\Delta
r_{i}\cdot\overrightarrow{u_{r}}+p_{Ti}\Delta\psi_{i}\cdot\overrightarrow{u_{cA}}+p_{Ti}\Delta\theta_{i}\cdot\overrightarrow{u_{cB}}$
(11)
$B_{i,ENU(i)}=\left[\begin{array}[]{ccc}\cos\theta_{i}\cos\psi_{i}&-\sin\psi_{i}&-\sin\theta_{i}\cos\psi_{i}\\\
\cos\theta_{i}\sin\psi_{i}&\cos\psi_{i}&-\sin\theta_{i}\sin\psi_{i}\\\
\sin\theta_{i}&0&\cos\theta_{i}\end{array}\right]\left[\begin{array}[]{c}\Delta
r_{i}\\\ p_{Ti}\Delta\psi_{i}\\\ p_{Ti}\Delta\theta_{i}\end{array}\right]$
$=\left[\begin{array}[]{c}\Delta
r_{i}\cos\theta_{i}\cos\psi_{i}-\Delta\psi_{i}\left(\sin\psi_{i}\right)\cdot
p_{Ti}-\Delta\theta_{i}\left(\sin\theta_{i}\cos\psi_{i}\right)\cdot p_{Ti}\\\
\Delta
r_{i}\cos\theta_{i}\sin\psi_{i}+\Delta\psi_{i}\left(\cos\psi_{i}\right)\cdot
p_{Ti}-\Delta\theta_{i}\left(\sin\theta_{i}\sin\psi_{i}\right)\cdot p_{Ti}\\\
\Delta r_{i}\sin\theta_{i}+\Delta\theta_{i}\left(\cos\theta_{i}\right)\cdot
p_{Ti}\end{array}\right]\text{ .}$ (12)
We can obviously obtain $B_{i,ENU(j)}$ (for $i$ not necessarily equal to $j$)
if needed. The quantities $\Delta r_{1},$ $\Delta\psi_{1},$
$\Delta\theta_{1},$ $\Delta r_{2},$ $\Delta\psi_{2},$ $\Delta\theta_{2}$ are
the ones we minimize. To refer to these as a group we on occasion write
$e=\left(\Delta r_{1},\Delta\psi_{1},\Delta\theta_{1},\Delta
r_{2},\Delta\psi_{2},\Delta\theta_{2}\right)$.
We need tolerances or costs for the sensor biases. These are expressed in
spherical coordinates.
$k_{r1}$ | Sensor 1 range bias cost, unitless
---|---
$k_{r2}$ | Sensor 2 range bias cost, unitless
$k_{\psi 1}$ | Sensor 1 azimuth bias cost, meters
$k_{\psi 2}$ | Sensor 2 azimuth bias cost, meters
$k_{\theta 1}$ | Sensor 1 elevation bias cost, meters
$k_{\theta 2}$ | Sensor 2 elevation bias cost, meters
### 2.1 Problem Statement
We want to compute the minimum (absolute) bias cost for the two sensors when
there are known (computed) expressions for the relative bias. The given
relative bias is expressed in ENU rectangular coordinates. We compute the
minimum absolute bias in spherical coordinates. The relative bias in
rectangular coordinates contrasted with the absolute bias in spherical
coordinates allows us to formulate this as a minimization problem. We view the
relative bias as a constraint. We use a quadratic cost:
$F=\frac{k_{r_{1}}^{2}}{2}\cdot\left(\Delta
r_{1}\right)^{2}+\frac{k_{\psi_{1}}^{2}}{2}\cdot\left(\Delta\phi_{1}\right)^{2}+\frac{k_{\theta_{1}}^{2}}{2}\cdot\left(\Delta\theta_{1}\right)^{2}+\frac{k_{r_{2}}^{2}}{2}\cdot\left(\Delta
r_{2}\right)^{2}+\frac{k_{\psi_{2}}^{2}}{2}\cdot\left(\Delta\phi_{2}\right)^{2}+\frac{k_{\theta_{2}}^{2}}{2}\cdot\left(\Delta\theta_{2}\right)^{2}\text{
.}$ (13)
So that the addition in (13) is permissible, we have that $k_{r_{1}}$,
$k_{r_{2}}$ are unitless and $k_{\psi_{1}}$, $k_{\theta_{1}}$, $k_{\psi_{2}}$,
$k_{\theta_{2}}$ are in meters. We note $F$ may be rewritten in the form
$F=\left[\begin{array}[]{ccc}\Delta
r_{1}&\Delta\psi_{1}&\Delta\theta_{1}\end{array}\right]\left[\begin{array}[]{ccc}2/k_{r_{1}}^{2}&0&0\\\
0&2/k_{\psi_{1}}^{2}&0\\\
0&0&2/k_{\theta_{1}}^{2}\end{array}\right]^{-1}\left[\begin{array}[]{c}\Delta
r_{1}\\\ \Delta\psi_{1}\\\ \Delta\theta_{1}\end{array}\right]$
$+\left[\begin{array}[]{ccc}\Delta
r_{2}&\Delta\psi_{2}&\Delta\theta_{2}\end{array}\right]\left[\begin{array}[]{ccc}2/k_{r_{2}}^{2}&0&0\\\
0&2/k_{\psi_{2}}^{2}&0\\\
0&0&2/k_{\theta_{2}}^{2}\end{array}\right]^{-1}\left[\begin{array}[]{c}\Delta
r_{2}\\\ \Delta\psi_{2}\\\ \Delta\theta_{2}\end{array}\right]\text{ ,}$ (14)
which we recognize as being in the form of a Mahalanobis distance. We note
that the Mahalanobis distance comes up in the Levedahl/Lincoln Labs work ([2]
and [3]) when the log is taken of the Gaussian distribution. The cost $F$ is
minimized subject to this equality constraint:
$G(B)=\left(B_{2,ENU(1)}-B_{1,ENU(1)}\right)-B_{R,ENU(1)}=0\text{ .}$ (15)
Thus, we have
$G(B)=\left[\begin{array}[]{c}\Delta
r_{2}\cos\theta_{2}\cos\psi_{2}-\Delta\psi_{2}\left(\sin\psi_{2}\right)\cdot
p_{T2}-\Delta\theta_{2}\left(\sin\theta_{2}\cos\psi_{2}\right)\cdot p_{T2}\\\
\Delta
r_{2}\cos\theta_{2}\sin\psi_{2}+\Delta\psi_{2}\left(\cos\psi_{2}\right)\cdot
p_{T2}-\Delta\theta_{2}\left(\sin\theta_{2}\sin\psi_{2}\right)\cdot p_{T2}\\\
\Delta r_{2}\sin\theta_{2}+\Delta\theta_{2}\left(\cos\theta_{2}\right)\cdot
p_{T2}\end{array}\right]$ $-\left[\begin{array}[]{c}\Delta
r_{1}\cos\theta_{1}\cos\psi_{1}-\Delta\psi_{1}\left(\sin\psi_{1}\right)\cdot
p_{T1}-\Delta\theta_{1}\left(\sin\theta_{1}\cos\psi_{1}\right)\cdot p_{T1}\\\
\Delta
r_{1}\cos\theta_{1}\sin\psi_{1}+\Delta\psi_{1}\left(\cos\psi_{1}\right)\cdot
p_{T1}-\Delta\theta_{1}\left(\sin\theta_{1}\sin\psi_{1}\right)\cdot p_{T1}\\\
\Delta r_{1}\sin\theta_{1}+\Delta\theta_{1}\left(\cos\theta_{1}\right)\cdot
p_{T1}\end{array}\right]-B_{R}=0$ (16)
where all of the terms in (16) reside entirely in one or the other of the two
$ENU$ coordinate systems. We see that $G(B)$ gives that the difference between
the two absolute biases (whatever they may be) is equal to the relative bias.
Also, we note that (16) is affine. Another equivalent representation for
$G(B)$ is
$G(B)=A\left(p_{T2},\psi_{2},\theta_{2}\right)\left[\begin{array}[]{c}\Delta
r_{2}\\\ \Delta\psi_{2}\\\
\Delta\theta_{2}\end{array}\right]-A\left(p_{T1},\psi_{1},\theta_{1}\right)\left[\begin{array}[]{c}\Delta
r_{1}\\\ \Delta\psi_{1}\\\ \Delta\theta_{1}\end{array}\right]-B_{R}$ (17)
where
$A\left(p_{T1},\psi_{1},\theta_{1}\right)=\left[\begin{array}[]{ccc}\cos\theta_{1}\cos\psi_{1}&-\sin\psi_{1}\cdot
p_{T1}&-\sin\theta_{1}\cos\psi_{1}\cdot p_{T1}\\\
\cos\theta_{1}\sin\psi_{1}&\cos\psi_{1}\cdot
p_{T1}&-\sin\theta_{1}\sin\psi_{1}\cdot p_{T1}\\\
\sin\theta_{1}&0&\cos\theta_{1}\cdot p_{T1}\end{array}\right]$ (18)
,
$A\left(p_{T2},\psi_{2},\theta_{2}\right)=\left[\begin{array}[]{ccc}\cos\theta_{2}\cos\psi_{2}&-\sin\psi_{2}\cdot
p_{T2}&-\sin\theta_{2}\cos\psi_{2}\cdot p_{T2}\\\
\cos\theta_{2}\sin\psi_{2}&\cos\psi_{2}\cdot
p_{T2}&-\sin\theta_{2}\sin\psi_{2}\cdot p_{T2}\\\
\sin\theta_{2}&0&\cos\theta_{2}\cdot p_{T2}\end{array}\right]\text{ .}$ (19)
Setting $G(B)=0$, we solve for $\Delta r_{2},\Delta\psi_{2},\Delta\theta_{2}$
$\left[\begin{array}[]{c}\Delta r_{2}\\\ \Delta\psi_{2}\\\
\Delta\theta_{2}\end{array}\right]=A^{-1}\left(p_{T2},\psi_{2},\theta_{2}\right)\left(A\left(p_{T1},\psi_{1},\theta_{1}\right)\left[\begin{array}[]{c}\Delta
r_{1}\\\ \Delta\psi_{1}\\\ \Delta\theta_{1}\end{array}\right]+B_{R}\right)$
(20)
(provided $p_{T2}\neq 0$.) This vector equality constraint (16) can be written
in the form of three scalar equality constraints
$G_{E}(B)=\Delta
r_{2}\cos\theta_{2}\cos\psi_{2}-\Delta\psi_{2}\sin\psi_{2}\cdot
p_{T2}-\Delta\theta_{2}\sin\theta_{2}\cos\psi_{2}\cdot p_{T2}$ $-\Delta
r_{1}\cos\theta_{1}\cos\psi_{1}+\Delta\psi_{1}\sin\psi_{1}\cdot
p_{T1}+\Delta\theta_{1}\sin\theta_{1}\cos\psi_{1}\cdot p_{T1}-B_{RE}$ (21)
$G_{N}(B)=\Delta
r_{2}\cos\theta_{2}\sin\psi_{2}+\Delta\psi_{2}\cos\psi_{2}\cdot
p_{T2}-\Delta\theta_{2}\sin\theta_{2}\sin\psi_{2}\cdot p_{T2}$ $-\Delta
r_{1}\cos\theta_{1}\sin\psi_{1}-\Delta\psi_{1}\cos\psi_{1}\cdot
p_{T1}+\Delta\theta_{1}\sin\theta_{1}\sin\psi_{1}\cdot p_{T1}-B_{RN}$ (22)
$G_{U}(B)=\Delta r_{2}\sin\theta_{2}+\Delta\theta_{2}\cos\theta_{2}\cdot
p_{T2}-\Delta r_{1}\sin\theta_{1}-\Delta\theta_{1}\cos\theta_{1}\cdot
p_{T1}-B_{RU}\text{ .}$ (23)
### 2.2 Solving The Minimization Problem
To solve this minimization problem, we need to take a few derivatives. We need
the gradient of the function to be minimized. We also need the gradient of the
constraint, which is an equality constraint in this case.
$\nabla F=\left[\begin{array}[]{c}\partial F/\partial\Delta r_{1}\\\ \partial
F/\partial\Delta\psi_{1}\\\ \partial F/\partial\Delta\theta_{1}\\\ \partial
F/\partial\Delta r_{2}\\\ \partial F/\partial\Delta\psi_{2}\\\ \partial
F/\partial\Delta\theta_{2}\end{array}\right]=\left[\begin{array}[]{c}k_{r_{1}}^{2}\cdot\Delta
r_{1}\\\ k_{\psi_{1}}^{2}\cdot\Delta\psi_{1}\\\
k_{\theta_{1}}^{2}\cdot\Delta\theta_{1}\\\ k_{r_{2}}^{2}\cdot\Delta r_{2}\\\
k_{\psi_{2}}^{2}\cdot\Delta\psi_{2}\\\
k_{\theta_{2}}^{2}\cdot\Delta\theta_{2}\end{array}\right]$ (24) $\nabla
G_{E}=\left[\begin{array}[]{c}\partial G_{E}/\partial\Delta r_{1}\\\ \partial
G_{E}/\partial\Delta\psi_{1}\\\ \partial G_{E}/\partial\Delta\theta_{1}\\\
\partial G_{E}/\partial\Delta r_{2}\\\ \partial
G_{E}/\partial\Delta\psi_{2}\\\ \partial
G_{E}/\partial\Delta\theta_{2}\end{array}\right]=\left[\begin{array}[]{c}-\cos\theta_{1}\cos\psi_{1}\\\
\sin\psi_{1}\cdot p_{T1}\\\ \sin\theta_{1}\cos\psi_{1}\cdot p_{T1}\\\
\cos\theta_{2}\cos\psi_{2}\\\ -\sin\psi_{2}\cdot p_{T2}\\\
-\sin\theta_{2}\cos\psi_{2}\cdot p_{T2}\end{array}\right]$ (25) $\nabla
G_{N}=\left[\begin{array}[]{c}\partial G_{N}/\partial\Delta r_{1}\\\ \partial
G_{N}/\partial\Delta\psi_{1}\\\ \partial G_{N}/\partial\Delta\theta_{1}\\\
\partial G_{N}/\partial\Delta r_{2}\\\ \partial
G_{N}/\partial\Delta\psi_{2}\\\ \partial
G_{N}/\partial\Delta\theta_{2}\end{array}\right]=\left[\begin{array}[]{c}-\cos\theta_{1}\sin\psi_{1}\\\
-\cos\psi_{1}\cdot p_{T1}\\\ \sin\theta_{1}\sin\psi_{1}\cdot p_{T1}\\\
\cos\theta_{2}\sin\psi_{2}\\\ \cos\psi_{2}\cdot p_{T2}\\\
-\sin\theta_{2}\sin\psi_{2}\cdot p_{T2}\end{array}\right]$ (26) $\nabla
G_{U}=\left[\begin{array}[]{c}\partial G_{U}/\partial\Delta r_{1}\\\ \partial
G_{U}/\partial\Delta\psi_{1}\\\ \partial G_{U}/\partial\Delta\theta_{1}\\\
\partial G_{U}/\partial\Delta r_{2}\\\ \partial
G_{U}/\partial\Delta\psi_{2}\\\ \partial
G_{u}/\partial\Delta\theta_{2}\end{array}\right]=\left[\begin{array}[]{c}-\sin\theta_{1}\\\
0\\\ -p_{T1}\cdot\cos\theta_{1}\\\ \sin\theta_{2}\\\ 0\\\
p_{T2}\cdot\cos\theta_{2}\end{array}\right]\text{ .}$ (27)
We are looking for an optimal solution located at the point
$e^{\ast}=\left(\Delta
r_{1}^{\ast},\Delta\psi_{1}^{\ast},\Delta\theta_{1}^{\ast},\Delta
r_{2}^{\ast},\Delta\psi_{2}^{\ast},\Delta\theta_{2}^{\ast}\right)$. We employ
the Kuhn-Tucker conditions that stipulate the optimal solution $e^{\ast}$
should satisfy these equality constraints for $e$ and there exist numbers
$a_{1}^{\ast},a_{2}^{\ast},a_{3}^{\ast}$ such that
$\nabla F\left(e^{\ast}\right)=a_{1}^{\ast}\cdot\nabla
G_{E}\left(e^{\ast}\right)+a_{2}^{\ast}\cdot\nabla
G_{N}\left(e^{\ast}\right)+a_{3}^{\ast}\cdot\nabla
G_{U}\left(e^{\ast}\right)\text{ .}$ (28)
The gradients $\nabla G_{E}\left(e^{\ast}\right),\nabla
G_{N}\left(e^{\ast}\right),\nabla G_{U}\left(e^{\ast}\right)$ are linearly
independent. Taking an inventory of the equations and unknowns, we see that
there are $9$ unknowns ($e,a_{1},a_{2},a_{3}$) and $9$ equations ($3$ from the
equality constraint and $6$ from the above equation). We may be able to find
the solution. Since the cost $F$ is quadratic and the constraint $G$ is
affine, the necessary conditions we give for optimality are also sufficient
conditions and an optimal solution $e^{\ast}$ is a global optimal solution.
Equation (28) in longhand is:
$\left[\begin{array}[]{c}k_{r_{1}}^{2}\cdot\Delta r_{1}\\\
k_{\psi_{1}}^{2}\cdot\Delta\psi_{1}\\\
k_{\theta_{1}}^{2}\cdot\Delta\theta_{1}\\\ k_{r_{2}}^{2}\cdot\Delta r_{2}\\\
k_{\psi_{2}}^{2}\cdot\Delta\psi_{2}\\\
k_{\theta_{2}}^{2}\cdot\Delta\theta_{2}\end{array}\right]=a_{1}\left[\begin{array}[]{c}-\cos\theta_{1}\cos\psi_{1}\\\
\sin\psi_{1}\cdot p_{T1}\\\ \sin\theta_{1}\cos\psi_{1}\cdot p_{T1}\\\
\cos\theta_{2}\cos\psi_{2}\\\ -\sin\psi_{2}\cdot p_{T2}\\\
-\sin\theta_{2}\cos\psi_{2}\cdot
p_{T2}\end{array}\right]+a_{2}\left[\begin{array}[]{c}-\cos\theta_{1}\sin\psi_{1}\\\
-\cos\psi_{1}\cdot p_{T1}\\\ \sin\theta_{1}\sin\psi_{1}\cdot p_{T1}\\\
\cos\theta_{2}\sin\psi_{2}\\\ \cos\psi_{2}\cdot p_{T2}\\\
-\sin\theta_{2}\sin\psi_{2}\cdot
p_{T2}\end{array}\right]+a_{3}\left[\begin{array}[]{c}-\sin\theta_{1}\\\ 0\\\
-p_{T1}\cdot\cos\theta_{1}\\\ \sin\theta_{2}\\\ 0\\\
p_{T2}\cdot\cos\theta_{2}\end{array}\right]\text{ .}$ (29)
The right hand side of (29) may be written in the form of the product of two
matrices $M_{1}$ and $M_{2}$,
$\left[\begin{array}[]{ccc}-\cos\theta_{1}\cos\psi_{1}&-\cos\theta_{1}\sin\psi_{1}&-\sin\theta_{1}\\\
\sin\psi_{1}\cdot p_{T1}&-\cos\psi_{1}\cdot p_{T1}&0\\\
\sin\theta_{1}\cos\psi_{1}\cdot p_{T1}&\sin\theta_{1}\sin\psi_{1}\cdot
p_{T1}&-p_{T1}\cdot\cos\theta_{1}\\\
\cos\theta_{2}\cos\psi_{2}&\cos\theta_{2}\sin\psi_{2}&\sin\theta_{2}\\\
-\sin\psi_{2}\cdot p_{T2}&\cos\psi_{2}\cdot p_{T2}&0\\\
-\sin\theta_{2}\cos\psi_{2}\cdot p_{T2}&-\sin\theta_{2}\sin\psi_{2}\cdot
p_{T2}&p_{T2}\cdot\cos\theta_{2}\end{array}\right]\left[\begin{array}[]{c}a_{1}\\\
a_{2}\\\
a_{3}\end{array}\right]=\left[\QDATOP{M_{1}}{M_{2}}\right]\left[\begin{array}[]{c}a_{1}\\\
a_{2}\\\ a_{3}\end{array}\right]\text{ ,}$ (30)
where
$M_{1}=\left[\begin{array}[]{ccc}-\cos\theta_{1}\cos\psi_{1}&-\cos\theta_{1}\sin\psi_{1}&-\sin\theta_{1}\\\
\sin\psi_{1}\cdot p_{T1}&-\cos\psi_{1}\cdot p_{T1}&0\\\
\sin\theta_{1}\cos\psi_{1}\cdot p_{T1}&\sin\theta_{1}\sin\psi_{1}\cdot
p_{T1}&-p_{T1}\cdot\cos\theta_{1}\end{array}\right]$ (31)
$M_{2}=\left[\begin{array}[]{ccc}\cos\theta_{2}\cos\psi_{2}&\cos\theta_{2}\sin\psi_{2}&\sin\theta_{2}\\\
-\sin\psi_{2}\cdot p_{T2}&\cos\psi_{2}\cdot p_{T2}&0\\\
-\sin\theta_{2}\cos\psi_{2}\cdot p_{T2}&-\sin\theta_{2}\sin\psi_{2}\cdot
p_{T2}&p_{T2}\cdot\cos\theta_{2}\end{array}\right]\text{ .}$ (32)
Note that $a_{1}$, $a_{2}$, $a_{3}$ have the units of meters. Let
$D_{1}=\left[\begin{array}[]{ccc}k_{r_{1}}^{2}&0&0\\\ 0&k_{\psi_{1}}^{2}&0\\\
0&0&k_{\theta_{1}}^{2}\end{array}\right]$ (33)
$D_{2}=\left[\begin{array}[]{ccc}k_{r_{2}}^{2}&0&0\\\ 0&k_{\psi_{2}}^{2}&0\\\
0&0&k_{\theta_{2}}^{2}\end{array}\right]\text{ .}$ (34)
Rewriting the left hand side of (29),
$\left[\begin{array}[]{cccccc}k_{r_{1}}^{2}&0&0&0&0&0\\\
0&k_{\psi_{1}}^{2}&0&0&0&0\\\ 0&0&k_{\theta_{1}}^{2}&0&0&0\\\
0&0&0&k_{r_{2}}^{2}&0&0\\\ 0&0&0&0&k_{\psi_{2}}^{2}&0\\\
0&0&0&0&0&k_{\theta_{2}}^{2}\end{array}\right]\left[\begin{array}[]{c}\Delta
r_{1}\\\ \Delta\psi_{1}\\\ \Delta\theta_{1}\\\ \Delta r_{2}\\\
\Delta\psi_{2}\\\
\Delta\theta_{2}\end{array}\right]=\left[\begin{array}[]{cc}D_{1}&0_{3,3}\\\
0_{3,3}&D_{2}\end{array}\right]\left[\begin{array}[]{c}\Delta r_{1}\\\
\Delta\psi_{1}\\\ \Delta\theta_{1}\\\ \Delta r_{2}\\\ \Delta\psi_{2}\\\
\Delta\theta_{2}\end{array}\right]$
$=\left[\QDATOP{M_{1}}{M_{2}}\right]\left[\begin{array}[]{c}a_{1}\\\ a_{2}\\\
a_{3}\end{array}\right]\text{ .}$ (35)
Hence, we have
$D_{1}\left[\begin{array}[]{c}\Delta r_{1}\\\ \Delta\psi_{1}\\\
\Delta\theta_{1}\end{array}\right]=M_{1}\left[\begin{array}[]{c}a_{1}\\\
a_{2}\\\ a_{3}\end{array}\right]\text{ ,}$ (36)
$D_{2}\left[\begin{array}[]{c}\Delta r_{2}\\\ \Delta\psi_{2}\\\
\Delta\theta_{2}\end{array}\right]=M_{2}\left[\begin{array}[]{c}a_{1}\\\
a_{2}\\\ a_{3}\end{array}\right]\text{ ,}$ (37)
or,
$\left[\begin{array}[]{c}\Delta r_{2}\\\ \Delta\psi_{2}\\\
\Delta\theta_{2}\end{array}\right]=D_{2}^{-1}M_{2}M_{1}^{-1}D_{1}\left[\begin{array}[]{c}\Delta
r_{1}\\\ \Delta\psi_{1}\\\ \Delta\theta_{1}\end{array}\right]\text{ .}$ (38)
Substituting (38) into (20) yields
$D_{2}^{-1}M_{2}M_{1}^{-1}D_{1}\left[\begin{array}[]{c}\Delta r_{1}\\\
\Delta\psi_{1}\\\
\Delta\theta_{1}\end{array}\right]=A^{-1}\left(p_{T2},\psi_{2},\theta_{2}\right)\left(A\left(p_{T1},\psi_{1},\theta_{1}\right)\left[\begin{array}[]{c}\Delta
r_{1}\\\ \Delta\psi_{1}\\\ \Delta\theta_{1}\end{array}\right]+B_{R}\right)$
(39)
so we get
$\left(D_{2}^{-1}M_{2}M_{1}^{-1}D_{1}-A^{-1}\left(p_{T2},\psi_{2},\theta_{2}\right)A\left(p_{T1},\psi_{1},\theta_{1}\right)\right)\left[\begin{array}[]{c}\Delta
r_{1}\\\ \Delta\psi_{1}\\\
\Delta\theta_{1}\end{array}\right]=A^{-1}\left(p_{T2},\psi_{2},\theta_{2}\right)B_{R}$
(40) $\left[\begin{array}[]{c}\Delta r_{1}\\\ \Delta\psi_{1}\\\
\Delta\theta_{1}\end{array}\right]=\left(D_{2}^{-1}M_{2}M_{1}^{-1}D_{1}-A^{-1}\left(p_{T2},\psi_{2},\theta_{2}\right)A\left(p_{T1},\psi_{1},\theta_{1}\right)\right)^{-1}A^{-1}\left(p_{T2},\psi_{2},\theta_{2}\right)B_{R}\text{
,}$ (41)
which allows us to obtain $\left(\Delta
r_{1},\Delta\psi_{1},\Delta\theta_{1}\right)$. Finally, substituting (41) into
(38) we get $\left(\Delta r_{2},\Delta\psi_{2},\Delta\theta_{2}\right)$.
### 2.3 Numerical Examples
We illustrate this idea with a few examples.
#### 2.3.1 a.
INPUT | OUTPUT
---|---
$B_{R}$ = [ 200 500 300]’ | Cost = 1.6250e+004
$p_{T1}$ = 25000 | $\Delta r_{1}$ = -1.7678e+002
$\psi_{1}$ = 0 | $\Delta\psi_{1}$ = -1.0000e-002
$\theta_{1}$ = 7.8540e-001 | $\Delta\theta_{1}$ = -1.4142e-003
$p_{T2}$ = 50000 | $\Delta r_{2}$ = 3.5355e+001
$\psi_{2}$ = 0 | $\Delta\psi_{2}$ = 5.0000e-003
$\theta_{2}$ = 2.3562e+000 | $\Delta\theta_{2}$ = -3.5355e-003
INPUT CONTINUED | OUTPUT______________
---|---
$k_{r1}^{2}$ = 2 |
$k_{\psi 1}^{2}$ = 1.2500e+009 = $2\ast\mathrm{PT1}^{2}$. |
$k_{\theta 1}^{2}$ = 1.2500e+009 |
$k_{r2}^{2}$ = 2 |
$k_{\psi 2}^{2}$ = 5.0000e+009 = $2\ast\mathrm{PT2}^{2}$. |
$k_{\theta 2}^{2}$ = 5.0000e+009 |
#### 2.3.2 b.
INPUT Same as with a. but with | OUTPUT
---|---
$B_{R}$ = [ 200 0 500]’ | Cost = 3.6250e+004
| $\Delta r_{1}$ = -2.4749e+002
| $\Delta\psi_{1}$ = 0
| $\Delta\theta_{1}$ = -4.2426e-003
| $\Delta r_{2}$ = 1.0607e+002
| $\Delta\psi_{2}$ = 0
| $\Delta\theta_{2}$ = -4.9497e-003
#### 2.3.3 c.
INPUT Same as with a. but with | OUTPUT
---|---
$\psi_{2}$ = $\pi$ | Cost = 1.6250e+004
$\theta_{2}$ = $\pi/4$ | $\Delta r_{1}$ = -1.7678e+002
| $\Delta\psi_{1}$ = -1.0000e-002
| $\Delta\theta_{1}$ = -1.4142e-003
| $\Delta r_{2}$ = 3.5355e+001
| $\Delta\psi_{2}$ = -5.0000e-003
| $\Delta\theta_{2}$ = 3.5355e-003
The input for this case is a variation of the input in a.
Note that the output is the same as with a except for a sign swap between
$\Delta\psi_{2}$ and $\Delta\theta_{2}$ to account for the orientation
difference of the “2” coordinates.
#### 2.3.4 d.
INPUT Same as with a. but with | OUTPUT
---|---
$\psi_{2}$ = $\pi/2$ | Cost = 5.5625e+004
$\theta_{2}$ = $\pi/4$ | $\Delta r_{1}$ = -1.7678e+002
| $\Delta\psi_{1}$ = -1.0000e-002
| $\Delta\theta_{1}$ = -1.4142e-003
| $\Delta r_{2}$ = 2.8284e+002
| $\Delta\psi_{2}$ = -2.0000e-003
| $\Delta\theta_{2}$ = -1.4142e-003
## 3 An Optimized Reduced-State Filter For Unknown Bias
A novel technique for calculating a steady-state reduced-order filter to track
a maneuvering target is presented by Mookerjee and Reifler [4]. The filter
they derive is optimized for performance with a stochastic acceleration. In
this paper, this technique is modified to derive a steady-state filter that is
optimized for performance with a stochastic measurement bias. Similar to [4],
the filter developed here is a reduced-state filter. We can see what a
reduced-state filter is by considering [6] and [5]. In these reports, we
estimate the position and velocity of an aircraft (a Beechcraft 1900) with
DMEs (distance measuring equipment), an INS (inertial navigation system) and a
barometric altimeter. The filter (in [6]) and the smoother (in [5]) were
designed with a state-to-estimate range bias in each DME (up to 5 were used),
a state-to-estimate INS drift, and a state-to-estimate bias in the baro. The
filter (or smoother) ran with these additional bias states in tow (i.e., in
addition to the position and velocity states). (The results in [6] and [5]
achieved the design goals in position and velocity accuracy.)
It appears likely the design goals of [4] and this paper are competing design
goals. The design methods discussed in [7], which are based on the Bode gain-
phase relationship, possibly could be brought to bear to quantify a possible
trade-off on the design goals of this paper and [4]. We don’t cover such
trade-offs in this paper, but it could be a problem for future investigations.
The classical control concepts of the sensitivity function and the
complimentary sensitivity function come to mind.
We use discrete time dynamical equations. It is fair to consider our state and
output (dynamical) equations to be the dual (in the control theory sense) of
the state and output equations, Equations (8)333In this section, we often
refer to equations from [4]. Hence we adopt the convention that all equation
references appearing in bold typeface are to equations in [4]. and (5).
Compared to the dynamical equations in [4], we eliminate the unknown
acceleration from the state equation and add an unknown bias in the output
(measurement) equation, the typical dual situation. We have:
$x\left(k+1\right)=\Phi x\left(k\right)+m\left(k\right)$ (42)
$z\left(k\right)=H\cdot
x\left(k\right)+n\left(k\right)+Wu\left(x\left(k\right),\lambda\right)\text{
.}$ (43)
The state $x(k)$ at time $k$ is of dimension $n$ and the state transition
matrix $\Phi$ is of dimension $n$ by $n$. The output $z(k)$ at time $k$ is of
dimension $q$ and the output matrix $H$ is of dimension $q$ by $n$. The
process noise term $m(k)$ is of dimension $n$ with covariance $Q$. The
measurement noise term $n(k)$ is of dimension $q$ with covariance $N$. The
bias matrix $W$ is $q$ by $m$. The bias function $u$ is
$\Re^{n}\times\Re^{p}\rightarrow\Re^{m}$, and we have that the bias $\lambda$
is a $p$-dimensional random vector with mean $\overline{\lambda}$ and
covariance $\Lambda$.
The time update equation, using (42), is simply
$\widehat{x}\left(k+1|k\right)=\Phi\widehat{x}\left(k|k\right)\text{ .}$ (44)
The measurement update equation becomes
$\widehat{x}\left(k+1|k+1\right)=\widehat{x}\left(k+1|k\right)+K\left(z\left(k+1\right)-H\widehat{x}\left(k+1|k\right)-Wu\left(\widehat{x}\left(k+1|k\right),\overline{\lambda}\right)\right)\text{
,}$ (45)
where $K$ is the $n$ by $q$ measurement, or Kalman, gain matrix. In the
steady-state case, which is discussed below, the position gain $\alpha$ and
velocity gain $\beta$ substitute for $K$.
### 3.1 Filter Development - General Case
In this subsection, we develop the filter equations for the general case. The
development in this section is (basically) dual (dual in the sense of control
theory) to Section III in [4]. The error is defined as (we develop the errors
analogous to (27) and (32)):
$\varepsilon\left(k+1|k+1\right)\equiv
x\left(k+1\right)-\widehat{x}\left(k+1|k+1\right)$ (46)
$=x\left(k+1\right)-\widehat{x}\left(k+1|k\right)-K\left(z\left(k+1\right)-H\widehat{x}\left(k+1|k\right)-Wu\left(\widehat{x}\left(k+1|k\right),\overline{\lambda}\right)\right)$
$=x\left(k+1\right)-\widehat{x}\left(k+1|k\right)$
$-K\left(Hx\left(k+1\right)+n\left(k+1\right)+Wu\left(x\left(k+1\right),\lambda\right)-H\widehat{x}\left(k+1|k\right)-Wu\left(\widehat{x}\left(k+1|k\right),\overline{\lambda}\right)\right)\text{
.}$
Continuing,
$\varepsilon\left(k+1|k+1\right)=x\left(k+1\right)-KHx\left(k+1\right)-Kn\left(k+1\right)-KWu\left(x\left(k+1\right),\lambda\right)$
$-\widehat{x}\left(k+1|k\right)+KH\widehat{x}\left(k+1|k\right)+KWu\left(\widehat{x}\left(k+1|k\right),\overline{\lambda}\right)$
$=\Phi x\left(k\right)+m\left(k\right)-KH\left(\Phi
x\left(k\right)+m\left(k\right)\right)-Kn\left(k+1\right)-KWu\left(\Phi
x\left(k\right),\lambda\right)$
$-\Phi\widehat{x}\left(k|k\right)+KH\Phi\widehat{x}\left(k|k\right)+KWu\left(\Phi\widehat{x}\left(k|k\right),\overline{\lambda}\right)$
$=\left(I-KH\right)\Phi\left(x\left(k\right)-\widehat{x}\left(k|k\right)\right)+\left(I-KH\right)m\left(k\right)-Kn\left(k+1\right)$
$-KW\left(u\left(\Phi
x\left(k\right),\lambda\right)-u\left(\Phi\widehat{x}\left(k|k\right),\overline{\lambda}\right)\right)\text{
.}$
So we have
$\varepsilon\left(k+1|k+1\right)=L\Phi\varepsilon\left(k|k\right)+Lm\left(k\right)-K\left(W\Delta
u_{k|k}+n\left(k+1\right)\right)\text{ ;}$ (47)
where
$L=\left(I-KH\right)$ (48)
an $n$ by $n$ matrix, and
$\Delta u_{k|k}\equiv u\left(\Phi
x\left(k\right),\lambda\right)-u\left(\Phi\widehat{x}\left(k|k\right),\overline{\lambda}\right)\text{
.}$ (49)
We can make the linear approximation
$\Delta u_{k|k}\approx\left.\frac{\partial u}{\partial
x}\right|_{x=\widehat{x}\left(k|k\right),\lambda=\overline{\lambda}}\Phi\Delta
x+\left.\frac{\partial
u}{\partial\lambda}\right|_{x=\widehat{x}\left(k|k\right),\lambda=\overline{\lambda}}\Delta\lambda$
(50)
so
$\Delta
x=\varepsilon\left(k|k\right)=x\left(k\right)-\widehat{x}\left(k|k\right)\text{
,}$ (51)
and
$\Delta\lambda=\lambda-\overline{\lambda}\text{ .}$ (52)
We obtain the result:
$\varepsilon\left(k+1|k+1\right)=L\Phi\varepsilon\left(k|k\right)+Lm\left(k\right)$
$-KW\left(\left.\frac{\partial u}{\partial
x}\right|_{x=\widehat{x}\left(k|k\right),\lambda=\overline{\lambda}}\Phi\varepsilon\left(k|k\right)+\left.\frac{\partial
u}{\partial\lambda}\right|_{x=\widehat{x}\left(k|k\right),\lambda=\overline{\lambda}}\Delta\lambda\right)-Kn\left(k+1\right)$
$=\left(L-KW\left.\frac{\partial u}{\partial
x}\right|_{x=\widehat{x}\left(k|k\right),\lambda=\overline{\lambda}}\right)\Phi\varepsilon\left(k|k\right)+Lm\left(k\right)-KW\left.\frac{\partial
u}{\partial\lambda}\right|_{x=\widehat{x}\left(k|k\right),\lambda=\overline{\lambda}}\Delta\lambda-
Kn\left(k+1\right)$
$=F\varepsilon\left(k|k\right)+Lm\left(k\right)+C\Delta\lambda-
Kn\left(k+1\right)$ (53)
where
$F=\left(L-KW\left.\frac{\partial u}{\partial
x}\right|_{x=\widehat{x}\left(k|k\right),\lambda=\overline{\lambda}}\right)\Phi\text{
,}$ (54)
an $n$ by $n$ matrix, and
$C=-KW\left.\frac{\partial
u}{\partial\lambda}\right|_{x=\widehat{x}\left(k|k\right),\lambda=\overline{\lambda}}\text{
,}$ (55)
an $n$ by $p$ matrix.
We now implement the observation made in [4] that the error
$\varepsilon\left(k|k\right)$ may be viewed as consisting of two components.
The first component of error, $\varepsilon^{\left(1\right)}$, is due to the
process noise $m$ and the measurement noise $n$. The second component of
error, $\varepsilon^{\left(2\right)}$, is due to the measurement bias. To the
extent that the linear approximation is valid, a linear analysis holds. That
is, the two error inputs may be treated in separate equations by applying the
superposition principle of linear analysis.
$\varepsilon^{\left(1\right)}\left(k+1|k+1\right)=F\varepsilon^{\left(1\right)}\left(k|k\right)+Lm\left(k\right)-Kn\left(k+1\right)$
(56)
$\varepsilon^{\left(2\right)}\left(k+1|k+1\right)=F\varepsilon^{\left(2\right)}\left(k|k\right)+C\cdot\Delta\lambda\text{
.}$ (57)
These equations are comparable to (33) and (34).
In addition, we require update equations for the total covariance and the
covariance of $\varepsilon^{\left(1\right)}\left(k|k\right)$. Using (42), the
time update equation for $\varepsilon^{\left(1\right)}\left(k|k\right)$ is
$M\left(k+1|k\right)\equiv
E\left[\varepsilon^{\left(1\right)}\left(k+1|k\right)\varepsilon^{\left(1\right)}\left(k+1|k\right)^{\prime}\right]$
(58) $=\Phi M\left(k|k\right)\Phi^{\prime}+Q\text{ .}$
From (56) the combined (measurement and time) update for the covariance of
$\varepsilon^{\left(1\right)}\left(k|k\right)$ is
$M\left(k+1|k+1\right)\equiv
E\left[\varepsilon^{\left(1\right)}\left(k+1|k+1\right)\varepsilon^{\left(1\right)}\left(k+1|k+1\right)^{\prime}\right]$
$=E\left[\left(F\varepsilon^{\left(1\right)}\left(k|k\right)+Lm\left(k\right)-Kn\left(k+1\right)\right)\left(F\varepsilon^{\left(1\right)}\left(k|k\right)+Lm\left(k\right)-Kn\left(k+1\right)\right)^{\prime}\right]$
and we use $E\left[n\left(k\right)n\left(l\right)^{\prime}\right]=0$ for
$k\neq l$ giving
$E\left[F\varepsilon^{\left(1\right)}\left(k|k\right)\left(Kn\left(k+1\right)\right)^{\prime}\right]=0$.
Hence,
$M\left(k+1|k+1\right)=E\left[F\varepsilon^{\left(1\right)}\left(k|k\right)\varepsilon^{\left(1\right)}\left(k|k\right)^{\prime}F^{\prime}+Lm\left(k\right)m\left(k\right)^{\prime}L^{\prime}+Kn\left(k+1\right)n\left(k+1\right)^{\prime}K^{\prime}\right]$
$=FM\left(k|k\right)F^{\prime}+LQL^{\prime}+KNK^{\prime}\text{ .}$ (59)
Working towards update equations for the total covariance, we firstly define
the $n$ by $p$ matrices $D\left(k|k\right)$ and $D\left(k+1|k\right)$ as
$\varepsilon^{\left(2\right)}\left(k|k\right)\equiv
D\left(k|k\right)\cdot\Delta\lambda\text{ .}$ (60)
We can define $D\left(k|k\right)$ in this way since in view of our linearized
analysis, the system output ($\varepsilon^{\left(2\right)}\left(k|k\right)$)
is a linear function of the system input ($\Delta\lambda$). We proceed by
defining
$D\left(k+1|k\right)\equiv FD\left(k|k\right)\text{ .}$ (61)
In (60), $\varepsilon^{\left(2\right)}$ and $\Delta\lambda$ are known
quantities (the equation defines $D\left(k|k\right)$). In (61), $F$ and
$D\left(k|k\right)$ are known quantities. Then, substituting (60) into (57),
we obtain
$D\left(k+1|k+1\right)\cdot\Delta\lambda=FD\left(k|k\right)\cdot\Delta\lambda+C\cdot\Delta\lambda$
$=D\left(k+1|k\right)\cdot\Delta\lambda+C\cdot\Delta\lambda\text{ ,}$ (62)
and subsequently (assuming (62) holds for all $\Delta\lambda$.)
$D\left(k+1|k+1\right)=D\left(k+1|k\right)+C\text{ .}$ (63)
Let $S$ be the total error covariance. By superposition, we get the total
error by the addition of the two error terms. We also make the observation
that since the two errors, $\varepsilon^{\left(1\right)}$ and
$\varepsilon^{\left(2\right)}$, originate from independent sources they remain
independent for all times $k$. Looking at $S$,
$S\left(k+1|k\right)\equiv
E\left[\varepsilon\left(k+1|k\right)\varepsilon\left(k+1|k\right)^{\prime}\right]$
$=E\left[\left(\varepsilon^{\left(1\right)}\left(k+1|k\right)+\varepsilon^{\left(2\right)}\left(k+1|k\right)\right)\left(\varepsilon^{\left(1\right)}\left(k+1|k\right)+\varepsilon^{\left(2\right)}\left(k+1|k\right)\right)^{\prime}\right]$
$=E\left[\varepsilon^{\left(1\right)}\left(k+1|k\right)\varepsilon^{\left(1\right)}\left(k+1|k\right)^{\prime}\right]+E\left[\varepsilon^{\left(2\right)}\left(k+1|k\right)\varepsilon^{\left(2\right)}\left(k+1|k\right)^{\prime}\right]$
$=M\left(k+1|k\right)+E\left[\varepsilon^{\left(2\right)}\left(k+1|k\right)\varepsilon^{\left(2\right)}\left(k+1|k\right)^{\prime}\right]$
$=M\left(k+1|k\right)+E\left[\Phi
D\left(k|k\right)\Delta\lambda\cdot\Delta\lambda^{\prime}D\left(k|k\right)^{\prime}\Phi^{\prime}\right]\text{
,}$
using (42), (44) and (60). Hence,
$S\left(k+1|k\right)=M\left(k+1|k\right)+\Phi
D\left(k|k\right)E\left[\Delta\lambda\Delta\lambda^{\prime}\right]D\left(k|k\right)^{\prime}\Phi^{\prime}\text{
.}$
Finally,
$S\left(k+1|k\right)=M\left(k+1|k\right)+\Phi D\left(k|k\right)\Lambda
D\left(k|k\right)^{\prime}\Phi^{\prime}\text{ .}$ (64)
Basically, this is the same result as (19).
We next obtain the measurement update for $S$:
$S\left(k+1|k+1\right)\equiv
E\left[\varepsilon\left(k+1|k+1\right)\varepsilon\left(k+1|k+1\right)^{\prime}\right]$
$=E\left[\left(x\left(k+1\right)-\widehat{x}\left(k+1|k+1\right)\right)\left(x\left(k+1\right)-\widehat{x}\left(k+1|k+1\right)\right)^{\prime}\right]$
$=E\left[\left(x\left(k+1\right)-\widehat{x}\left(k+1|k\right)-K\left[z\left(k+1\right)-H\widehat{x}\left(k+1|k\right)-Wu\left(\widehat{x}\left(k+1|k\right),\overline{\lambda}\right)\right]\right)\left(\mathrm{ditto}\right)^{\prime}\right]$
$=E\left[\left(x\left(k+1\right)-\widehat{x}\left(k+1|k\right)\right.\right.$
$\left.\left.-K\left[Hx\left(k+1\right)+n\left(k+1\right)+Wu\left(x\left(k+1\right),\lambda\right)-H\widehat{x}\left(k+1|k\right)-Wu\left(\widehat{x}\left(k+1|k\right),\overline{\lambda}\right)\right]\right)\left(\mathrm{ditto}\right)^{\prime}\right]$
$=E\left[\left(x\left(k+1\right)-\widehat{x}\left(k+1|k\right)-KH\left[x\left(k+1\right)-\widehat{x}\left(k+1|k\right)\right]\right.\right.$
$\left.\left.-K\left[n\left(k+1\right)+Wu\left(x\left(k+1\right),\lambda\right)-Wu\left(\widehat{x}\left(k+1|k\right),\overline{\lambda}\right)\right]\right)\left(\mathrm{ditto}\right)^{\prime}\right]$
$=E\left[\left\\{x\left(k+1\right)-\widehat{x}\left(k+1|k\right)-KH\left[x\left(k+1\right)-\widehat{x}\left(k+1|k\right)\right]\right.\right.$
$\left.\left.-K\left[n\left(k+1\right)+W\left(u\left(x\left(k+1\right),\lambda\right)-u\left(\widehat{x}\left(k+1|k\right),\overline{\lambda}\right)\right)\right]\right\\}\left\\{\mathrm{ditto}\right\\}^{\prime}\right]$
$=E\left[\left\\{x\left(k+1\right)-\widehat{x}\left(k+1|k\right)-KH\left(x\left(k+1\right)-\widehat{x}\left(k+1|k\right)\right)\right.\right.$
$\left.\left.-K\left[n\left(k+1\right)+W\left(u\left(\Phi
x\left(k\right),\lambda\right)-u\left(\Phi\widehat{x}\left(k|k\right),\overline{\lambda}\right)\right)\right]\right\\}\left\\{\mathrm{ditto}\right\\}^{\prime}\right]$
$=E\left[\left\\{x\left(k+1\right)-\widehat{x}\left(k+1|k\right)-KH\left(x\left(k+1\right)-\widehat{x}\left(k+1|k\right)\right)-K\left[n\left(k+1\right)+W\left(\Delta
u_{k|k}\right)\right]\right\\}\left\\{\mathrm{ditto}\right\\}^{\prime}\right]\text{
.}$
We take this next step only to the extent of the approximation,
$S\left(k+1|k+1\right)=E\left[\left\\{\left(I-KH\right)\left(x\left(k+1\right)-\widehat{x}\left(k+1|k\right)\right)-K\left[n\left(k+1\right)+W\left(\frac{\partial
u}{\partial x}\Phi\varepsilon\left(k|k\right)+\frac{\partial
u}{\partial\lambda}\Delta\lambda\right)\right]\right\\}\right.$
$\times\left.\left\\{\mathrm{ditto}\right\\}^{\prime}\right]$
$=E\left[\left\\{\left(I-KH\right)\varepsilon\left(k+1|k\right)-K\left[n\left(k+1\right)+W\left(\frac{\partial
u}{\partial x}\varepsilon\left(k+1|k\right)+\frac{\partial
u}{\partial\lambda}\Delta\lambda\right)\right]\right\\}\left\\{\mathrm{ditto}\right\\}^{\prime}\right]$
$=E\left[\left\\{\left(I-KH-KW\frac{\partial u}{\partial
x}\right)\varepsilon\left(k+1|k\right)-K\left[n\left(k+1\right)+W\frac{\partial
u}{\partial\lambda}\Delta\lambda\right]\right\\}\left\\{\mathrm{ditto}\right\\}^{\prime}\right]\text{
.}$
Let
$\widetilde{H}=H+W\frac{\partial u}{\partial x}$ (65)
and
$\widetilde{N}=N+W\frac{\partial u}{\partial\lambda}\Lambda\frac{\partial
u}{\partial\lambda}^{\prime}W^{\prime}\text{ .}$ (66)
Then
$S\left(k+1|k+1\right)=\left(I-K\widetilde{H}\right)S\left(k+1|k\right)\left(I-K\widetilde{H}\right)^{\prime}+K\widetilde{N}K^{\prime}$
$-\left(I-K\widetilde{H}\right)E\left[\varepsilon\left(k+1|k\right)\Delta\lambda^{\prime}\right]\left(W\frac{\partial
u}{\partial\lambda}\right)^{\prime}K^{\prime}$ $-K\left(W\frac{\partial
u}{\partial\lambda}\right)E\left[\Delta\lambda\varepsilon\left(k+1|k\right)^{\prime}\right]\left(I-K\widetilde{H}\right)^{\prime}\text{
.}$
Now
$E\left[\varepsilon\left(k+1|k\right)\Delta\lambda^{\prime}\right]=E\left[\left(\varepsilon^{(1)}\left(k+1|k\right)+\varepsilon^{(2)}\left(k+1|k\right)\right)\Delta\lambda^{\prime}\right]$
$=E\left[\varepsilon^{(2)}\left(k+1|k\right)\Delta\lambda^{\prime}\right]=\Phi
E\left[\varepsilon^{(2)}\left(k|k\right)\Delta\lambda^{\prime}\right]$ $=\Phi
E\left[D\left(k|k\right)\Delta\lambda\Delta\lambda^{\prime}\right]=\Phi
D\left(k|k\right)\Lambda\text{ .}$
Hence,
$S\left(k+1|k+1\right)=\left(I-K\widetilde{H}\right)S\left(k+1|k\right)\left(I-K\widetilde{H}\right)^{\prime}+K\widetilde{N}K^{\prime}$
$-\left(I-K\widetilde{H}\right)\Phi
D\left(k|k\right)\Lambda\left(W\frac{\partial
u}{\partial\lambda}\right)^{\prime}K^{\prime}-K\left(W\frac{\partial
u}{\partial\lambda}\right)\Lambda
D\left(k|k\right)^{\prime}\Phi^{\prime}\left(I-K\widetilde{H}\right)^{\prime}\text{
.}$ (67)
Equation (67) is similar in form as (37). We select $K$ so as to minimize the
trace of $S\left(k+1|k+1\right)$: $tr\left(S\left(k+1|k+1\right)\right)$. We
minimize $tr\left(S\left(k+1|k+1\right)\right)$ because for positive definite
matrices trace going to zero implies the matrix $L_{2}$ norm goes to zero. Let
$P$ be a positive definite $n$ by $n$ matrix, we have
$\left\|P\right\|\leq tr(P)\leq n\,\left\|P\right\|\text{ .}$
So minimizing $tr\left(P\right)$, gives a smaller upper bound for
$\left\|P\right\|$. We find the optimal $K$ ($K$ that minimizes
$tr\left(S\left(k+1|k+1\right)\right)$) by taking derivatives with respect to
$K$, setting the result to zero and solving for $K$. We recall the following
facts: Let $A$ be a matrix independent of $K$,
$\frac{\partial}{\partial
K}tr\left(KAK^{\prime}\right)=K\left(A+A^{\prime}\right)\text{ ;}$
$\frac{\partial}{\partial K}tr\left(KA\right)=A^{\prime}\text{ ;}$
$\frac{\partial}{\partial K}tr\left(K^{\prime}A\right)=A\text{ ;}$
$\frac{\partial}{\partial K}tr\left(AK\right)=A^{\prime}\text{ ;}$
$\frac{\partial}{\partial K}tr\left(AK^{\prime}\right)=A\text{ .}$
Using these facts on the terms of (67),
$\frac{\partial}{\partial
K}tr\left(K\widetilde{H}S\left(k+1|k\right)\widetilde{H}^{\prime}K^{\prime}\right)=2K\widetilde{H}S\left(k+1|k\right)\widetilde{H}^{\prime}\text{
;}$ $\frac{\partial}{\partial
K}tr\left(K\widetilde{N}K^{\prime}\right)=2K\widetilde{N}\text{ ;}$
$\frac{\partial}{\partial K}tr\left(S\left(k+1|k\right)\right)=0\text{ ;}$
$\frac{\partial}{\partial
K}tr\left(S\left(k+1|k\right)\widetilde{H}^{\prime}K^{\prime}\right)=S\left(k+1|k\right)\widetilde{H}^{\prime}\text{
;}$ $\frac{\partial}{\partial
K}tr\left(K\widetilde{H}S\left(k+1|k\right)\right)=S\left(k+1|k\right)\widetilde{H}^{\prime}\text{
;}$ $\frac{\partial}{\partial K}tr\left(\Phi
D\left(k|k\right)\Lambda\left(W\frac{\partial
u}{\partial\lambda}\right)^{\prime}K^{\prime}\right)=\Phi
D\left(k|k\right)\Lambda\left(W\frac{\partial
u}{\partial\lambda}\right)^{\prime}\text{ ;}$ $\frac{\partial}{\partial
K}tr\left(K\widetilde{H}\Phi D\left(k|k\right)\Lambda\left(W\frac{\partial
u}{\partial\lambda}\right)^{\prime}K^{\prime}\right)=K\left(\widetilde{H}\Phi
D\left(k|k\right)\Lambda\left(W\frac{\partial
u}{\partial\lambda}\right)^{\prime}+\left(W\frac{\partial
u}{\partial\lambda}\right)\Lambda
D\left(k|k\right)^{\prime}\Phi^{\prime}\widetilde{H}^{\prime}\right)\text{ ;}$
$\frac{\partial}{\partial K}tr\left(K\left(W\frac{\partial
u}{\partial\lambda}\right)\Lambda
D\left(k|k\right)^{\prime}\Phi^{\prime}\right)=\Phi
D\left(k|k\right)\Lambda\left(W\frac{\partial
u}{\partial\lambda}\right)^{\prime}\text{ ;}$ $\frac{\partial}{\partial
K}tr\left(K\left(W\frac{\partial u}{\partial\lambda}\right)\Lambda
D\left(k|k\right)^{\prime}\Phi^{\prime}\widetilde{H}^{\prime}K^{\prime}\right)=K\left(\left(W\frac{\partial
u}{\partial\lambda}\right)\Lambda
D\left(k|k\right)^{\prime}\Phi^{\prime}\widetilde{H}^{\prime}+\widetilde{H}\Phi
D\left(k|k\right)\Lambda\left(W\frac{\partial
u}{\partial\lambda}\right)^{\prime}\right)\text{ .}$
We differentiate the trace of $S\left(k+1|k+1\right)$ as it is represented in
(67) by $K$ and set the result equal to zero,
$\frac{\partial}{\partial
K}tr\left(S\left(k+1|k+1\right)\right)=2K\widetilde{H}S\left(k+1|k\right)\widetilde{H}^{\prime}+2K\widetilde{N}+0-S\left(k+1|k\right)\widetilde{H}^{\prime}-S\left(k+1|k\right)\widetilde{H}^{\prime}$
$-\Phi D\left(k|k\right)\Lambda\left(W\frac{\partial
u}{\partial\lambda}\right)^{\prime}+K\left(\widetilde{H}\Phi
D\left(k|k\right)\Lambda\left(W\frac{\partial
u}{\partial\lambda}\right)^{\prime}+\left(W\frac{\partial
u}{\partial\lambda}\right)\Lambda
D\left(k|k\right)^{\prime}\Phi^{\prime}\widetilde{H}^{\prime}\right)$ $-\Phi
D\left(k|k\right)\Lambda\left(W\frac{\partial
u}{\partial\lambda}\right)^{\prime}+K\left(\left(W\frac{\partial
u}{\partial\lambda}\right)\Lambda
D\left(k|k\right)^{\prime}\Phi^{\prime}\widetilde{H}^{\prime}+\widetilde{H}\Phi
D\left(k|k\right)\Lambda\left(W\frac{\partial
u}{\partial\lambda}\right)^{\prime}\right)=0\text{ .}$
After some algebra,
$K\left(\widetilde{H}S\left(k+1|k\right)\widetilde{H}^{\prime}+\widetilde{N}+\widetilde{H}\Phi
D\left(k|k\right)\Lambda\left(W\frac{\partial
u}{\partial\lambda}\right)^{\prime}+\left(W\frac{\partial
u}{\partial\lambda}\right)\Lambda
D\left(k|k\right)^{\prime}\Phi^{\prime}\widetilde{H}^{\prime}\right)$
$=S\left(k+1|k\right)\widetilde{H}^{\prime}+\Phi
D\left(k|k\right)\Lambda\left(W\frac{\partial
u}{\partial\lambda}\right)^{\prime}\text{ .}$ (68)
The optimal filter gain is
$K=\left(\widetilde{H}S\left(k+1|k\right)\widetilde{H}^{\prime}+\widetilde{N}+\widetilde{H}\Phi
D\left(k|k\right)\Lambda\left(W\frac{\partial
u}{\partial\lambda}\right)^{\prime}+\left(W\frac{\partial
u}{\partial\lambda}\right)\Lambda
D\left(k|k\right)^{\prime}\Phi^{\prime}\widetilde{H}^{\prime}\right)^{-1}$
$\times\left(S\left(k+1|k\right)\widetilde{H}^{\prime}+\Phi
D\left(k|k\right)\Lambda\left(W\frac{\partial
u}{\partial\lambda}\right)^{\prime}\right)\text{ ,}$ (69)
which is an $n$ by $q$ matrix.
### 3.2 Filter Development - Steady-State Case
We next examine the steady-state case of our problem. Referencing equations
(41-43) we have
$\Phi=\left[\begin{array}[]{cc}1&T\\\ 0&1\end{array}\right]$ (70)
$H=\left[\begin{array}[]{cc}1&0\end{array}\right]$ (71) $W=1\text{ .}$ (72)
Hence, with $p$ as position, $v$ as velocity, and $z$ as the measurement, the
state transition and output equations for the steady-state case are
$\left[\begin{array}[]{c}p\left(k+1\right)\\\
v\left(k+1\right)\end{array}\right]=\left[\begin{array}[]{cc}1&T\\\
0&1\end{array}\right]\left[\begin{array}[]{c}p\left(k\right)\\\
v\left(k\right)\end{array}\right]+\left[\begin{array}[]{c}0\\\
1\end{array}\right]m\left(k\right)$ (73)
$z\left(k\right)=\left[\begin{array}[]{cc}1&0\end{array}\right]\left[\begin{array}[]{c}p\left(k\right)\\\
v\left(k\right)\end{array}\right]+n\left(k\right)+u\left(x\left(k\right),\lambda\left(k\right)\right)\text{
.}$ (74)
Setting $u\left(x,\lambda\right)=\lambda$, our linear approximations from (50)
become
$\left.\frac{\partial u}{\partial
x}\right|_{x=\widehat{x}\left(k|k\right),\lambda=\overline{\lambda}}=\left[\begin{array}[]{cc}0&0\end{array}\right]$
(75)
and
$\left.\frac{\partial
u}{\partial\lambda}\right|_{x=\widehat{x}\left(k|k\right),\lambda=\overline{\lambda}}=1\text{
.}$ (76)
Then, substituting (75) into (65)
$\widetilde{H}=H+W\frac{\partial u}{\partial
x}=\left[\begin{array}[]{cc}1&0\end{array}\right]+1\cdot\left[\begin{array}[]{cc}0&0\end{array}\right]=\left[\begin{array}[]{cc}1&0\end{array}\right]=H$
(77)
and substituting (76) into (66)
$\widetilde{N}=N+W\frac{\partial u}{\partial\lambda}\Lambda\frac{\partial
u}{\partial\lambda}^{\prime}W^{\prime}=N+1\cdot 1\cdot\Lambda\cdot 1\cdot
1=N+\Lambda\text{ .}$ (78)
We have that the steady-state filter gain is
$\overline{K}\equiv\left[\begin{array}[]{c}\alpha\\\
\beta/T\end{array}\right]\text{ .}$ (79)
As mentioned previously, (79) is where $\alpha$ and $\beta$ fit in for the
Kalman gain matrix $K$ as given by (69). These gains are obtained by computing
the steady-state values for all the variables in (69). The major objective of
this section is to find a relationship between $\alpha$ and $\beta$.
The steady-state version of $L$ from (48), $\overline{L}$, is
$\overline{L}=\left(I-\overline{K}H\right)=\left[\begin{array}[]{cc}1&0\\\
0&1\end{array}\right]-\left[\begin{array}[]{c}\alpha\\\
\beta/T\end{array}\right]\left[\begin{array}[]{cc}1&0\end{array}\right]=\left[\begin{array}[]{cc}1-\alpha&0\\\
-\beta/T&1\end{array}\right]\text{ .}$ (80)
The steady-state version of $F$ from (54), $\overline{F}$, is
$\overline{F}=\left(\left[\begin{array}[]{cc}1-\alpha&0\\\
-\beta/T&1\end{array}\right]-\left[\begin{array}[]{c}\alpha\\\
\beta/T\end{array}\right]\cdot
1\cdot\left[\begin{array}[]{cc}0&0\end{array}\right]\right)\left[\begin{array}[]{cc}1&T\\\
0&1\end{array}\right]=\left[\begin{array}[]{cc}1-\alpha&\left(1-\alpha\right)T\\\
-\beta/T&1-\beta\end{array}\right]\text{ .}$ (81)
The eigenvalues of $\overline{F}$ are
$\lambda_{1,2}=1-\frac{\left(\alpha+\beta\right)}{2}\pm\frac{1}{2}\sqrt{2\alpha\beta-4\beta+\alpha^{2}+\beta^{2}}\text{
.}$ (82)
Then, referring to (55), the steady-state version of $C$, $\overline{C}$, is
$\overline{C}=-\left[\begin{array}[]{c}\alpha\\\
\beta/T\end{array}\right]\cdot 1\cdot 1=-\left[\begin{array}[]{c}\alpha\\\
\beta/T\end{array}\right]\text{ .}$ (83)
The measurement updated steady-state covariance $M$, referring to (59), is
$\overline{M}\equiv\lim_{k\rightarrow\infty}M\left(k|k\right)=\overline{F}\,\overline{M}\,\overline{F}^{\prime}+\overline{L}Q\overline{L}^{\prime}+\overline{K}\,N\,\overline{K}^{\prime}$
(84)
Superimpose the two noise terms by letting
$\overline{M}=\overline{M}_{Q}+\overline{M}_{N}$ (85)
and solve:
$\overline{M}_{N}=\overline{F}\,\overline{M}_{N}\,\overline{F}^{\prime}+\overline{K}\,N\,\overline{K}^{\prime}$
(86)
$\overline{M}_{Q}=\overline{F}\,\overline{M}_{Q}\,\overline{F}^{\prime}+\overline{L}Q\overline{L}^{\prime}\text{
.}$ (87)
Comparing (79), (81) and (86) to (46), (47) and (48), we see that our solution
for $\overline{M}_{N}$ is of the same form as (49). Consequently,
$\overline{M}_{N}=\frac{N}{\alpha\left(4-2\alpha-\beta\right)}\left[\begin{array}[]{cc}2\alpha^{2}+2\beta-3\alpha\beta&\beta\left(2\alpha-\beta\right)/T\\\
\beta\left(2\alpha-\beta\right)/T&2\beta^{2}/T^{2}\end{array}\right]\text{ .}$
(88)
The solution of $\overline{M}_{Q}$ remains to be determined. Substituting
(80), (81) and (73) into (87) gives
$\overline{M}_{Q}=\left[\begin{array}[]{cc}1-\alpha&\left(1-\alpha\right)T\\\
-\beta/T&1-\beta\end{array}\right]\overline{M}_{Q}\left[\begin{array}[]{cc}1-\alpha&-\beta/T\\\
\left(1-\alpha\right)T&1-\beta\end{array}\right]+\left[\begin{array}[]{cc}1-\alpha&0\\\
-\beta/T&1\end{array}\right]\left[\begin{array}[]{cc}0&0\\\
0&q_{22}\end{array}\right]\left[\begin{array}[]{cc}1-\alpha&-\beta/T\\\
0&1\end{array}\right]\text{ .}$
In longhand,
$\overline{M}_{Q}=\left[\begin{array}[]{cc}\overline{m}_{11Q}&\overline{m}_{12Q}\\\
\overline{m}_{12Q}&\overline{m}_{22Q}\end{array}\right]$
$=\left[\begin{array}[]{c}\left(1-\alpha\right)^{2}\left(\overline{m}_{11Q}+2T\overline{m}_{12Q}+T^{2}\overline{m}_{22Q}\right)\\\
\left(1-\alpha\right)\left(-\beta\overline{m}_{11Q}/T+\left(1-2\beta\right)\overline{m}_{12Q}+T\left(1-\beta\right)\overline{m}_{22Q}\right)\end{array}\right.$
$\left.\begin{array}[]{c}\left(1-\alpha\right)\left(-\beta\overline{m}_{11Q}/T+\left(1-2\beta\right)\overline{m}_{12Q}+T\left(1-\beta\right)\overline{m}_{22Q}\right)\\\
\left(\beta^{2}/T^{2}\right)\overline{m}_{11Q}+\left(2\beta\left(\beta-1\right)/T\right)\overline{m}_{12Q}+\left(1-2\beta+\beta^{2}\right)\overline{m}_{22Q}\end{array}\right]+\allowbreak\left[\begin{array}[]{cc}0&0\\\
0&q_{22}\end{array}\right]\text{ .}$ (89)
Hence,
$\left[\begin{array}[]{c}\overline{m}_{11Q}-\left(1-\alpha\right)^{2}\left(\overline{m}_{11Q}+2T\overline{m}_{12Q}+T^{2}\overline{m}_{22Q}\right)\\\
\overline{m}_{12Q}-\left(1-\alpha\right)\left(-\beta\overline{m}_{11Q}/T+\left(1-2\beta\right)\overline{m}_{12Q}+T\left(1-\beta\right)\overline{m}_{22Q}\right)\end{array}\right.$
$\left.\begin{array}[]{c}\overline{m}_{12Q}-\left(1-\alpha\right)\left(-\beta\overline{m}_{11Q}/T+\left(1-2\beta\right)\overline{m}_{12Q}+T\left(1-\beta\right)\overline{m}_{22Q}\right)\\\
\overline{m}_{22Q}-\left(\beta^{2}/T^{2}\right)\overline{m}_{11Q}-\left(2\beta\left(\beta-1\right)/T\right)\overline{m}_{12Q}-\left(1-2\beta+\beta^{2}\right)\overline{m}_{22Q}\end{array}\right]$
$=\left[\begin{array}[]{cc}0&0\\\ 0&q_{22}\end{array}\right]\text{ .}$ (90)
We get three equations in the three unknowns $m_{11Q}$, $m_{12Q}$ and
$m_{22Q}$,
$\left[\begin{array}[]{ccc}1-\left(1-\alpha\right)^{2}&-2\left(1-\alpha\right)^{2}T&-\left(1-\alpha\right)^{2}T^{2}\\\
\left(1-\alpha\right)\beta/T&1-\left(1-\alpha\right)\left(1-2\beta\right)&-\left(1-\alpha\right)\left(1-\beta\right)T\\\
-\left(\beta^{2}/T^{2}\right)&-2\beta\left(\beta-1\right)/T&1-\left(1-2\beta+\beta^{2}\right)\end{array}\right]\left[\begin{array}[]{c}\overline{m}_{11Q}\\\
\overline{m}_{12Q}\\\ \overline{m}_{22Q}\end{array}\right]\text{ }$
$=\left[\begin{array}[]{c}0\\\ 0\\\ q_{22}\end{array}\right]\text{ .}$
Taking the matrix inverse to solve for $\overline{M}_{Q}$,
$\left[\begin{array}[]{c}\overline{m}_{11Q}\\\ \overline{m}_{12Q}\\\
\overline{m}_{22Q}\end{array}\right]=\left[\begin{array}[]{ccc}1-\left(1-\alpha\right)^{2}&-2\left(1-\alpha\right)^{2}T&-\left(1-\alpha\right)^{2}T^{2}\\\
\left(1-\alpha\right)\beta/T&1-\left(1-\alpha\right)\left(1-2\beta\right)&-\left(1-\alpha\right)\left(1-\beta\right)T\\\
-\left(\beta^{2}/T^{2}\right)&-2\beta\left(\beta-1\right)/T&1-\left(1-2\beta+\beta^{2}\right)\end{array}\right]^{-1}\left[\begin{array}[]{c}0\\\
0\\\ q_{22}\end{array}\right]\text{ .}$
The determinant of this matrix,
$4\alpha\beta-\alpha\beta^{2}-2\alpha^{2}\beta=\alpha\beta\left(4-\beta-2\alpha\right)$,
should not be zero for the inverse to exist. This is satisfied by these
conditions:
$\begin{array}[]{ll}\mathrm{1.}&\alpha\neq 0\\\ \mathrm{2.}&\beta\neq 0\\\
\mathrm{3.}&\beta\neq 4-2\alpha\end{array}\text{ .}$ (91)
If the determinant is not zero, we can obtain the solution:
$\left[\begin{array}[]{c}\overline{m}_{11Q}\\\ \overline{m}_{12Q}\\\
\overline{m}_{22Q}\end{array}\right]=q_{22}\cdot\left[\begin{array}[]{c}T^{2}\left(-2+5\alpha-4\alpha^{2}+\alpha^{3}\right)\\\
T\left(-2\alpha+\beta-\alpha\beta+3\alpha^{2}-\alpha^{3}\right)\\\
\left(-2\beta+2\alpha\beta-2\alpha^{2}+\alpha^{3}\right)\end{array}\right]/\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)\text{
.}$ (92)
In matrix form,
$\overline{M}_{Q}=\frac{q_{22}}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}\left[\begin{array}[]{cc}T^{2}\left(-2+5\alpha-4\alpha^{2}+\alpha^{3}\right)&T\left(-2\alpha+\beta-\alpha\beta+3\alpha^{2}-\alpha^{3}\right)\\\
T\left(-2\alpha+\beta-\alpha\beta+3\alpha^{2}-\alpha^{3}\right)&\left(-2\beta+2\alpha\beta-2\alpha^{2}+\alpha^{3}\right)\end{array}\right]\text{
.}$ (93)
And finally $\overline{M}$ is obtained from (85), (88) and (93):
$\overline{M}=\frac{N}{\alpha\left(4-2\alpha-\beta\right)}\left[\begin{array}[]{cc}2\alpha^{2}+2\beta-3\alpha\beta&\beta\left(2\alpha-\beta\right)/T\\\
\beta\left(2\alpha-\beta\right)/T&2\beta^{2}/T^{2}\end{array}\right]$
$+\frac{q_{22}}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}\left[\begin{array}[]{cc}T^{2}\left(-2+5\alpha-4\alpha^{2}+\alpha^{3}\right)&T\left(-2\alpha+\beta-\alpha\beta+3\alpha^{2}-\alpha^{3}\right)\\\
T\left(-2\alpha+\beta-\alpha\beta+3\alpha^{2}-\alpha^{3}\right)&\left(-2\beta+2\alpha\beta-2\alpha^{2}+\alpha^{3}\right)\end{array}\right]\text{
.}$ (94)
We see that
$\overline{m}_{11}=\overline{m}_{11}\left(\alpha,\beta,T,N,q_{22}\right)$,
$\overline{m}_{12}=\overline{m}_{12}\left(\alpha,\beta,T,N,q_{22}\right)$ and
$\overline{m}_{22}=\overline{m}_{22}\left(\alpha,\beta,T,N,q_{22}\right)$. The
usual technique for solving the Liapunov equation (84) is by algebraic
manipulation and using the symmetry of the matrix, as demonstrated with the
solution (92). Numerical solutions may be obtained by repeated propagation
until steady-state is arrived at.
The time updated steady-state covariance $M$, referring to (58), is
$\overset{\cdot}{M}\equiv\lim_{k\rightarrow\infty}M\left(k+1|k\right)=\lim_{k\rightarrow\infty}\Phi
M\left(k|k\right)\Phi^{\prime}+Q=\Phi\left(\overline{M}_{N}+\overline{M}_{Q}\right)\Phi^{\prime}+\allowbreak
Q\text{ .}$ (95)
We get
$\Phi\overline{M}_{N}\Phi^{\prime}=\frac{N}{\alpha\left(4-2\alpha-\beta\right)}\left[\begin{array}[]{cc}1&T\\\
0&1\end{array}\right]\left[\begin{array}[]{cc}2\alpha^{2}+2\beta-3\alpha\beta&\beta\left(2\alpha-\beta\right)/T\\\
\beta\left(2\alpha-\beta\right)/T&2\beta^{2}/T^{2}\end{array}\right]\left[\begin{array}[]{cc}1&0\\\
T&1\end{array}\right]$
$=\frac{N}{\alpha\left(4-2\alpha-\beta\right)}\left[\begin{array}[]{cc}2\alpha^{2}+2\beta+\alpha\beta&\beta\left(2\alpha+\beta\right)/T\\\
\beta\left(2\alpha+\beta\right)/T&2\beta^{2}/T^{2}\end{array}\right]$
and
$\Phi\overline{M}_{Q}\Phi^{\prime}=$
$=\frac{q_{22}}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}\left[\begin{array}[]{cc}1&T\\\
0&1\end{array}\right]\left[\begin{array}[]{cc}T^{2}\left(-2+5\alpha-4\alpha^{2}+\alpha^{3}\right)&T\left(-2\alpha+\beta-\alpha\beta+3\alpha^{2}-\alpha^{3}\right)\\\
T\left(-2\alpha+\beta-\alpha\beta+3\alpha^{2}-\alpha^{3}\right)&\left(-2\beta+2\alpha\beta-2\alpha^{2}+\alpha^{3}\right)\end{array}\right]\left[\begin{array}[]{cc}1&0\\\
T&1\end{array}\right]$
$=\frac{q_{22}}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}\left[\begin{array}[]{cc}T^{2}\left(-2+\alpha\right)&T\left(-2\alpha-\beta+\alpha\beta+\alpha^{2}\right)\\\
T\left(-2\alpha-\beta+\alpha\beta+\alpha^{2}\right)&\left(-2\beta+2\alpha\beta-2\alpha^{2}+\alpha^{3}\right)\end{array}\right]\allowbreak\text{
.}$
Hence,
$\overset{\cdot}{M}=\frac{N}{\alpha\left(4-2\alpha-\beta\right)}\left[\begin{array}[]{cc}2\alpha^{2}+2\beta+\alpha\beta&\beta\left(2\alpha+\beta\right)/T\\\
\beta\left(2\alpha+\beta\right)/T&2\beta^{2}/T^{2}\end{array}\right]$
$+\frac{q_{22}}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}\left[\begin{array}[]{cc}T^{2}\left(-2+\alpha\right)&T\left(-2\alpha-\beta+\alpha\beta+\alpha^{2}\right)\\\
T\left(-2\alpha-\beta+\alpha\beta+\alpha^{2}\right)&\left(-2\beta+2\alpha\beta-2\alpha^{2}+\alpha^{3}\right)\end{array}\right]+\left[\begin{array}[]{cc}0&0\\\
0&q_{22}\end{array}\right]\text{ .}$ (96)
We need steady-state versions of these: designating
$\overline{D}\equiv\lim_{k\rightarrow\infty}D\left(k|k\right)$
$\overset{\cdot}{D}\equiv\lim_{k\rightarrow\infty}D\left(k+1|k\right)\text{
,}$
we then have, referring to (61) and (63),
$\overset{\cdot}{D}=\overline{F}\,\overline{D}$
$\overline{D}=\overset{\cdot}{D}+\overline{C}$
then
$\overset{\cdot}{D}=\overline{D}-\overline{C}\text{ .}$
Continuing,
$\overline{F}\,\overline{D}=\overline{D}-\overline{C}$
$\overline{F}\,\overline{D}-\overline{D}=\left(\overline{F}-I\right)\overline{D}=-\overline{C}\text{
;}$
hence,
$\overline{D}=-\left(\overline{F}-I\right)^{-1}\overline{C}\text{ .}$
Using (81), and (83)
$\overline{D}=-\left(\left[\begin{array}[]{cc}1-\alpha&\left(1-\alpha\right)T\\\
-\beta/T&1-\beta\end{array}\right]-\left[\begin{array}[]{cc}1&0\\\
0&1\end{array}\right]\right)^{-1}\left(-\left[\begin{array}[]{c}\alpha\\\
\beta/T\end{array}\right]\right)$
$=\left[\begin{array}[]{cc}-\alpha&\left(1-\alpha\right)T\\\
-\beta/T&-\beta\end{array}\right]^{-1}\left[\begin{array}[]{c}\alpha\\\
\beta/T\end{array}\right]$
$=\frac{\left[\begin{array}[]{cc}-\beta&-\left(1-\alpha\right)T\\\
\beta/T&-\alpha\end{array}\right]}{\beta}\left[\begin{array}[]{c}\alpha\\\
\beta/T\end{array}\right]=\left[\begin{array}[]{c}-1\\\
0\end{array}\right]\text{ .}$
Hence,
$\overset{\cdot}{D}=\overline{F}\,\overline{D}=\left[\begin{array}[]{cc}1-\alpha&\left(1-\alpha\right)T\\\
-\beta/T&1-\beta\end{array}\right]\left[\begin{array}[]{c}-1\\\
0\end{array}\right]=\left[\begin{array}[]{c}\alpha-1\\\
\beta/T\end{array}\right]\text{ .}$
Finally, the steady-state time updated total covariance is obtained by
substituting into (64),
$\overset{\cdot}{S}=\left[\begin{array}[]{cc}\overset{\cdot}{S}_{11}&\overset{\cdot}{S}_{12}\\\
\overset{\cdot}{S}_{21}&\overset{\cdot}{S}_{22}\end{array}\right]=\overset{\cdot}{M}+\Phi\overline{D}\Lambda\overline{D}^{\prime}\Phi^{\prime}$
$=\left[\begin{array}[]{cc}\overset{\cdot}{M}_{11}&\overset{\cdot}{M}_{12}\\\
\overset{\cdot}{M}_{21}&\overset{\cdot}{M}_{22}\end{array}\right]+\left[\begin{array}[]{cc}1&T\\\
0&1\end{array}\right]\left[\begin{array}[]{c}-1\\\
0\end{array}\right]\cdot\Lambda\cdot\left[\begin{array}[]{cc}-1&0\end{array}\right]\left[\begin{array}[]{cc}1&0\\\
T&1\end{array}\right]$
$=\left[\begin{array}[]{cc}\overset{\cdot}{M}_{11}&\overset{\cdot}{M}_{12}\\\
\overset{\cdot}{M}_{21}&\overset{\cdot}{M}_{22}\end{array}\right]+\allowbreak\left[\begin{array}[]{cc}\Lambda&0\\\
0&0\end{array}\right]\text{ .}$
So,
$\left[\begin{array}[]{cc}\overset{\cdot}{S}_{11}&\overset{\cdot}{S}_{12}\\\
\overset{\cdot}{S}_{21}&\overset{\cdot}{S}_{22}\end{array}\right]=\frac{N}{\alpha\left(4-2\alpha-\beta\right)}\left[\begin{array}[]{cc}2\alpha^{2}+2\beta+\alpha\beta&\beta\left(2\alpha+\beta\right)/T\\\
\beta\left(2\alpha+\beta\right)/T&2\beta^{2}/T^{2}\end{array}\right]$
$+\frac{q_{22}}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}\left[\begin{array}[]{cc}T^{2}\left(-2+\alpha\right)&T\left(-2\alpha-\beta+\alpha\beta+\alpha^{2}\right)\\\
T\left(-2\alpha-\beta+\alpha\beta+\alpha^{2}\right)&\left(-2\beta+2\alpha\beta-2\alpha^{2}+\alpha^{3}\right)\end{array}\right]$
$+\left[\begin{array}[]{cc}0&0\\\ 0&q_{22}\end{array}\right]\text{
}+\allowbreak\left[\begin{array}[]{cc}\Lambda&0\\\ 0&0\end{array}\right]\text{
.}$
In particular,
$\overset{\cdot}{S}_{11}=\overset{\cdot}{M}_{11}+\Lambda$
$=\frac{N\left(2\alpha^{2}+2\beta+\alpha\beta\right)}{\alpha\left(4-2\alpha-\beta\right)}+\frac{q_{22}T^{2}\left(-2+\alpha\right)}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}+\Lambda$
(97) $\overset{\cdot}{S}_{21}=\overset{\cdot}{M}_{21}$
$=\frac{N\beta\left(2\alpha+\beta\right)}{\alpha\left(4-2\alpha-\beta\right)T}+\frac{q_{22}T\left(-2\alpha-\beta+\alpha\beta+\alpha^{2}\right)}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}\text{
.}$ (98)
We turn our attention to (68). In steady-state,
$\widetilde{H}S\left(k+1|k\right)\widetilde{H}^{\prime}+\widetilde{N}=\left[\begin{array}[]{cc}1&0\end{array}\right]\left[\begin{array}[]{cc}\overset{\cdot}{S}_{11}&\overset{\cdot}{S}_{12}\\\
\overset{\cdot}{S}_{21}&\overset{\cdot}{S}_{22}\end{array}\right]\left[\begin{array}[]{c}1\\\
0\end{array}\right]+N+\Lambda$ $=\overset{\cdot}{S}_{11}+N+\Lambda\text{ .}$
(99)
Also,
$\widetilde{H}\Phi D\left(k|k\right)\Lambda\left(W\frac{\partial
u}{\partial\lambda}\right)^{\prime}=\left[\begin{array}[]{cc}1&0\end{array}\right]\left[\begin{array}[]{cc}1&T\\\
0&1\end{array}\right]\left[\begin{array}[]{c}-1\\\
0\end{array}\right]\cdot\Lambda\cdot 1\cdot 1=-\Lambda\text{ ,}$ (100)
and
$\left(W\frac{\partial u}{\partial\lambda}\right)\Lambda
D\left(k|k\right)^{\prime}\Phi^{\prime}\widetilde{H}^{\prime}=-\Lambda\text{
,}$ (101)
and
$S\left(k+1|k\right)\widetilde{H}^{\prime}=\left[\begin{array}[]{cc}\overset{\cdot}{S}_{11}&\overset{\cdot}{S}_{12}\\\
\overset{\cdot}{S}_{12}&\overset{\cdot}{S}_{22}\end{array}\right]\left[\begin{array}[]{c}1\\\
0\end{array}\right]=\left[\begin{array}[]{c}\overset{\cdot}{S}_{11}\\\
\overset{\cdot}{S}_{12}\end{array}\right]\text{ ,}$ (102)
and
$\Phi D\left(k|k\right)\Lambda\left(W\frac{\partial
u}{\partial\lambda}\right)^{\prime}=\left[\begin{array}[]{cc}1&T\\\
0&1\end{array}\right]\left[\begin{array}[]{c}-1\\\
0\end{array}\right]\cdot\Lambda\cdot 1\cdot
1=\left[\begin{array}[]{c}-\Lambda\\\ 0\end{array}\right]\text{ .}$ (103)
Substituting (99), (100), (101), (102) and (103) into (68) gives
$\left[\begin{array}[]{c}\alpha\\\
\beta/T\end{array}\right]\left(\overset{\cdot}{S}_{11}+N+\Lambda-\Lambda-\Lambda\right)=\left[\begin{array}[]{c}\overset{\cdot}{S}_{11}\\\
\overset{\cdot}{S}_{12}\end{array}\right]+\left[\begin{array}[]{c}-\Lambda\\\
0\end{array}\right]\text{ .}$
This, written as two scalar equations,
$\alpha\left(\overset{\cdot}{S}_{11}+N+\Lambda-\Lambda-\Lambda\right)=\alpha\left(\overset{\cdot}{S}_{11}+N-\Lambda\right)=\overset{\cdot}{S}_{11}-\Lambda$
$\frac{\beta}{T}\left(\overset{\cdot}{S}_{11}+N+\Lambda-\Lambda-\Lambda\right)=\frac{\beta}{T}\left(\overset{\cdot}{S}_{11}+N+-\Lambda\right)=\overset{\cdot}{S}_{12}\text{
.}$
Substituting (97) and (98)
$\alpha\left(\left(\frac{N\left(2\alpha^{2}+2\beta+\alpha\beta\right)}{\alpha\left(4-2\alpha-\beta\right)}+\frac{q_{22}T^{2}\left(-2+\alpha\right)}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}+\Lambda\right)+N-\Lambda\right)$
$=\frac{N\left(2\alpha^{2}+2\beta+\alpha\beta\right)}{\alpha\left(4-2\alpha-\beta\right)}+\frac{q_{22}T^{2}\left(-2+\alpha\right)}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}+\Lambda-\Lambda$
$\beta\left(\left(\frac{N\left(2\alpha^{2}+2\beta+\alpha\beta\right)}{\alpha\left(4-2\alpha-\beta\right)}+\frac{q_{22}T^{2}\left(-2+\alpha\right)}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}+\Lambda\right)+N+-\Lambda\right)$
$=\left(\frac{N\beta\left(2\alpha+\beta\right)}{\alpha\left(4-2\alpha-\beta\right)T}+\frac{q_{22}T\left(-2\alpha-\beta+\alpha\beta+\alpha^{2}\right)}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}\right)\cdot
T$
$\alpha\left(\frac{N\left(2\alpha^{2}+2\beta+\alpha\beta\right)}{\alpha\left(4-2\alpha-\beta\right)}+\frac{q_{22}T^{2}\left(-2+\alpha\right)}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}+N\right)=\frac{N\left(2\alpha^{2}+2\beta+\alpha\beta\right)}{\alpha\left(4-2\alpha-\beta\right)}+\frac{q_{22}T^{2}\left(-2+\alpha\right)}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}$
$\beta\left(\frac{N\left(2\alpha^{2}+2\beta+\alpha\beta\right)}{\alpha\left(4-2\alpha-\beta\right)}+\frac{q_{22}T^{2}\left(-2+\alpha\right)}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}+N\right)=\frac{N\beta\left(2\alpha+\beta\right)}{\alpha\left(4-2\alpha-\beta\right)}+\frac{q_{22}T^{2}\left(-2\alpha-\beta+\alpha\beta+\alpha^{2}\right)}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}\text{
.}$
We define $\rho=q_{22}T^{2}/N$. With $q_{22}$ in $\left(m/\sec\right)^{2}$,
$T$ in seconds and $N$ in $m^{2}$, $\rho$ is unitless. Substituting this in
the previous two equations, we obtain
$\alpha\left(\frac{\left(2\alpha^{2}+2\beta+\alpha\beta\right)}{\alpha\left(4-2\alpha-\beta\right)}+\frac{\rho\left(-2+\alpha\right)}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}+1\right)=\frac{\left(2\alpha^{2}+2\beta+\alpha\beta\right)}{\alpha\left(4-2\alpha-\beta\right)}+\frac{\rho\left(-2+\alpha\right)}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}$
$\beta\left(\frac{\left(2\alpha^{2}+2\beta+\alpha\beta\right)}{\alpha\left(4-2\alpha-\beta\right)}+\frac{\rho\left(-2+\alpha\right)}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}+1\right)=\frac{\beta\left(2\alpha+\beta\right)}{\alpha\left(4-2\alpha-\beta\right)}+\frac{\rho\left(-2\alpha-\beta+\alpha\beta+\alpha^{2}\right)}{\left(-4\alpha\beta+\alpha\beta^{2}+2\alpha^{2}\beta\right)}\text{
.}$
We divide the previous two equations and cancel an $\alpha$,
$\frac{\alpha}{\beta}=\frac{\left(2\alpha^{2}+2\beta+\alpha\beta\right)\left(-4\beta+\beta^{2}+2\alpha\beta\right)+\rho\left(-2+\alpha\right)\left(4-2\alpha-\beta\right)}{\beta\left(2\alpha+\beta\right)\left(-4\beta+\beta^{2}+2\alpha\beta\right)+\rho\left(-2\alpha-\beta+\alpha\beta+\alpha^{2}\right)\left(4-2\alpha-\beta\right)}\text{
.}$
Cross multiplying gives:
$\allowbreak\allowbreak
2\beta^{4}+\left(4\alpha-8\right)\beta^{3}+\rho\left(\allowbreak\left(\alpha^{2}-2\alpha+2\right)\beta^{2}+\left(3\alpha^{3}-10\alpha^{2}\allowbreak+12\alpha-8\right)\beta+\left(2\alpha^{4}-8\alpha^{3}+8\alpha^{2}\right)\right)=0\text{
.}$ (104)
Equation (104) gives our relationship between $\alpha$ and $\beta$. The noise
ratio $\rho$ is a parameter in the equation which is known, or at least known
to be within a range. Equation (104) may be factored
$\left(\beta+\left(2\alpha-4\right)\right)\left(2\beta^{3}+\rho\left(\left(\alpha^{2}-2\alpha+2\right)\beta+\alpha^{2}\left(\alpha-2\right)\right)\right)=0\text{
.}$ (105)
Hence, $\beta=(4-2\alpha)$ is a solution that is independent of $\rho$. This
solution is not permitted however since it violates condition 3. of (91).
There is a second real solution for $\beta$ given $\alpha$ (which depends on
$\rho$.) The remaining two solutions for $\beta$ given $a$ may be a complex
conjugate pair. The table below gives some representative solutions to (104).
The two real solutions are presented. The second one listed we don’t use
because of the condition 3. The Newton-Raphson method may be used to compute
all of the solutions to (104).
$\rho$ | $\alpha$ | $\beta$
---|---|---
2 | 0.2 | 0.04385, 3.6
4 | 0.2 | 0.04386, 3.6
6 | 0.2 | 0.04389, 3.6
6 | 0.4 | 0.1866, 3.2
8 | 0.2 | 0.04389, 3.6
8 | 0.4 | 0.1870, 3.2
10 | 0.2 | 0.04389, 3.6
10 | 0.4 | 0.1873, 3.2
10 | 0.5 | 0.2959, 3.0
These $\alpha$ and $\beta$ give that the eigenvalues of $\overline{F}$, in
(82), have norm less than $1$. Hence, by Theorem 2.1, page 64 of [1], $M_{N}$
and $M_{Q}$, the solutions to (86) and (87) respectively, exist are unique and
are positive definite.
## 4 Summary and Conclusions
In this paper we considered some topics in radar sensor bias. We presented an
algorithm that estimates the absolute bias of two sensors when the relative
bias between the sensors is given. The algorithm uses the relative bias, which
is given in rectangular coordinates, as a constraint. The absolute biases, in
spherical coordinates, for the sensors are obtained by the solution to an
optimization problem that exploits the spherical-to-rectangular coordinate
conversion. We presented a reduced-state filter that is designed for
performance with sensor bias. The filter is reduced-state since it does not
contain additional bias states. The filter design is influenced by the filter
in [4]. It may be viewed as a dual design (in the control theory sense) to the
filter in [4].
A flow diagram for processing radar data with bias may contain these stages:
1\. Estimate state with the $\alpha-\beta$ filter optimized for measurement
bias, as presented in Section 3.
2\. For a multi-sensor problem, estimate the relative sensor bias using an
optimized algorithm such as in [2].
3\. Continue by estimating the absolute bias for each sensor using the
algorithm presented in Section 2.
## References
* [1] B.D.O. Anderson and J.B. Moore, Optimal Filtering, Prentice-Hall, Inc., 1979
* [2] M. Levedahl, “An Explicit Pattern Matching Assignment Algorithm” Signal and Data Processing of Small Targets, Oliver Drummond, Editor, Proceedings of SPIE Vol. 4728, 2002.
* [3] W.J. Brown, W.J. Weisman and L.M. Brock, “Multi-Sensor Data Fusion for Object Association and Discrimination,” M.I.T. Lincoln Laboratory Technical Report,27 January 2006.
* [4] P. Mookerjee and F. Reifler, “Reduced State Estimator for Systems with Parametric Inputs,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 40, No. 2, April 2004.
* [5] D. Serakos, “SIGINT Mapping System Navigation Using Kalman Smoothing,” MIT Lincoln Laboratory Technical Memorandum 44L-0625, February 27, 1990.
* [6] D. Gustafson and D. Serakos, “SIGINT Mapping System Navigation Using Kalman Filtering,” MIT Lincoln Laboratory Technical Memorandum 44L-0610, October 12, 1989.
* [7] M.M. Seron, J.H. Braslavsky and G.C. Goodwin, Fundamental Limitations in Filtering and Control, Springer, 1997.
## 5 Appendix: Transformation from $ENU(1)$ to $ENU(2)$
In this appendix we present the transformation from the $ENU(1)$ to the
$ENU(2)$ coordinate systems. But first, consider the transformation from $ECI$
to $ENU$. Consider an $ENU$ coordinate axis located at longitude-latitude
$\Omega-L$ and define the rotation matrix
$T_{ECI2ENU}=\left[\begin{array}[]{ccc}1&0&0\\\ 0&\cos L&-\sin L\\\ 0&\sin
L&\cos
L\end{array}\right]\left[\begin{array}[]{ccc}\cos\Omega&0&-\sin\Omega\\\
0&1&0\\\
\sin\Omega&0&\cos\Omega\end{array}\right]\left[\begin{array}[]{ccc}0&1&0\\\
0&0&1\\\ 1&0&0\end{array}\right]$
$=\left[\begin{array}[]{ccc}-\sin\Omega&\cos\Omega&0\\\ -\sin
L\cos\Omega&-\sin L\sin\Omega&\cos L\\\ \cos L\cos\Omega&\cos L\sin\Omega&\sin
L\end{array}\right]\allowbreak$
with $T_{ENU2ECI}=T_{ECI2ENU}^{\prime}$. For position, we need to include a
translation, so that for a given position vector in $ENU$ coordinates
$P_{ECI}=\frac{r_{ee}}{\sqrt{1-e^{2}\sin^{2}L}}\left[\begin{array}[]{c}\cos
L\cos\Omega\\\ \cos L\sin\Omega\\\ \left(1-e^{2}\right)\sin
L\end{array}\right]+T_{ENU2ECI}P_{ENU}$
where $r_{ee}$ is the earth’s equatorial radius and $e$ is the earth’s
eccentricity. For velocity, use the rotation alone.
Let the $ENU(1)$, $ENU(2)$ coordinate system be located at longitude-latitude
$\Omega_{1}-L_{1}$ and $\Omega_{2}-L_{2}$ respectively. Next we consider our
transformation going from $\Omega_{1}-L_{1}$ to $\Omega_{2}-L_{2}$. The
rotation part of this transformation can be represented by the matrix below
(going from $ENU(1)$ to $ENU(2)$). There are $3$ steps. First, the $ENU(1)$
coordinates are rotated down to the equator. Second, these coordinates are
rotated along the equator by the longitude difference. Third is the rotation
up to the latitude of the $ENU(2)$ system.
$T_{ENU(1)2ENU(2)}=\left[\begin{array}[]{ccc}1&0&0\\\ 0&\cos L_{2}&-\sin
L_{2}\\\ 0&\sin L_{2}&\cos
L_{2}\end{array}\right]\left[\begin{array}[]{ccc}\cos\left(\Omega_{2}-\Omega_{1}\right)&0&-\sin\left(\Omega_{2}-\Omega_{1}\right)\\\
0&1&0\\\
\sin\left(\Omega_{2}-\Omega_{1}\right)&0&\cos\left(\Omega_{2}-\Omega_{1}\right)\end{array}\right]\left[\begin{array}[]{ccc}1&0&0\\\
0&\cos L_{1}&\sin L_{1}\\\ 0&-\sin L_{1}&\cos L_{1}\end{array}\right]$
$=\left[\begin{array}[]{ccc}\cos\left(\Omega_{2}-\Omega_{1}\right)&\sin
L_{1}\sin\left(\Omega_{2}-\Omega_{1}\right)&-\cos
L_{1}\sin\left(\Omega_{2}-\Omega_{1}\right)\\\ -\sin
L_{2}\sin\left(\Omega_{2}-\Omega_{1}\right)&\cos L_{1}\cos L_{2}+\sin
L_{1}\sin L_{2}\cos\left(\Omega_{2}-\Omega_{1}\right)&\cos L_{2}\sin
L_{1}-\cos L_{1}\sin L_{2}\cos\left(\Omega_{2}-\Omega_{1}\right)\\\ \cos
L_{2}\sin\left(\Omega_{2}-\Omega_{1}\right)&\cos L_{1}\sin L_{2}-\cos
L_{2}\sin L_{1}\cos\left(\Omega_{2}-\Omega_{1}\right)&\sin L_{1}\sin
L_{2}+\cos L_{1}\cos
L_{2}\cos\left(\Omega_{2}-\Omega_{1}\right)\end{array}\right]\allowbreak$
(Note
$\left[\begin{array}[]{ccc}\cos\left(\Omega_{2}-\Omega_{1}\right)&0&-\sin\left(\Omega_{2}-\Omega_{1}\right)\\\
0&1&0\\\
\sin\left(\Omega_{2}-\Omega_{1}\right)&0&\cos\left(\Omega_{2}-\Omega_{1}\right)\end{array}\right]=\left[\begin{array}[]{ccc}\cos\Omega_{1}\cos\Omega_{2}+\sin\Omega_{1}\sin\Omega_{2}&0&-\cos\Omega_{1}\sin\Omega_{2}+\cos\Omega_{2}\sin\Omega_{1}\\\
0&1&0\\\
\cos\Omega_{1}\sin\Omega_{2}-\cos\Omega_{2}\sin\Omega_{1}&0&\cos\Omega_{1}\cos\Omega_{2}+\sin\Omega_{1}\sin\Omega_{2}\end{array}\right]\allowbreak\allowbreak$
$=\left[\begin{array}[]{ccc}\cos\Omega_{2}&0&-\sin\Omega_{2}\\\ 0&1&0\\\
\sin\Omega_{2}&0&\cos\Omega_{2}\end{array}\right]\left[\begin{array}[]{ccc}\cos\Omega_{1}&0&\sin\Omega_{1}\\\
0&1&0\\\ -\sin\Omega_{1}&0&\cos\Omega_{1}\end{array}\right]$
so that the rotation $T_{ENU(1)2ENU(2)}$ is a rotation from the first
coordinates down to $ECI$ and then up to the second coordinates.) We have
$T_{ENU(2)2ENU(1)}=T_{ENU(1)2ENU(2)}^{\prime}$
The position vector from the $ENU(1)$ to the $ENU(2)$ coordinate axes (in
$ECI$ coordinates) is
$P_{ENU(1)2ENU(2),ECI}=r_{ee}\frac{\left[\begin{array}[]{c}\cos
L_{2}\cos\Omega_{2}\\\ \cos L_{2}\sin\Omega_{2}\\\ \left(1-e^{2}\right)\sin
L_{2}\end{array}\right]}{\sqrt{1-e^{2}\sin^{2}L_{2}}}-r_{ee}\frac{\left[\begin{array}[]{c}\cos
L_{1}\cos\Omega_{1}\\\ \cos L_{1}\sin\Omega_{1}\\\ \left(1-e^{2}\right)\sin
L_{1}\end{array}\right]}{\sqrt{1-e^{2}\sin^{2}L_{1}}}$
and in the other coordinates this vector is
$P_{ENU(1)2ENU(2),ENU\left(i\right)}=T_{ECI2ENU(i)}P_{ENU(1)2ENU(2),ECI}$
The total position coordinate transformation, including translation can be
represented by
$P_{ENU(2)}=-P_{ENU(1)2ENU(2),ENU\left(2\right)}+T_{ENU(1)2ENU(2)}P_{ENU(1)}$
The total velocity coordinate transformation is given by the rotation alone.
|
arxiv-papers
| 2008-10-23T19:33:13 |
2024-09-04T02:48:58.416888
|
{
"license": "Public Domain",
"authors": "Demetrios Serakos, John E. Gray and Hazim Youssef",
"submitter": "Demetrios Serakos",
"url": "https://arxiv.org/abs/0810.4326"
}
|
0810.4430
|
# The staircase structure of the Southern Brazilian Continental Shelf
M. S. Baptista Centro de Matemática da Universidade do Porto, Rua do Campo
Alegre 687, 4169-007 Porto, Portugal L. A. Conti Escola de Artes Ciências e
Humanidades, Universidade de São Paulo, Av. Arlindo Bettio 1000, 03828-000,
São Paulo, Brasil
###### Abstract
We show some evidences that the Southeastern Brazilian Continental Shelf
(SBCS) has a devil’s staircase structure, with a sequence of scarps and
terraces with widths that obey fractal formation rules. Since the formation of
these features are linked with the sea level variations, we say that the sea
level changes in an organized pulsating way. Although the proposed approach
was applied in a particular region of the Earth, it is suitable to be applied
in an integrated way to other Shelves around the world, since the analyzes
favor the revelation of the global sea level variations.
## I Introduction
During the late quaternary period, after the last glacial maximum (LGM), from
18 kyears ago till present, a global warming was responsible for the melting
of the glaciers leading to a fast increase in the sea level. In approximately
13 kyears, the sea level rised up to about 120 meters, reaching the actual
level. However, the sea level did not go up in a continuous fashion, but
rather, it has evolved in a pulsatile way, leaving behind a signature of what
actually happened, the Continental Shelf, i.e. the seafloor.
Continental shelves are located at the boundary with the land so that they are
shaped by both marine and terrestrial processes. Sea-level oscillations
incessantly transform terrestrial areas in marine environments and vice-versa,
thus increasing the landscape complexity lambeck:2001 . The presence of
regions with abnormal slope as well as the presence of terraces on a
Continental Shelf are indicators of sea level positions after the Last Glacial
Maximum (LGM), when large ice sheets covered high latitudes of Europe and
North America, and sea levels stood about 120-130m lower than today
heinrich:1988 . Geomorphic processes responsible for the formation of these
terraces and discontinuities on the bottom of the sea topography are linked to
the coastal dynamics during eustatic processes associated with both erosional
or depositional forcing (wave cut and wave built terraces respectively
goslar:2000 ).
The irregular distribution of such terraces and shoreface sediments is mainly
controlled by the relationship between shelf paleo-physiography and changes on
the sea level and sediment supply which reflect both global and local
processes. Several works have dealt with mapping and modeling the distribution
of shelf terraces in order to understand the environmental consequences of
climate change and sea level variations after the LGM adams:1999 ;
andrews:1998 ; broecker:1998 .
In this period of time the sea-level transgression was punctuated by at least
six relatively short flooding events that collectively accounted for more than
90m of the 120m rise. Most (but not all) of the floodings appear to correspond
with paleoclimatic events recorded in Greenland and Antarctic ice-cores,
indicative of the close coupling between rapid climate change, glacial melt,
and corresponding sea-level rise taylor:1997 .
In this work, we analyze data from the Southeastern Brazilian Continental
Shelf (SBCS) located in a typical sandy passive margin with the predominance
of palimpsests sediments. The mean length is approximately 250km and the
shelfbreak is located at 150m depth. It is a portion of a greater
geomorphologic region of the southeastern Brazilian coast called São Paulo
Bight, an arc-shaped part of the southeastern Brazilian margin. The geology
and topography of the immersed area are very peculiar, represented by the
Mesozoic/Cenozoic tectonic processes that generated the mountainous landscapes
known as ”Serra do Mar”. These landscapes (with mean altitudes of 800m) have a
complex pattern that characterize the coastal morphology, and leads to several
scarps intercalated with small coastal plains and pocket beaches.
This particular characteristic determines the development of several small
size fluvial basins and absence of major rivers conditioning low sediment
input, what tends to preserve topographic signatures of the sea-level
variations.
For the purpose of the present study, we select three parallel profiles
acquired from echo-sounding surveys, since for all the considered profiles,
the same similar series of sequences of terraces were found. These profiles
furtado:1992 ; conti:2001 ; correa:1996 are transversal to the coastline and
the isobaths trend, and they extend from a 20m to a 120m depth.
The importance of understanding the formation of these ridges is that it can
tell us about the coastal morphodynamic conditions, inner shelf processes and
about the characteristics of periods of the sea level regimes standstills
(paleoshores). In particular, the widths of the terraces are related to the
time the sea level ”stabilized”. All this information is vital for the better
understanding of the late quaternary climate changes dynamic.
We find relations between the widths of the terraces that follow a self-affine
pattern description. These relations are given by a mathematical model, which
describes an order of appearance for the terraces. Our results suggest that
this geomorphological structure for the terraces can be described by a devil’s
staircase mandelbrot:1977 , a staircase with infinitely many steps in between
two steps. This property gives the name ”devil” to the staircase, once an
idealized being would take an infinite time to go from one step to another.
So, the seafloor morphology is self-affine (fractal structure) as reported in
Ref. herzfeld ; goff , but according to our findings, it has a special kind of
self-affine structure, the devil’s staircase structure.
A devil’s staircase as well as other self-affine structure are the response of
an oscillatory system when excited by some external force. The presence of a
step means that while varying some internal parameter, the system preserves
some averaged regular behavior, a consequence of the stable frequency-locking
regime between a natural frequency of the system and the frequency of the
excitation. This staircase as well as other self-affine structures are
characterized by the presence of steps whose widths are directly related to
the rational ratio between the natural frequency of the system and the
frequency of the excitation.
In a similar fashion, we associate the widths of the terraces with rational
numbers that represent two hypothetical frequencies of oscillation which are
assumed to exist in the system that creates the structure of the SBCS, here
regarded as the sea level dynamics (SLD), also known as the sea level
variations. Then, once these rational numbers are found, we show that the
relative distances between triples of terraces (associated with some
hypothetical frequencies) follow similar scalings found in the relative
distance between triples of plateaus (associated with these same frequencies)
observed in the devil’s staircase.
The seafloor true structure, apart from the dynamics that originated it, is
also a very relevant issue, specially for practical applications. For example,
one can measure the seafloor with one resolution and then reconstruct the rest
based on some modeling mareschal . As we show in this work (Sec. V), a devil’s
staircase structure fits remarkably well the experimental data.
Our paper is organized as follows. In Sec. II, we describe the data to be
analyzed. In Sec. III, we describe which kind of dynamical systems can create
a devil’s staircase and how one can detect its presence in experimental data
based on only a few observations. In Sec. IV, we show the evidences that led
us to characterize the SBCS as a devil’s staircase, and in Sec. V we show how
to construct seafloor profiles based on the devil’s staircase geometry.
Finally, in Sec. VI, we present our conclusions, discussing also possible
scenarios for the future of the sea level dynamics under the perspective of
our findings.
## II Data
Figure 1: [Color online] Profiles (depth versus the distance to the cost) of
the Southeastern Brazilian Continental Shelf. The arrows indicate the terraces
considered in our analyzes. The profile shown with a thick black line is the
profile chosen for our derivations, reproduced also in (B). The other two
profiles had their original position of the two axes shifted by a constant
value such that one can also identify the terraces observed in the chosen
profile in these other two. Note that a translation of the profiles by a
constant value has no effect on any of the scalings observed. The reason of
this mismatch between the profiles is due to the local geometry of the cost at
the time the sea reached that level.
The data consists of the tree profiles given in Fig. 1(A-B). The profile
considered for our analyzes is shown in Fig. 1(B), where we show the
Continental Shelf of the State of São Paulo, in a transversal cut in the
direction: inner shelf (”cost”) $\rightarrow$ shelfbreak (”open sea”). The
horizontal axis represents the distance to the cost and the vertical axis, the
sea level (depth), $d$. We are interested in the terraces widths and their
respective depths.
The profiles shown in Fig. 1 were the result of a smoothing (filtering)
process from the original data collected by Sonar note3 . The smoothing
process is needed to eliminate from the measured data the influence of the
oscillations of the ship where the sonar is located and local oscillations on
the sea floor probably due to the stream flows.
Smaller topographic terraces could be smoothed or masked due to several
processes such as: coastal dynamic erosional during sea-level rising, Holocene
sediment cover, erosional processes associated with modern hidrodynamic
pattern (geostrophic currents). For that reason we only consider the largest
ones, as the ones shown in Fig. 2, (located at $d=-30.01m$ with the width of
$l=6.06km$). As one can see, the edges of the terraces are not so sharp as one
would expect from a staircase plateau. Again, this is due to the action of the
sea waves and stream flows throughout the time. To reconstruct what we believe
to be the original terrace, we consider that its depth is given by the depth
of the middle point, and its width is given by the minimal distance between
two parallel lines placed along the scarps of the terrace edges. Using this
procedure, we construct Table 1 with the largest and more relevant terraces
found.
We identify a certain terrace introducing a lower index $n$ in $l$ and $d$,
according to their chronological order of appearance. More recent appearance
(closer to the cost, less deep) smaller is the index $n$. We consider the more
recent data to have a zero distance from the cost, but in fact, this data is
positioned at about 15km away from the shore, where the bottom of the sea is
not affected by the turbulent zone caused by the break out of the waves. The
profile of Fig. 1(B) was the one chosen among the other tree profiles because
from it we could more clearly identify the largest number of relevant terraces
note3 .
Figure 2: Reconstruction of the terraces. The width of the terrace is given by the minimal distance between the two parallel dashed lines. $n$ | $d_{n}(m)$ | $l_{n}(km)$
---|---|---
1 | -30.01 | 6.06 $\pm$ 0.05
2 | -41.86 | 1.59 $\pm$ 0.05
3 | -54.01 | 2.93 $\pm$ 0.05
4 | -61.14 | 1.73 $\pm$ 0.05
5 | -66.69 | 2.21 $\pm$ 0.05
6 | -74.33 | 0.80 $\pm$ 0.1
7 | -79.75 | 0.80 $\pm$ 0.1
8 | -85.30 | 0.80 $\pm$ 0.1
Table 1: Terrace widths and depths. While the depths present no representative
deviation, the deviation in the widths become larger for deeper terraces. The
deviation in the widths is estimated by calculating the widths assuming many
possible configurations between the placement of the two parallel lines used
to calculate the widths.
## III The Devil’s Staircase
Frequency-locking is a resonant response occurring in systems of coupled
oscillators or oscillators coupled to external forces. The first relevant
report about this phenomenon was given by Christian Huygens in the 17${}^{t}h$
century. He observed that two clocks back to back in the wall, set initially
with slightly different frequencies, would have their oscillations coupled by
the energy transfer throughout the wall, and then they would eventually have
their frequencies synchronized. Usually, we expect that an harmonic, $P_{w1}$,
of one oscillatory system locks with an harmonic, $Q_{w2}$, of the other
oscillatory system, leading to a locked system working in the rational ratio
$P/Q$ jensen:1984 .
To understand what is the dynamics responsible for the onset of a frequency-
locked oscillation, that is, the reasons for which a system either locks or
unlocks, we present the simplest model one can come up with to describe a more
general oscillator. This model is described by an angle $\theta$, which is
changed (after one period) to the angle $f(\theta)$. So,
$f(\theta)=\theta+\Omega$. In order to introduce an external force in the
oscillator modeling also possible physical interactions with other
oscillators, a resonant term, $g$, is added into this model, resulting in the
following model
$f(\theta)=\theta+\Omega-g(\theta,K)\mbox{\ \ \ }(mod\mbox{\ }1),$ (1)
where
$g(\theta,K)=\frac{K}{2\pi}\sin{2\pi\theta}.$ (2)
Despite this map simplicity, the same can not be said about its complexity
argyris:1994 . Arnold (see ref. arnold:1965 ) studied this map in detail
aiming to understand how an oscillatory system would undergo into periodic
stable state when perturbed by an external perturbation.
For $K=0$, Eq. (1) represents a pure rotation, which is topological equivalent
to a twice continuously differentiable, orienting preserving mapping of the
circle onto itself [Theorem of Denjoy, see Ref. arnold:1988 ]. Therefore, the
simple Eq. (1) can be considered as a model for many types of oscillatory
systems. In fact, Eq. (1) represents a more complicated system, a three-
dimensional torus with frequencies $w_{1}$ and $w_{2}$, when viewed by a
Poincaré map. Thus, $\Omega$ in Eq. (1) represents the ratio $w_{1}/w_{2}$.
When $w_{1}/w_{2}=p/q$ (with $p\leq q$) is rational, this map has a period $p$
motion and its trajectory, i.e. the value of $\theta$, assumes the same value
after $q$ iterations. For $K=0$, the so called winding number $W$ is exactly
equal to $\Omega$, i.e. $W$=$p/q$.
For $K\neq 0$ (non-linear case) $W$ is defined by
$W(\Omega,K)=\lim_{n\to\infty}\frac{h(\theta_{0},K)+h(\theta_{1},K)+\ldots+h(\theta_{n-1},K)}{n},$
(3)
where
$h(\theta,K)=\Omega+g(\theta,K).$ (4)
For $K<1$, Eq. (1) is monotonic and invertible. For $K=1$, it develops a cubic
inflection point at $\theta=0$. The map is still invertible but the inverse
has a singularity. For $K>1$ the map is non-invertible.
Figure 3: A complete devil’s staircase, obtained from Eq. (1), for $K$=1.
Arnold wanted to understand how periodic oscillations would appear as one
increases $K$ from zero to positive values. He observed that a quasi-periodic
oscillation, for an irrational $\Omega$ and $K=0$, would turn into a periodic
oscillation as one varies $K$, from zero to positive values. He demonstrated
that a periodic oscillation has probability zero of being found for $K=0$
(rational numbers set is countable while the irrational numbers set is
uncountable) and positive probability of being found for $K>0$. He also
observed that fixing a positive value $K$, the winding number $W$ [Eq. (3)] is
a continuous but not differentiable function of $\Omega$, as one can see in
Fig. 3, forming a stair-like structure.
If $W(\Omega,K)$ is rational, it can be represented by the ratio between two
integer numbers as $W=\frac{P}{Q}$. At this point, the frequency $\Omega$ and
the frequency of the function $g$ are locked, producing the phenomenon of
frequency-locking, when $W(K,\Omega)$ does not change its value within an
interval $\Delta\Omega$ (a plateau) of values of $\Omega$. In fact, smaller is
the denominator $Q$, larger is the interval $\Delta\Omega$.
As one changes $\Omega$, plateaus for $W$ rational appear following a natural
order described by the Farey mediant. Given two plateaus that represent a
$\frac{P_{1}}{Q_{1}}$ and a $\frac{P_{3}}{Q_{3}}$ winding numbers, with
plateau widths of $\Delta\Omega_{1}$ and $\Delta\Omega_{3}$, respectively,
there exists another plateau positioned at a winding number $W$ within the
interval $[\frac{P_{1}}{Q_{1}},\frac{P_{3}}{Q_{3}}]$ given by
$\frac{P_{2}}{Q_{2}}=\frac{P_{1}+P_{3}}{Q_{1}+Q_{3}}.$ (5)
The Farey mediant gives the rational with the smallest integer denominator
that is within the interval $[\frac{P_{1}}{Q_{1}},\frac{P_{3}}{Q_{3}}]$.
Therefore, the $\Delta\Omega(\frac{P_{2}}{Q_{2}})$ plateau is smaller than
$\Delta\Omega(\frac{P_{1}}{Q_{1}})$ and $\Delta\Omega(\frac{P_{3}}{Q_{3}})$,
but is bigger than any other possible plateau. Organizing the rationals
according to the Farey mediant creates an hierarchical level of rationals,
which are called Farey Tree. The plateaus $\Delta\Omega_{1}$ and
$\Delta\Omega_{3}$ are regarded as the parents, and $\Delta\Omega_{2}$ as the
daughter plateau.
The interesting case of Eq. (1) for our purpose, is exactly when $K$=1. For
that case, one can find periodic orbits with any possible rational winding
number, as one varies $\Omega$. What results in a zero (probability) measure
of finding quasi-periodic oscillation in Eq. (1), by a random chosen of the
$\Omega$ value. Also, for $K>1$, due to the overlap of resonances (periodic
oscillation), chaos is possible.
The devil’s staircase can be fully characterized by the relations between the
plateaus widths, and the relations between the gaps between two of them. While
the plateau widths are linked to the probability one has to find periodic
oscillations, the gaps widths between plateaus are linked to the probability
one has to find quasi-periodic oscillations, in Eq. (1).
There are many scaling laws relating the plateau widths jensen:1983 ;
jensen:1984 ; cvitanovic:1985 . There are local scalings, which relate the
widths of plateaus that appear close to a specific winding number, for example
the famous golden mean $W_{G}=\frac{\sqrt{5}-1}{2}$. However, we will focus
our attention in the global scalings, which can be experimentally observed,
and only for the case where $K$=1. For this case, we are interested in two
scalings. The one that relates the plateau widths with the respective winding
numbers in the form $\frac{1}{Q}$ (the largest plateaus), and the one that
describes the structure of the complementary set to the plateaus
$\Delta\Omega$, i.e. the structure of the gaps between plateaus. The structure
of the plateaus is a Cantor set as well as the structure of the complementary
set. Therefore, a characterization of these sets can be done in terms of the
fractal dimension $D_{0}$ farmer:1983 of the complementary set.
The first scaling is jensen:1984
$\Delta\Omega(\frac{1}{Q})\propto\frac{1}{Q^{\gamma}}\mbox{\ \ }(\gamma>3),$
(6)
The second scaling relates the widths of the complementary set as one goes to
smaller and smaller scales. These widths are related to a power-scaling law
whose coefficient $D_{0}$ is the fractal dimension of the complementary set.
For $K=1$, we have that the fractal dimension of the complementary set is
$D_{0}\cong 0.87$. This is an universal scaling. Since the complementary set
of the plateaus represents the irrational rotations, the smaller is its
fractal dimension, the smaller is the probability of finding quasi-periodic
oscillation.
For experimental data, the determination of $D_{0}$ is difficult to obtain
because the dimension measures a microscopic quantity of the plateaus widths,
and in experimental data one can only observe the largest plateaus.
Fortunately, an approximation $D^{\prime}$ for $D_{0}$ can be obtained from
the largest plateaus by using the idea proposed in hentschel:1983 ,
$\left(\frac{S^{\prime}}{S}\right)^{D^{\prime}}+\left(\frac{S^{\prime\prime}}{S}\right)^{D^{\prime}}=1.$
(7)
where $S^{\prime}$, $S$, and $S^{\prime\prime}$ are represented in Fig. 4.
Figure 4: Representation of the gaps $S^{\prime\prime}$, $S^{\prime}$, and
$S$, used to estimate the fractal dimension $D_{0}$ of the complementary set,
using Eq. (7).
In case one has $K\cong 1$ ($K<1$), we do not have a complete devil’s
staircase. In other words, winding numbers with denominators larger than a
given ${\widetilde{Q}}$ are cut off from the Farey Tree. Using this
information we can estimate the value of $K$ through the largest denominator
observed jensen:1984
${\widetilde{Q}}\geq\frac{1}{1-K}.$ (8)
Finally, we would like to stress that while in Eq. (1) the plateaus of the
devil’s staircase are positioned at winding numbers defined by Eq. (3), in
nature, devil’s staircases have plateaus positioned at some accessible
measurement.
In the driven Rayleigh-Bénard experiment stavans:1985 ; jensen:1985 ,
convection rolls appear in a small brick-shaped cell filled with mercury, for
a critical temperature difference between the upper and lower plates. As one
perturbs the cell by a constant external magnetic field parallel to the axes
of the rolls and by the introduction of an AC electrical current sheet
pulsating with a frequency $f_{e}$ and amplitude $B$, a devil’s staircase is
found in the variable $f_{i}$, the main frequency of the power spectra of the
fluid velocity. As one varies the external frequency $f_{e}$, stable
oscillations take place at a frequency ratio $f_{i}/f_{e}$, for a given value
of $f_{e}$. In analogy to the devil’s staircase of Eq. (1), $f_{e}$ should be
thought as playing the same role of $\Omega$ in Eq. (1), and the ratio
$f_{i}/f_{e}$ as playing the same role of the winding number $W$.
A devil’s staircase can also be observed (see Ref. baptista:2004 ) in the
amount of information $H$ (topological entropy) that an unstable chaotic set
has in terms of an interval of size $\epsilon$, used to create the set. To
generate the unstable chaotic set, we eliminate all possible trajectories of a
stable chaotic set that visits this interval $\epsilon$. In analogy with the
devil’s staircase of the circle map, $\epsilon$ should be related to
$\Delta\Omega$, while $H$ to $W$.
The first proof of a complete devil’s staircase in a physical model was given
in Ref. bak:1982 , in the one-dimensional ising model with convex long-range
antiferromagnetic interactions.
In Ref. jin:1994 , it was found that a model for the El Niño, a phenomenon
that is the result of a tropical ocean-atmosphere interaction when coupled
nonlinearly with the Earth’s annual cycle, could undergo a transition to chaos
through a series of frequency-locked steps. The overlapping of these
resonances, which are the steps of the devil’s staircase, leads to the chaotic
behavior.
## IV A devil’s staircase in the Southern Brazilian Continental Shelf
The main premise that guides the application of the devil’s staircase model to
the Shelf is that the rules found in quantities related to the widths and
depths of the terraces obey the same rules found in a complete devil’s
staircase for the frequency-locked intervals $\Delta\Omega$ and their rational
winding number, $W$. Thus, we assume that the terrace widths $l_{n}$ play the
same role of the frequency-locked intervals $\Delta\Omega$, and the terrace
depths play the same role of the rational winding number $W$.
In order to interpret the Shelf as a devil’s staircase, we have to show that
the terraces appear in positions which respect the Farey mediant, the rule
that describes the winding number ”positions” of the many plateaus. For that,
we verify whether the position of the terraces at $d_{n}$ can be associated to
hypothetical frequency ratios, denoted by $w_{n}=\frac{p_{n}}{q_{n}}$, which
respects the Farey mediant. In doing so, we want that the metric of three
adjacent terraces respects the Farey mediant. In addition, we also assume that
the larger terraces are the parents of the Farey Tree, while the smaller
terraces between two larger ones are the daughters. Thus, for each triple of
terrace, we want that
$\frac{d_{n+2}+d_{n}}{d_{n+1}}=\frac{w_{n+2}+w_{n}}{w_{n+1}}.$ (9)
One could have considered other ways to relate the depths and the frequency
ratios. The one chosen in Eq. (9) is used in order to account for the fact
that while $d_{n}$ is negative $w_{n}$ is not.
From Eq. (9) it becomes clear that for the further analysis the depth of a
particular terrace does not play a so important role as the ratio between the
depths of triple of terraces that contains this particular terrace. These
ratios may eliminate the influence of the local morphology and the influence
of the local sea level dynamics into the formation of the Shelf. Therefore,
the proposed quantity might be suitable for an integrate analysis of the
different Shelfs all over the world, specially the ones affected by local
geomorphological characteristics.
From the Farey mediant, we have a way to obtain the frequency ratios
associated to each terrace,
$w_{n+1}=\frac{p_{n}+p_{n+2}}{q_{n}+q_{n+2}}.$ (10)
Therefore, combining Eq. (9) and Eq. (10), we obtain
$\frac{p_{n}+p_{n+2}}{q_{n}+q_{n+2}}=\frac{\frac{p_{n+2}}{q_{n+2}}+\frac{p_{n}}{q_{n}}}{E},$
(11)
which results in
$(p_{n}+p_{n+2})(E-1)q_{n}q_{n+2}p_{n}=p_{n}q_{n+2}^{2}+q_{n}^{2}p_{n+2},$
(12)
where $E$ is defined by
$\frac{d_{n+2}+d_{n}}{d_{n+1}}=E.$ (13)
$n$ | $p_{n}$ | $q_{n}$
---|---|---
1 | 1 | 8
2 | 2 | 17
3 | 1 | 9
4 | - | -
5 | - | -
6 | 1 | 17
7 | 2 | 35
8 | 1 | 18
Table 2: Integers associated with the $n$ considered terraces, with
$n=1,\ldots,8$.
We do not expect to have Eq. (12) satisfied. We only require that the
difference between the left and right hand sides of this equation, regarded as
$\delta\epsilon$, is the lowest possible, among all possible values for
$p_{m}$ and $q_{m}$ (with $m=n,n+2$), for a given $E$, with the restriction
that the considered largest terraces are related to largest plateaus of Eq.
(1), and thus $p_{m+2}$=$p_{m}$ and $q_{m+2}$=$q_{m}+1$, and
$\delta\epsilon\ll 1$. Doing so, we find the rationals associated with the
terraces, which are shown in table 2. The minimal value of $\delta\epsilon$,
denoted by $\min{[\delta\epsilon]}$, is
$\min{[\delta\epsilon(d_{1},d_{2},d_{3})]}$=0.032002, with $p_{1}/q_{1}$=$1/8$
for the terrace 1, and $p_{3}/q_{3}$=$1/9$, for the terrace 3. We also find
that $\min{[\delta\epsilon(d_{6},d_{7},d_{8})]}$=0.002344, with
$p_{6}/q_{6}$=$1/17$, for the terrace 6, and $p_{8}/q_{8}$=$1/18$, for the
terrace 8\. These minimal values can be seen in Figs. 5(A-B), where we show
the values of $\delta\epsilon(d_{1},d_{2},d_{3})$ [in (A)] and the values of
$\delta\epsilon(d_{6},d_{7},d_{8})$ [in (B)], for different values of $p$ and
$q$. Using bigger values for $p$ has the only effect to increase the value of
$\delta\epsilon$.
We have not identified rationals that can be associated with the terraces $4$
and $5$, which means that for $p$ and $q$ within $p=[1,50]$ and $q=[1,400]$,
we find that $\delta\epsilon>1$. We have assumed that they could be either a
daughter or a parent.
From now on, when convenient, we will drop the index $n$ and represent each
terrace by the associated frequency ratio. So, the terrace 1, for $n$=1, is
represented as the terrace with $w=1/8$.
Figure 5: [Color online] (A) Values of $\delta\epsilon(d_{1},d_{2},d_{3})$ and
(B) $\delta\epsilon(d_{6},d_{7},d_{8})$, for different values of $p$ and $q$.
$\delta\epsilon$ is the difference between the left and the right hand sides
of Eq. (12).
Table 2 can be represented in the form of the Farey Tree as shown in Fig. 6.
The branch of rationals in the Farey Tree in the form $1/q$ belongs to the
most stable branch, which means that the observed terraces should have the
largest widths. We believe that the other less important branches of the
complete devil’s staircase present in the data were smoothed out by the action
of the waves and the flow streams throughout the time, and at the present time
cannot be observed.
Notice that as the time goes by, the frequency ratios are increasing their
absolute value, which means that if this tendency is preserved in the future,
we should expect to see larger terraces.
Figure 6: Farey Tree representing the frequency ratios associated with the
major terraces.
Figure 7: Scaling between the $1/q$-terrace widths and the value of $q$.
In the following, we will try to recover in the experimental profile, the
universal scaling laws of Eqs. (6) and (7). Regarding Eq. (6), we find that
$l$ scales as $1/q^{-3.60}$, as shown in Fig. 7, which is the expected global
universal scaling for a complete devil’s staircase. Regarding Eq. (7), and
calculating $S^{\prime}$, $S^{\prime\prime}$, and $S$ using the triple of
terraces with widths $l(w=1/8)$, $l(w=2/17)$, and $l(w=1/9)$, as represented
in Fig. 4, we find $D^{\prime}$=0.89. Using the triple of terraces
($n$=3,$n$=4,$n$=5), we find that $D^{\prime}$=0.87. Both results are very
close from the universal fractal dimension $D_{0}\cong 0.87$, found for a
complete devil’s staircase.
## V Fitting the SBCS
Figure 8: Magnifications of the small box of Fig. 3, showing the plateaus of
the devil’s staircase of Eq. (1) that appear for the same frequency ratios
associated with the triple of terraces $w$=(1/8,2/17,1/9), in (A), and
$w$=(1/17,2/35,1/18), in (B).
Motivated by our previous results, we fit the observed Shelf as a complete
devil’s staircase, using Eq. (1). Notice that the only requirement for Eq. (1)
to generate a complete devil’s staircase is that the function $g$ has a cubic
inflection point at the critical parameter $K=1$. Whether Eq. (1) is indeed an
optimal modeling for the Shelf is beyond the scope of the present study. We
only chose this map because it is a well known system and it captures most of
the relevant characteristic a dynamical systems needs to fulfill in order to
create a devil’s staircase.
We model the SBCS as a complete devil’s staircase, but we rescale the winding
number $W$ into the observed terrace depth. So, we transform the complete
devil’s staircase of Fig. 8 as good as possible into the profile of Fig. 1(B),
by rescaling the vertical axis of the staircase in Fig. 8.
We do that by first obtaining the function $F$ (see Fig. 9) whose application
into the terrace depth $d(w)$ gives the frequency ratio
$w_{n}=\frac{p_{n}}{q_{n}}$ associated with the terrace.
For the triple of terraces $w$=(1/8,2/17,1/9), we obtain
$F(d[km])=0.14219+0.00057853d[km],$ (14)
and for the triple of terraces $w$=(1/17,2/35,1/18), we obtain
$F(d[km])=0.080941+0.00029786d[km].$ (15)
Therefore, we assume that, locally, the frequency ratios are linearly related
to the depth of the terraces.
Then, we rescale the vertical axis of the staircases in Figs. 8(A-B) and
calculate an equivalent depth, $d$, for the winding number $W$ by using
$d=F^{-1}(W),$ (16)
Figure 9: The function $F$, which is a linear relation between the frequency
ratios associated with the terraces and their depths for the triple of
terraces $w$=(1/8,2/17,1/9) and $w$=(1/17,2/35,1/18).
Figure 10: [Color online] (A) Rescaling of Fig. 8(A) in black, showing that
the devil’s staircase fits well the terraces with $w$=(1/8,2/17,1/9) of the
profile of Fig. 1(B), in gray. (B) Rescaling of Fig. 8(B) in black, showing
that the devil’s staircase fits well the terraces with $w$=(1/17,2/35,1/18) of
the profile of Fig. 1(B), in gray.
We also allow tiny adjustments in the axes for a best fitting. The result is
shown in Fig. 10(A) for the triple of terraces $w$=(1/8,2/17,1/9) and in Fig.
10(B) for the triple of terraces $w$=(1/17,2/35,1/18). We see that locally,
for a short time interval, we can have a good agreement of the terrace widths
and positions, with the rescaled devil’s staircase. However, globally, the
fitting in (A) does not do well, as it is to be expected since the function
$F$ is only locally well defined and it changes depending on the depths of the
terraces. Notice however that this short time interval is not so short since
the time interval correspondent to a triple of terraces is of the order of a
few hundred years.
The assumption made that $K=1$ is also supported from Eq. (8). Using this
equation, we can obtain an estimation of the maximum value of $K$ from a
terrace with a frequency ratio that has the largest denominator. In our case,
we observed $w=2/35$. Using ${\widetilde{Q}}$=35 in Eq. (8), we obtain $K\leq
0.97$.
In Fig. 10(A), we see a 1/7 plateau positioned in the zero sea level. That is
the current level. Thus, the model predicts that nowadays we should have a
large terrace, which might imply in an average stabilization of the sea level
for a large period of time. However, this prediction might not correspond to
reality if the sea dynamics responsible for the creation of the observed
Continental Shelf suffered structurally modifications.
## VI Conclusions
We have shown some experimental evidences that the Southern Brazilian
Continental Shelf (SBCS) has a structure similar to the devil’s staircase.
That means that the terraces found in the bottom of the sea are not randomly
distributed but they occur following a dynamical rule. This finding lead us to
model the SBCS as a complete devil’s staircase, in which, between two real
terraces, we suppose an infinite number of virtual (smaller) ones. We do not
find these later ones, either because they have been washed out by the stream
flow or simply due to the fact that the time period in which the sea level
dynamics (SLD) stayed locked was not sufficient to create a terrace. By our
hypothesis, the SLD creates a terrace if it is a dynamics in which two
relevant frequencies are locked in a rational ratio.
This special phase-locked dynamics possesses a critical characteristic: large
changes in some parameter responsible for a relevant natural frequency of the
SLD might not destroy the phase-locked regime, which might imply that the
averaged sea level would remain still. On the other hand, small changes in the
parameter associated with an external forcing of the SLD could be
catastrophic, inducing a chaotic SLD, what would mean a turbulent averaged sea
level rising/regression.
In order to interpret the Shelf as a devil’s staircase, we have shown that the
terraces appear in an organized way according to the Farey mediant, the rule
that describes the way plateaus appear in the devil’s staircase. That allow us
to ”name” each terrace depth, $d_{n}$, by a rational number, $w_{n}$, regarded
as the hypothetical frequency ratio. Arguably, these ratios represent the
ratio between real frequencies that are present in the SLD. It is not the
scope of the present work to verify this hypothesis, however, one way to check
if the hypothetical frequency ratios are more than just a mathematical
artifact would be to check if the SLD has, nowadays, two relevant frequencies
in a ratio 1/7, as predicted.
The newly proposed approach to characterize the SBCS rely mainly on the ratios
between terraces widths and between terraces depths. While single terrace
widths and depths are strongly influenced by local properties of the costal
morphology and the local sea level variations, the ratios between terrace
widths and depths should be a strong indication of the global sea level
variations. Therefore, the newly proposed approach has a general character and
it seems to be appropriated as a tool of analysis to other Continental Shelves
around the world.
Reminding that the local morphology of the studied area, the ”Serra do Mar”
does not have a strong impact in the formation of the Shelf and assuming that
the local SLD is not directed involved in the formation of the large terraces
considered in our analyses, thus, our results should reflect mainly the action
of the global SLD.
If the characteristics observed locally in the São Paulo Bight indeed reflect
the effect of the global SLD, then the global SLD might be a critical system.
Hopefully, the environmental changes caused by the modern men have not yet
made any significant change in a relevant parameter of this global system.
Acknowledgements: MSB was partially supported by the “Fundação para a Ciência
e Tecnologia” (FCT) and the Max-Planck Institute für die Physik komplexer
Systeme.
## References
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* (5) J. T. Andrews, J. of Quaternary Science, 13, 3 (1988).
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* (9) L. A. Conti and V. V. Furtado, IGCP. Abstracts, 12 (2001).
* (10) I. C. S. Correa, Marine Geology 130, 163 (1996).
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* (12) U. C. Herzfeld, I. I. Kim, and J. A. Orcutt, Mathematical Geology, 27, 3 (1995).
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* (15) The profiles used in this work were obtained by the use of polinomial spilines in a two dimensional grip of points 10km appart. This also imposes limits in the indentification of terraces that have small widths.
* (16) M. H. Jensen, P. Bak, and T. Bohr, Phys. Rev. A 30, 1960 (1984).
* (17) J. Argyris, G. Faust, and M. Haase, An exploration of chaos (North-Holland, New York, 1994).
* (18) V. I. Arnold, Trans. of the Am. Math. Soc. Ser. 2 46, 213 (1965).
* (19) V. I. Arnold, Geometrical methods in the theory of ordinary differential equations (Springer, Berlin 1988).
* (20) M. H. Jensen, P. Bak, and T. Bohr, Phys. Rev. Lett. 50, 1637 (1983).
* (21) P. Cvitanovic, B. Shraiman, and B. Söderberg, “Scaling laws for mode locking in circle maps”, Phys. Scripta A 32, 263 (1985).
* (22) J. D. Farmer, E. Ott, and J. A. Yorke, Physica D 7, 153 (1983).
* (23) H. G. E. Hentschel and I. Procaccia, Physica D 8, 435 (1983).
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|
arxiv-papers
| 2008-10-24T11:14:13 |
2024-09-04T02:48:58.426246
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. S. Baptista and L. A. Conti",
"submitter": "Murilo Baptista S.",
"url": "https://arxiv.org/abs/0810.4430"
}
|
0810.4501
|
# Dispersion and fidelity in quantum interferometry
D.S. Simon Dept. of Electrical and Computer Engineering, Boston University, 8
St. Mary’s St., Boston, MA 02215 A.V. Sergienko Dept. of Electrical and
Computer Engineering, Boston University, 8 St. Mary’s St., Boston, MA 02215
Dept. of Physics, Boston University, 590 Commonwealth Ave., Boston, MA 02215
T.B. Bahder Charles M. Bowden Research Facility, US Army RDECOM, Aviation and
Missile Research, Development, and Engineering Center, Redstone Arsenal, AL
35898
###### Abstract
We consider Mach-Zehnder and Hong-Ou-Mandel interferometers with nonclassical
states of light as input, and study the effect that dispersion inside the
interferometer has on the sensitivity of phase measurements. We study in
detail a number of different one- and two-photon input states, including Fock,
dual Fock, N00N states, and photon pairs from parametric downconversion.
Assuming there is a phase shift $\phi_{0}$ in one arm of the interferometer,
we compute the probabilities of measurement outcomes as a function of
$\phi_{0}$, and then compute the Shannon mutual information between $\phi_{0}$
and the measurements. This provides a means of quantitatively comparing the
utility of various input states for determining the phase in the presence of
dispersion. In addition, we consider a simplified model of parametric
downconversion for which probabilities can be explicitly computed
analytically, and which serves as a limiting case of the more realistic
downconversion model.
###### pacs:
42.50.St,42.50.Dv,07.60.Ly,03.67.-a
## I Introduction
Interferometry is both an important tool for practical measurements and a
useful testing ground for fundamental physical principles. As a result, the
search for methods to improve the resolution of interferometers forms an
active area of study. It has been shown by a number of authors (yurke1 (1, 2,
3, 4)) that nonclassical states, in particular those with high degrees of
entanglement, when used as input to an interferometer can lead to resolutions
that approach the Heisenberg limit, the fundamental physical limit imposed by
the uncertainty principle. Most of this previous work has dealt with idealized
interferometers, with no dispersion or photon losses. Before quantum
interferometry may become a useful practical tool the question must be asked
as to how well the conclusions of these previous studies hold up in more
realistic and less idealized situations. In this paper, we will attempt to
take the next step along this road by adding dispersion to the apparatus and
examining what effect this has on the phase sensitivity of interferometry with
nonclassical input. The motivation for this work is the desire to ultimately
construct quantum sensors that can measure the values of external fields by
measuring the phases shifts they produce in an interferometer.
In particular, the nonclassical input states we will consider are (i) Fock
states $|N,0\rangle$ which have a fixed number of photons incident on one
input port, (ii) dual or twin Fock states $|N,N\rangle$ which have the same
number of photons incident on each input port, and (iii) N00N states
${1\over\sqrt{2}}\left[|N,0\rangle+|0,N\rangle\right]$. Here,
$|N_{a},N_{b}\rangle$ denotes a state with numbers $N_{a}$ and $N_{b}$ of
photons entering each of the two interferometer input ports.
There has been a great deal of recent work on the production of nonclassical
states of light with large ($N>2$) numbers of photons by means of
postselection (for example, knill (5, 6, 7, 8)), however at present the
utility of these postselection schemes for application to practical situations
is not clear. Although this work is useful for clarifying the scientific
issues involved, it is not technologically feasible at present to use these
methods to produce the desired states on demand. Rather, postselection
produces states statistically, at random times, and therefore can not be
relied upon to produce states on demand for a quantum sensor. In addition, for
large photon number, great care must be taken to distinguish between states of
$N$ photons and those of $N-1$ photons, making it difficult to prevent mixed
states from appearing, which would change the physics involved. In contrast,
two-photon entangled states with well-defined properties can be easily
produced by parametric downconversion or other methods.
Due to the current practical difficulties of producing on demand entangled
photon states with large, well-defined $N$, we save the large $N$ case for
later study and restrict ourselves in this current paper to situations which
are both simpler and of more immediate practical interest, namely the cases of
one or two photons. Furthermore, for the two-photon case, we consider two
possibilities: (i) the photons may be uncorrelated in frequency, or (ii) the
pair may be produced through spontaneous parametric downconversion (SPDC),
resulting in anticorrelation between the two frequencies.
Our goal is to compare the usefulness of each of these cases for making phase
measurements in the presence of dispersion, so we will need a means of
quantifying the sensitivity of the interferometer with respect to these
measurements. Consider a single shot consisting of a nonclassical state of
light with a fixed number of photons being injected into the input ports of
the interferometer. Suppose some phase-dependent observable $M(\phi)$ is
measured during this shot. The usual way to define the phase sensitivity of
the measurement is by computing
$\Delta\phi=\left|{{dM}\over{d\phi}}\right|^{-1}\Delta M.$ (1)
However, this is correct only if the probability distribution of the phases
has a single peak and is approximately Gaussian in shape. A more general
strategy is to take an information-theoretical approach and to define the
quantum fidelity by means of the Shannon mutual information bahdlap (9)
$H(\Phi:M)={1\over{2\pi}}\sum_{m}\int_{-\pi}^{\pi}d\phi\;P(m|\phi)\log_{2}\left[{{2\pi
P(m|\phi)}\over{\int_{-\pi}^{\pi}P(m|\phi^{\prime})d\phi^{\prime}}}\right].$
(2)
Here, $m$ and $\phi$ are the measured values of the random variables $M$ and
$\Phi$, while $P(m|\phi)$ is the conditional probability of obtaining
measurement $m$ given the phase $\phi$ on a particular shot. In this formula,
we have also assumed maximum ignorance of the phase, i.e., we have assumed a
uniform distribution for $\phi$, $p(\phi)={1\over{2\pi}}$. Suppose that the
detectors have a characteristic time-scale $T_{D}$. Then in this context, a
single shot will consist of a well-defined number of photons entering the
apparatus simultaneously (i.e. within a temporal window much smaller than
$T_{D}$) and seperated in time from any other entering photons by a time $\geq
T_{D}$. The mutual information is a measure of the information gained per shot
about the phase $\Phi$ from a measurement of the observable $M$. In our case,
the role of $M$ will be played by the number of photons detected at each of
the output ports. For $N$ input photons, output detector C will count $l$
photons, detector $D$ will detect the remaining $N-l$ photons, and the sum in
equation 2 will become a sum over $l$, where $l=0,1,\dots N$. Throughout this
paper we will use the quantum fidelity as our measure of phase sensitivity.
Besides being of very general applicability and giving a precise, calculable
measure for the utility of a measurement, the introduction of the mutual
information provides a link to the theory of quantum information processing.
Bahder and Lopata bahdlap (9) have computed the quantum fidelity as a function
of $N$ for idealized lossless and dispersionless interferometers with Fock and
N00N state inputs. In the following sections, we will see how their results
change for the cases of $N=1$ and $N=2$ when dispersion is present.
Although not the principal focus of this paper, it should be noted that the
existence of multiple peaks in the output probability distributions invalidate
the assumptions used to derive the Heisenberg bound from the Cramer-Rao lower
bound, which makes input states with multimodal distributions especially
interesting from the point of view of the study of phase sensitivity. Note
that violations of the Heisenberg limit have recently been shown to exist in
another context, distinct from the situation examined in this paper, namely in
the context of nonlinear interferometry luis (10, 11, 12).
We will assume one branch of the interferometer has a dispersive element which
gives the photon wavenumber $k$ a frequency dependence of the form
$k(\omega)=k_{0}+\alpha(\omega-\omega_{0})+\beta(\omega-\omega_{0})^{2},$ (3)
ignoring the possibility of higher order terms. The other interferometer arm
will be assumed to be of negligible dispersion. Here, $\alpha$ is the inverse
of the group velocity, and $\beta$ is the group delay dispersion per unit
length.
In addition to the Mach-Zehnder interferometer, we will examine the fidelity
of an alternate setup used in dowl (6), in which N00N states are incident on a
single beam-splitter used as a Hong-Ou-Mandel (HOM) interferometer. We will
then be in a position to compare the possible input states and interferometer
setups, with a view to gaining insight into their relative usefulness in
practical measurements. In the two-photon cases, we must distinguish between
situations in which the photon energies (or frequencies) are correlated and
those in which they are independent. Thus, after we examine the case of
energy-uncorrelated photons, we look at photon pairs anticorrelated in energy.
We further consider two subcases of the latter: (i) a simple model which can
be solved analytically and which amounts to a simplified version of
spontaneous downconversion, and (ii) a more realistic but less analytically
tractable version of downconversion.
The plan of this paper is as follows: in section II we consider the setup for
the dispersive Mach-Zehnder interferometer and define the input states we will
use in more detail. In section III, we apply the possible one-photon inputs to
the interferometer and compute the probabilities for the various possible
outcomes. In sections IV and V respectively, we do the same for the Mach-
Zehnder interferometer with several different two-photon inputs and for the
HOM interferometer with $N=2$ N00N state input. In section VI we compute and
plot the mutual information for each of the preceeding cases as functions of
bandwidth and dispersion levels; we then compare and discuss the results for
the various cases. Finally, in section VII we repeat the same calculation for
input consisting of a photon pair produced via spontaneous parametric
downconversion before arriving at final conclusions in section VIII.
For ease of reference later, table I summarizes the specific cases we will
examine over the following sections.
Table 1: Summary of the special cases examined in the later sections of this paper. Case | # of | Interferometer | Input | Frequency
---|---|---|---|---
No. | photons | Type | State | Correlation
A | 1 | MZ | Fock | not applicable
B | 1 | MZ | N00N | not applicable
C | 2 | MZ | Fock | none
D | 2 | MZ | Dual Fock | none
E | 2 | MZ | N00N | none
F | 2 | MZ | Fock | anticorrelated
G | 2 | MZ | N00N | anticorrelated
H | 2 | MZ | Dual Fock | anticorrelated
I | 1 | HOM | N00N | none
J | 2 | HOM | N00N | none
K | 2 | HOM | N00N | anticorrelated
L | 2 | MZ | SPDC Fock | anticorrelated
## II The Dispersive Mach-Zehnder Interferometer
Consider the Mach-Zehnder interferometer of figure 1, with 50/50
beamsplitters. Assume for the moment that there is no dispersion in the
apparatus. Let $\hat{a}_{\omega}$ and $\hat{b}_{\omega}$ be operators that
annihilate photon states in the two input ports A and B. They obey the usual
canonical commutation relations with the corresponding creation operators
$\hat{a}_{\omega}^{\dagger}$ and $\hat{b}_{\omega}^{\dagger}$:
$\left[\hat{a}_{\omega},\hat{a}_{\omega^{\prime}}^{\dagger}\right]=\left[\hat{b}_{\omega},\hat{b}_{\omega^{\prime}}^{\dagger}\right]=\delta(\omega-\omega^{\prime}),$
(4)
with all other commutators vanishing. For independent photons, the input
states to the interferometer can be described in terms of the number of
photons entering the two ports:
$\displaystyle|N_{a},N_{b};\omega_{1},\dots,\omega_{N_{a}};\omega^{\prime}_{1}\dots,\omega^{\prime}_{N_{b}}\rangle$
$\displaystyle\qquad={1\over\sqrt{N_{a}!N_{b}!}}\hat{a}^{\dagger}_{\omega_{1}}\dots\hat{a}^{\dagger}_{\omega_{N_{a}}}\hat{b}^{\dagger}_{\omega^{\prime}_{1}}\dots\hat{b}^{\dagger}_{\omega^{\prime}_{N_{b}}}|0\rangle,$
(5)
where $N_{a}$ and $N_{b}$ are the number of photons in ports A and B,
respectively, and $|0\rangle$ is the vacuum state with no photons. Similarly,
$N_{c}$, $N_{d}$, $\hat{c}_{\omega}$, and $\hat{d}_{\omega}$ will represent
the photon numbers and annihilation operators at output ports C and D.
Figure 1: Mach-Zehnder interferometer with dispersion in one arm. There is
also a phase shift $\phi_{0}$ of nondispersive origin in the same arm.
The effect of the Mach-Zehnder interferometer on a given input state may be
described in terms of the scattering matrix, $S(\phi)$. The initial and final
annihilation operators are related by a scattering matrix $S(\phi)$:
$\left(\begin{array}[]{c}\hat{c}_{\omega}(\phi)\\\
\hat{d}_{\omega}(\phi)\end{array}\right)=S(\phi)\left(\begin{array}[]{c}\hat{a}_{\omega}\\\
\hat{b}_{\omega}\end{array}\right),$ (6)
where $\phi$ is the relative phase difference experienced by photons in the
two arms. In the absence of photon losses in the system, the scattering matrix
will be unitary. Then, for an ideal Mach-Zehnder interferometer, the
scattering matrix is given by
$\displaystyle S(\phi)$ $\displaystyle=$ $\displaystyle{1\over
2}\left[e^{i\phi}e^{ikL_{1}}-e^{ikL_{2}}\right]\sigma_{z}-{i\over
2}\left[e^{i\phi}e^{ikL_{1}}+e^{ikL_{2}}\right]\sigma_{x}$ (9)
$\displaystyle=$ $\displaystyle-
ie^{ikL_{1}}e^{i\phi/2}\left(\begin{array}[]{cc}-\sin{\phi\over
2}&\cos{\phi\over 2}\\\ \cos{\phi\over 2}&\sin{\phi\over
2}\end{array}\right),$
where the Pauli matrices are
$\sigma_{x}=\left(\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right)\qquad\mbox{and}\qquad\sigma_{z}=\left(\begin{array}[]{cc}1&0\\\
0&-1\\\ \end{array}\right).$ (10)
In this scattering matrix we have assumed (as we will assume henceforth) that
the lengths of the two interferometer arms are equal, $L_{1}=L_{2}$. Using
this matrix in equation 6, we can invert the equation and take adjoints to
arrive at the following result:
$\displaystyle\hat{a}_{\omega}^{\dagger}$ $\displaystyle=$ $\displaystyle
i\left[\hat{c}_{\omega}^{\dagger}\sin{\phi\over
2}-\hat{d}_{\omega}^{\dagger}\cos{\phi\over 2}\right]e^{i\phi/2}$ (11)
$\displaystyle\hat{b}_{\omega}^{\dagger}$ $\displaystyle=$
$\displaystyle-i\left[\hat{c}_{\omega}^{\dagger}\cos{\phi\over
2}+\hat{d}_{\omega}^{\dagger}\sin{\phi\over 2}\right]e^{i\phi/2}$ (12)
We assume that the frequency distribution for each incoming photon is Gaussian
and that each Gaussian has the same width and central frequency, of the form
$e^{-{1\over 2}\sigma(\omega-\omega_{0})^{2}}$. Input and output states will
either be states of definite photon number in the sense that they are
eigenstates of number operators of the form $\hat{N}_{j}=\int d\omega
a^{(j)\dagger}_{\omega}a^{(j)}_{\omega}$ (where $a_{\omega}^{(j)}$ is the
annihilation operator for photons at the jth port), or else superpositions of
such states.
We introduce dispersion to the upper branch of the interferometer by giving
the wavenumber $k$ a frequency dependence of the form in equation 3. We assume
that the dispersion in the other branch of the interferometer is negligible,
i.e. that $k(\omega)=k_{0}$ in that branch. The length of the portion of the
upper arm for which $k(\omega)$ differs from $k_{0}$ will be denoted $L$,
where $0\leq L\leq L_{1}$. In addition to any phase difference resulting from
the asymmetric dispersion, we also assume that photons travelling through the
upper branch of the interferometer gain an additional phase difference
$\phi_{0}$ relative to the lower branch. $\phi_{0}$ is any phase difference of
nondispersive origin that may be present in the setup; this may be due to a
difference in path length, or an interaction of one arm of the interferometer
with an external field. Note that for our setup, the assumption of a balanced
interferometer entails no loss of generality; to account for an unbalanced
interferometer, it suffices to simply include a term of the form
$k_{0}(L_{1}-L_{2})$ inside the phase factor $\phi_{0}$.
In the presence of the dispersion, the scattering matrix will now be of the
form
$\displaystyle S(\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}e^{ik_{0}L_{1}}\left(\begin{array}[]{cc}e^{i\phi(\omega)}-1&-i\left(e^{i\phi(\omega)}+1\right)\\\
-i\left(e^{i\phi(\omega)}+1\right)&-\left(e^{i\phi(\omega)}-1\right)\end{array}\right)$
(15) $\displaystyle=$ $\displaystyle-
ie^{ik_{0}L_{1}}e^{i\phi(\omega)/2}\left(\begin{array}[]{cc}-\sin{{\phi(\omega)}\over
2}&\cos{{\phi(\omega)}\over 2}\\\ \cos{{\phi(\omega)}\over
2}&\sin{{\phi(\omega)}\over 2}\end{array}\right)$ (18)
where for future convenience we have shifted the frequency dependence into a
new phase angle by defining
$\phi(\omega)=\phi_{0}+\alpha L(\omega-\omega_{0})+\beta
L(\omega-\omega_{0})^{2}.$ (19)
Consider $N$ photons entering the interferometer and assume for now that their
frequencies are independent variables. The Fock, dual Fock, and N00N input
states are of the form:
$\displaystyle|N,0\rangle_{\sigma}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{N!}}\left({{\sigma}\over\pi}\right)^{N/4}\int
d\omega_{1}\dots d\omega_{N}e^{-{\sigma\over
2}\sum_{j=1}^{N}(\omega_{j}-\omega_{0})^{2}}$ (20)
$\displaystyle\qquad\qquad\qquad\times\hat{a}^{\dagger}_{\omega_{1}}\dots\hat{a}^{\dagger}_{\omega_{N}}|0\rangle$
$\displaystyle|N,N\rangle_{\sigma}$ $\displaystyle=$
$\displaystyle{1\over{N!}}\left({{\sigma}\over\pi}\right)^{N/2}\int
d\omega_{1}\dots d\omega_{2N}e^{-{\sigma\over
2}\sum_{j=1}^{2N}(\omega_{j}-\omega_{0})^{2}}$ (21)
$\displaystyle\qquad\qquad\times\hat{a}^{\dagger}_{\omega_{1}}\dots\hat{a}^{\dagger}_{\omega_{N}}\hat{b}^{\dagger}_{\omega_{N+1}}\dots\hat{b}^{\dagger}_{\omega_{2N}}|0\rangle$
and
$\displaystyle{1\over\sqrt{2}}\left[|N,0\rangle+|0,N\rangle\right]_{\sigma}$
$\displaystyle\quad={1\over\sqrt{N!}}{1\over\sqrt{2}}\left({{\sigma}\over\pi}\right)^{N/4}\;\int
d\omega_{1}\dots d\omega_{N}e^{-{\sigma\over
2}\sum_{j=1}^{N}(\omega_{j}-\omega_{0})^{2}}$
$\displaystyle\quad\qquad\times\left[\hat{a}^{\dagger}_{\omega_{1}}\dots\hat{a}^{\dagger}_{\omega_{N}}+\hat{b}^{\dagger}_{\omega_{1}}\dots\hat{b}^{\dagger}_{\omega_{N}}\right]|0\rangle,$
(22)
where the bandwidth of the incident beams is given by
$\Delta\omega\equiv\sigma^{-1/2}$. If the photons are produced by SPDC, then
the frequencies must occur in pairs with the photons in each pair being equal
distances above or below the pump frequency; we will consider this situation
in simplified form in section IV.2 and in a more realistic form in section
VII.
Suppose that one of the $N$-photon or $2N$-photon states described above is
input to the interferometer. Write this input state as $|\psi_{in}\rangle$.
Then, assuming that the frequencies of the final photons are not measured, we
want the joint probabilities to find $N_{c}$ photons at detector $C$ and
$N_{d}$ photons at detector $D$ (with $N_{c}+N_{d}=N_{a}+N_{b}$) for a given
nondispersive phase shift $\phi_{0}$ in the upper interferometer arm. These
probabilities can be expressed in the form
$P(N_{c},N_{d}|\phi_{0})=\langle\psi_{in}|\hat{\pi}(N_{c},N_{d};\phi_{0})|\psi_{in}\rangle,$
(23)
where the projective operator $\hat{\pi}(N_{c},N_{d};\phi_{0})$ is defined as
$\hat{\pi}(N_{c},N_{d},\phi_{0})=\int\;d\Omega|N_{c},N_{d};\Omega,\phi_{0}\rangle\langle
N_{c},N_{d};\Omega,\phi_{0}|,$ (24)
with
$|N_{c},N_{d};\Omega,\phi_{0}\rangle={1\over\sqrt{N_{c}!\;N_{d}!}}\hat{c}_{\Omega_{1}}^{\dagger}\dots\hat{c}_{\Omega_{N_{c}}}^{\dagger}\hat{d}_{\Omega_{1}^{\prime}}^{\dagger}\dots\hat{d}_{\Omega_{N_{d}}^{\prime}}^{\dagger}|0\rangle.$
(25)
Here we have suppressed the $\phi_{0}$-dependence of the $\hat{c}_{\Omega}$
and $\hat{d}_{\Omega^{\prime}}$ operators for notational simplicity, and have
represented the collection of output frequencies
$\left\\{\Omega_{1},\dots,\Omega_{N_{c}},\Omega_{1}^{\prime},\dots,\Omega_{N_{d}}^{\prime}\right\\}$
by the single symbol $\Omega$. Similarly, $d\Omega$ is being used as shorthand
for the full frequency integration measure $d\Omega_{1}\dots
d\Omega_{N_{c}}d\Omega_{1}^{\prime}\dots d\Omega_{N_{d}}^{\prime}$. These
probabilities may also be expressed in the form
$P(N_{c},N_{d}|\phi_{0})=\int d\Omega\left|\langle
N_{c},N_{d};\Omega,\phi_{0}|\psi_{in}\rangle\right|^{2}.$ (26)
From equation 2 the mutual information between the phase $\Phi$ and the output
photon numbers $M$ is then
$H(\Phi:M)={1\over{2\pi}}\sum_{N_{c},N_{d}}\int_{-\pi}^{\pi}d\phi_{0}\;P(N_{c},N_{d}|\phi_{0})\log_{2}\left[{{2\pi
P(N_{c},N_{d}|\phi_{0})}\over{\int_{-\pi}^{\pi}d\phi_{0}\;P(N_{c},N_{d}|\phi_{0})}}\right].$
(27)
We note that the probabilities $P(N_{c},N_{d},|\phi_{0})$ are also conditional
upon the values of $\alpha$, $\beta$, and $\sigma$, although we do not
explicitly include these parameters in the notation for the probabilities for
the sake of notational simplicity. We now restrict ourselves to the cases
$N=1$ and $N=2$, and proceed in the following sections to compute the mutual
information $H$ for a number of different possible input states.
## III MZ Interferometry With One-Photon Input
In this section, we begin with the cases in which there is only one photon in
the initial state.
Case A: One-photon Fock state. We introduce the normalized input state
$|\psi_{in}\rangle_{\sigma}=|10\rangle_{\sigma}=\sqrt[4]{\sigma\over\pi}\int
d\omega\;e^{-{1\over
2}\sigma(\omega-\omega_{0})^{2}}\hat{a}_{\omega}^{\dagger}|0\rangle,$ (28)
representing a single photon incident on port A. Using relations 11 and 12,
this is equivalent to
$\displaystyle|10\rangle_{\sigma}$ $\displaystyle=$ $\displaystyle{1\over
2}\sqrt[4]{\sigma\over\pi}\int d\omega\;e^{-{1\over
2}\sigma(\omega-\omega_{0})^{2}}$ (29)
$\displaystyle\times\left[\hat{c}^{\dagger}_{\omega}\left(e^{i\phi(\omega)}-1\right)-i\hat{d}^{\dagger}_{\omega}\left(e^{i\phi(\omega)}+1\right)\right]|0\rangle.$
The photon may leave the interferometer via either port C or port D. We assume
that the detectors count the number of photons leaving the apparatus but do
not measure their frequencies. Therefore, we must integrate over the final
frequencies. The output state is then measured using the projective operators
$\displaystyle\hat{\pi}(1,0)$ $\displaystyle=$ $\displaystyle\int
d\Omega\;\hat{c}_{\Omega}^{\dagger}|0\rangle\langle 0|\hat{c}_{\Omega}$ (30)
$\displaystyle\hat{\pi}(0,1)$ $\displaystyle=$ $\displaystyle\int
d\Omega\;\hat{d}_{\Omega}^{\dagger}|0\rangle\langle 0|\hat{d}_{\Omega}.$ (31)
Expectation values of these operators give the probabilities of measurement
outcomes:
$\displaystyle P(1,0|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left[1-{{e^{-{{\alpha^{2}L^{2}}\over{4r_{1}^{2}\sigma}}}}\over\sqrt{r_{1}}}\cos\left(\phi_{0}+{\theta_{1}\over
2}-{{\alpha^{2}\beta L^{3}}\over{4r_{1}^{2}\sigma^{2}}}\right)\right]$ (32)
$\displaystyle P(0,1|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left[1+{{e^{-{{\alpha^{2}L^{2}}\over{4r_{1}^{2}\sigma}}}}\over\sqrt{r_{1}}}\cos\left(\phi_{0}+{\theta_{1}\over
2}-{{\alpha^{2}\beta L^{3}}\over{4r_{1}^{2}\sigma^{2}}}\right)\right]$ (33)
In the previous two lines, we have introduced some notation that will be
convenient for simplifying the results of this and the following sections. The
parameters $r_{1},r_{2},\theta_{1},\theta_{2}$ are defined by (see figure 2):
$\displaystyle r_{1}^{2}$ $\displaystyle=$ $\displaystyle 1+\left({{\beta
L}\over\sigma}\right)^{2}\qquad\qquad\tan\theta_{1}\;=\;{{\beta
L}\over\sigma}$ (34) $\displaystyle r_{2}^{2}$ $\displaystyle=$ $\displaystyle
1+\left({{\beta
L}\over{2\sigma}}\right)^{2}\qquad\qquad\tan\theta_{2}\;=\;{{\beta
L}\over{2\sigma}}.$ (35)
Note that these parameters depend on the second order dispersion coefficient
$\beta$, but not on $\alpha$, and that when $\beta$ vanishes we then have
$r_{1}=r_{2}=1$ and $\theta_{1}=\theta_{2}=0$.
Figure 2: Definitions of $r_{1}$, $r_{2}$, $\theta_{1}$, and $\theta_{2}$.
Case B: One-photon N00N state. The input state is
${1\over\sqrt{2}}\left[|10\rangle+|01\rangle\right]_{\sigma}={1\over\sqrt{2}}\sqrt[4]{\sigma\over\pi}\int
d\omega\;e^{-{\sigma\over
2}(\omega-\omega_{0})^{2}}\left(\hat{a}_{\omega}^{\dagger}+\hat{b}_{\omega}^{\dagger}\right)|0\rangle,$
(36)
where
$\displaystyle{1\over\sqrt{2}}\left(\hat{a}_{\omega}^{\dagger}+\hat{b}_{\omega}^{\dagger}\right)$
$\displaystyle=$
$\displaystyle{i\over{2\sqrt{2}}}\left\\{\hat{c}^{\dagger}_{\omega}\left[(i-1)-(i+1)e^{i\phi(\omega)}\right]\right.$
$\displaystyle\left.\qquad+\;\hat{d}^{\dagger}_{\omega}\left[e^{i\phi(\omega)}(i-1)-(i+1)\right]\right\\}.$
The resulting output probabilities in this case turn out to be
$\displaystyle P(1,0|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left[1-{{e^{-{{\alpha^{2}\beta
L^{3}}\over{4r_{1}^{2}\sigma}}}}\over\sqrt{r_{1}}}\sin\left(\phi_{0}+{\theta_{1}\over
2}-{{\alpha^{2}\beta L^{3}}\over{4r_{1}^{2}\sigma^{2}}}\right)\right]$ (38)
$\displaystyle P(0,1|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left[1+{{e^{-{{\alpha^{2}\beta
L^{3}}\over{4r_{1}^{2}\sigma}}}}\over\sqrt{r_{1}}}\sin\left(\phi_{0}+{\theta_{1}\over
2}-{{\alpha^{2}\beta L^{3}}\over{4r_{1}^{2}\sigma^{2}}}\right)\right].$ (39)
## IV MZ Interferometry With Two-Photon Input
We now consider input states with two photons distributed in assorted ways
among the input ports. However, now we must make a distinction as to whether
the two photon frequencies are independent or correlated in some manner. We
treat the uncorrelated version first. Then we will examine one particular case
of frequency-correlated photons which is of special interest for experiment:
that of photon pairs created through spontaneous parametric downconversion
(SPDC). In this section we treat only a simplified version of SPDC which will
allow us to obtain simple exact expressions for the probabilities of all of
the output states. In a later section we will compare this simplified SPDC to
a more realistic version for which only numerical results are available.
### IV.1 Two-Photon Input with Uncorrelated Energies
Case C: Energy-uncorrelated two-photon Fock state. Sending a two-particle Fock
state into input A,
$|2,0\rangle_{\sigma}=\sqrt{\sigma\over{2\pi}}\int
d\omega_{1}d\omega_{2}\;e^{-{\sigma\over
2}\left[(\omega_{1}-\omega_{0})^{2}+(\omega_{2}-\omega_{0})^{2}\right]}\hat{a}^{\dagger}_{\omega_{1}}\hat{a}^{\dagger}_{\omega_{2}}|0\rangle,$
(40)
where we use equations 11 and 12 to write each $\hat{a}^{\dagger}$ factor in
terms of the output operators $\hat{c}^{\dagger}$ and $\hat{d}^{\dagger}$.
After a straightforward calculation, this leads to the following output
probabilities:
$\displaystyle P(2,0|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
4}\left[1-{{e^{-{{\alpha^{2}L^{2}}\over{4r_{1}^{2}\sigma}}}}\over\sqrt{r_{1}}}\cos\left(\phi_{0}+{{\theta_{1}}\over
2}-{{\alpha^{2}\beta L^{3}}\over{4r_{1}^{2}\sigma^{2}}}\right)\right]^{2}$
(41) $\displaystyle P(0,2|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
4}\left[1+{{e^{-{{\alpha^{2}L^{2}}\over{4r_{1}^{2}\sigma}}}}\over\sqrt{r_{1}}}\cos\left(\phi_{0}+{{\theta_{1}}\over
2}-{{\alpha^{2}\beta L^{3}}\over{4r_{1}^{2}\sigma^{2}}}\right)\right]^{2}$
(42) $\displaystyle P(1,1|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left[1-{{e^{-{{\alpha^{2}L^{2}}\over{2r_{1}^{2}\sigma}}}}\over{r_{1}}}\cos^{2}\left(\phi_{0}+{{\theta_{1}}\over
2}-{{\alpha^{2}\beta L^{3}}\over{4r_{1}^{2}\sigma^{2}}}\right)\right]$ (43)
Case D: Energy-uncorrelated two-photon dual Fock input. The normalized input
state is
$\displaystyle|1,1\rangle_{\sigma}$ $\displaystyle=$
$\displaystyle\sqrt{\sigma\over\pi}\int
d\omega_{1}\omega_{2}\;e^{-{\sigma\over
2}\left[(\omega_{1}-\omega_{0})^{2}+(\omega_{2}-\omega_{0})^{2}\right]}\hat{a}_{\omega_{1}}^{\dagger}\hat{a}_{\omega_{2}}^{\dagger}|0\rangle$
(44) $\displaystyle=$ $\displaystyle{1\over 4}\sqrt{\sigma\over\pi}\int
d\omega_{1}\omega_{2}\;e^{-{\sigma\over
2}\left[(\omega_{1}-\omega_{0})^{2}+(\omega_{2}-\omega_{0})^{2}\right]}$
$\displaystyle\times\left\\{-i\hat{c}_{\omega_{1}}^{\dagger}\hat{c}_{\omega_{2}}^{\dagger}\left(e^{i\phi(\omega_{1})}-1\right)\left(e^{i\phi(\omega_{2})}+1\right)+i\hat{d}_{\omega_{1}}^{\dagger}\hat{d}_{\omega_{2}}^{\dagger}\left(e^{i\phi(\omega_{2})}-1\right)\left(e^{i\phi(\omega_{1})}+1\right)\right.$
$\displaystyle\qquad\left.+\hat{c}_{\omega_{1}}^{\dagger}\hat{d}_{\omega_{2}}^{\dagger}\left(e^{i\phi(\omega_{1})}-1\right)\left(e^{i\phi(\omega_{2})}-1\right)-\hat{c}_{\omega_{2}}^{\dagger}\hat{d}_{\omega_{1}}^{\dagger}\left(e^{i\phi(\omega_{1})}+1\right)\left(e^{i\phi(\omega_{2})}+1\right)\right\\}|0\rangle,$
which gives the results
$\displaystyle P(2,0)$ $\displaystyle=$ $\displaystyle P(0,2)$ (46)
$\displaystyle=$ $\displaystyle{1\over
4}\left\\{1-{{e^{-{{\alpha^{2}L^{2}}\over{2r_{1}^{2}\sigma}}}}\over{r_{1}}}\cos\left[2\phi_{0}+\theta_{1}-{{\alpha^{2}\beta
L^{3}}\over{2r_{1}^{2}\sigma^{2}}}\right]\right\\}$ $\displaystyle P(1,1)$
$\displaystyle=$ $\displaystyle{1\over
2}\left\\{1+{{e^{-{{\alpha^{2}L^{2}}\over{2r_{1}^{2}\sigma}}}}\over{r_{1}}}\cos\left[2\phi_{0}+\theta_{1}-{{\alpha^{2}\beta
L^{3}}\over{2r_{1}^{2}\sigma^{2}}}\right]\right\\}$ (47)
Case E: Energy-uncorrelated two-photon N00N state.
For the input state
$\displaystyle|20\rangle_{\sigma}+|02\rangle_{\sigma}$ $\displaystyle=$
$\displaystyle\sqrt{\sigma\over{2\pi}}\int
d\omega_{1}\;d\omega_{2}e^{-{\sigma\over
2}[(\omega_{1}-\omega_{0})^{2}+(\omega_{2}-\omega_{0})^{2}]}$ (48)
$\displaystyle\qquad\times{1\over\sqrt{2}}\left(\hat{a}_{\omega_{1}}^{\dagger}\hat{a}_{\omega_{2}}^{\dagger}+\hat{b}_{\omega_{1}}^{\dagger}\hat{b}_{\omega_{2}}^{\dagger}\right)|0\rangle,$
the output probabilities are
$\displaystyle P(2,0|\phi_{0})$ $\displaystyle=$ $\displaystyle
P(0,2|\phi_{0})\;=\;{1\over
4}\left\\{1+{1\over{r_{1}}}e^{-{{\alpha^{2}L^{2}}\over{2r_{1}^{2}\sigma}}}\right\\}$
(49) $\displaystyle P(1,1|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left\\{1-{1\over{r_{1}}}e^{-{{\alpha^{2}L^{2}}\over{2r_{1}^{2}\sigma}}}\right\\}.$
(50)
In the absence of dispersion ($\alpha=\beta=0$) or in the narrow bandwidth
limit ($\sigma\to\infty$), we see that the coincidence rate $P(1,1|\phi_{0})$
vanishes, while the other two probabilities are both equal to ${1\over 2}$.
Note that there is no dependence on $\phi_{0}$. We will see later that this
fact manifests itself in a vanishing mutual information.
### IV.2 Two-Photon Input with Anticorrelated Energies: Simplified SPDC model
Case F: Simplified SPDC Fock states. We now examine a case with two photons
incident on the same input port and anticorrelated in energy. We do this in
the context of a simplified model of spontaneous parametric downconversion
(SPDC). Energy conservation requires that the two downconverted photons have
frequencies $\omega_{\pm}=\omega_{0}\pm\Omega$, where $2\omega_{0}$ is the
pump frequency. We again assume a Gaussian distribution of frequencies,
centered around $\omega_{0}$, of the form $e^{-{1\over
2}\sigma(\omega_{\pm}-\omega_{0})^{2}}=e^{-{\sigma\over 2}\Omega^{2}}$. We
follow essentially the same calculational procedure as before, except now we
enforce the requirement that the incoming photon frequencies satisfy
$\omega_{1}+\omega_{2}=2\omega_{0}$. In this section we impose this condition
in a manner that will allow us to obtain analytic solutions for the output
probabilities. This will serve us as a simplified version of SPDC, and we will
see in section VII that this model seems to give an upper bound to the mutual
information obtained from a more realistic model of SPDC. The input state in
this model is taken to be of the form
$|20\rangle_{\sigma}=\sqrt[4]{{2\sigma}\over\pi}\int_{-\infty}^{\infty}d\Omega\int_{-\infty}^{\infty}d\epsilon\;e^{-\sigma\Omega^{2}}f(\epsilon)a_{\omega_{+}}^{\dagger}a_{\omega_{-}}^{\dagger}|0\rangle,$
(51)
where now $\omega_{+}=\omega_{0}+\Omega$ and
$\omega_{-}=\omega_{0}-\Omega+\epsilon$. We can choose $f(\epsilon)$ to be any
function sharply peaked at zero with normalized integral (unit area under its
graph). We then compute the output probabilities according to
$P(N_{c},N_{d}|\phi_{0})=\int d\omega_{1}d\omega_{2}\left|\langle
N_{c},N_{d}|\psi_{in}\rangle\right|^{2},$ (52)
or equivalently, by applying the projection operators
$\hat{\pi}(N_{c},N_{d})=\int
d\omega_{1}\;d\omega_{2}|N_{c},N_{d}\rangle\langle N_{c},N_{d}|.$ (53)
The auxiliary function $f_{\lambda}(\epsilon)$ is necessary in this model in
order to impose the constraint $\omega_{1}+\omega_{2}=2\omega_{0}$ without
causing squares of delta functions to arise in the probability calculations. A
more correct treatment of SPDC will follow in section VII.
The measurement outcomes, integrated over final frequency, are then given by
$\displaystyle P(2,0|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
8}\left\\{2+e^{-{{\alpha^{2}L^{2}}\over{2\sigma}}}+{1\over\sqrt{r_{1}}}\cos\left(2\phi_{0}+{{\theta_{1}}\over
2}\right)\right.$
$\displaystyle\qquad\left.-{4\over\sqrt{r_{2}}}e^{-{{\alpha^{2}L^{2}}\over{8r_{2}^{2}\sigma}}}\cos\left[\phi_{0}+{{\theta_{2}}\over
2}-{{\alpha^{2}\beta L^{3}}\over{16r_{2}^{2}\sigma^{2}}}\right]\right\\}$
$\displaystyle P(0,2|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
8}\left\\{2+e^{-{{\alpha^{2}L^{2}}\over{2\sigma}}}+{1\over\sqrt{r_{1}}}\cos\left(2\phi_{0}+{{\theta_{1}}\over
2}\right)\right.$
$\displaystyle\qquad\left.+{4\over\sqrt{r_{2}}}e^{-{{\alpha^{2}L^{2}}\over{8r_{2}^{2}\sigma}}}\cos\left[\phi_{0}+{{\theta_{2}}\over
2}-{{\alpha^{2}\beta L^{3}}\over{16r_{2}^{2}\sigma^{2}}}\right]\right\\}$
$\displaystyle P(1,1|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
4}\left\\{2-e^{-{{\alpha^{2}L^{2}}\over{2\sigma}}}-{1\over\sqrt{r_{1}}}\cos\left(2\phi_{0}+{{\theta_{1}}\over
2}\right)\right\\}.$ (56)
As $\beta$ increases, $r_{1}$ and $r_{2}$ increase, leading to decreased
visibility of all of the oscillating terms.
Note also that in the case of zero dispersion ($\alpha=\beta=0$), the exact
expressions for energy-uncorrelated (Case C, section 4.1) and energy-
anticorrelated (downconverted) Fock states (Case F) are identical to each
other. However the probabilities begin to diverge when dispersion is turned
on. The same effect will be seen to occur for the uncorrelated and
anticorrelated N00N states in the HOM interferometer (cases J and K, below).
Case G: Simplified SPDC N00N states. Now the input state is taken to be a N00N
state,
${1\over\sqrt{2}}\left\\{|20\rangle_{\lambda,\sigma}+|02\rangle_{\lambda,\sigma}\right\\}$.
We find the measurement outcomes to be:
$\displaystyle P(2,0|\phi_{0})$ $\displaystyle=$ $\displaystyle
P(0,2)\;=\;{1\over 4}\left[1+e^{-{{\alpha^{2}L^{2}}\over{2\sigma}}}\right]$
(57) $\displaystyle P(1,1|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left[1-e^{-{{\alpha^{2}L^{2}}\over{2\sigma}}}\right]$ (58)
As in the uncorrelated case, the probabilities show no dependence on
$\phi_{0}$, and so have vanishing mutual information.In this case we also see
that there is no dependence on the 2nd order dispersion coefficient $\beta$.
It is interesting to note what happens if we shift the phase of the photons in
one input port by $\pi\over 2$ before they hit the first beamsplitter. The
input to the interferometer is now proportional to
$\left|2,0\rangle\right.-\left|0,2\rangle\right..$ In this case, the
interference in $\phi_{0}$ reemerges, and the result is independent of
$\alpha$ instead of $\beta$. In fact, the counting probabilities turn out to
be very similar to those of the N00N state incident on an HOM interferometer
presented in the next section (case K). Moreover, these two cases have
identical values for the mutual information.
Case H: Simplified SPDC dual Fock state. The frequency-anticorrelated dual
Fock input state
$|1,1\rangle_{\sigma}=\sqrt[4]{\sigma\over{2\pi}}\int d\Omega\int
d\epsilon\;e^{-\sigma\Omega^{2}}f(\epsilon)\hat{a}_{\omega_{+}}^{\dagger}\hat{b}_{\omega_{-}}^{\dagger}|0\rangle$
(59)
gives the results
$\displaystyle P(2,0|\phi_{0})$ $\displaystyle=$ $\displaystyle
P(0,2|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
4}\left[1-{1\over\sqrt{r_{1}}}\cos\left(2\phi_{0}+{{\theta_{1}}\over
2}\right)\right]$ $\displaystyle P(1,1|\phi_{0})$ $\displaystyle=$
$\displaystyle{1\over
2}\left[1+{1\over\sqrt{r_{1}}}\cos\left(2\phi_{0}+{{\theta_{1}}\over
2}\right)\right].$ (61)
## V Dispersive Hong-Ou-Mandel Interferometer With N00N Input
An alternative setup has been proposed to improve phase resolution(dowl (6)).
In this section we examine this alternate version and compare it to the
previous results.
In this version, it is assumed that the N00N state is created inside the
interferometer, rather than at the input ports. Effectively, we need to remove
the first beam splitter from the interferometer and use the N00N state as
input to the remaining beamsplitter, which now acts as a Hong-Ou-Mandel (HOM)
interferometer hompaper (13). The setup is shown in figure 3. We again assume
dispersion and phase-shift $\phi_{0}$ along one of the lines entering the beam
splitter, neglecting absorption. Ignoring an overall constant phase of
$e^{ik_{0}L}$, the scattering matrix now has the form
$S(\phi_{0})={1\over\sqrt{2}}\left(\begin{array}[]{cc}e^{i\phi(\omega)}&i\\\
ie^{i\phi(\omega)}&+1\end{array}\right),$ (62)
where $\phi(\omega)$ is again given by equation 19. Thus,
$\displaystyle\hat{c}_{\omega}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}\left[e^{i\phi(\omega)}\hat{a}_{\omega}+i\hat{b}_{\omega}\right]$
(63) $\displaystyle\hat{d}_{\omega}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}\left[ie^{i\phi(\omega)}\hat{a}_{\omega}+\hat{b}_{\omega}\right].$
(64)
Figure 3: Hong-Ou-Mandel interferometer with dispersion and non-dispersive
phase shift $\phi_{0}$ in one arm.
Case I: Single-photon N00N state in HOM interferometer. By the same methods as
before, we can compute the counting rates for a given N00N state input. The
one-photon N00N input state is
$|\psi\rangle=C\int d\omega\;e^{-{\sigma\over
2}(\omega-\omega_{0})^{2}}(a_{\omega}^{\dagger}+b_{\omega}^{\dagger})|0\rangle,$
(65)
for which we find
$\displaystyle P(1,0|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left\\{1+{{e^{-{{\alpha^{2}L^{2}}\over{4r_{1}^{2}\sigma}}}}\over\sqrt{r_{1}}}\sin\left[\phi_{0}+{\theta_{1}\over
2}-{{\alpha^{2}\beta L^{3}}\over{4r_{1}^{2}\sigma^{2}}}\right]\right\\}$ (66)
$\displaystyle P(0,1|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left\\{1-{{e^{-{{\alpha^{2}L^{2}}\over{4r_{1}^{2}\sigma}}}}\over\sqrt{r_{1}}}\sin\left[\phi_{0}+{\theta_{1}\over
2}-{{\alpha^{2}\beta L^{3}}\over{4r_{1}^{2}\sigma^{2}}}\right]\right\\}$ (67)
Finally, assuming two-photons, with no correlation or with anticorrelation, we
arrive at two additional cases (J and K).
Case J: Energy-uncorrelated two-photon N00N state in HOM interferometer.
The input state is
$|\psi\rangle=C\int d\omega_{1}d\omega_{2}\;e^{-{\sigma\over
2}\left[(\omega_{1}-\omega_{0})^{2}+(\omega_{2}-\omega_{0})^{2}\right]}(a_{\omega_{1}}^{\dagger}a_{\omega_{2}}^{\dagger}+b_{\omega_{1}}^{\dagger}b_{\omega_{2}}^{\dagger})|0\rangle.$
(68)
From this state, we arrive at the results:
$\displaystyle P(2,0|\phi_{0})$ $\displaystyle=$ $\displaystyle
P(0,2|\phi_{0})$ (69) $\displaystyle=$ $\displaystyle{1\over
4}\left[1-{{e^{-{{\alpha^{2}L^{2}}\over{2r_{1}^{2}\sigma}}}}\over{r_{1}}}\cos\left(2\phi_{0}+\theta_{1}-{{\alpha^{2}\beta
L^{3}}\over{2r_{1}^{2}\sigma^{2}}}\right)\right]$ $\displaystyle
P(1,1|\phi_{0})$ $\displaystyle=$ $\displaystyle{1\over
2}\left[1+{{e^{-{{\alpha^{2}L^{2}}\over{2r_{1}^{2}\sigma}}}}\over{r_{1}}}\cos\left(2\phi_{0}+\theta_{1}-{{\alpha^{2}\beta
L^{3}}\over{2r_{1}^{2}\sigma^{2}}}\right)\right]$ (70)
Case K: Simplified SPDC two-photon N00N state in HOM interferometer.
For the input
$|\psi\rangle=C\int
d\Omega\;e^{-\sigma\Omega^{2}}(a_{\omega_{+}}^{\dagger}a_{\omega_{-}}^{\dagger}+b_{\omega_{+}}^{\dagger}b_{\omega_{-}}^{\dagger})|0\rangle,$
(71)
we compute:
$\displaystyle P(2,0|\phi_{0})$ $\displaystyle=$ $\displaystyle
P(0,2|\phi_{0})$ (72) $\displaystyle=$ $\displaystyle{1\over
4}\left[1-{1\over\sqrt{r_{1}}}\cos\left(2\phi_{0}+{{\theta_{1}}\over
2}\right)\right]$ $\displaystyle P(1,1|\phi_{0})$ $\displaystyle=$
$\displaystyle{1\over
2}\left[1+{1\over\sqrt{r_{1}}}\cos\left(2\phi_{0}+{{\theta_{1}}\over
2}\right)\right]$ (73)
In this last case, the results turn out to be independent of the first order
dispersion coefficient, $\alpha$.
## VI Comparison and Discussion of Cases A to K
The detection probabilities of the previous sections can now be combined with
equation 2 to compute the mutual information for each of the experimental
setups and inputs states. Plotting the results as functions of $\alpha$,
$\beta$, and $\sigma$, we find the results in figures 4-6 for single photon
input and figures 7-9 for two photons. $\alpha L$ is given in units of
$\omega_{0}^{-1}$, while $\beta L$ and $\sigma$ are in units of
$\omega_{0}^{-2}$. In the dispersionless limit, $\alpha,\beta\to 0$, we find
Shannon mutual information values that agree with those previously calculated
in bahdlap (9).
Figure 4: (color online). Mutual information versus alpha for single photon
cases (cases A, B, and I), plotted for the values $\beta=0$, $\sigma=1$. The
mutual information is the same for all three cases. ($\alpha$, $\beta$, and
$\sigma$ are in units of $L^{-1}\omega_{0}^{-1}$, $L^{-1}\omega_{0}^{-2}$, and
$\omega_{0}^{-2}$, respectively.) Figure 5: (color online). Mutual information
versus $\beta$ for single photon cases (cases A, B, and I), for the values
$\alpha=.5$, $\sigma=1$. ($\alpha$, $\beta$, and $\sigma$ are in units of
$L^{-1}\omega_{0}^{-1}$, $L^{-1}\omega_{0}^{-2}$, and $\omega_{0}^{-2}$,
respectively.) Figure 6: (color online). Mutual information versus squared
inverse bandwidth $\sigma$ for single photon cases (cases A, B, and I), for
the values $\alpha=1$, $\beta=.1$. ($\alpha$, $\beta$, and $\sigma$ are in
units of $L^{-1}\omega_{0}^{-1}$, $L^{-1}\omega_{0}^{-2}$, and
$\omega_{0}^{-2}$, respectively.) Figure 7: (color online). Mutual information
versus alpha for two-photon cases, for the values $\sigma=1$, $\beta=0$. Cases
E and G vanish identically. Cases J and D are identical, as are cases H and K.
($\alpha$, $\beta$, and $\sigma$ are in units of $L^{-1}\omega_{0}^{-1}$,
$L^{-1}\omega_{0}^{-2}$, and $\omega_{0}^{-2}$, respectively.) Figure 8:
(color online). Mutual information versus beta for two-photon cases, for the
values $\sigma=1$, $\alpha=.5$. ($\alpha$, $\beta$, and $\sigma$ are in units
of $L^{-1}\omega_{0}^{-1}$, $L^{-1}\omega_{0}^{-2}$, and $\omega_{0}^{-2}$,
respectively.) Figure 9: (color online). Mutual information versus squared
inverse bandwidth $\sigma$ for two-photon cases, for the values $\alpha=1$,
$\beta=.1$. ($\alpha$, $\beta$, and $\sigma$ are in units of
$L^{-1}\omega_{0}^{-1}$, $L^{-1}\omega_{0}^{-2}$, and $\omega_{0}^{-2}$,
respectively.)
Only positive values of $\beta$ were graphed. However, the formulas of the
previous sections work equally in the anomalous dispersion (negative $\beta$)
region.
Note also that the four parameters $\alpha,\beta,\sigma,L$ appear in all
equations only through the dimensionless quantities
$\Lambda_{1}={\sigma\over{\beta
L}}\qquad\qquad\mbox{and}\qquad\qquad\Lambda_{2}={\sigma\over{\alpha^{2}L^{2}}}.$
(74)
Thus, other parameter ranges can easily be obtained from those graphed here
via appropriate rescaling of variables with the dimensionless ratios held
fixed.
A few conclusions are immediately clear from these graphs and from the
equations of the previous sections. (i) First, the dual Fock states entering
the Mach-Zehnder interferometer give identical results as the N00N states
entering the Hong-Ou Mandel interferometer (compare equations IV.2 and 61 to
72 and 73, or compare 46 and 47 to 69 and 70). This is to be expected, since
for $N=2$ the two cases are equivalent: the first beam splitter in the Mach-
Zehnder interferometer turns a dual Fock input state into a N00N output state,
which then strikes the second beamsplitter. The second beamsplitter can then
be viewed as an HOM interferometer. Thus cases J and D are equivalent, as are
cases H and K. (This equivalence will not for $N>2$.) (ii) Second, the single-
photon cases (cases A, B, and I) all give identical curves for the mutual
information as functions of $\alpha$, $\beta$, and $\sigma$. The explanation
for this is clear if the action of the first beam splitter on the input is
examined. Cases B and I are equivalent for the same reason mentioned in the
previous point: they both lead to a one-particle N00N state in the portion of
the interferometer before the dispersive element is reached, and so give the
same output. Meanwhile, in case A, the output of the first beam splitter is
the state ${1\over\sqrt{2}}\left(|01\rangle+i|10\rangle\right);$ this is
similar to a N00N state, except one term is shifted in phase by $\pi\over 2$
relative to the other. This converts the sines in the probabilities of cases B
and I into the cosines of case A (equations 32 and 33), but has no other
effect. Since the mutual information involves integrals from $-\pi$ to $\pi$,
interchanging sines and cosines inside the integrals has no effect on the
mutual information. Unsurprisingly, the single photon cases generally result
in lower mutual information than the two-photon cases. (iii) We see from the
graphs for the two-photon states that the energy-uncorrelated and energy-
anticorrelated version of each input give identical results for zero
dispersion or zero bandwidth ($\sigma=\infty$); however, the uncorrelated
versions all drop off rapidly to zero fidelity as the dispersion increases,
whereas the anticorrelated (downconverted) input leads to a much slower drop.
(iv) Two-photon N00N states incident on the MZ interferometer (cases E and G)
have zero mutual information as anticipated earlier. (v) For fixed bandwidth
and fixed quadratic dispersion coefficient $\beta$, the two-photon
downconverted N00N state in the HOM interferometer (case K) is independent of
the linear coefficient $\alpha$. However, it decays rapidly with increasing
$\beta$. (vi) Overall, the simplified SPDC-generated Fock states (case F) seem
to hold up best in the presence of dispersion. This case starts off with a
higher value of $H$ at zero dispersion and decays more slowly as $\alpha$ and
$\beta$ increase. The only exception to this statement is when $\beta$ is
small, in which case the anticorrelated HOM N00N state (case K) works better
at large $\alpha$.
A bit of insight into some of the properties of the 2-photon results may be
obtained by considering the exponential decay factor
$\zeta\equiv
e^{-{{\alpha^{2}L^{2}}\over{4r_{1}^{2}\sigma}}}=e^{-{{\Lambda_{2}}\over{4r_{1}^{2}}}}=e^{-{{\Lambda_{2}}\over{4(1+\Lambda_{1}^{-2})}}}.$
(75)
$\Lambda_{1}$ and $\Lambda_{2}$ are the dimensionless quantities defined in
equation 74. In frequency-uncorrelated cases such as cases C and D, all of the
$\phi_{0}$-dependent terms are multiplied by a factor of $\zeta$ which arises
from interference between $e^{i\phi(\omega_{1})}$ and $e^{i\phi(\omega_{2})}$
terms, where $\omega_{1}$ and $\omega_{2}$ are the frequencies of the photons
entering the input ports. The relevant term is of the form
$e^{i[\phi(\omega_{1})+\phi(\omega_{2})]}$. As $\alpha\to\infty$ or $\sigma\to
0$, we find that $\Lambda_{2}\to 0$ and $\zeta\to 0$, so that only constant
($\phi_{0}$-independent) terms survive in the limit. Thus, for large $\alpha$
or small $\sigma$, the dependence of the probability distributions on
$\phi_{0}$ decays exponentially, causing the mutual information to also decay
rapidly.
In contrast, for the frequency-anticorrelated cases, such as F and H, the term
$e^{i(\phi(\omega_{1})+\phi(\omega_{2}))}$ becomes
$e^{i[\phi(\omega_{+})+\phi(\omega_{-})]}=e^{i[2\phi_{0}+2\beta\omega^{2}]},$
(76)
with the $\alpha$-dependence cancelling. As a result, $\phi_{0}$-dependent
terms occur without the exponentially decaying $\zeta$ factor, allowing much
slower decay of $H$ at large $\alpha$ (or even no decay at all, as in case H).
The slower decay at large dispersion is therefore a direct consequence of the
quantum-mechanical correlations present in the initial state.
As for the $\beta$ dependence, we see that as $\beta$ becomes large, both
$\zeta$ and $r_{1}$ become $\beta$ independent, with $\zeta\to
e^{-{{\Lambda_{2}}\over 4}}$ and $r_{1}\to 1$; thus all the curves approach
constant values at large $\beta$, with slopes ${{dH}\over{d\beta}}$ of
comparable order of magnitude.
We turn now to one additional case, that of more realistic SPDC photon pairs,
which we then proceed to compare with the simplified SPDC model already
examined.
## VII Case L: SPDC
Now we present results for the mutual information using a more realistic model
for the parametric downconversion process. Numerically, the results turn out
qualitatively (and for some parameter ranges quantitatively as well) to be
very similar to those of the simplified SPDC model in the previous section;
however we no longer will be able to present explicit analytic expressions for
the measurement outcomes.
There are many possible cases that could be considered, but we restrict
ourselves here to the single case of collinear type-II SPDC in a nonlinear
crystal, with both of the outgoing photons entering port A of the dispersive
Mach-Zehnder interferometer. We now have to consider the parameters of both
the interferometer and the crystal. We allow the pump frequency to vary around
central frequency $2\omega_{0}$, with the deviation from the center of the
distribution represented by $2\Omega_{p}$; in other words, the pump frequency
is represented as
$\omega_{p}=2(\omega_{0}+\Omega_{p}).$
We once again assume a Gaussian distribution of frequencies, in this case
represented by a weighting factor $e^{-{{\sigma}\over
2}(\omega_{p}-2\omega_{0})^{2}}=e^{-2\sigma\Omega_{p}^{2}}$. The signal and
idler frequencies are then
$\displaystyle\omega_{s}$ $\displaystyle=$ $\displaystyle{{\omega_{p}}\over
2}+\Omega=\omega_{0}+\Omega_{p}+\Omega$ (77) $\displaystyle\omega_{i}$
$\displaystyle=$ $\displaystyle{{\omega_{p}}\over
2}-\Omega=\omega_{0}+\Omega_{p}-\Omega,$ (78)
with $\omega_{s}+\omega_{i}=\omega_{p}$. Suppose that the crystal is cut so
that exact phase matching occurs at the central frequency
$k_{p}(2\omega_{0})=k_{s}(\omega_{0})+k_{i}(\omega_{0}).$ (79)
Then, assuming that terms quadratic and higher in the frequencies are small,
the phase matching condition for the crystal gives us a condition on the wave-
vectors of the form ou (14)
$\Delta k\equiv
k_{p}(\omega_{p})-k_{s}(\omega_{s})-k_{i}(\omega_{i})=\Lambda_{p}\Omega_{p}+\Lambda\Omega,$
(80)
where
$\Lambda_{p}=2k_{p}^{\prime}(2\omega_{0})-k_{s}^{\prime}(\omega_{0})-k_{i}^{\prime}(\omega_{0})$
and $\Lambda=k_{i}^{\prime}(\omega_{0})-k_{s}^{\prime}(\omega_{0})$.
The wavefunction for the biphoton state entering the interferometer is now
$|\psi_{in}\rangle=\int
d\Omega\;d\Omega_{p}\Phi(\Omega_{p},\Omega)\hat{a}_{\omega_{0}+\Omega_{p}+\Omega}^{\dagger}\hat{a}_{\omega_{0}+\Omega_{p}-\Omega}^{\dagger}|0\rangle,$
(81)
where
$\Phi(\Omega_{p},\Omega)={\cal
N}e^{-2\sigma\Omega_{p}^{2}}\left({{\sin{{\Delta kL_{c}}\over 2}}\over{{\Delta
kL_{c}}\over 2}}\right)e^{-i\Delta k\;L_{c}/2},$ (82)
with normalization constant ${\cal N}$. Here, $L_{c}$ is the length of the
nonlinear crystal. Using this wavefunction, we can compute output
probabilities as before. Denoting the frequencies at the detectors by $\omega$
and $\omega^{\prime}$, we have
$\displaystyle P(2,0|\phi_{0})$ $\displaystyle=$ $\displaystyle\int
d\omega\;d\omega^{\prime}\left|\Phi({{\omega-\omega^{\prime}}\over
2},\omega_{0}-{{\omega+\omega^{\prime}}\over
2})+\Phi(-{{\omega-\omega^{\prime}}\over
2},\omega_{0}-{{\omega+\omega^{\prime}}\over 2})\right|^{2}$ (83)
$\displaystyle\times\left|1+e^{i[\phi(\omega)+\phi(\omega^{\prime})]}-e^{i\phi(\omega)}-e^{i\phi(\omega^{\prime})}\right|^{2}$
$\displaystyle P(1,1|\phi_{0})$ $\displaystyle=$ $\displaystyle\int
d\omega\;d\omega^{\prime}\left|\Phi({{\omega-\omega^{\prime}}\over
2},\omega_{0}-{{\omega+\omega^{\prime}}\over
2})\left[1-e^{i[\phi(\omega)+\phi(\omega^{\prime})]}-e^{i\phi(\omega)}+e^{i\phi(\omega^{\prime})}\right.\right.$
(84) $\displaystyle+\left.\Phi({{\omega-\omega^{\prime}}\over
2},\omega_{0}-{{\omega+\omega^{\prime}}\over
2})\left[1-e^{i[\phi(\omega)+\phi(\omega^{\prime})]}+e^{i\phi(\omega)}-e^{i\phi(\omega^{\prime})}\right]\right|^{2}$
$\displaystyle P(0,2|\phi_{0})$ $\displaystyle=$ $\displaystyle\int
d\omega\;d\omega^{\prime}\left|\Phi({{\omega-\omega^{\prime}}\over
2},\omega_{0}-{{\omega+\omega^{\prime}}\over
2})+\Phi(-{{\omega-\omega^{\prime}}\over
2},\omega_{0}-{{\omega+\omega^{\prime}}\over 2})\right|^{2}$ (85)
$\displaystyle\times\left|1+e^{i[\phi(\omega)+\phi(\omega^{\prime})]}+e^{i\phi(\omega)}+e^{i\phi(\omega^{\prime})}\right|^{2},$
where $\phi(\omega)$ is as defined in equation 19. Note that
$\Phi\left({{\omega-\omega^{\prime}}\over
2},\omega_{0}-{{\omega+\omega^{\prime}}\over 2}\right)$ depends only on the
crystal properties, while $\phi(\omega)$ depends only on the properties of the
interferometer. The integrands of $P(0,2|\phi_{0})$ and $P(2,0|\phi_{0})$
factor in their dependence on these two sets of parameters; that of
$P(1,1|\phi_{0})$ does not, indicating the entangled nature of the
$|11\rangle$ state.
Given these output probabilities, the mutual information can once again be
computed. In contrast to the previous sections, the analytic forms of the
probabilities are too complicated to be enlightening, so we proceed to
numerical calculations. Some examples are graphed in figures 10 to 14. The
plots are expressed in terms of the new parameters $b=\Lambda_{p}L_{c}$ and
$\lambda={\Lambda\over{\Lambda_{p}}}$.
Examples of the dependence of $H$ on the parameters of the pump beam
($\sigma$), interferometer ($\alpha$, $\beta$), and nonlinear crystal
($\lambda$, $b$) are given in figures 10 through 14. We see that, although $H$
decays overall with increasing values of the dispersion paramaters in the
interferometer, $\alpha$ and $\beta$, there are oscillations superimposed on
the decay, which are especially noticeable at low values of $\alpha$ and
$\beta$. This effect was in fact also present in the simplified SPDC model of
the previous sections, but in the latter case the oscillations were too weak
to be visible on the graphs. We see also that as either $b$ or $\lambda$
increases (or equivalently, as $\Lambda$ or $\Lambda_{p}$ increases), the
plots approach those of the simplified SPDC model. Since $\Lambda$ and $b$ are
proportional to the crystal length, this means that the simplified SPDC model
is an increasingly better approximation to real SPDC for longer crystals. It
also appears from the numerical simulations that for a given set of parameter
values $\alpha$, $\beta$, $L$, and $\sigma$, the simplified SPDC model
provides an upper bound to $H$ for the real SPDC cases with the same parameter
values. The maximum information content clearly occurs for low dispersion in
the interferometer, long nonlinear crystals, and large mismatch at
$\omega_{0}$ between signal and idler inverse group velocities in the crystal
(large $\Lambda$).
## VIII Conclusions
In this paper, we have examined the effect of dispersion on the mutual
information that interferometric photon-detection measurements carry about
phase shifts. We have looked at a number of different situations involving two
interferometer set-ups and several different types of non-classical input
states. Comparing the results, we now have a precise and quantitative means to
measure the relative merits of different input states for various input-
parameter ranges. As a by-product, we have shown that in some circumstances,
parametric downconversion can be approximated by a much simpler model that is
amenable to exact analytical analysis.
Returning to the original question of which input state yields the most
information about the phase shift, the graphs of the previous sections yield
fairly clear results. Restricting discussion to MZ interferometers for
simplicity, we can see that for quantum interferometry in the presence of
dispersion the entangled photon pair produced by downconversion has a clear
advantage over other cases when input to a single port (Fock state input).
This advantage does not exist in the case of an dispersionless interferometer,
in which case the presence or absence of frequency correlations becomes
irrelevant for the information content. The only situation we have found in
which another input is superior to the frequency-anticorrelated Fock input is
when $\alpha$ is large but $\beta$ small, in which case the anticorrelated
dual Fock input is superior. These conclusions all hold when the simplified
downconversion model of section IV.2 is a good approximation; the results of
section VII imply that such conclusions weaken as the crystal becomes shorter.
Figure 10: (color online). Mutual information versus squared inverse bandwidth
$\sigma$ for SPDC. ($\alpha$, $\beta$, and $\sigma$ are in units of
$L^{-1}\omega_{0}^{-1}$, $L^{-1}\omega_{0}^{-2}$, and $\omega_{0}^{-2}$,
respectively. $b$ is in units of $\omega_{0}^{-2}$, while $\lambda$ is
dimensionless.) Figure 11: (color online). Mutual information versus alpha for
SPDC with $\sigma=1$, $\beta=.1$. ($\alpha$, $\beta$, and $\sigma$ are in
units of $L^{-1}\omega_{0}^{-1}$, $L^{-1}\omega_{0}^{-2}$, and
$\omega_{0}^{-2}$, respectively. $b$ is in units of $\omega_{0}^{-2}$, while
$\lambda$ is dimensionless.) Figure 12: (color online). Mutual information
versus beta for SPDC with $\sigma=1$, $\alpha=.3$. ($\alpha$, $\beta$, and
$\sigma$ are in units of $L^{-1}\omega_{0}^{-1}$, $L^{-1}\omega_{0}^{-2}$, and
$\omega_{0}^{-2}$, respectively. $b$ is in units of $\omega_{0}^{-2}$, while
$\lambda$ is dimensionless.) Figure 13: (color online). Mutual information
versus lambda ($\lambda=\Lambda/\Lambda_{p}$) for SPDC. ($\alpha$, $\beta$,
and $\sigma$ are in units of $L^{-1}\omega_{0}^{-1}$, $L^{-1}\omega_{0}^{-2}$,
and $\omega_{0}^{-2}$, respectively. $b$ is in units of $\omega_{0}^{-2}$,
while $\lambda$ is dimensionless.) Figure 14: (color online). Mutual
information versus b ($b={{L_{c}}\over 2}\Lambda_{p}$) for SPDC. ($\alpha$,
$\beta$, and $\sigma$ are in units of $L^{-1}\omega_{0}^{-1}$,
$L^{-1}\omega_{0}^{-2}$, and $\omega_{0}^{-2}$, respectively. $b$ is in units
of $\omega_{0}^{-2}$, while $\lambda$ is dimensionless.)
###### Acknowledgements.
This work was supported by a U. S. Army Research Office (ARO)
Multidisciplinary University Research Initiative (MURI) Grant; by the Bernard
M. Gordon Center for Subsurface Sensing and Imaging Systems (CenSSIS), an NSF
Engineering Research Center; by the Intelligence Advanced Research Projects
Activity (IARPA) and ARO through Grant No. W911NF-07-1-0629.
## References
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* (7) P. Walther, J.W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, A. Zeilinger, Nature 429, 158 (2004).
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|
arxiv-papers
| 2008-10-24T16:34:40 |
2024-09-04T02:48:58.433220
|
{
"license": "Public Domain",
"authors": "D.S. Simon, A.V. Sergienko, T.B. Bahder",
"submitter": "Thomas B. Bahder",
"url": "https://arxiv.org/abs/0810.4501"
}
|
0810.4579
|
by S. Simons
Abstract
In this paper, we unify the theory of SSD spaces, part of the theory of
strongly representable multifunctions, and the theory of the equivalence of
various classes of maximally monotone multifunctions.
0 Introduction
In this paper, we unify three different lines of investigation: the theory of
SSD spaces as expounded in [11] and [13], part of the theory of strongly
representable multifunctions as expounded in [15] and [4], and the equivalence
of various classes of maximally monotone multifunctions, as expounded in [5].
The purely algebraic concepts of SSD space and $q$–positive set are introduced
in Definition 1.2. These were originally defined in [11], and the development
of the theory was continued in [13]. Apart from the fact that we write “${\cal
P}$” instead of “pos”, we use the notation of the latter of these references.
We show in Lemma 1.9 how certain proper convex functions $f$ on an SSD space
lead to a $q$–positive set, ${\cal P}(f)$. In Definition 1.10, we define the
intrinsic conjugate, $f^{@}$, of a proper convex function on an SSD space, and
we end Section 1 by proving in Lemma 1.11 a simple, but useful, property of
intrinsic conjugates.
In Definition 2.1, we introduce the concept of a Banach SSD space, which is an
SSD space with a Banach space structure satisfying the compatibility
conditions (2.1.1) and (2.1.2). A proper convex function on a Banach SSD space
may be a VZ function, which is introduced in Definition 2.5. Our main result
on VZ functions, established in Theorem 2.9(c,d), is that if $f$ is a lower
semicontinuous VZ function then ${\cal P}(f)$ is maximally $q$–positive,
$f^{@}$ is also a VZ function, and ${\cal P}\big{(}f^{@}\big{)}={\cal P}(f)$.
Lemma 2.7(b) is an important stepping–stone to Theorem 2.9. In Definition 2.12
and Lemma 2.13, we introduce and discuss the properties of various convex
functions on a Banach SSD space and its dual, and show in Theorem 2.15(c) that
if $f$ is a lower semicontinuous VZ function on a Banach SSD then there is a
whole family of VZ functions $h$ associated with $f$ such that ${\cal
P}(h)={\cal P}(f)$.
If $E$ is a nonzero Banach space then it is shown in Examples 1.4, 2.3, and
2.4 that $E\times E^{*}$ is a Banach SSD space under various different norms.
We show in Section 3 how the definitions and results of Section 2 specialize
to this case. Theorem 3.1 extends some concepts and results from [1] and [5].
The definition of VZ function involves the norm of $B$ in an essential way.
Looking ahead, we will see in Theorem 5.3 that there is a large class of norms
on $E\times E^{*}$ for which the classes of VZ functions coincide. This
follows from the analysis in Section 4, which we will now discuss.
In Definition 4.1, we introduce the concept of a Banach SSD dual space, which
is the dual of a Banach SSD space which has an SSD structure in its own right,
satisfying the compatibility conditions (4.1.1) and (4.1.2). In this
situation, a proper convex function on the (original) Banach SSD space may be
an MAS function, which is introduced in Definition 4.8. The main result here
is Theorem 4.9(c), in which we prove that, under the $\widetilde{p}$–density
condition (4.2.1), a function is an MAS function if, and only if, it is a VZ
function. The main stepping stone to Theorem 4.9 is Lemma 4.7, which relies on
Rockafellar’s formula for the conjugate of the sum of two convex functions.
The subtlety of the analysis outlined in the previous paragraph is that
definition of MAS function does not use the norm of $B$ explicitly — it only
uses the knowledge of $B^{*}$. In a certain sense, the analysis of Section 2
is isometric, while the analysis of Section 4 is isomorphic, though it would
be a mistake to push this analogy too far because, despite the fact that the
definition of MAS function does not use the norm of $B$, the conditions
(4.1.2) and (4.2.1) referred to above do use the norm very strongly.
In Section 5, we show how the results of Section 4 specialize to the $E\times
E^{*}$ case. In Theorem 5.5, we show how the negative alignment analysis
introduced in [10, Section 8, pp. 274–280] and [13, Section 42, pp. 161–167]
can be used to obtain, and in some cases strengthen, results from [4] and
[15]. In Theorem 5.8, we generalize some equivalencies from [5, Theorem 1.2].
In particular, we give a proof of the very nice result from [5] that a
maximally monotone multifunction is strongly representable if, and only if, it
is of type (NI).
At one point in this paper, we will use the Fenchel–Moreau theorem for a not
necessarily Hausdorff locally convex space. For the convenience of the reader,
we give a proof of this result in the Appendix, Section 6.
The author would like to thank Constantin Zălinescu for making him aware of
the preprints [4] and [15], and Benar Svaiter for making him aware of the
preprint [5]. He would also like to thank Constantin Zălinescu for some very
perceptive comments on an earlier version of this paper.
1 SSD spaces
We first introduce the concepts of an SSD space and $q$–positive set. As
pointed out in the introduction, these were introduced in [11] and [13]. The
first of these references has a detailed discussion of the finite dimensional
case.
Definition 1.1. If $X$ is a nonzero vector space and $f\colon\
X\to\,]{-}\infty,\infty]$, we write $\hbox{\rm dom}\,f$ for the set
$\big{\\{}x\in X\colon\ f(x)\in\hbox{\tenmsb R}\big{\\}}$. $\hbox{\rm dom}\,f$
is the effective domain of $f$. We say that $f$ is proper if $\hbox{\rm
dom}\,f\neq\emptyset$. We write ${\cal PC}(X)$ for the set of all proper
convex functions from $X$ into $\,]{-}\infty,\infty]$. If $X$ is a nonzero
Banach space, we write ${\cal PCLSC}(X)$ for the set
$\\{f\in{\cal PC}(X)\colon\ f\ \hbox{is lower semicontinuous on}\ X\\},$
and ${\cal PCLSC}^{*}(X^{*})$ for the set
$\\{f\in{\cal PC}(X^{*})\colon\ f\ \hbox{is $w(X^{*},X)$--lower semicontinuous
on}\ X^{*}\\}.$
Definition 1.2. We will say that $\big{(}B,\lfloor\cdot,\cdot\rfloor\big{)}$
is a symmetrically self–dual space (SSD space) (if there is no risk of
confusion, we will say simply “$B$ is an SSD space”) if $B$ is a nonzero real
vector space and $\lfloor\cdot,\cdot\rfloor\colon B\times B\to\hbox{\tenmsb
R}$ is a symmetric bilinear form. We define the quadratic form $q$ on $B$ by
$q(b):={\textstyle{1\over 2}}\lfloor b,b\rfloor$. Let $A\subset B$. We say
that $A$ is $q$–positive if $A\neq\emptyset$ and
$b,c\in A\Longrightarrow q(b-c)\geq 0.$
We say that $A$ is maximally $q$–positive if $A$ is $q$–positive and $A$ is
not properly contained in any other $q$–positive set. We make the elementary
observation that if $b\in B$ and $q(b)\geq 0$ then the linear span
$\hbox{\tenmsb R}b$ of $\\{b\\}$ is $q$–positive.
We now give some examples of SSD spaces and their associated $q$–positive
sets.
Example 1.3. Let $B$ be a Hilbert space with inner product
$(b,c)\mapsto\langle b,c\rangle$ and $T\colon B\to B$ be a self–adjoint linear
operator. Then $B$ is an SSD space with $\lfloor b,c\rfloor:=\langle
Tb,c\rangle$, and then $q(b)={\textstyle{1\over 2}}\langle Tb,b\rangle$. Here
are three special cases of this example:
(a) If, for all $b\in B$, $Tb=b$ then $\lfloor b,c\rfloor:=\langle
b,c\rangle$, $q(b)={\textstyle{1\over 2}}\|b\|^{2}$ and every subset of $B$ is
$q$–positive
(b) If, for all $b\in B$, $Tb=-b$ then $\lfloor b,c\rfloor:=-\langle
b,c\rangle$, $q(b)=-{\textstyle{1\over 2}}\|b\|^{2}$ and the $q$–positive sets
are the singletons.
(c) If $B=\hbox{\tenmsb R}^{3}$ and $T(b_{1},b_{2},b_{3})=(b_{2},b_{1},b_{3})$
then
$\big{\lfloor}(b_{1},b_{2},b_{3}),(c_{1},c_{2},c_{3})\big{\rfloor}:=b_{1}c_{2}+b_{2}c_{1}+b_{3}c_{3},$
and $q(b_{1},b_{2},b_{3})=b_{1}b_{2}+{\textstyle{1\over 2}}b_{3}^{2}$. Here,
If $M$ is any nonempty monotone subset of $\hbox{\tenmsb R}\times\hbox{\tenmsb
R}$ (in the obvious sense) then $M\times\hbox{\tenmsb R}$ is a $q$–positive
subset of $B$. The set $\hbox{\tenmsb R}(1,-1,2)$ is a $q$–positive subset of
$B$ which is not contained in a set $M\times\hbox{\tenmsb R}$ for any monotone
subset of $\hbox{\tenmsb R}\times\hbox{\tenmsb R}$. The helix
$\big{\\{}(\cos\theta,\sin\theta,\theta)\colon\theta\in\hbox{\tenmsb
R}\big{\\}}$ is a $q$–positive subset of $B$, but if $0<\lambda<1$ then the
helix
$\big{\\{}(\cos\theta,\sin\theta,\lambda\theta)\colon\theta\in\hbox{\tenmsb
R}\big{\\}}$ is not.
Example 1.4. Let $E$ be a nonzero Banach space and $B:=E\times E^{*}$. For all
$b=(x,x^{*})$ and $c=(y,y^{*})\in B$, we set $\lfloor b,c\rfloor:=\langle
x,y^{*}\rangle+\langle y,x^{*}\rangle$. Then $B$ is an SSD space and
$q(b)={\textstyle{1\over 2}}\big{[}\langle x,x^{*}\rangle+\langle
x,x^{*}\rangle\big{]}=\langle x,x^{*}\rangle.$
Consequently, if $b=(x,x^{*})\ \hbox{and}\ c=(y,y^{*})\in B$ then
$\langle
x-y,x^{*}-y^{*}\rangle=q(x-y,x^{*}-y^{*})=q\big{(}(x,x^{*})-(y,y^{*})\big{)}=q(b-c).$
Thus if $A\subset B$ then $A$ is $q$–positive exactly when $A$ is a nonempty
monotone subset of $B$ in the usual sense, and $A$ is maximally $q$–positive
exactly when $A$ is a maximally monotone subset of $B$ in the usual sense. We
point out that any finite dimensional SSD space of the form described here
must have even dimension. Thus cases of Example 1.3 with finite odd dimension
cannot be of this form.
Example 1.5. $\hbox{\tenmsb R}^{3}$ is not an SSD space with
$\big{\lfloor}(b_{1},b_{2},b_{3}),(c_{1},c_{2},c_{3})\big{\rfloor}:=b_{1}c_{2}+b_{2}c_{3}+b_{3}c_{1}.$
(The bilinear form $\lfloor\cdot,\cdot\rfloor$ is not symmetric.)
Lemma 1.6. Let $B$ be an SSD space, $f\in{\cal PC}(B)$, $f\geq q$ on $B$ and
$b,c\in B$. Then
$-q(b-c)\leq\Big{[}\sqrt{(f-q)(b)}+\sqrt{(f-q)(c)}\Big{]}^{2}.$
Proof. We can and will suppose that $0\leq(f-q)(b)<\infty$ and
$0\leq(f-q)(c)<\infty$. Let $\sqrt{(f-q)(b)}<\beta<\infty$ and
$\sqrt{(f-q)(c)}<\gamma<\infty$, so that $\beta^{2}+q(b)>f(b)$ and
$\gamma^{2}+q(c)>f(c)$. Then
$\eqalign{\beta\gamma+{\gamma q(b)+\beta
q(c)\over\beta+\gamma}&={\gamma\over\beta+\gamma}\big{(}\beta^{2}+q(b)\big{)}+{\beta\over\beta+\gamma}\big{(}\gamma^{2}+q(c)\big{)}\cr&>{\gamma\over\beta+\gamma}f(b)+{\beta\over\beta+\gamma}f(c)\geq
f\bigg{(}{\gamma b+\beta c\over\beta+\gamma}\bigg{)}\cr&\geq q\bigg{(}{\gamma
b+\beta c\over\beta+\gamma}\bigg{)}={\gamma^{2}q(b)+\gamma\beta\lfloor
b,c\rfloor+\beta^{2}q(c)\over(\beta+\gamma)^{2}}.}$
Clearing of fractions, we obtain
$(\beta+\gamma)^{2}\beta\gamma+(\beta+\gamma)\big{(}\gamma q(b)+\beta
q(c)\big{)}>\gamma^{2}q(b)+\gamma\beta\lfloor b,c\rfloor+\beta^{2}q(c),$
from which $(\beta+\gamma)^{2}\beta\gamma>-\beta\gamma q(b)+\beta\gamma\lfloor
b,c\rfloor-\beta\gamma q(c)=-\beta\gamma q(b-c)$. If we now divide by
$\beta\gamma$, we obtain $(\beta+\gamma)^{2}>-q(b-c)$, and the result follows
by letting $\beta\to\sqrt{(f-q)(b)}$ and $\gamma\to\sqrt{(f-q)(c)}$. □
Remark 1.7. It follows from Lemma 1.6 and the Cauchy–Schwarz inequality that
$-q(b-c)\leq 2(f-q)(b)+2(f-q)(c).$
In the situation of Example 1.4, we recover [15, Proposition 1].
Definition 1.8. If $B$ be an SSD space, $f\in{\cal PC}(B)$ and $f\geq q$ on
$B$, we write
${\cal P}(f):=\big{\\{}b\in B\colon\ f(b)=q(b)\big{\\}}.$
The following result is suggested by Burachik–Svaiter, [1, Theorem 3.1, pp.
2381–2382] and Penot, [7, Proposition 4(h)$\Longrightarrow$(a), pp. 860–861].
Lemma 1.9. Let $B$ be an SSD space, $f\in{\cal PC}(B)$, $f\geq q$ on $B$ and
${\cal P}(f)\neq\emptyset$. Then ${\cal P}(f)$ is a $q$–positive subset of
$B$.
Proof. This is immediate from Lemma 1.6. □
We now introduce a concept of conjugate that is intrinsic to an SSD space
without any topological conditions.
Definition 1.10. If $B$ is an SSD space and $f\in{\cal PC}(B)$, we write
$f^{@}$ for the Fenchel conjugate of $f$ with respect to the pairing
$\lfloor\cdot,\cdot\rfloor$, that is to say,
$\hbox{for all}\ c\in B,\qquad f^{@}(c):=\sup\nolimits_{b\in B}\big{[}\lfloor
b,c\rfloor-f(b)\big{]}.$
Our next result represents an improvement of the result proved in [13, Lemma
19.12, p. 82], and uses a disguised differentiability argument. See Remark
1.12 below for another proof of Lemma 1.11, due to Constantin Zălinescu.
Lemma 1.11. Let $B$ be an SSD space, $f\in{\cal PC}(B)$ and $f\geq q$ on $B$.
Then:
$\leqalignno{a\in{\cal P}(f)\ \hbox{and}\ b\in
B&\quad\Longrightarrow\quad\lfloor b,a\rfloor\leq q(a)+f(b).&(a)\cr a\in{\cal
P}(f)&\quad\Longrightarrow\quad f^{@}(a)=q(a).&(b)}$
Proof. Let $a\in{\cal P}(f)$ and $b\in B$. Let $\lambda\in\,]0,1[\,$. For
simplicity in writing, let $\mu:=1-\lambda\in\,]0,1[\,$. Then
$\eqalign{\lambda^{2}q(b)+\lambda\mu\lfloor
b,a\rfloor+\mu^{2}q(a)&=q\big{(}\lambda b+\mu a\big{)}\leq f(\lambda b+\mu
a)\cr&\leq\lambda f(b)+\mu f(a)=\lambda f(b)+\mu q(a).}$
Thus $\lambda^{2}q(b)+\lambda\mu\lfloor b,a\rfloor\leq\lambda f(b)+\lambda\mu
q(a)$. We now obtain (a) by dividing by $\lambda$ and letting $\lambda\to 0$.
Now let $a\in{\cal P}(f)$. From (a), $b\in B\Longrightarrow\lfloor
a,b\rfloor-f(b)\leq q(a)$, and it follows by taking the supremum over $b\in B$
that $f^{@}(a)\leq q(a)$. On the other hand, $f^{@}(a)\geq\lfloor
a,a\rfloor-f(a)=2q(a)-q(a)=q(a)$, completing the proof of (b). □
Remark 1.12. The author is grateful to Constantin Zălinescu for pointing out
to him the following alternative proof of Lemma 1.11(a). From Lemma 1.6, with
$c$ replaced by $a$, $-q(b)+\lfloor b,a\rfloor-q(a)=-q(b-a)\leq(f-q)(b)$. Thus
$\lfloor b,a\rfloor-q(a)\leq f(b)$, as required.
2 Banach SSD spaces
Definition 2.1. We say that $B$ is a Banach SSD space if $B$ is an SSD space
and $\|\cdot\|$ is a norm on $B$ with respect to which $B$ is a Banach space
with norm–dual $B^{*}$,
${\textstyle{1\over 2}}\|\cdot\|^{2}+q\geq 0\ \hbox{on}\ B$ $None$
and there exists $\iota\in L(B,B^{*})$ such that
$\hbox{for all}\ b,c\in B,\quad\big{\langle}b,\iota(c)\big{\rangle}=\lfloor
b,c\rfloor\hbox{,\quad\big{(}from which }\big{|}\lfloor
b,c\rfloor\big{|}\leq\|\iota\|\|b\|\|c\|\hbox{\big{)}.}$ $None$
Then, for all $d,e\in B$,
$|q(d)-q(e)|={\textstyle{1\over 2}}\big{|}\lfloor d,d\rfloor-\lfloor
e,e\rfloor\big{|}={\textstyle{1\over 2}}\big{|}\lfloor
d-e,d+e\rfloor\big{|}\leq{\textstyle{1\over 2}}\|\iota\|\|d-e\|\|d+e\|.$
$None$
We define the continuous convex functions $g$ and $p$ on $B$ by
$g:={\textstyle{1\over 2}}\|\cdot\|^{2}$ and $p:=g+q$, so that $p\geq 0$ on
$B$. Since $p(0)=0$, in fact
$\inf\nolimits_{B}p=0.$ $None$
Also, for all $d,e\in B$, $|g(d)-g(e)|={\textstyle{1\over
2}}\big{|}\|d\|-\|e\|\big{|}\big{(}\|d\|+\|e\|\big{)}\leq{\textstyle{1\over
2}}\|d-e\|\big{(}\|d\|+\|e\|\big{)}$. Combining this with (2.1.3), for all
$d,e\in B$,
$|p(d)-p(e)|\leq{\textstyle{1\over
2}}\big{(}1+\|\iota\|\big{)}\|d-e\|\big{(}\|d\|+\|e\|\big{)}.$ $None$
(2.1.2) implies that, for all $f\in{\cal PC}(B)$ and $c\in B$,
$f^{@}(c)=\sup\nolimits_{b\in
B}\big{[}\big{\langle}b,\iota(c)\big{\rangle}-f(b)\big{]}=f^{*}\big{(}\iota(c)\big{)}$,
that is to say,
$f^{@}=f^{*}\circ\iota\ \hbox{on}\ B.$ $None$
Remark 2.2. Example 1.3 is a Banach SSD space provided that $\|T\|\leq 1$.
This is the case with (a), (b) and (c) of Example 1.3.
Example 2.3. We now continue our discussion of Example 1.4. We suppose that
$B=E\times E^{*}$ and $\big{(}B,\|\cdot\|\big{)}$ is a Banach SSD space such
that $B^{*}=E^{**}\times E^{*}$, under the pairing
$\langle b,c^{*}\rangle:=\langle x,y^{*}\rangle+\langle
x^{*},y^{**}\rangle\quad\big{(}b=(x,x^{*})\in B,\ c^{*}=(y^{**},y^{*})\in
B^{*}\big{)}.$ $None$
We recall that, for all $(x,x^{*})\in B$, $q(x,x^{*})=\langle x,x^{*}\rangle$.
It is clear that, for all $(x,x^{*})\in B$,
$\iota(x,x^{*}):=(\widehat{x},x^{*})$ where $\widehat{x}$ is the canonical
image of $x$ in $E^{**}$. We note that if $\big{(}B,\|\cdot\|\big{)}$ is a
Banach SSD space and $\|\cdot\|^{\prime}$ is a larger norm on $B$ such that
$\big{(}B,\|\cdot\|^{\prime}\big{)}^{*}=E^{**}\times E^{*}$ then
$\big{(}B,\|\cdot\|^{\prime}\big{)}$ is also a Banach SSD space.
Example 2.4. We now discuss some specific examples of the above concepts. Here
it is convenient to introduce a parameter $\tau>0$. ($\tau$ stands for
“torsion”.) Then$E\times E^{*}$ is a Banach SSD space if we use the norm
$\|(x,x^{*})\|_{1,\tau}:=\textstyle{1\over\sqrt{2}}\big{(}\tau\|x\|+\|x^{*}\|/\tau\big{)}$
or $\|(x,x^{*})\|_{2,\tau}:=\sqrt{\tau^{2}\|x\|^{2}+\|x^{*}\|^{2}/\tau^{2}}$
or
$\|(x,x^{*})\|_{\infty,\tau}:=\sqrt{2}\big{(}\tau\|x\|\vee\|x^{*}\|/\tau\big{)}$.
(These are arranged in order of increasing size.) Then the dual norm of
$\big{(}B,\|\cdot\|_{1,\tau}\big{)}$ is given by
$\|(y^{**},y^{*})\|_{\infty,\tau}:=\sqrt{2}\big{(}\tau\|y^{**}\|\vee\|y^{*}\|/\tau\big{)}$,
the dual norm of $\big{(}B,\|\cdot\|_{2,\tau}\big{)}$ is given by
$\|(y^{**},y^{*})\|_{2,\tau}:=\sqrt{\tau^{2}\|y^{**}\|^{2}+\|y^{*}\|^{2}/\tau^{2}}$,
and the dual norm of $\big{(}B,\|\cdot\|_{\infty,\tau}\big{)}$ is given by
$\|(y^{**},y^{*})\|_{1,\tau}:=\textstyle{1\over\sqrt{2}}\big{(}\tau\|y^{**}\|+\|y^{*}\|/\tau\big{)}$.
Definition 2.5. Let $X$ be a vector space and $h,k\colon
X\to\,]{-}\infty,\infty]$. The inf–convolution of $h$ and $k$ is defined by
$(h\mathop{\nabla}k)(x):=\inf\nolimits_{y\in X}\big{[}h(y)+k(x-y)\big{]}$
($x\in X$). It is clear that
$\inf\nolimits_{X}k=0\quad\Longrightarrow\quad\inf\nolimits_{X}\big{[}h\mathop{\nabla}k\big{]}=\inf\nolimits_{X}h.$
$None$
Now let $\big{(}B,\|\cdot\|\big{)}$ be a Banach SSD space and $f\in{\cal
PC}(B)$. We say that $f$ is a VZ function (with respect to $\|\cdot\|\big{)}$
if
$(f-q)\mathop{\nabla}p=0\ \hbox{on}\ B.$ $None$
It follows from (2.1.4) and (2.5.1) that
$\hbox{if}\ f\ \hbox{is a VZ function with respect to}\ \|\cdot\|\
\hbox{then}\ \inf\nolimits_{B}[f-q]=0.$ $None$
“VZ” stands for “Voisei–Zălinescu”, since (2.5.2) is an extension to Banach
SSD spaces of a condition introduced in [15, Proposition 3].
Definition 2.6. Let $A$ be a subset of a Banach SSD space $B$. We say that $A$
is $p$–dense if, for all $c\in B$, $\inf p(c-A)=0$.
We now come to our main results on Banach SSD spaces. Lemma 2.7(b) is
interesting since it tells us that we can determine whether $f$ is a VZ
function by inspecting ${\cal P}(f)$.
Lemma 2.7. Let $B$ be a Banach SSD space and $f\in{\cal PCLSC}(B)$.
(a) Let $f$ be a VZ function. Then ${\cal P}(f)$ is a $q$–positive subset of
$B$ and
$c\in B\quad\Longrightarrow\quad\hbox{\rm dist}(c,{\cal
P}(f))\leq\sqrt{2}\sqrt{(f-q)(c)}.$ $None$
(b) The following three conditions are equivalent:
(i) $f$ is a VZ function.
(ii) $f\geq q$ on $B$ and, for all $c\in B$ there exists a bounded sequence
$\\{a_{n}\\}_{n\geq 1}$ of elements of ${\cal P}(f)$ such that
$\lim\nolimits_{n\to\infty}p(c-a_{n})=0$.
(iii) $f\geq q$ on $B$ and ${\cal P}(f)$ is $p$–dense.
Proof. (a) (2.5.3) implies that $f\geq q$ on $B$, and so ${\cal P}(f)$ is
defined. Since (2.7.1) is trivial if $c\in B\setminus\hbox{\rm dom}\,f$, we
can and will suppose that $c\in\hbox{\rm dom}\,f$. Let
$\varepsilon\in\,]0,1[\,$. We first prove that there exists a Cauchy sequence
$\\{b_{n}\\}_{n\geq 1}$ such that, for all $n\geq 1$,
$(f-q)(b_{n})\leq(f-q)(c)/16^{n}\quad\hbox{and}\quad\|c-b_{n}\|\leq(1+\varepsilon)\sqrt{2}\sqrt{(f-q)(c)}.$
$None$
Since we can take $b_{n}=c$ if $(f-q)(c)=0$, we can and will suppose that
$\alpha:=\sqrt{(f-q)(c)}>0.$ $None$
Let $\lambda:=\varepsilon/(3+\varepsilon)\in\,]0,1/4[\,$ and write $b_{0}:=c$.
Then we can choose inductively $b_{1},b_{2},\dots\in B$ such that, for all
$n\geq 1$, $(f-q)(b_{n})+p(b_{n-1}-b_{n})\leq\lambda^{2n}\alpha^{2}$. It
follows from this and (2.5.3) that,
$\hbox{for all}\ n\geq 1,\qquad p(b_{n-1}-b_{n})\leq\lambda^{2n}\alpha^{2},$
$None$
and, combining with (2.1.4),
$\hbox{for all}\ n\geq
0,\qquad(f-q)(b_{n})\leq\lambda^{2n}\alpha^{2}\leq\alpha^{2}/16^{n}.$ $None$
Substituting the first inequality of (2.7.5) into Lemma 1.6, for all $n\geq
1$,
$-q(b_{n-1}-b_{n})\leq\Big{[}\sqrt{(f-q)(b_{n-1})}+\sqrt{(f-q)(b_{n})}\Big{]}^{2}\leq(1+\lambda)^{2}\lambda^{2n-2}\alpha^{2}.$
Consequently, since $g(b_{n-1}-b_{n})=p(b_{n-1}-b_{n})-q(b_{n-1}-b_{n})$,
(2.7.4) gives,
$\hbox{for all}\ n\geq 1,\qquad
g(b_{n-1}-b_{n})\leq(1+\lambda)^{2}\lambda^{2n-2}\alpha^{2}+\lambda^{2n}\alpha^{2}\leq(1+2\lambda)^{2}\lambda^{2n-2}\alpha^{2},$
and so, for all $n\geq 1$,
$\|b_{n-1}-b_{n}\|\leq\sqrt{2}(1+2\lambda)\lambda^{n-1}\alpha$. Adding up this
inequality for $n=1,\dots,m$, we derive that, for all $m\geq 1$,
$\|c-b_{m}\|\leq\sqrt{2}(1+2\lambda)\alpha/(1-\lambda)$. Since
$(1+2\lambda)/(1-\lambda)=1+\varepsilon$, this and (2.7.5) give (2.7.2). Now
set $a=\lim_{n}b_{n}$, so that
$\|c-a\|\leq(1+\varepsilon)\sqrt{2}\sqrt{(f-q)(c)}$. (2.7.5) and the lower
semicontinuity of $f-q$ now imply that $(f-q)(a)\leq 0$, that is to say,
$a\in{\cal P}(f)$. Since $\hbox{\rm dom}\,f\neq\emptyset$, it follows that
${\cal P}(f)\neq\emptyset$ and so, from Lemma 1.9, ${\cal P}(f)$ is a
$q$–positive subset of $B$. We also have
$\hbox{\rm dist}(c,{\cal P}(f))\leq(1+\varepsilon)\sqrt{2}\sqrt{(f-q)(c)},$
and so if we now let $\varepsilon\to 0$, we obtain (2.7.1). This completes the
proof of (a). (b) Suppose first that (i) is satisfied. (2.5.3) implies that
$f\geq q$ on $B$. Let $c\in B$. We choose inductively $b_{1},b_{2},\dots\in B$
such that, for all $n\geq 1$,
$f(b_{n})+g(c-b_{n})+q(c)-\lfloor
c,b_{n}\rfloor=(f-q)(b_{n})+p(c-b_{n})<1/n^{2}.$
Consequently, using (2.1.4) and (2.1.2), for all $n\geq 1$,
$(f-q)(b_{n})<1/n^{2},\ p(c-b_{n})<1/n^{2}$ $None$
and
$f(b_{n})+g(c-b_{n})+q(c)-\|\iota\|\|c\|\|b_{n}\|<1/n^{2}.$ $None$
Since $f\in{\cal PCLSC}(B)$, $f$ dominates a continuous affine function, and
so (2.7.7) and the usual coercivity argument imply that $K:=\sup_{n\geq
1}\|b_{n}\|<\infty$. From (a) and (2.7.6), there exists $a_{n}\in{\cal P}(f)$
such that $\|a_{n}-b_{n}\|\leq\sqrt{2}/n$. Now, from (2.1.5), for all $n\geq
1$,
$\eqalign{|p(c-a_{n})-p(c-b_{n})|&\leq{\textstyle{1\over
2}}(1+\|\iota\|)\|a_{n}-b_{n}\|(2\|c\|+\|a_{n}\|+\|b_{n}\|)\cr&\leq{\textstyle{1\over
2}}(1+\|\iota\|)\big{(}2\|c\|+\big{(}K+\sqrt{2}\big{)}+K\big{)}\sqrt{2}/n.}$
Thus $\lim_{n\to\infty}\big{[}p(c-a_{n})-p(c-b_{n})\big{]}=0$, and (ii)
follows by combining this with (2.7.6). It is trivial that
(ii)$\Longrightarrow$(iii). Suppose, finally, that (iii) is satisfied. Then,
for all $c\in B$,
$\big{(}(f-q)\mathop{\nabla}p\big{)}(c)\leq\inf\nolimits_{a\in{\cal
P}(f)}\big{[}(f-q)(a)+p(c-a)\big{]}=\inf p(c-{\cal P}(f))=0,$
from which $(f-q)\mathop{\nabla}p\leq 0$ on $B$. On the other hand, since
$f-q\geq 0$ on $B$ and, from (2.1.4), $p\geq 0$ on $B$, we have
$(f-q)\mathop{\nabla}p\geq 0$ on $B$. Thus $f$ is a VZ function, giving (i). □
Lemma 2.8. Let $A$ be a closed, $p$–dense and $q$–positive subset of a Banach
SSD space $B$. (a) For all $c\in B$, $\inf q(c-A)\leq 0$ and $\hbox{\rm
dist}(c,A)\leq\sqrt{2}\sqrt{{-}\inf q(c-A)}$. (b) Let $h\in{\cal PC}(B)$,
$h\geq q$ on $B$, and ${\cal P}(h)\supset A$. Then $h$ is a VZ function. (c)
$A$ is a maximally $q$–positive subset of $B$.
Proof. (a) Let $c\in B$. Then $\inf g(c-A)+\inf q(c-A)\leq\inf p(c-A)=0$. Thus
${\textstyle{1\over 2}}\hbox{\rm dist}(c,A)^{2}=\inf g(c-A)\leq-\inf q(c-A)$,
from which (a) is an immediate consequence. (b) Clearly, ${\cal P}(h)$ is also
$p$–dense, and it follows as in Lemma 2.7(b)((iii)$\Longrightarrow$(i)) (which
does not use any semicontinuity) that $h$ is a VZ function, which gives (b).
(c) We suppose that $c\in B$ and $\inf q(c-A)\geq 0$, and we must prove that
$c\in A$. From (a), in fact $\inf q(c-A)=0$ and $\hbox{\rm dist}(c,A)=0$.
Since $A$ is closed, $c\in A$. This completes the proof of (c). □
Theorem 2.9. Let $B$ be a Banach SSD space and $f\in{\cal PCLSC}(B)$ be a VZ
function. Then: (a) For all $c\in B$, $\inf q(c-{\cal P}(f))\leq 0$ and
$\hbox{\rm dist}(c,{\cal P}(f))\leq\sqrt{2}\sqrt{{-}\inf q(c-{\cal P}(f))}$.
(b) Let $h\in{\cal PC}(B)$, $h\geq q$ on $B$, and ${\cal P}(h)\supset{\cal
P}(f)$. Then $h$ is a VZ function. (c) ${\cal P}(f)$ is a maximally
$q$–positive subset of $B$. (d) $f^{@}\in{\cal PCLSC}(B)$, $f^{@}$ is a VZ
function and ${\cal P}\big{(}f^{@}\big{)}={\cal P}(f)$.
Proof. (a), (b) and (c) are immediate from Lemma
2.7(b)((i)$\Longrightarrow$(iii)) and the corresponding parts of Lemma 2.8.
(d) Let $c\in B$. Then, since $q\leq p$ on $B$, Definition 1.10 gives
$q(c)-f^{@}(c)=\inf\nolimits_{b\in B}\big{[}f(b)-\lfloor
b,c\rfloor+q(c)\big{]}=\big{(}(f-q)\mathop{\nabla}q\big{)}(c)\leq\big{(}(f-q)\mathop{\nabla}p\big{)}(c)=0,$
and so $f^{@}\geq q$ on $B$. It now follows from Lemma 1.11(b) that ${\cal
P}\big{(}f^{@}\big{)}\supset{\cal P}(f)$, and so (b) and (c) imply that
$f^{@}$ is a VZ function and ${\cal P}\big{(}f^{@}\big{)}={\cal P}(f)$. Since
${\cal P}(f)\neq\emptyset$, it is evident that $f^{@}\in{\cal PCLSC}(B)$.
(${\cal P}(f)$ is closed because $f$ is lower semicontinuous.) □
Remark 2.10. In general, Theorem 2.9(a) is strictly stronger than Lemma
2.7(a). While this can be proved directly, we will see in Remark 2.17 that it
follows easily from the properties of the $\Phi$–functions. We will also see
in Remark 2.17 that the constant $\sqrt{2}$ in (2.7.1) is sharp.
The proof of Theorem 2.9 relies heavily on the lower semicontinuity of $f$. We
will show in Corollary 2.11 below that part of Theorem 2.9(d) can be recovered
even if $f$ is not assumed to be lower semicontinuous.
Corollary 2.11. Let $B$ be a Banach SSD space and $f\in{\cal PC}(B)$ be a VZ
function. Then $f^{@}\in{\cal PCLSC}(B)$, $f^{@}$ is a VZ function and ${\cal
P}\big{(}f^{@}\big{)}$ is a maximally $q$–positive subset of $B$.
Proof. Let $\overline{f}$ be the lower semicontinuous envelope of $f$. Since
$q$ is continuous and $f\geq q$ on $B$, it follows that $f\geq\overline{f}\geq
q$ on $B$. Thus, from (2.1.4),
$0=(f-q)\mathop{\nabla}p\geq(\overline{f}-q)\mathop{\nabla}p\geq
0\mathop{\nabla}p=0\ \hbox{on}\ B,$
and so $\overline{f}$ is a VZ function. Since $\overline{f}\in{\cal
PCLSC}(B)$, Theorem 2.9(d) implies that $\overline{f}^{@}$ is a VZ function
also. It is well known that $\overline{f}^{*}=f^{*}$ on $B^{*}$ thus,
composing with $\iota$ and using (2.1.6), $\overline{f}^{@}=f^{@}$ on $B$. The
result now follows from Theorem 2.9(d,c), with $f$ replaced by $f^{@}$. □
Definition 2.12. Let $B$ be a Banach SSD space and $A$ be a nonempty
$q$–positive subset of $B$. We define the function $\Theta_{A}\colon\
B^{*}\to\,]{-}\infty,\infty]$ by: for all $b^{*}\in B^{*}$,
$\Theta_{A}(b^{*}):=\sup\nolimits_{a\in A}\big{[}\langle
a,b^{*}\rangle-q(a)\big{]}.$
We define the function $\Phi_{A}\colon\ B\to\,]{-}\infty,\infty]$ by
$\Phi_{A}:=\Theta_{A}\circ\iota$.
We define the function ${}^{*}\Theta_{A}\colon\ B\to\,]{-}\infty,\infty]$ by:
for all $c\in B$,
${}^{*}\Theta_{A}(c):=\sup\nolimits_{b^{*}\in B^{*}}\big{[}\langle
c,b^{*}\rangle-\Theta_{A}(b^{*})\big{]}.$
We collect together in Lemma 2.13 some elementary properties of $\Theta_{A}$,
$\Phi_{A}$, ${}^{*}\Theta_{A}$, and $\Phi_{A}^{@}$. The properties of
$\Phi_{A}$ and $\Phi_{A}^{@}$ have already appeared in [13].
Lemma 2.13. Let $B$ be a Banach SSD space and $A$ be a nonempty $q$–positive
subset of $B$. (a) For all $b\in B$, $\Phi_{A}(b)=\sup\nolimits_{a\in
A}\big{[}\lfloor a,b\rfloor-q(a)\big{]}=q(b)-\inf q(b-A)$. (b)
$\Phi_{A}\in{\cal PCLSC}(B)\quad\hbox{and}\quad\Phi_{A}=q\ \hbox{on}\ A$. (c)
$\Theta_{A}\in{\cal PCLSC}^{*}(B^{*})$. (d) $(^{*}\Theta_{A})^{*}=\Theta_{A}$
and $(^{*}\Theta_{A})^{@}=\Phi_{A}$. (e) ${}^{*}\Theta_{A}\leq q$ on $A$.
Consequently, ${}^{*}\Theta_{A}\in{\cal PCLSC}(B)$. (f)
${}^{*}\Theta_{A}\geq{\Phi_{A}}^{@}\geq\Phi_{A}\vee q\ \hbox{on}\ B$. (g)
${}^{*}\Theta_{A}={\Phi_{A}}^{@}=q\ \hbox{on}\ A$. (h) Let $A$ be maximally
$q$–positive. Then ${}^{*}\Theta_{A}\geq{\Phi_{A}}^{@}\geq\Phi_{A}\geq q$ on
$B$ and $A\subset{\cal P}\big{(}^{*}\Theta_{A}\big{)}$. (i) Let $A$ be
maximally $q$–positive. Then ${\cal P}\big{(}^{*}\Theta_{A}\big{)}={\cal
P}\big{(}{\Phi_{A}}^{@}\big{)}={\cal P}\big{(}\Phi_{A}\big{)}=A$.
Proof. (a) is immediate from (2.1.2), (b) from (a), and (c) from (b) and the
definition of $\Theta_{A}$. The first assertion in (d) follows from (c) and
the Fenchel–Moreau theorem for the locally convex space
$\big{(}B^{*},w(B^{*},B)\big{)}$, while the second assertion follows from the
first by composing with $\iota$, and using (2.1.6) and the definition of
$\Phi_{A}$. (e) Let $a\in A$. The definition of $\Theta_{A}$ implies that, for
all $b^{*}\in B^{*}$, $\langle a,b^{*}\rangle-\Theta_{A}(b^{*})\leq q(a)$.
Taking the supremum over $b^{*}\in B^{*}$, ${}^{*}\Theta_{A}(a)\leq q(a)$, as
required. (f) Let $c\in B$. Then, from (2.1.2), the definition of $\Phi_{A}$
and (b),
$\eqalignno{{}^{*}\Theta_{A}(c)&\geq\sup\nolimits_{b\in B}\big{[}\langle
c,\iota(b)\rangle-\Theta_{A}(\iota(b))\big{]}\cr&=\sup\nolimits_{b\in
B}\big{[}\lfloor
c,b\rfloor-\Phi_{A}(b)\big{]}\quad\big{(}={\Phi_{A}}^{@}(c)\big{)}\cr&\geq\big{[}\lfloor
c,c\rfloor-\Phi_{A}(c)\big{]}\vee\sup\nolimits_{a\in A}\big{[}\lfloor
c,a\rfloor-\Phi_{A}(a)\big{]}\cr&=\big{[}2q(c)-\Phi_{A}(c)\big{]}\vee\sup\nolimits_{a\in
A}\big{[}\lfloor
c,a\rfloor-q(a)\big{]}\cr&=\big{[}2q(c)-\Phi_{A}(c)\big{]}\vee\Phi_{A}(c).}$
Now if $\Phi_{A}(c)=\infty$ then obviously
$\big{[}2q(c)-\Phi_{A}(c)\big{]}\vee\Phi_{A}(c)\geq q(c)$, while if
$\Phi_{A}(c)\in\hbox{\tenmsb R}$ then
$\big{[}2q(c)-\Phi_{A}(c)\big{]}\vee\Phi_{A}(c)\geq{\textstyle{1\over
2}}[2q(c)-\Phi_{A}(c)\big{]}+{\textstyle{1\over 2}}\Phi_{A}(c)=q(c)$. Thus
${\Phi_{A}}^{@}(c)\geq\Phi(c)\vee q(c)$. This completes the proof of (f). (g)
is immediate from (e) and (f). (h) In this case, for all $b\in B\setminus A$,
there exists $a\in A$ such that $q(b-a)<0$, and so $\inf q(b-A)<0$. Thus, from
(a), $\Phi_{A}>q$ on $B\setminus A$. Combining this with (b), $\Phi_{A}\geq q$
on $B$ and ${\cal P}\big{(}\Phi_{A}\big{)}=A$. Thus (h) follows from (f) and
(g). It is clear from (h) that $A\subset{\cal
P}\big{(}^{*}\Theta_{A}\big{)}\subset{\cal
P}\big{(}{\Phi_{A}}^{@}\big{)}\subset{\cal P}\big{(}\Phi_{A}\big{)}$, and so
(i) follows from the maximality of $A$. □
Remark 2.14. We will see in (2.15.2) and (2.15.3) that ${}^{*}\Theta$ and
${\Phi_{A}}^{@}$ are both “upper limiting” functions in various situations, so
the question arises whether these two functions are identical. If
${}^{*}\Theta_{A}={\Phi_{A}}^{@}$ then ${}^{*}\Theta_{A}$ is obviously
$w(B,B)$–lower semicontinuous. If, conversely, ${}^{*}\Theta_{A}$ is
$w(B,B)$–lower semicontinuous then the Fenchel–Moreautheorem for the (possibly
nonhausdorff) locally convex space $\big{(}B,w(B,B)\big{)}$ and Lemma 2.13(d)
imply that
${}^{*}\Theta_{A}=\big{(}^{*}\Theta_{A}\big{)}^{@@}={\Phi_{A}}^{@}$. The
author is grateful to Constantin Zălinescu for the following example showing
that, in general, the functions ${}^{*}\Theta$ and ${\Phi_{A}}^{@}$ are not
identical. Let $B$ be a Banach space, $\lfloor\cdot,\cdot\rfloor=0$ on
$B\times B$ and $A$ be a nonempty proper closed convex subset of $B$. Then
${}^{*}\Theta_{A}$ is the indicator function of $A$ and ${\Phi_{A}}^{@}=0$ on
$B$. We do not know what the situation is if $A$ is maximally $q$–positive, or
in the special situation of Example 2.4. For the convenience of the reader, we
will give a proof of the Fenchel–Moreau theorem for nonhausdorff locally
convex spaces in Theorem 6.1.
Theorem 2.15. Let $B$ be a Banach SSD space.
(a) Let $f\in{\cal PCLSC}(B)$, $f\geq q$ on $B$ and $A:={\cal
P}(f)\neq\emptyset$. Then ${}^{*}\Theta_{A}\geq f\geq\Phi_{A}$ on $B$ and
${\Phi_{A}}^{*}\geq f^{*}\geq\Theta_{A}$ on $B^{*}$. (b) Let $A$ be a
maximally $q$–positive subset of $B$, $h\in{\cal PC}(B)$ and
${}^{*}\Theta_{A}\geq h\geq\Phi_{A}$ on $B$. Then $h\geq q$ on $B$, $h^{@}\geq
q$ on $B$ and ${\cal P}(h)={\cal P}\big{(}h^{@}\big{)}=A$. (c) Let $f\in{\cal
PCLSC}(B)$ be a VZ function and $A:={\cal P}(f)$. Then
${}^{*}\Theta_{A}\geq f\geq\Phi_{A}\geq q\ \hbox{on}\
B\quad\hbox{and}\quad{\Phi_{A}}^{*}\geq f^{*}\geq\Theta_{A}\ \hbox{on}\
B^{*}.$ $None$
Now let $h\in{\cal PC}(B)$ and ${}^{*}\Theta_{A}\geq h\geq\Phi_{A}$ on $B$.
Then $h$ and $h^{@}$ are VZ functions. In particular, ${\cal
P}\big{(}^{*}\Theta_{A}\big{)}={\cal P}\big{(}{\Phi_{A}}^{@}\big{)}={\cal
P}\big{(}\Phi_{A}\big{)}={\cal P}(f)$ and $\Phi_{A}$, ${\Phi_{A}}^{@}$ and
${}^{*}\Theta_{A}$ are all VZ functions.
Proof. (a) Let $b\in B$ and $a\in{\cal P}(f)$. Then, from Lemma 1.11(a),
$f(b)\geq\lfloor b,a\rfloor-q(a)$. Taking the supremum over $a\in{\cal P}(f)$
and using Lemma 2.13(a), $f(b)\geq\Phi_{A}(b)$. Thus $f\geq\Phi_{A}$ on $B$
and, taking conjugates, ${\Phi_{A}}^{*}\geq f^{*}$ on $B^{*}$. Now, for all
$b^{*}\in B^{*}$,
$\eqalign{f^{*}(b^{*})&=\sup\nolimits_{b\in B}\big{[}\langle
b,b^{*}\rangle-f(b)\big{]}\geq\sup\nolimits_{a\in{\cal P}(f)}\big{[}\langle
a,b^{*}\rangle-f(a)\big{]}\cr&=\sup\nolimits_{a\in{\cal P}(f)}\big{[}\langle
a,b^{*}\rangle-q(a)\big{]}=\Theta_{A}(b^{*}).}$
Thus $f^{*}\geq\Theta_{A}$ on $B^{*}$. Taking conjugates and using the
Fenchel–Moreau theorem for the normed space $B$, ${}^{*}\Theta_{A}\geq f$ on
$B$. This completes the proof of (a). (b) From Lemma 2.13(h),
${}^{*}\Theta_{A}\geq h\geq\Phi_{A}\geq q\ \hbox{on}\ B,\quad\hbox{from
which}\quad{\cal P}\big{(}^{*}\Theta_{A}\big{)}\subset{\cal P}(h)\subset{\cal
P}\big{(}\Phi_{A}\big{)}.$ $None$
It is clear from our assumptions that ${\Phi_{A}}^{@}\geq
h^{@}\geq(^{*}\Theta_{A})^{@}$ on $B$. If we now combine this with Lemma
2.13(d,h), we derive that
${\Phi_{A}}^{@}\geq h^{@}\geq\Phi_{A}\geq q\ \hbox{on}\ B,\quad\hbox{from
which}\quad{\cal P}\big{(}{\Phi_{A}}^{@}\big{)}\subset{\cal
P}\big{(}h^{@}\big{)}\subset{\cal P}\big{(}\Phi_{A}\big{)}.$ $None$
(b) now follows from (2.15.2), (2.15.3) and Lemma 2.13(i). (c) The assertions
about $f$ follow from (2.5.3), Theorem 2.9(c), (a) and Lemma 2.13(h), the
assertions about $h$ and $h^{@}$ follow from Theorem 2.9(c,b) and (b), and
then the assertions about $\Phi_{A}$, ${\Phi_{A}}^{@}$ and ${}^{*}\Theta_{A}$
follow from Theorem 2.9(c) and Lemma 2.13(h,i). □
In Theorem 2.16 below, we show that ${}^{*}\Theta_{A}$ has a certain maximal
property. This result was motivated by results originally proved by Burachik
and Svaiter in [1] for maximally monotone multifunctions.
Theorem 2.16. Let $A$ be a nonempty $q$–positive subset of a Banach SSD space
$B$ and
$\sigma_{A}:=\sup\big{\\{}h\colon\ h\in{\cal PCLSC}(B),\ h\leq q\ \hbox{on}\
A\big{\\}}.$
Then ${}^{*}\Theta_{A}=\sigma_{A}$ on $B$.
Proof. Let $h\in{\cal PCLSC}(B)$ and $h\leq q$ on $A$. The Fenchel Young
inequality implies that, for all $b^{*}\in B^{*}$ and $a\in A$,
$h^{*}(b^{*})\geq\langle a,b^{*}\rangle-h(a)\geq\langle a,b^{*}\rangle-q(a)$.
Thus, taking the supremum over $a\in A$, $h^{*}(b^{*})\geq\Theta_{A}(b^{*})$.
In other words, $h^{*}\geq\Theta_{A}$ on $B^{*}$. Taking conjugates and using
the Fenchel Moreau theorem for the normed space $B$, ${}^{*}\Theta_{A}\geq h$
on $B$. It follows by taking the supremum over $h$ that
${}^{*}\Theta_{A}\geq\sigma_{A}$ on $B$. On the other hand, it is clear from
Lemma 2.13(e) that $\sigma_{A}\geq{{}^{*}\Theta_{A}}\ \hbox{on}\ B$. □
Remark 2.17. Let $B$ be a Banach SSD space and $f\in{\cal PCLSC}(B)$ be a VZ
function. We know from Theorem 2.15(c) that ${\cal P}\big{(}\Phi_{{\cal
P}(f)}\big{)}={\cal P}(f)$, $\Phi_{{\cal P}(f)}$ is a VZ function and
$\Phi_{{\cal P}(f)}\leq f$ on $B$. Thus Lemma 2.7(a) implies that, for all
$c\in B$,
$\hbox{\rm dist}(c,{\cal P}(f))=\hbox{\rm dist}\big{(}c,{\cal
P}\big{(}\Phi_{{\cal P}(f)}\big{)}\big{)}\leq\sqrt{2}\sqrt{\big{(}\Phi_{{\cal
P}(f)}-q\big{)}(c)}\leq\sqrt{2}\sqrt{(f-q)(c)}.$
From Lemma 2.13(a), $\big{(}\Phi_{{\cal P}(f)}-q\big{)}(c)=\Phi_{{\cal
P}(f)}(c)-q(c)=-\inf q(c-{\cal P}(f))$, thus we have
$\hbox{\rm dist}(c,{\cal P}(f))\leq\sqrt{2}\sqrt{-\inf q(c-{\cal
P}(f))}\leq\sqrt{2}\sqrt{(f-q)(c)}.$
This shows that Theorem 2.9(a) is as least as strong as Lemma 2.7(a). Now let
$E:=\hbox{\tenmsb R}$ and $B$ be the Banach SSD space $\hbox{\tenmsb R}^{2}$
as in Example 2.4, using the norm $\|\cdot\|_{2,1}$. Define $f\in{\cal
PCLSC}(B)$ by $f(x_{1},x_{2}):={\textstyle{1\over 2}}(x_{1}^{2}+x_{2}^{2})$.
Then $(f-q)(x_{1},x_{2})={\textstyle{1\over
2}}(x_{1}^{2}+x_{2}^{2})-x_{1}x_{2}={\textstyle{1\over 2}}(x_{1}-x_{2})^{2}$
and $p(x_{1},x_{2})={\textstyle{1\over
2}}(x_{1}^{2}+x_{2}^{2})+x_{1}x_{2}={\textstyle{1\over 2}}(x_{1}+x_{2})^{2}$.
Let $c:=(z_{1},z_{2})\in B$ and $b:=\big{(}{\textstyle{1\over
2}}(z_{1}+z_{2}),{\textstyle{1\over 2}}(z_{1}+z_{2})\big{)}\in B$. Then
$(f-q)(b)=0$ and $p(c-b)=0$. Consequently, $f$ is a VZ function. Now ${\cal
P}(f)$ is the diagonal of $\hbox{\tenmsb R}^{2}$ and so, by direct
computation, for all $c=(x_{1},x_{2})\in\hbox{\tenmsb R}^{2}$, $-\inf
q(c-{\cal P}(f))=\textstyle{1\over 4}(x_{1}-x_{2})^{2}$. Since
$\textstyle{1\over 4}(x_{1}-x_{2})^{2}<{\textstyle{1\over
2}}(x_{1}-x_{2})^{2}$ when $x_{1}\neq x_{2}$, Theorem 2.9(a) is strictly
stronger than Lemma 2.7(a) in this case.
Now let $h:=\Phi_{{\cal P}(f)}$. Lemma 2.13(a) gives us that, for all
$(x_{1},x_{2})\in B$,
$\sqrt{(h-q)(x_{1},x_{2})}=\sqrt{\textstyle{1\over
4}(x_{1}-x_{2})^{2}}={\textstyle{1\over 2}}|x_{1}-x_{2}|.$
On the other hand, by direct computation, $\hbox{\rm
dist}\big{(}(x_{1},x_{2}),{\cal
P}(h)\big{)}=\textstyle{1\over\sqrt{2}}|x_{1}-x_{2}|$. Thus the constant
$\sqrt{2}$ in (2.7.1) is sharp. The genesis of this argument and example can
be found in the results of Martínez-Legaz and Théra in [6].
Remark 2.18. We note that the inequalities for $B$ in (2.15.1) have four
functions, while the inequality for $B^{*}$ has only three. The reason for
this is that we do not have a function on $B^{*}$ that plays the role that the
function $q$ plays on $B$. We will introduce such a function in Definition
4.1.
3 Applications of Section 2 to $E\times E^{*}$
In this section, we suppose that $E$ is a nonzero Banach space, and follow the
notation of Example 2.3. Let $A$ be a nonempty monotone subset of $E\times
E^{*}$. In this case, the definitions and results obtained in Definition 2.12
and Lemma 2.13 specialize as follows. The function $\Theta_{A}\in{\cal
PCLSC}^{*}(E^{**}\times E^{*})$ is defined by:
$\Theta_{A}(x^{**},x^{*}):=\sup\nolimits_{(s,s^{*})\in A}\big{[}\langle
s,x^{*}\rangle+\langle s^{*},x^{**}\rangle-\langle s,s^{*}\rangle\big{]}.$
The function $\Phi_{A}\in{\cal PCLSC}(E\times E^{*})$ is defined by:
$\Phi_{A}(x,x^{*})=\sup\nolimits_{(s,s^{*})\in A}\big{[}\langle
x,s^{*}\rangle+\langle s,x^{*}\rangle-\langle s,s^{*}\rangle\big{]}.$
$\Phi_{A}$ is the Fitzpatrick function of $A$, first introduced in [2], which
has been discussed by many authors in recent years. The function
${}^{*}\Theta_{A}\in{\cal PCLSC}(E\times E^{*})$ is defined by:
${}^{*}\Theta_{A}(y,y^{*}):=\sup\nolimits_{(x^{**},x^{*})\in E^{**}\times
E^{*}}\big{[}\langle y,x^{*}\rangle+\langle
y^{*},x^{**}\rangle-\Theta_{A}(x^{**},x^{*})\big{]}.$
Then $(^{*}\Theta_{A})^{*}=\Theta_{A}$ and $(^{*}\Theta_{A})^{@}=\Phi_{A}$.
Furthermore,
${}^{*}\Theta_{A}\geq{\Phi_{A}}^{@}\geq\Phi_{A}\vee q\ \hbox{on}\ E\times
E^{*}\quad\hbox{and}\quad^{*}\Theta_{A}={\Phi_{A}}^{@}=\Phi_{A}=q\ \hbox{on}\
A.$
If $f\in{\cal PC}(E\times E^{*})$ and $f\geq q$ on $E\times E^{*}$ then we
define ${\cal M}f$ to be the monotone set $\\{(x,x^{*})\in E\times E^{*}\colon
f(x,x^{*})=\langle x,x^{*}\rangle\\}$. ${\cal M}f$ is identical with ${\cal
P}(f)$ as in Definition 1.8, but the “${\cal M}$” notation seems more
appropriate in this case. Continuing with the consequences of Lemma 2.13, we
have:
${}^{*}\Theta_{A}\geq{\Phi_{A}}^{@}\geq\Phi_{A}\geq q\ \hbox{on}\ E\times
E^{*}\quad\hbox{and}\quad{\cal M}\big{(}^{*}\Theta_{A}\big{)}={\cal
M}\big{(}{\Phi_{A}}^{@}\big{)}={\cal M}\big{(}\Phi_{A}\big{)}=A.$
The following results are then immediate from Theorems 2.15 and 2.16. The
expression $\sup\big{\\{}h\colon\ h\in{\cal PCLSC}(E\times E^{*}),\ h\leq q\
\hbox{on}\ A\big{\\}}$ that appears in Theorem 3.1(b) was first introduced by
Burachik and Svaiter in [1] (for $A$ maximally monotone) and further studied
by Marques Alves and Svaiter in [5]. The analysis of Lemma 2.13 and Theorem
2.15 suggests that the natural framework in which to consider these results is
that of Banach SSD spaces.
Theorem 3.1. Let $E$ be a nonzero Banach space, $E\times E^{*}$ be normed as
in Example 2.3, and $A$ be a nonempty monotone subset of $E\times E^{*}$.
(a) Let $f\in{\cal PCLSC}(E\times E^{*})$, $f\geq q$ on $E\times E^{*}$ and
$A:={\cal M}f\neq\emptyset$. Then
${}^{*}\Theta_{A}\geq f\geq\Phi_{A}\ \hbox{on}\ E\times
E^{*},\quad\hbox{and}\quad{\Phi_{A}}^{*}\geq f^{*}\geq\Theta_{A}\ \hbox{on}\
E^{**}\times E^{*}.$
(b) Let $A$ be maximally monotone, $h\in{\cal PC}(E\times E^{*})$ and
${}^{*}\Theta_{A}\geq h\geq\Phi_{A}$ on $E\times E^{*}$. Then $h\geq q$ on
$E\times E^{*}$, $h^{@}\geq q$ on $E\times E^{*}$ and ${\cal M}h={\cal
M}\big{(}h^{@}\big{)}=A$. (c) Let $f\in{\cal PCLSC}(E\times E^{*})$ be a VZ
function and $A:={\cal M}f$. Then
${}^{*}\Theta_{A}\geq f\geq\Phi_{A}\geq q\ \hbox{on}\ E\times
E^{*}\quad\hbox{and}\quad{\Phi_{A}}^{*}\geq f^{*}\geq\Theta_{A}\ \hbox{on}\
E^{**}\times E^{*}$
Now let $h\in{\cal PC}(E\times E^{*})$ and ${}^{*}\Theta_{A}\geq
h\geq\Phi_{A}$ on $E\times E^{*}$. Then $h$ and $h^{@}$ are VZ functions. In
particular, ${\cal M}\big{(}^{*}\Theta_{A}\big{)}={\cal
M}\big{(}{\Phi_{A}}^{@}\big{)}={\cal M}\big{(}\Phi_{A}\big{)}=A$, and
$\Phi_{A}$, ${\Phi_{A}}^{@}$ and ${}^{*}\Theta_{A}$ are all VZ functions.
${}^{*}\Theta_{A}=\sup\big{\\{}h\colon\ h\in{\cal PCLSC}(E\times E^{*}),\
h\leq q\ \hbox{on}\ A\big{\\}}.$ $None$
4 Banach SSD dual spaces
Definition 4.1. Let $(B,\|\cdot\|)$ be a Banach SSD space and
$(B^{*},\|\cdot\|)$ be the norm–dual of $B$. We say that
$(B^{*},\lceil\cdot,\cdot\rceil)$ is a Banach SSD dual of $B$ if
$\lceil\cdot,\cdot\rceil\colon\ B^{*}\times B^{*}\to\hbox{\tenmsb R}$ is a
symmetric bilinear form,
$\hbox{for all}\ b\in B\ \hbox{and}\ c^{*}\in
B^{*},\qquad\lceil\iota(b),c^{*}\rceil=\langle b,c^{*}\rangle.$ $None$
Writing $\widetilde{q}(c^{*}):={\textstyle{1\over 2}}\lceil c^{*},c^{*}\rceil$
and $\widetilde{p}(c^{*}):={\textstyle{1\over
2}}\|c^{*}\|^{2}+\widetilde{q}(c^{*})$, we suppose also that
$\widetilde{p}\geq 0\ \hbox{on}\ B^{*}.$ $None$
Now if we take $c^{*}=\iota(c)$ in (4.1.1) and use (2.1.2), we obtain
$\hbox{for all}\ b,c\in
B,\qquad\lceil\iota(b),\iota(c)\rceil=\big{\langle}b,\iota(c)\big{\rangle}=\lfloor
b,c\rfloor,$ $None$
from which
$\widetilde{q}\circ\iota=q.$ $None$
It is easy to see from these definitions that,
$\hbox{for all}\ b^{*}\in
B^{*},\qquad\Theta_{A}(b^{*})=\widetilde{q}(b^{*})-\inf\widetilde{q}(b^{*}-\iota(A)).$
$None$
This should be compared with Lemma 2.13(a).
Definition 4.2. Let $(B,\|\cdot\|)$ be a Banach SSD space and
$(B^{*},\lceil\cdot,\cdot\rceil)$ be a Banach SSD dual of $B$. We say that
$\iota(B)$ is $\widetilde{p}$–dense in $B^{*}$ if
$\hbox{for all}\ b^{*}\in
B^{*},\quad\inf\widetilde{p}\big{(}b^{*}-\iota(B)\big{)}=0.$ $None$
Remark 4.3. In Example 1.3 with $\|T\|\leq 1$ (see also Remark 2.2), for all
$c\in B$, $\iota(c)=Tc$. Suppose now that $T^{2}$ is the identity on $B$.
Since $B^{*}=B$,
$\hbox{for all}\ b\in B\ \hbox{and}\ c^{*}\in
B^{*}=B,\qquad\lfloor\iota(b),c^{*}\rfloor=\lfloor Tb,c^{*}\rfloor=\langle
T^{2}b,c^{*}\rangle=\langle b,c^{*}\rangle.$
Thus (4.1.1) is satisfied with
$\lceil\cdot,\cdot\rceil:=\lfloor\cdot,\cdot\rfloor$, and so
($B,\lfloor\cdot,\cdot\rfloor$) is its own Banach SSD dual. We note that
$T^{2}$ is the identity on $B$ in (a), (b) and (c) of Example 1.3.
Example 4.4. We now continue our discussion of Examples 1.4, 2.3 and 2.4. We
recall that $B=E\times E^{*}$, $B^{*}=E^{**}\times E^{*}$ and, for all
$(x,x^{*})\in B$, $\iota(x,x^{*})=(\widehat{x},x^{*})$. We define the
symmetric bilinear form $\lceil\cdot,\cdot\rceil\colon\ B^{*}\times
B^{*}\to\hbox{\tenmsb R}$ by
$\lceil b^{*},c^{*}\rceil:=\langle y^{*},x^{**}\rangle+\langle
x^{*},y^{**}\rangle\quad\big{(}b^{*}=(x^{**},x^{*})\in B^{*},\
c^{*}=(y^{**},y^{*})\in B^{*}\big{)}.$
It is then easily checked from (2.3.1) that (4.1.1) is satisfied and, for all
$c^{*}=(y^{**},y^{*})\in B^{*}$, $\widetilde{q}(c^{*})={\textstyle{1\over
2}}\big{[}\langle y^{*},y^{**}\rangle+\langle
y^{*},y^{**}\rangle\big{]}=\langle y^{*},y^{**}\rangle$. We now discuss
briefly the limitations of this definition. Let $E:=\hbox{\tenmsb R}$ and $B$
be the SSD space $\hbox{\tenmsb R}^{2}$ as in Example 1.4, using the norm
$2\|\cdot\|_{2,1}$. As we observed in Example 2.4, $\hbox{\tenmsb R}^{2}$ is a
Banach SSD space under $\|\cdot\|_{2,1}$, and consequently also a Banach SSD
space under the larger norm $2\|\cdot\|_{2,1}$. Since $\iota$ is the identity
on $\hbox{\tenmsb R}^{2}$, (4.1.3) implies that
$\lceil\cdot,\cdot\rceil:=\lfloor\cdot,\cdot\rfloor$. Now the norm on
$B^{*}=B$ dual to $2\|\cdot\|_{2,1}$ is ${\textstyle{1\over
2}}\|\cdot\|_{2,1}$. Since $\widetilde{p}(1,-1)={1\over 8}(2)+(1)(-1)=-{3\over
4}<0$, $B$ does not admit a Banach SSD dual. We now return to the general
case. If $c^{*}=(y^{**},y^{*})\in B^{*}$ then
${\textstyle{1\over
2}}\|c^{*}\|_{1,\tau}^{2}+\widetilde{q}(c^{*})\geq\textstyle{1\over
4}\big{(}\tau\|y^{**}\|+\|y^{*}\|/\tau\big{)}^{2}-\|y^{**}\|\|y^{*}\|=\textstyle{1\over
4}\big{(}\tau\|y^{**}\|-\|y^{*}\|/\tau\big{)}^{2}\geq 0.$
Consequently, $\big{(}B^{*},\|\cdot\|_{1,\tau}\big{)}$ is a Banach SSD dual of
$\big{(}B,\|\cdot\|_{\infty,\tau}\big{)}$. Since
$\|\cdot\|_{\infty,\tau}\geq\|\cdot\|_{2,\tau}\geq\|\cdot\|_{1,\tau}$ on
$B^{*}$, $\big{(}B^{*},\|\cdot\|_{2,\tau}\big{)}$ is a Banach SSD dual of
$\big{(}B,\|\cdot\|_{2,\tau}\big{)}$ and
$\big{(}B^{*},\|\cdot\|_{\infty,\tau}\big{)}$ is a Banach SSD dual of
$\big{(}B,\|\cdot\|_{1,\tau}\big{)}$. Next, if $b^{*}=(y^{**},y^{*})\in B^{*}$
and $\varepsilon>0$ then there exists $z^{*}\in E^{*}$ such that
$\|z^{*}\|\leq\|\tau y^{**}\|$ and $\langle z^{*},\tau y^{**}\rangle\geq\|\tau
y^{**}\|^{2}-\varepsilon$. Let $c:=(0,y^{*}+\tau z^{*})\in B$, so that
$b^{*}-\iota(c)=(y^{**},-\tau z^{*})\in B^{*}$. Thus
$\eqalign{{\textstyle{1\over
2}}\|b^{*}-\iota(c)\|_{\infty,\tau}^{2}&+\widetilde{q}(b^{*}-\iota(c))=\big{(}\tau\|y^{**}\|\vee\|z^{*}\|\big{)}^{2}-\langle\tau
z^{*},y^{**}\rangle\cr&=\big{(}\|\tau y^{**}\|\vee\|z^{*}\|\big{)}^{2}-\langle
z^{*},\tau y^{**}\rangle=\|\tau y^{**}\|^{2}-\langle z^{*},\tau
y^{**}\rangle\leq\varepsilon.}$
Consequently, if $B$ is normed by $\|\cdot\|_{1,\tau}$ then $\iota(B)$ is
$\widetilde{p}$–dense in $B^{*}$. Since
$\|\cdot\|_{1,\tau}\leq\|\cdot\|_{2,\tau}\leq\|\cdot\|_{\infty,\tau}$ on
$B^{*}$, the same is true if $B$ is normed by $\|\cdot\|_{2,\tau}$ or
$\|\cdot\|_{\infty,\tau}$.
We now recall Rockafellar’s formula for the conjugate of a sum:
Lemma 4.5. Let $X$ be a nonzero real Banach space and $f\in{\cal PC}(X)$, and
let $h\in{\cal PC}(X)$ be real–valued and continuous. Then, for all $x^{*}\in
X^{*}$,
$(f+h)^{*}(x^{*})=\min\nolimits_{y^{*}\in
X^{*}}\big{[}f^{*}(y^{*})+h^{*}(x^{*}-y^{*})\big{]}.$
Proof. See Rockafellar, [8, Theorem 3(a), p. 85], Zălinescu, [16, Theorem
2.8.7(iii), p. 127], or [13, Corollary 10.3, p. 52]. □
Remark 4.6. [13, Theorem 7.4, p. 43] contains a version of the Fenchel duality
theorem with a sharp lower bound on the functional obtained.
Our next result exhibits a certain pleasing symmetry between $B$ and $B^{*}$.
Lemma 4.7. Let $B$ be a Banach SSD space with a Banach SSD dual $B^{*}$ and
$f\in{\cal PC}(B)$. Then
$\big{(}(f-q)\mathop{\nabla}p\big{)}+\big{(}(f^{*}-\widetilde{q})\mathop{\nabla}\widetilde{p}\big{)}\circ\iota=0$
on $B$.
Proof. Let $c\in B$. Define $h\colon\ B\to\hbox{\tenmsb R}$ by $h(b):=g(c-b)$.
Then, by direct computation using the fact that $g$ is an even function,
$\hbox{for all}\ c^{*}\in B^{*},\qquad h^{*}(c^{*})=g^{*}(c^{*})+\langle
c,c^{*}\rangle.$ $None$
Then, using (2.1.2), the continuity of $h$, Lemma 4.5, (4.7.1), (4.1.4) and
the fact that, for all $c^{*}\in B^{*}$, $g^{*}(c^{*})={\textstyle{1\over
2}}\|c^{*}\|^{2}$,
$\eqalign{-\big{(}(f-q)\mathop{\nabla}p\big{)}(c)&=\sup\nolimits_{b\in
B}\big{[}-(f-q)(b)-p(c-b)\big{]}\cr&=\sup\nolimits_{b\in B}\big{[}\langle
b,\iota(c)\rangle-f(b)-h(b)\big{]}-q(c)=(f+h)^{*}\big{(}\iota(c)\big{)}-q(c)\cr&=\min\nolimits_{b^{*}\in
B^{*}}\big{[}f^{*}(b^{*})+h^{*}\big{(}\iota(c)-b^{*}\big{)}\big{]}-q(c)\cr&=\min\nolimits_{b^{*}\in
B^{*}}\big{[}f^{*}(b^{*})+g^{*}\big{(}\iota(c)-b^{*}\big{)}+\big{\langle}c,\iota(c)-b^{*}\big{\rangle}\big{]}-q(c)\cr&=\min\nolimits_{b^{*}\in
B^{*}}\big{[}f^{*}(b^{*})+g^{*}\big{(}\iota(c)-b^{*}\big{)}-\lceil\iota(c),b^{*}\rceil+\widetilde{q}\big{(}\iota(c)\big{)}\big{]}\cr&=\min\nolimits_{b^{*}\in
B^{*}}\big{[}(f^{*}-\widetilde{q})(b^{*})+\widetilde{p}\big{(}\iota(c)-b^{*}\big{)}\big{]}\cr&=\big{(}(f^{*}-\widetilde{q})\mathop{\nabla}\widetilde{p}\big{)}\big{(}\iota(c)\big{)}.}$
This completes the proof of Lemma 4.7. □
Definition 4.8. Let $B$ be a Banach SSD space with Banach SSD dual $B^{*}$ and
$f\in{\cal PC}(B)$. We say that $f$ is an MAS function if $f\geq q$ on $B$ and
$f^{*}\geq\widetilde{q}$ on $B^{*}$. This is an extension to Banach SSD spaces
of the concept introduced by Marques Alves and Svaiter in [4] for the
situation described in Example 4.4.
Theorem 4.9. Let $B$ be a Banach SSD space with Banach SSD dual $B^{*}$ and
$f\in{\cal PC}(B)$.
(a) Let $f$ be an MAS function. Then $f$ is a VZ function.
(b) Let $\iota(B)$ be $\widetilde{p}$–dense in $B^{*}$ and $f$ be a VZ
function. Then $f$ is an MAS function.
(c) Let $\iota(B)$ be $\widetilde{p}$–dense in $B^{*}$. Then $f$ is a VZ
function if, and only if, $f$ is an MAS function.
Proof. (a) We have (using (2.1.4)) $f-q\geq 0$ and $p\geq 0$ on $B$, and
(using (4.1.2)), $f^{*}-\widetilde{q}\geq 0$ and $\widetilde{p}\geq 0$ on
$B^{*}$. Thus $(f-q)\mathop{\nabla}p\geq 0$ and
$\big{(}(f^{*}-\widetilde{q})\mathop{\nabla}\widetilde{p}\big{)}\circ\iota\geq
0$ on $B$. It now follows from Lemma 4.7 that $f$ is a VZ function.
(b) Let $b^{*}\in B^{*}$ and $c\in B$. Then, from Lemma 4.7 again,
$(f^{*}-\widetilde{q})(b^{*})+\widetilde{p}\big{(}\iota(c)-b^{*}\big{)}\geq\big{(}(f^{*}-\widetilde{q})\mathop{\nabla}\widetilde{p}\big{)}\big{(}\iota(c)\big{)}=-\big{(}(f-q)\mathop{\nabla}p\big{)}(c)=0.$
Taking the infimum over $c\in B$ and using (4.2.1),
$(f^{*}-\widetilde{q})(b^{*})\geq 0$ on $B^{*}$. Since this holds for all
$b^{*}\in B^{*}$, $f$ is an MAS function.
(c) is immediate from (a) and (b). □
In Theorem 4.10, we shift the emphasis from the properties of a given function
$f\in{\cal PC}(B)$ to the properties of a given maximally $q$–positive subset
$A$ of $B$. We note that (a), (b), (c), (f) and (g) of Theorem 4.10 do not
involve any functions on $B$ other than those introduced in Definition 2.12.
Theorem 4.10. Let $(B,\|\cdot\|)$ be a Banach SSD space with Banach SSD dual
$B^{*}$ and $\iota(B)$ be $\widetilde{p}$–dense in $B^{*}$. Let $A$ be a
maximally $q$–positive subset of $B$. Then the following conditions are
equivalent:
(a) For all $b^{*}\in B^{*}$,
$\inf\widetilde{q}\big{(}b^{*}-\iota(A)\big{)}\leq 0$. (b)
$\Theta_{A}\geq\widetilde{q}$ on $B^{*}$. (c)
${\Phi_{A}}^{*}\geq\widetilde{q}$ on $B^{*}$. (d) There exists an MAS function
$f\in{\cal PCLSC}(B)$ such that ${\cal P}(f)=A$. (e) There exists a VZ
function $f\in{\cal PCLSC}(B)$ such that ${\cal P}(f)=A$. (f) $\Phi_{A}$ is a
VZ function. (g) ${}^{*}\Theta_{A}$ is a VZ function. (b1) If $h\in{\cal
PC}(B)$ and ${}^{*}\Theta_{A}\geq h$ on $B$ then $h^{*}\geq\widetilde{q}$ on
$B^{*}$. (b2) If $h\in{\cal PCLSC}(B)$ and ${}^{*}\Theta_{A}\geq
h\geq\Phi_{A}$ on $B$ then $h^{*}\geq\widetilde{q}$ on $B^{*}$. (c1) There
exists $h\in{\cal PCLSC}(B)$ such that ${}^{*}\Theta_{A}\geq h\geq\Phi_{A}$ on
$B$ and $h^{*}\geq\widetilde{q}$ on $B^{*}$. (c2) There exists $h\in{\cal
PC}(B)$ such that $h\geq\Phi_{A}$ on $B$ and $h^{*}\geq\widetilde{q}$ on
$B^{*}$.
Proof. The equivalence of (a) and (b) is immediate from (4.1.5). Taking the
conjugate of the inequality in Lemma 2.13(f) and using Lemma 2.13(d) implies
that ${\Phi_{A}}^{*}\geq\Theta_{A}$ on $B^{*}$. Thus (b)$\Longrightarrow$(c).
If (c) is satisfied then Lemma 2.13(b,h,i) give (d) with $f:=\Phi_{A}$. It is
immediate from Theorem 4.9(c) that (d)$\Longrightarrow$(e). If (e) is
satisfied then Theorem 2.15(c) gives (f) and (g). If (f) or (g) is satisfied
then, from Theorem 4.9(c) again, $\Phi_{A}$ or ${}^{*}\Theta_{A}$
(respectively) are MAS functions. The first of these possibilities implies
(c), and the second of these possibilities together with Lemma 2.13(d) implies
(b). Thus (a), (b), (c), (d), (e), (f) and (g) are equivalent.
If $h\in{\cal PC}(B)$ and ${}^{*}\Theta_{A}\geq h$ on $B$ then, from Lemma
2.13(d), $h^{*}\geq(^{*}\Theta_{A})^{*}=\Theta_{A}$ on $B^{*}$, thus (b)
implies (b1). It is trivial that (b1) implies (b2), and it follows by taking
$h:={{}^{*}\Theta_{A}}$ and using Lemma 2.13(f,d) that (b) is true. Thus (b),
(b1) and (b2) are equivalent.
If (c) is true then (c1) follows by taking $h:=\Phi_{A}$ and using Lemma
2.13(b,f). It is trivial that (c1) implies (c2). If $h\in{\cal PC}(B)$ and
$h\geq\Phi_{A}$ on $B$ then ${\Phi_{A}}^{*}\geq h^{*}$ on $B^{*}$, and (c2)
implies (c). Thus (c), (c1) and (c2) are equivalent. □
5 Applications of Section 4 to $E\times E^{*}$
In this section, we suppose that $E$ is a nonzero Banach space, and show how
the results of Section 4 can be applied to Example 4.4. We refer the reader to
Section 3 for the definitions of $\Theta_{A}$ and $\Phi_{A}$ in this case.
Remark 5.1. Before proceeding with our analysis, we make some remarks about
the essential difference between the concepts of MAS function introduced in
Definition 4.8 and VZ function introduced in Definition 2.5. As observed in
Example 4.4, we have $(E\times E^{*})^{*}=E^{**}\times E^{*}$ and
$\widetilde{q}(x^{**},x^{*})=\langle x^{*},x^{**}\rangle$, so we have all the
information needed to decide whether a function $f\in{\cal PC}(E\times E^{*})$
is an MAS function. The situation with VZ functions is different since that
involves the function $g$ in an essential way, and this is determined by the
precise norm we are using on $E\times E^{*}$. In order to clarify the
situation, we make the following definition.
Definition 5.2. We say that the norm $\|\cdot\|$ on $E\times E^{*}$ is special
if, for some $\tau>0$, $\|\cdot\|$ is identical with one of the norms
$\|\cdot\|_{1,\tau}$, $\|\cdot\|_{2,\tau}$ or $\|\cdot\|_{\infty,\tau}$
introduced in Example 2.4. As we pointed out in the comments in Example 4.4,
if $E\times E^{*}$ is normed by a special norm then $\iota(E\times E^{*})$ is
$\widetilde{p}$–dense in $(E\times E^{*})^{*}$.
Theorem 5.3. Let $E$ be a nonzero Banach space and $f\in{\cal PC}(E\times
E^{*})$ be a VZ function with respect to a given special norm on $E\times
E^{*}$. Then $f$ is a VZ function with respect to all special norms on
$E\times E^{*}$.
Proof. This is clear from the comments above and Theorem 4.9(c). □
Definition 5.4. Let Let $E$ be a nonzero Banach space and $f\in{\cal
PC}(E\times E^{*})$. We say that $f$ is a VZ function on $E\times E^{*}$ if
$f$ is a VZ function with respect to any one special norm on $E\times E^{*}$
or, equivalently, with respect to all special norms on $E\times E^{*}$. This
is also equivalent to the statement that $f$ is an MAS function.
Theorem 5.5(a) was obtained in [15, Theorem 8] under the VZ hypothesis and, in
[4, Theorem 4.2(2)] under the MAS hypothesis.
Theorem 5.5(c) extends the result proved in [15, Corollary 25] that ${\cal
M}f$ is of type (ANA).
Theorem 5.5(d) extends the result proved in [4, Theorem 4.2(2)].
Theorem 5.5(f) was obtained in [15, Corollary 7]. This is a very significant
result, because maximally monotone sets $A$ of $E\times E^{*}$ are known such
that $\overline{\pi_{E^{*}}(A)}$ is not convex. (The first such example was
given by Gossez in [3, Proposition, p. 360]). Thus (as was first observed in
[15]) Theorem 5.5(f) implies that there exist maximally monotone sets $A$ that
are not of the form ${\cal M}f$ for any lower semicontinuous VZ function on
$E\times E^{*}$ or, equivalently, not of the form ${\cal M}f$ for any lower
semicontinuous MAS function on $E\times E^{*}$. Theorem 5.5(f) can also be
proved directly from Lemma 2.7(a) rather than from the more circuitous
argument given here.
The techniques used in Theorem 5.5 originated in the negative alignment
analysis of [10, Section 8, pp. 274–280] and [13, Section 42, pp. 161–167].
Theorem 5.5. Let $E$ be a nonzero Banach space and $f\in{\cal PCLSC}(E\times
E^{*})$. Assume either that $f$ is a VZ function on $E\times E^{*}$ or,
equivalently, that $f$ is an MAS function. Then:
(a) ${\cal M}f$ is a maximally monotone subset of $E\times E^{*}$. (b) Let
$(x,x^{*})\in E\times E^{*}$ and $\alpha,\beta>0$. Then there exists a unique
value of $\omega\geq 0$ for which there exists a bounded sequence
$\big{\\{}(y_{n},y_{n}^{*})\big{\\}}_{n\geq 1}$ of elements of ${\cal M}f$
such that,
$\lim_{n\to\infty}\|y_{n}-x\|=\alpha\omega,\quad\lim_{n\to\infty}\|y_{n}^{*}-x^{*}\|=\beta\omega\quad\hbox{and}\quad\lim_{n\to\infty}\langle
y_{n}-x,y_{n}^{*}-x^{*}\rangle=-\alpha\beta\omega^{2}.$
(c) Let $(x,x^{*})\in E\times E^{*}\setminus{\cal M}f$ and $\alpha,\beta>0$.
Then there exists a bounded sequence
$\big{\\{}(y_{n},y_{n}^{*})\big{\\}}_{n\geq 1}$ of elements of ${\cal
M}f\cap\big{[}(E\setminus\\{x\\})\times(E^{*}\setminus\\{x^{*}\\})\big{]}$
such that,
$\lim_{n\to\infty}{\|y_{n}-x\|\over\|y_{n}^{*}-x^{*}\|}={\alpha\over\beta}\quad\hbox{and}\quad\lim_{n\to\infty}{\langle
y_{n}-x,y_{n}^{*}-x^{*}\rangle\over\|y_{n}-x\|\|y_{n}^{*}-x^{*}\|}=-1.$ $None$
In particular, ${\cal M}f$ is of type (ANA) (see [13, Definition 36.11, p.
152]). (d) Let $(x,x^{*})\in E\times E^{*}\setminus{\cal M}f$,
$\alpha,\beta>0$ and $\inf_{(y,y^{*})\in{\cal M}f}\langle
y-x,y^{*}-x^{*}\rangle>-\alpha\beta$. Then there exists a bounded sequence
$\big{\\{}(y_{n},y_{n}^{*})\big{\\}}_{n\geq 1}$ in ${\cal
M}f\cap\big{[}(E\setminus\\{x\\})\times(E^{*}\setminus\\{x^{*}\\})\big{]}$
such that (5.5.1) is satisfied, $\lim_{n\to\infty}\|y_{n}-x\|<\alpha$ and
$\lim_{n\to\infty}\|y_{n}^{*}-x^{*}\|<\beta$. In particular, ${\cal M}f$ is of
type (BR) (see [13, Definition 36.13, p. 153]). (e) Let $(x,x^{*})\in E\times
E^{*}\setminus{\cal M}f$, $\alpha,\beta>0$ and $f(x,x^{*})<\langle
x,x^{*}\rangle+\alpha\beta$. Then there exists a bounded sequence
$\big{\\{}(y_{n},y_{n}^{*})\big{\\}}_{n\geq 1}$ of elements of ${\cal
M}f\cap\big{[}(E\setminus\\{x\\})\times(E^{*}\setminus\\{x^{*}\\})\big{]}$
such that (5.5.1) is satisfied, $\lim_{n\to\infty}\|y_{n}-x\|<\alpha$ and
$\lim_{n\to\infty}\|y_{n}^{*}-x^{*}\|<\beta$. (f) We define the projection
maps $\pi_{E}\colon E\times E^{*}\to E$ and $\pi_{E^{*}}\colon E\times
E^{*}\to E^{*}$ by $\pi_{E}(x,x^{*}):=x$ and $\pi_{E^{*}}(x,x^{*}):=x^{*}$.
Then $\overline{\pi_{E}({\cal M}f)}=\overline{\pi_{E}(\hbox{\rm dom}\,f)}$ and
$\overline{\pi_{E^{*}}({\cal M}f)}=\overline{\pi_{E^{*}}(\hbox{\rm dom}\,f)}$.
Consequently, the sets $\overline{\pi_{E}({\cal M}f)}$ and
$\overline{\pi_{E^{*}}({\cal M}f)}$ are convex.
Proof. (a) is immediate from Theorem 2.9(c).
(b) Let $\tau:=\sqrt{\beta/\alpha}$ and use the norm $\|\cdot\|_{\infty,\tau}$
on $E\times E^{*}$. Lemma 2.7(b) provides us with a bounded sequence
$\big{\\{}(y_{n},y_{n}^{*})\big{\\}}_{n\geq 1}$ of elements of ${\cal M}f$
such that
$\lim_{n\to\infty}\big{[}\beta\|y_{n}-x\|^{2}/\alpha\vee\alpha\|y_{n}^{*}-x^{*}\|^{2}/\beta+\langle
y_{n}-x,y_{n}^{*}-x^{*}\rangle\big{]}=0.$
By passing to an appropriate subsequence, we can and will suppose that the
three limits $\rho:=\lim_{n\to\infty}\|y_{n}-x\|$,
$\sigma:=\lim_{n\to\infty}\|y_{n}^{*}-x^{*}\|$ and $\lim_{n\to\infty}\langle
y_{n}-x,y_{n}^{*}-x^{*}\rangle$ all exist. Consequently,
$\beta\rho^{2}/\alpha\vee\alpha\sigma^{2}/\beta+\lim_{n\to\infty}\langle
y_{n}-x,y_{n}^{*}-x^{*}\rangle=0$, from which
$\beta\rho^{2}/\alpha\vee\alpha\sigma^{2}/\beta=-\lim\nolimits_{n\to\infty}\langle
y_{n}-x,y_{n}^{*}-x^{*}\rangle\leq\rho\sigma=\sqrt{\beta\rho^{2}/\alpha}\sqrt{\alpha\sigma^{2}/\beta}.$
It follows easily from this that $\beta\rho^{2}/\alpha=\alpha\sigma^{2}/\beta$
and $\lim_{n\to\infty}\langle y_{n}-x,y_{n}^{*}-x^{*}\rangle=-\rho\sigma$. The
first of these equalities implies that $\rho/\alpha=\sigma/\beta$. We take
$\omega:=\rho/\alpha=\sigma/\beta$, and it is immediate that $\omega$ has the
required properties. The uniqueness of $\omega$ was established in [10,
Theorem 8.4(b), p. 276] and [13, Theorem 42.2(b), pp. 163–164].
(c) Following on from (b), if $\omega=0$ then $(\rho,\sigma)=(0,0)$, that is
to say $\lim_{n\to\infty}y_{n}=x$ in $E$ and
$\lim_{n\to\infty}y_{n}^{*}=x^{*}$ in $E^{*}$. Since ${\cal M}f$ is closed,
this would contradict the hypothesis that $(x,x^{*})\not\in{\cal M}f$. Thus
$\omega>0$, from which $\rho>0$ and $\sigma>0$. (c) now follows by truncating
the sequences so that, for all $n$, $\|y_{n}-x\|>0$ and
$\|y_{n}^{*}-x^{*}\|>0$.
(d) Continuing with the notation of (c), we have
$-\alpha\beta<\inf\nolimits_{(y,y^{*})\in{\cal M}f}\langle
y-x,y^{*}-x^{*}\rangle\leq\lim\nolimits_{n\to\infty}\langle
y_{n}-x,y_{n}^{*}-x^{*}\rangle=-\rho\sigma,$
from which $(\rho/\alpha)(\sigma/\beta)<1$. Since $\rho/\alpha=\sigma/\beta$,
in fact $\rho/\alpha<1$. and $\sigma/\beta<1$, that is to say
$\rho=\lim_{n\to\infty}\|y_{n}-x\|<\alpha$ and
$\sigma=\lim_{n\to\infty}\|y_{n}^{*}-x^{*}\|<\beta$. This gives (d). (e) is
immediate from (d) and the comment in Remark 2.17 that, for all $(x,x^{*})\in
E\times E^{*}$, $-\inf\nolimits_{(y,y^{*})\in{\cal M}f}\langle
y-x,y^{*}-x^{*}\rangle\leq f(x,x^{*})-\langle x,x^{*}\rangle$. . (f) If
$x\in\pi_{E}(\hbox{\rm dom}\,f)$ then there exists $x^{*}\in E^{*}$ such that
$f(x,x^{*})<\infty$, and so it follows from (e) that there exists
$(y,y^{*})\in{\cal M}f$ such that $\|y-x\|<1/n$. Consequently,
$x\in\overline{\pi_{E}({\cal M}f)}$. Thus we have proved that
$\pi_{E}(\hbox{\rm dom}\,f)\subset\overline{\pi_{E}({\cal M}f)}$. On the other
hand, ${\cal M}f\subset\hbox{\rm dom}\,f$, and so $\overline{\pi_{E}({\cal
M}f)}=\overline{\pi_{E}(\hbox{\rm dom}\,f)}$. We can prove in an exactly
similar way that $\overline{\pi_{E^{*}}({\cal
M}f)}=\overline{\pi_{E^{*}}(\hbox{\rm dom}\,f)}$. The convexity of the sets
$\overline{\pi_{E}({\cal M}f)}$ and $\overline{\pi_{E^{*}}({\cal M}f)}$ now
follows immediately. □
Remark 5.6. If we combine Theorem 2.9(a) (using the norm $\|\cdot\|_{2,1}$ on
$E\times E^{*}$) with the comments made in the proof of Theorem 5.5(e) we
obtain the following result: Let $E$ be a nonzero Banach space, $f\in{\cal
PCLSC}(E\times E^{*})$, and $f$ be a VZ function on $E\times E^{*}$. Then, for
all $(x,x^{*})\in E\times E^{*}$,
$\eqalign{\inf\nolimits_{(y,y^{*})\in{\cal
M}f}\sqrt{\|y-x\|^{2}+\|y^{*}-x^{*}\|^{2}}&\leq\sqrt{2}\sqrt{-\inf\nolimits_{(y,y^{*})\in{\cal
M}f}\langle y-x,y^{*}-x^{*}\rangle}\cr&\leq\sqrt{2}\sqrt{f(x,x^{*})-\langle
x,x^{*}\rangle}.}$
This strengthens the result proved in [15, Theorem 4], namely that
$\inf\nolimits_{(y,y^{*})\in{\cal
M}f}\sqrt{\|y-x\|^{2}+\|y^{*}-x^{*}\|^{2}}\leq 2\sqrt{f(x,x^{*})-\langle
x,x^{*}\rangle}.$
As we observed in Remark 2.17, the constant $\sqrt{2}$ is sharp.
Definition 5.7. Let $E$ be a nonzero Banach space and $A$ be a nonempty
monotone subset of $E\times E^{*}$. We say that $A$ is of type (NI) if, for
all $(x^{**},x^{*})\in E^{**}\times E^{*}$,
$\inf\nolimits_{(s,s^{*})\in A}\langle
x^{*}-s^{*},x^{**}-\widehat{s}\rangle\leq 0.$
This concept was introduced in [9, Definition 10, p. 183]. We say that $A$ is
strongly representable if there exists $f\in{\cal PCLSC}(E\times E^{*})$ such
that $f\geq q$ on $E\times E^{*}$, $f^{*}\geq\widetilde{q}$ on $E^{**}\times
E^{*}$ (i.e., $f$ is a lower semicontinuous MAS function) and ${\cal M}f=A$.
This concept was introduced and studied in [4], [5] and [15].
Theorem 5.8 was motivated by and extends that proved in [5, Theorem 1.2]. The
most significant part of it is the fact that (a) implies (d) and (a) implies
(e). In particular, if $A$ is maximally monotone of type (NI), then the
conclusions of Theorem 5.5(b–f) hold (with ${\cal M}f$ replaced by $A$). This
leads to a substantial generalization of [10, Theorem 8.6, pp. 277–278] and
[13, Theorem 42.6, pp. 163–164]. The fact that $\overline{\pi_{E}A}$ and
$\overline{\pi_{E^{*}}A}$ are convex whenever $A$ is of type (NI) was first
proved by Zagrodny in [14].
Theorem 5.8. Let $E$ be a nonzero Banach space and $A$ be a maximally monotone
subset of $E\times E^{*}$. Then the following conditions are equivalent: (a)
$A$ is of type (NI). (b) For all $(x^{**},x^{*})\in E^{**}\times E^{*}$,
$\sup\nolimits_{(s,s^{*})\in A}\big{[}\langle s,x^{*}\rangle+\langle
s^{*},x^{**}\rangle-\langle s,s^{*}\rangle\big{]}\geq\langle
x^{*},x^{**}\rangle$. (c) For all $(x^{**},x^{*})\in E^{**}\times E^{*}$,
$\sup\nolimits_{(y,y^{*})\in E\times E^{*}}\big{[}\langle
y,x^{*}\rangle+\langle y^{*},x^{**}\rangle-\Phi_{A}(y,y^{*})\big{]}\geq\langle
x^{*},x^{**}\rangle$. (d) $A$ is strongly representable. (e) There exists a
lower semicontinuous VZ function on $E\times E^{*}$ such that ${\cal M}f=A$.
(f) $\Phi_{A}$ is a VZ function on $E\times E^{*}$. (g) ${}^{*}\Theta_{A}$ is
a VZ function on $E\times E^{*}$. (b1) If $h\in{\cal PC}(E\times E^{*})$ and
${}^{*}\Theta_{A}\geq h$ on $E\times E^{*}$ then, for all $(x^{**},x^{*})\in
E^{**}\times E^{*}$,
$h^{*}(x^{**},x^{*})=\sup\nolimits_{(y,y^{*})\in E\times E^{*}}\big{[}\langle
y,x^{*}\rangle+\langle y^{*},x^{**}\rangle-h(y,y^{*})\big{]}\geq\langle
x^{*},x^{**}\rangle.$ $None$
(b2) If $h\in{\cal PCLSC}(E\times E^{*})$ and ${}^{*}\Theta_{A}\geq
h\geq\Phi_{A}$ on $E\times E^{*}$ then, for all $(x^{**},x^{*})\in
E^{**}\times E^{*}$, (5.8.1) is satisfied. (c1) There exists $h\in{\cal
PCLSC}(E\times E^{*})$ such that ${}^{*}\Theta_{A}\geq h\geq\Phi_{A}$ on
$E\times E^{*}$ and, for all $(x^{**},x^{*})\in E^{**}\times E^{*}$, (5.8.1)
is satisfied. (c2) There exists $h\in{\cal PC}(E\times E^{*})$ such that
$h\geq\Phi_{A}$ on $E\times E^{*}$ and, for all $(x^{**},x^{*})\in
E^{**}\times E^{*}$, (5.8.1) is satisfied.
Proof. These results are all immediate from the corresponding parts of Theorem
4.10. □
6 Appendix: a nonhausdorff Fenchel–Moreau theorem
In Remark 2.14, we referred to the Fenchel–Moreau theorem for (possibly
nonhausdorff) locally convex spaces. We shall give a proof of this result in
Theorem 6.1. When we say that $X$ is a locally convex space, we mean that $X$
is a nonzero real vector space endowed with a topology compatible with its
vector structure with a base of neighborhoods of $0$ of the form
$\big{\\{}x\in X\colon\ S(x)\leq 1\big{\\}}_{S\in{\cal S}(X)}$, where ${\cal
S}(X)$ is a family of seminorms on $X$ such that if $S_{1}\in{\cal S}(X)$ and
$S_{2}\in{\cal S}(X)$ then $S_{1}\vee S_{2}\in{\cal S}(X)$; and if $S\in{\cal
S}(X)$ and $\lambda\geq 0$ then $\lambda S\in{\cal S}(X)$. If $L$ is a linear
functional on $X$ then $L$ is continuous if, and only if, there exists
$S\in{\cal S}(X)$ such that $L\leq S$ on $X$.
As an example of the construction above, we can suppose that $X$ and $Y$ are
vector spaces paired by a bilinear form $\langle\cdot,\cdot\rangle$. Then
$\big{(}X,w(X,Y)\big{)}$ is a locally convex space with determining family of
seminorms
$\big{\\{}|\langle\cdot,y_{1}\rangle|\vee\cdots\vee|\langle\cdot,y_{n}\rangle|\big{\\}}_{n\geq
1,\ y_{1},\dots,y_{n}\in Y}$.
The author is grateful to Constantin Zălinescu for showing him a proof of
Theorem 6.1 based on the standard (Hausdorff) result and a quotient
construction. The proof we give here is a simplification of the result on
“Fenchel–Moreau points” of [12, Theorem 5.3, pp. 157–158] or [13, Theorem
12.2, pp. 59–60], which is also valid in the nonhausdorff setting.
Theorem 6.1. Let $X$ be a locally convex space and $f\in{\cal PC}(X)$ be lower
semicontinuous. Write $X^{*}$ for the set of continuous linear functionals on
$X$. If $L\in X^{*}$, define $f^{*}(L):=\sup_{X}\big{[}L-f\big{]}$. Let $y\in
X$. Then
$f(y)=\sup\nolimits_{L\in X^{*}}\big{[}L(y)-f^{*}(L)\big{]}.$ $None$
Proof. Since, for all $L\in X^{*}$, $L(y)-f^{*}(L)=\inf\nolimits_{x\in
X}\big{[}L(y)-L(x)+f(x)\big{]}=(f\mathop{\nabla}L)(y)$ and the inequality
“$\geq$” in (6.1.1) is obvious from the definition of $f^{*}(L)$, we only have
to prove that
$f(y)\leq\sup\nolimits_{L\in X^{*}}(f\mathop{\nabla}L)(y)\big{]}.$ $None$
Let $\lambda\in\hbox{\tenmsb R}$ and $\lambda<f(y)$. Since $f$ is proper,
there exists $z\in\hbox{\rm dom}\,f$. Choose $Q\in{\cal S}(X)$ such that
$Q(z-x)\leq 1\quad\Longrightarrow\quad f(x)>f(z)-1$ $None$
and
$Q(y-x)\leq 1\quad\Longrightarrow\quad f(x)>\lambda.$ $None$
We first prove that
$(f\mathop{\nabla}Q)(z)\geq f(z)-1.$ $None$
To this end, let $x$ be an arbitrary element of $X$. If $Q(z-x)\leq 1$ then
(6.1.3) implies that $f(x)+Q(z-x)\geq f(x)>f(z)-1$. If, on the other hand,
$Q(z-x)>1$, let $\gamma:=1/Q(z-x)\in\,]0,1[\,$ and put $u:=\gamma
x+(1-\gamma)z$. Then $Q(z-u)=\gamma Q(z-x)=1$ and so, from the convexity of
$f$, and (6.1.3) with $x$ replaced by $u$,
$\gamma f(x)+(1-\gamma)f(z)\geq f\big{(}\gamma
x+(1-\gamma)z\big{)}=f(u)>f(z)-1.$
Substituting in the formula for $\gamma$ and clearing of fractions yields
$f(x)+Q(z-x)\geq f(z)$. This completes the proof of (6.1.5).
Now let $M\geq 1$ and $M\geq\lambda+2+Q(z-y)-f(z)$. We will prove that
$(f\mathop{\nabla}MQ)(y)\geq\lambda.$ $None$
To this end, let $x$ be an arbitrary element of $X$. If $Q(y-x)\leq 1$ then
(6.1.4) implies that $f(x)+MQ(y-x)\geq f(x)>\lambda$. If, on the other hand,
$Q(y-x)>1$ then, from (6.1.5),
$\eqalign{f(x)+MQ(y-x)&=f(x)+Q(y-x)+(M-1)Q(y-x)\cr&\geq
f(x)+Q(z-x)-Q(z-y)+(M-1)\cr&\geq f(z)-1-Q(z-y)+M-1\geq\lambda,}$
which completes the proof of (6.1.6). The “Hahn–Banach–Lagrange theorem”
of[12, Theorem 2.9, p. 153] or [13, Theorem 1.11, p. 21] now provides us with
a linear functional $L$ on $X$ such that $L\leq MQ$ on $X$ and
$(f\mathop{\nabla}L)(y)\geq\lambda$. (6.1.2) now follows by letting
$\lambda\to f(y)$. □
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2008, to appear in J. of Convex Anal.
[5] M. Marques Alves and B. F. Svaiter, A new old class of maximal monotone
operators., http://arxiv.org/abs/0805.4597v1, posted May 29, 2008.
[6] J.–E. Martínez-Legaz and M. Théra, $\varepsilon$–Subdifferentials in terms
of subdifferentials, Set–Valued Anal. 4 (1996), 327–332.
[7] J.–P. Penot, The relevance of convex analysis for the study of
monotonicity, Nonlinear Anal. 58 (2004), 855–871.
[8] R. T. Rockafellar, Extension of Fenchel’s duality theorem for convex
functions, Duke Math. J. 33 (1966), 81–89.
[9] S. Simons, The range of a monotone operator, J. Math. Anal. Appl. 199
(1996), 176–201.
[10] —–, Maximal monotone multifunctions of Brøndsted–Rockafellar type,
Set–Valued Anal. 7 (1999), 255–294.
[11] —–, Positive sets and Monotone sets, J. of Convex Anal., 14 (2007),
297–317.
[11] —–, The Hahn–Banach–Lagrange theorem, Optimization, 56 (2007), 149–169.
[13] —–, From Hahn–Banach to monotonicity, Lecture Notes in Mathematics,
1693,second edition, (2008), Springer–Verlag.
[14] D. Zagrodny, The convexity of the closure of the domain and the range of
a maximal monotone multifunction of Type NI, to appear in Set–Valued Anal.
[15] M. D. Voisei and C. Zălinescu, Strongly–representable operators,
http://arxiv.org/ abs/0802.3640v1, posted February 25, 2008.
[16] C. Zălinescu, Convex analysis in general vector spaces, (2002), World
Scientific.
Department of Mathematics
University of California
Santa Barbara
CA 93106-3080
U. S. A.
email: simons@math.ucsb.edu
|
arxiv-papers
| 2008-10-25T05:11:40 |
2024-09-04T02:48:58.442013
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Stephen Simons",
"submitter": "Stephen Simons",
"url": "https://arxiv.org/abs/0810.4579"
}
|
0810.4585
|
# A Cornucopia of Experiment at RHIC
Dept. of Physics, Brookhaven National Laboratory, Upton, NY 11973 USA
E-mail I thank P. Sorenson, M. Tannenbaum, and G. Torrieri for their comments.
This work was supported under U.S. Department of Energy grant #DE-
AC02-98CH10886. I also thank the Alexander von Humboldt Foundation for their
support.
###### Abstract:
I outline experimental results on heavy ion collisions at the Relativistic
Heavy Ion Collider for a non-technical audience. This includes: elliptic flow
and nearly ideal hydrodynamics; the suppression of hard particles and the
ratio $R_{\rm AA}$; and electromagnetic signals, including dileptons and
direct photons. Especially puzzling is why the behavior of heavy (charm)
quarks appears to be so similar to that of light quarks.
## 1 Introduction
The study of the collisions of heavy nuclei at high energies has a simple
motivation: heavy nuclei are big. Either gold or lead nuclei have $A\sim 200$
nucleons, where $A$ is the atomic number. The diameter of such a nucleus is
$A^{1/3}\sim 6$ larger than that of a proton; the transverse area,
$A^{2/3}\sim 34$ times larger. At high energies, one might hope to study the
phase transition(s) possible in QCD, to a deconfined, chirally symmetric state
of matter, the Quark Gluon Plasma (QGP). For big nuclei, one might close to a
system in thermal equilibrium.
As in other areas of hadronic physics [1], an essential insight was due to
Bjorken [2], who suggested that it would be useful going to energies where a
plateau in rapidity first emerges. This is the reason why the maximum energy
of the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National
Laboratory (BNL) was chosen to be what it is, as results from the ISR at CERN
had shown that in proton-proton ($pp$) collisions, a central plateau should
emerge by then. He also suggested that the study of hard particles, with
momentum much larger than the temperature, would be especially useful.
The results from RHIC have demonstrated, far beyond expectation, signs for a
novel phase at high energy density [3, 4, 5, 6]. Whatever has been created in
the collisions of two nuclei ($AA$ collisions), it is — experimentally — very
unlike what happens in $pp$ collisions. Indeed, there has been such a
profusion of experimental results that one may speak of a “cornucopia” of
data, whence my title. In this talk I try to give a brief overview of the
experimental situation to date. I generally assume that the reader is familiar
with concepts from high energy physics, such as rapidity and the like, but is
unfamiliar with the concepts essential for understanding $AA$ collisions, such
as the difference between central and peripheral collisions. For reasons of
space, I could not discuss many interesting (and still puzzling!) features of
the data. I have tried to show the standard plots which have come to define
the field since RHIC turned on in 2000.
While I concentrate on results from RHIC, there is continuity of results from
the SPS at CERN, to RHIC. This includes those for $J/\Psi$ suppression and the
dilepton enhancement at low invariant mass. What is gained by the higher
energies at RHIC is that the production of hard particles is much more common.
That, and having a dedicated machine and experiments which are able to
intensively study the physics.
Results from RHIC will continue with an increase in the luminosity by an order
of magnitude, and upgrades to the PHENIX [3] and STAR [4] detectors. In the
next year or so there will also be results for heavy ions at the Large Hadron
Collider (LHC) at CERN, which will probe a significantly higher regime in
energy. As I mention later, the physics for $AA$ collisions at the LHC might
be very different from that at RHIC.
While I suggest that RHIC is manifestly a triumph for experiment, the
theoretical situation is still most unsettled [7], and so I only discuss it in
passing. In some ways, the results are analogous to those for high-$T_{c}$
superconductivity, where experiment also continues to confound theory. I do
think that with the intense study possible at RHIC and the LHC, that a common
theoretical basis will eventually emerge. In all of this, results from
numerical simulations on the lattice form the absolute bedrock upon which our
understanding is based [8].
## 2 Basics of $AA$ collisions
At RHIC one can study $pp$, $AA$, and $dA$ collisions, where the latter are
the collisions of deuterons with nuclei. (Deuterons are used instead of
protons because the charge/mass ratio is closer to that of a large nucleus.)
For $pp$ and $AA$ collisions, the basic variable is the energy per nucleon,
$\sqrt{s}/A$. At the AGS at Brookhaven, this goes up to $5$ GeV; up to $17$
GeV at the SPS at CERN; and from $20$ to $200$ GeV at RHIC. When I quote
results from RHIC, typically I shall quote values from the highest energies,
$200$ GeV. To date, there do not appear to be dramatic differences in going
from the lowest, to the highest energies at RHIC. This will be clarified in
the coming years with low energy runs at RHIC down to $\sim 5$ GeV.
At the highest energies at RHIC there is no nuclear stopping: the incident
nucleons go down the beam pipe. Instead of the momentum along the beam,
$p_{z}$, it is better to use the rapidity, $y=1/2\log((E+p_{z})/(E-p_{z}))$,
which transforms additively under Lorentz boosts along the beam. Thus one
considers the distribution of particles at a given rapidity, $y$, versus
transverse momentum, $p_{t}$. Typically I concentrate on results at zero
rapidity, $p_{z}=y=0$.
The AGS and SPS are fixed target machines; RHIC and LHC are colliders. Fixed
target machines allow for much higher luminosities, but it is then difficult
to study zero rapidity, since that it somewhere in the forward direction. For
colliders, zero rapidity is at 90o to the beam, facilitating detector
construction. The central plateau, being essentially free of the incident
baryons, is the most natural place to produce a system at nonzero temperature,
and (almost) zero quark chemical potential [2]
Figure 1: Central versus peripheral collisions for large nuclei.
At RHIC, the particles are spread out over $\pm 5$ units of rapidity. At zero
rapidity there are $\sim 900$ particles per unit rapidity, versus $\sim 600$
at the highest energies at the SPS. This sounds like a large number, but in
fact, it is not. The total number of particles in a central $AA$ collision
should scale like $A$: $A^{2/3}$ for the area of one nucleus, times $A^{1/3}$
in length as it goes through the other nucleus. Starting with the total number
of particles in a $pp$ collision at these energies, and multiplying by $A$,
one finds that, proportionally, there are only about $30\%$ more particles
produced in $AA$ collisions than a trivial extrapolation from $pp$. This is a
strong constraint on the physics, as it shows that at these energies, there is
a small amount of entropy generated in $AA$ collisions, relative to $pp$.
There are two large experiments at RHIC: PHENIX [3] and STAR [4], each with
about 400 people; and two smaller ones, BRAHMS [5] and PHOBOS [6], each with
about 50. An amusing but specious observation is that the total number of
experimentalists working on the associated experiments nearly equals the
particle multiplicity (per unit rapidity). This increases slowly, only
logarithmically, with energy; the number of theorists grows much slower,
perhaps as the log of a log…
At RHIC, STAR [4] and BRAHMS [5] have shown there is a narrow plateau in
rapidity, in which the multiplicity, $dN/dy$, and the average transverse
momentum, $\langle p_{t}\rangle$, of identified particles are both constant
over $\pm 0.5$ units of rapidity.
Given the large transverse size of large nuclei, as illustrated in fig. 1 one
can distinguish between “central” collisions, where the nuclei overlap
completely, and “peripheral” collisions, where they only partially overlap;
the direction of the beam is into the page. Experimentalists speak of the
number of particpants in a collision: for a central collision with $A\sim
200$, this is $\sim 400$. The number of participants can be determined down to
about $\sim 30$, especially by using Zero Degree Calorimeters to measure what
goes down the beam pipe.
## 3 Soft particles: elliptic flow and nearly ideal hydrodynamics
Numerical simulations on the lattice indicate that at zero quark chemical
potential, there is a crossover to a new regime at $T_{c}\sim 150-200$ MeV
[8]. It is natural to think that the most obvious signals for a new state of
matter would be from soft particles, whose characteristic transverse momentum
is of order $T_{c}$.
Figure 2: Ratio of particle species, assuming chemical equilibrium.
The first thing one can ask is about total particle multiplicities, integrated
over $p_{t}$. This is illustrated in fig. 2 [9], which is a fit to over a
dozen particle species with only two parameters, a temperature $T_{\rm
chemical}\sim 165$ MeV, and a baryon chemical potential, $\mu_{\rm baryon}\sim
38$ MeV. It does not include short lived resonances, such as the $\Delta$,
$\phi$, $K^{*}$, etc., but with a $\chi^{2}$ per d.o.f. of $4/11$, is an
amazingly efficient summary of the data, using a trivial calculation. I remark
that this is unlike analogous fits to $pp$ or $e^{+}e^{-}$ collisions, where
it is necessary to include other parameters which are not standard in textbook
thermodynamics. I stress that I do not claim that chemical equilibrium has
been reached in $AA$ collisions; theoretically, I do not know an unambiguous
way of verifying this. Experimentally, though, overall ratios do appear to
look like it.
Figure 3: Average transverse momentum, $p_{t}$, for different species.
Instead of total multiplicity, integrated over $p_{t}$, the next thing one can
ask about is the average $p_{t}$, versus particle species. This is illustrated
in the two figures of fig. 3.
The figure on the left shows the change in the average $p_{t}$ for pions,
kaons, and protons, as one goes from $pp$ (on the left) to the large nuclei in
central $AA$, with $A\sim 200$ (on the right). One sees a large increase in
the average $p_{t}$ for kaons, and especially, protons. This is taken as
evidence of radial flow in the collisions of large nuclei: if a particle of
mass $\rm m$ flows with a velocity $\rm v$, its average transverse momentum
should scale as $\langle p_{t}\rangle\sim{\rm mv}$. Fits to the spectra
indicate that one needs a flow velocity ${\rm v}\sim 0.6\,\rm c$. The effect
is more dramatic the heavier the particle is, because light particles, such as
pions, already have an average velocity near the speed of light.
What I find striking about this figure, however, is that the average momentum
of pions does not increase significantly in going from $pp$ to $AA$ collsions,
with $A\sim 200$. In a hydrodynamic description, there is no reason for it to,
but in the Color Glass model, the saturation momentum $Q_{s}^{2}\sim A^{1/3}$,
and so one would expect the average $p_{t}$ to increase by a factor of
$A^{1/6}\sim 2.4$. One could easily imagine having further increases in the
average $p_{t}$ for kaons and protons on top of that, due to radial flow. But
this doesn’t happen: the average $p_{t}$ for pions barely budges. Isentropic
expansion decreases the average $p_{t}$ without increasing the multiplicity,
so models of the Color Glass plus hydrodynamics fit this result, Fig. 5 of
Ref. [10]. It will be very interesting to see if this changes at the LHC.
One can then turn to the average $p_{t}$ of heavier species, shown in the
figure on the right hand side of fig. 3; this plot is due originally to Nu Xu.
It shows the average $p_{t}$ for central $AA$ collisions with $A\sim 200$ at
the highest energies at RHIC. As seen in the figure on the left hand side,
there is a linear increase in the mean $p_{t}$ between pions, kaons, and
protons; for heavier species, though, the $\Lambda$, $\Xi$, and $\Omega$, they
all appear to have nearly constant $p_{t}\sim 1.1$ GeV, like that of the
proton. These species are all baryons, but it is also found to be true of the
$\phi$ meson.
The usual explanation is that hadrons composed of strange quarks decouple
earlier. Even if one assumes that strange particles bunch up into colorless
hadrons sooner, though, I would expect that in a graph of $\langle
p_{t}\rangle$ versus mass, that there is one value of the slope of $\pi$, $K$,
and $p$, and another, smaller value, for strange particles. Instead, it
appears as if starting with the proton, that it and all heavier hadrons are
emitted with essentially constant $p_{t}$. This is very difficult to
understand from any hydrodynamic description, where $\langle
p_{t}\rangle\sim{\rm mv}$.
Figure 4: Elliptic flow in a peripheral collision: evolution in both
coordinate and momentum space
A fundamental quantity to measure in heavy ion collisions is that of elliptic
flow. To understand this, consider the hot “almond” of the overlap region in a
peripheral collision, fig. 1. This is shown also in the upper left hand corner
of fig. 4, as a region in coordinate space. The corresponding region in
momentum space is shown in the upper right hand corner of fig. 4: it is
spherical, because by causality particles can’t start out knowing the shape of
such a large collision region. As the system evolves, and fields scatter off
of one another, in coordinate space the final distribution tends toward one
which is spherical; this is shown in the lower left hand corner of fig. 4. At
the same time, as the particles scatter, the distribution of particles in
momentum space becomes distorted, into an ellipse: particles along the
$x$-axis, where the almond is narrow, move a lot, while those along the
$y$-axis, move less. This is characterized by the quantity
${\rm v}_{2}=\frac{\langle p_{x}^{2}-p_{y}^{2}\rangle}{\langle
p_{x}^{2}+p_{y}^{2}\rangle}\;.$ (1)
This quantity is well defined and so can be measured experimentally. The main
problem is determining the reaction plane; i.e., what are the $x$ and $y$
axes. One can also define and measure higher moments, etc.
Nuclear physicists who work on collisions at lower energies are well familiar
with elliptic flow: then the two nuclei experience a lot of nuclear stopping,
form a big blob that lasts a long time, and thus naturally transform the
initial anisotropy in coordinate space into one in momentum space.
At high energies, however, the mere existence of elliptic flow tells one that
there are significant interactions in $AA$ collisions. The great question
about $AA$ collisions at high energies is whether there is anything
interesting beyond $A$ times a $pp$ collision. Especially in an asymptotically
free theory, it is certainly conceivable that the particles, while originally
in a almond, just free stream isotropically. In this case, there would be no
significant ${\rm v}_{2}$ generated.
Figure 5: Elliptic flow versus multiplicity
One way of computing elliptic flow is to use a hydrodynamic description. Given
the large particle multiplicities, to zeroth order such a description is
reasonable, as hydrodynamics is a simple way of encoding the conservation of
energy and momentum in a causal manner. Hydrodynamics requires an equation of
state; this one can take, for example, from numerical simulations on the
lattice [8]. It is also necessary to specify the transport coefficients of the
medium, such as the shear and bulk viscosity. For a relativistic medium, there
are other transport coefficients, but we concentrate on the shear viscosity,
as that appears to be largest and most important.
Shear viscosity is familiar from the non-relativistic example of two parallel
plates, in the $x$ and $z$ planes, separated by some distance in $y$. If one
plate is held fixed, and the other is moved with constant velocity along the
$x$-direction, then the shear stress is proportional to the viscosity times
the gradient of the velocity in $y$. That is, the more viscous the fluid, the
harder it is to move one plate parallel to the other.
The simplest thing one can do is to compute using ideal hydrodynamics,
assuming that the shear viscosity vanishes. This is shown in fig. 5, which
shows the elliptic flow versus multiplicity in $AA$ collisions. The elliptic
flow is divided by the eccentricity, which allows one to compare the
collisions of copper nuclei, $Cu$, with $A\sim 60$, to the largest nuclei,
where $A\sim 200$. Plotting versus multiplicity (divided by the transverse
area) allows one to plots results from energies at the AGS, SPS, and RHIC. The
basic point of this figure is that only for the collisions of the largest
nuclei, at the highest energies, that agreement between data and (nearly)
ideal hydrodynamics is found. a nearly ideal hydrodynamics agrees with the
data. The best fit to the data is obtained with an equation of state that
includes a phase transition to a deconfined phase.
Hydrodynamics predicts both single particle distributions (versus $p_{t}$) and
elliptic flow. It is found that elliptic flow provides a strong constraint on
the ratio of the shear viscosity, $\eta$, to the entropy density, $s$ [11]:
$\eta/s\approx 0.1\pm 0.1({\rm theory})\pm 0.1({\rm experiment})\;.$ (2)
The experimental errors arise from uncertainty as to the direction of the
event plane; there are many sources of error from theory. The value quoted is
for $\eta/s$, because this enters naturally in hydrodynamics, and is related
to an inverse mean free path.
Figure 6: Shear viscosity in various non-relativistic systems
A comparison to various non-relativistic systems is given in fig. 6 [12]. The
quantity plotted is again $\eta/s$, but for non-relativistic systems, $s$
doesn’t change significantly near $T_{c}$, unlike for QCD, where it drops
dramatically [8]. Taking this ratio does eliminate a trivial dependence on the
overall number of the degrees of freedom.
Even given the large error bars in eq. 2, this is an extremely small value for
$\eta/s$. (The points from a hadronic gas and the QGP in fig. 6 are
theoretical extrapolations.) The value at RHIC is almost an order of magnitude
smaller than the smallest value for non-relativistic systems, which is liquid
$He$. Thus RHIC produces “the most perfect fluid on earth”.
As a transport coefficient, the shear viscosity vanishes in the limit of weak
coupling, as $\eta\sim T^{3}/\alpha_{s}^{2}$, where $T$ is the temperature,
and $\alpha_{s}$ the QCD coupling constant. The fact that $\eta$ is inversely
proportional to a coupling constant sounds peculiar, but it’s not. Transport
coefficients measure how quickly a system, perturbed from thermal equilibrium,
goes back. It takes longer for a weakly coupled system, than a strongly
coupled system, because the particles interact less. Technically, it is
easiest computing $\eta$ from a Boltzmann equation. There one finds that
$\eta$ is the ratio of a source term (squared), divided by a collision term:
for small $\alpha_{s}$, the source term is of order one, and the collision
term $\sim\alpha_{s}^{2}$. I do not discuss values of $\eta$ is weak coupling.
To date, one cannot reliably compute either $\eta$ or the entropy near
$T_{c}$. The situation is not hopeless, though [7, 13, 14, 15, 16].
Since $\eta\sim 1/\alpha_{s}^{2}$, a small value for $\eta$ suggests that the
QCD coupling is very large near $T_{c}$. This is part of the motivation for
what is known as a “strong” QGP [7]. One case where one can compute at
infinite coupling is for a theory with ${\cal N}=4$ supersymmetry and an
infinite number of colors, where $\eta/s=1/4\pi$ [14]. This is conjectured to
be a universal bound, but string theory provides examples which are $16/25$
smaller, and may be the true bound [15].
As illustrated in fig. 5, the really interesting question is what the elliptic
flow will be like at the LHC. Straightforward extrapolations of ideal
hydrodynamics can be done, and predict a large increase in ${\rm v}_{2}$ [17].
In this, there appears to be real dichotomy. In a strong QGP [7], if the
plasma is strongly coupled near $T_{c}$, at RHIC, shouldn’t it remain so at
the higher temperatures at the LHC? Another example is provided by ${\cal
N}=4$ gauge theories: by modifying the theory, they can be adjusted to fit the
pressure, as computed from numerical simulations on the lattice for three
colors [8], down to $T_{c}$ [16]. In all of these models, however, $\eta/s$
remains small, $=1/4\pi$.
In contrast, as shown in fig. 6, non-relativistic models universally show that
while the shear viscosity has a minimum at the critical temperature, that it
also increases away from $T_{c}$. The question is really, is the QGP like
$He$, where the increase from $T_{c}$ to $2T_{c}$ is only a factor of two, or
like $H_{2}0$, where it is an order of magnitude? A weak coupling analysis of
a “semi”-QGP suggests that a large rise in $\eta/s$ is possible as $T$
increases from $T_{c}$ [13].
Measurements of the elliptic flow at the LHC will tell us from day one of
running $AA$ collisions. I note that detailed theoretical predictions in non-
ideal hydrodynamics need to be carried out, since even if collisions at the
LHC start out in a highly viscous regime, at say $\sim 2T_{c}$, one still
cools into a system which has a small viscosity near $T_{c}$.
Figure 7: Elliptic flow per quark, versus the transverse energy per quark.
Returning to experiment, in fig. 7 I show a plot of the elliptic flow per
quark, versus the transverse energy of a hadron, per quark. By per quark, I
simply mean that one divides by two for a meson, and three for a baryon. This
shows that at low $p_{t}$, there appears to be a universal scaling of elliptic
flow for all particle species. Dividing by the number of quarks in the hadron
is reasonable, but it is astounding that the correct variable to plot against
is the kinetic energy (and not, say, the transverse momentum; then one does
not find a universal curve). This is typical of the results from RHIC: there
are many results which are simply totally unexpected, and hint at some
universal mechanism(s), which we do not yet understand.
Figure 8: Elliptic flow for charm quarks
One can also ask about the elliptic flow of heavy quarks. Here experiment uses
single electrons, which arise from the decay of a charm quark, to tag their
flow. Now theoretically, one would expect that heavy quarks would not flow as
easily as light quarks: it should take heavy quarks longer to thermalize, and
they should interact in a characteristically different manner. Instead, as
shown in fig. 8, the elliptic flow for charm quarks appears to be just as
large as that of light quarks!
This is one of the truly astounding results from RHIC. As we shall see again
in the next section, heavy quarks appear to interact much more strongly with
the “stuff” in central $AA$ collisions than we would have expected: $AA$
collisions are manifestly not a trivial superposition of $pp$ collisions.
## 4 Hard particles: suppression and the ratio $R_{\rm AA}$
Figure 9: The ratio $R_{\rm AA}$ for photons and pions.
One of the great experimental surprises of RHIC is that while most of the
particles are down at low $p_{t}$, the clearest signs for something new in
central $AA$ collisions comes from high momentum, $p_{t}>2$ GeV. This is
typically referred to in the high energy nuclear community as “jets”, but is
far lower in energy than what most high energy physicists are used to.
Consequently, I eschew this term, and just refer to hard particles.
A basic quantity is the ratio $R_{\rm AA}$: this is the ratio of the number of
particles in a central $AA$ collision to that in $pp$, both measured at the
same $p_{t}$ (and rapidity):
${\rm R}_{\rm AA}(p_{t})=\frac{\\#\;{\rm
particles\;in\;central\;AA}(p_{t})}{{\rm A}^{4/3}\;\\#\;{\rm
particles\;in\;pp}(p_{t})}.$ (3)
The crucial question is how one normalizes. As I discussed above, soft
particles scale as $A$. For hard collisions, the number of binary collisions
is $A$, from the incident nucleus, times $A^{1/3}$ from the width of the
target, or $A^{4/3}$. This is only approximate; experimentally, this is
modeled by Glauber and Monte Carlo calculations.
However, one doesn’t need to understand (or believe) this normalization
factor, since one can directly appeal to experiment. The ratio $R_{\rm AA}$
can be measured for any particle species. In fig. 9, I show the plot for
photons and neutral pions. Since photons only interact electromagnetically, if
the normalization is performed correctly, then $R_{\rm AA}$ should be one.
While the error bars are large, $\sim 10\%$, this is true for photons with
$p_{t}>2$ GeV.
In contrast,one finds that above $p_{t}\sim 2$ GeV, there are only about
$20\%$ of the number of neutral pions expected. (Experimentally, at high
$p_{t}$ it is easiest to pick out neutral pions, by looking for two hard
photons with the right invariant mass.) This $20\%$ is a very small number.
From fig. 1, even in a central collision, there is a contribution from the
surface; at least half the hard particles emitted from the surface should
escape without interaction. This is another reason why people speak of a
strong QGP at RHIC [7].
Indeed, the really surprising thing is that $R_{\rm AA}$ is so flat to such a
high $p_{t}$. It is easy to imagine that effects in a medium would suppress
hard particles: they will scatter off of the medium, lose energy, and so emit
more soft particles. Theoretically, this is known as energy loss [7]. But at
high enough $p_{t}$, scattering off of the medium should go away. Fig. 9 shows
that this isn’t true for neutral pions with a $p_{t}$ as high as $20$ GeV!
Eventually, $R_{\rm AA}$ must go back up to one, or one will question whether
it is correctly normalized.
It is reasonable to ask if this suppression is due to some initial state
effect in nuclei. Here measurements in $dA$ collisions were crucial: the
normalization changes to $2A$, and experimentally one observes not
suppression, but enhancement [3, 4, 5, 6], with ${\rm R}_{\rm dA}\approx
1.4\pm 0.1$ at $p_{t}\sim 3$ GeV. This is due to what is known as the Cronin
effect; all that matters for us is that $R_{\rm dA}$ goes in the opposite
direction from $R_{\rm AA}$, and so $R_{\rm AA}$ is manifestly a final state
effect.
Figure 10: Geometrical suppression of hard particles
The suppression of hard particles can also be observed on a purely geometrical
basis, as shown in fig. 10. Consider a peripheral collision, and trigger on a
hard particle, with $p_{t}:4\rightarrow 6$ GeV. Then look for a hard particle
on the away side, $p_{t}>2$ GeV, as a function of the angle to the trigger
particle. In $pp$ or $dA$ collisions, this is peaked at $180^{o}$. Now in a
peripheral collision, one can look at a hard particle either in the plane of
the collision, or out of plane. If the hard particle is in the reaction plane,
it goes a small distance through the “hot” almond, and a long ways through the
cold nuclear spectators. If out of the plane, it goes a long way through the
almond, and little through the spectators. Fig. 10 shows that when the hard
particle is in the reaction plane, one does see the away side particle at
$180^{o}$; when the hard particle is perpendicular to the reaction plane, one
doesn’t see the away side particle. That is, the more particles go through the
almond, the more the “stuff” there affects their propagation. This is
consistent with the small value of $R_{\rm AA}$.
Figure 11: Away side correlations for peripheral to central $AA$ collisions.
There is interesting structure seen in the angular correlations of the away
side particle. Fig. 11 shows results for a trigger particle of
$p_{t}:2.5\rightarrow 4$ GeV, and an away side particle of $p_{t}:2\rightarrow
3$ GeV, integrated over all angles to the reaction plane. There are three
curves shown, going from most peripheral to most central. How one defines
centrality is in this case secondary. What one can see is that for peripheral
collisions, the angular distribution for the away side particle is peaked at
$180^{o}$, as in a $pp$ collision. For the most central collisions, the
angular distribution at $180^{o}$ is suppressed, as seen in fig. 10. What one
also sees, however, is an enhancement in the distribution of away side
particle away from $180^{o}$. This looks very like Cerenkov radiation, or
perhaps a Mach cone in a medium [7]. This is really a correlation between
three particles, as has been verified by both the PHENIX [3] and STAR [4]
collaborations.
Especially with planned upgrades to RHIC, one will also be able to measure
correlations between a hard photon and a hard particle. Measuring a hard
photon will tell one unambiguously what the incident energy of the hard
particle is, and so one will be able to understand the details of how fast
particles are affected by the medium in central $AA$ collisions.
Figure 12: A Lego plot of two jets in central $AA$, $p_{t}>20$ GeV.
All of these figures have triggered on “hard” particles with relatively low
$p_{t}$. In fig. 12 I show a plot from the STAR collaboration, which is a Lego
plot familiar in high energy physics. The trigger is $p_{t}>20$ GeV, for the
most central $AA$ collisions. Even given the high multiplicity of particles at
low $p_{t}$, if the trigger is sufficiently high, then jets just stick out. At
LHC energies, true jets, with transverse momenta of order $50$, $100$ GeV and
higher, will be (relatively) plentiful. This will enable one to really pin
down the mechanism which is responsible for $R_{\rm AA}$ and the like.
Figure 13: The ratio $R_{\rm AA}$ for charm quarks.
One can form the ratio $R_{\rm AA}$ for any particle species. In fig. 13 I
show the result for charm quarks from the PHENIX collaboration. Here one
observes charm by measuring direct electrons. The mass of the charm quark is
$\sim 1.5$ GeV, and the temperature is something like $T_{c}\sim 200$ MeV [8].
In perturbation theory, the scattering of a heavy quark is very different from
that of a light quark: emission of gluon radiation is suppressed in the
forward direction (“dead cone” effect). Even without detailed calculation, it
would be astonishing if one found that the behavior of a heavy quark were
anything like that of a light quark; one expects that heavy quarks are not
suppressed as much as light quarks, with so $R_{\rm AA}$ is larger.
This is not what experiment shows: fig. 13 shows that for transverse momenta a
couple of times the charm quark mass, $p_{t}\sim 4$ GeV, that $R_{\rm AA}\sim
0.2$, like $\pi^{0}$’s! This is a remarkable result, and completely unlike any
perturbative understanding. Perhaps energy loss is not the whole story.
## 5 Electromagnetic signals: dileptons and direct photons
Since dileptons only interact weakly with a hadronic medium, they provide
essential insight into $AA$ collisions. In fig. 14 I show the dielectron
spectra below the $J/\Psi$, as a function of the invariant mass of the
dielectron pair, $m_{\rm ee}$. It is necessary to normalize the spectrum from
central $AA$ collisions to that of a “cocktail” from $pp$ collisions.
Figure 14: Dilepton spectra for central $AA$ and $pp$ collisions.
As seen in collisions at SPS energies, at RHIC energies there is a striking
excess in dileptons below the $\rho$ meson. There is a smaller, but still
significant excess, above the $\rho$ meson as well. Any excess appears to have
disappeared for dileptons above the $J/\Psi$.
A crucial question is whether the normalization to $pp$ collisions is done
correctly. One can show that for the dilepton excess below the $\rho$ meson,
for $150<m_{ee}<750$ MeV, that the excess first appears when the number of
participants is greater than $\sim 200$, and that it increases as the number
of participants increases. This is dramatic evidence that the “stuff” created
in central $AA$ collisions is uniquely responsible for the excess at low
invariant mass.
Figure 15: The ratio $R_{\rm AA}$ for $J/\Psi$’s.
In fig. 15 I show the ratio $R_{\rm AA}$ for $J/\Psi$ production in central
$AA$ collisions at both RHIC and the SPS. When plotted in this way, one finds
that the behavior at these two energies is essentially identical. This was
absolutely unexpected. Various theoretical models had predicted that $J/\Psi$
production might be less at RHIC than the SPS, due to greater scattering in a
thermal medium, or greater, due to regeneration. But no model predicted
exactly the same behavior for $R_{\rm AA}$.
This year, PHENIX has also shown how low mass dielectron pairs can be used to
get direct photons from internal conversion [18]. They see a clear excess for
photon $p_{t}:1\rightarrow 3$ GeV, which they fit to an exponential. This
gives a temperature for photon production of $T_{\rm photon}\sim 223$ MeV,
with statistical errors of $\pm 23$ MeV and systematic errors of $\pm 18$ MeV.
This is a fundamental result, and gives us a lower bound on the temperatures
at which the photons were produced.
## 6 Summary
The results at RHIC have conclusively demonstrated that central $AA$
collisions have produced matter at high energy density which is very unlike
that produced in $pp$ collisions at the same energy.
There are numerous interesting phenomenon which I didn’t have space to cover:
the baryon/meson enhancement at intermediate $p_{t}:2\rightarrow 6$ GeV;
Hanbury-Brown-Twiss interferometry, which shows “explosive” behavior; and the
ridge in rapidity. I have emphasized that one of the most mystifying aspects
of the data is that the behavior of charm quarks — as seen in their elliptic
flow, and the ratio $R_{\rm AA}$ — is essentially identical to that of light
quarks. This is very difficult to understand theoretically. This, and the
nearly ideal behavior of hydrodynamics, has given rise to the suggestion that
the region near $T_{c}$ is behaving unexpectedly: either a strong [7, 14, 15,
16, 17], or maybe a semi- [13], QGP.
Of course we eagerly await results for $AA$ collisions at the LHC. Collisions
at the LHC will produce many more jets, and produce a medium in which the
temperatures are significantly (twice?) as high as at RHIC. One might hope
that LHC probes a perturbative (or complete [13]) QGP. We will know very soon
if LHC produces a nearly ideal fluid, as at RHIC [7, 16], or one which is
viscous [13]. The study of bottom quarks will also be very interesting, given
the unexpected behavior of charm quarks at RHIC.
I stress, however, that RHIC is uniquely set to intensively study the region
about $T_{c}$. In the end, I feel no hestitation whatsoever in saying that
once RHIC turned on, we entered what is clearly a golden age in high energy
nuclear physics, one which is well deserving of the highest possible
recognition [1, 2, 3, 4].
## References
* [1] J. D. Bjorken, _Applications Of The Chiral U(6) X (6) Algebra Of Current Densities_ , Phys. Rev. 148, 1467 (1966); _Asymptotic Sum Rules At Infinite Momentum_ , ibid. 179, 1547 (1969).
* [2] J. D. Bjorken, _Highly Relativistic Nucleus-Nucleus Collisions: The Central Rapidity Region_ , Phys. Rev. D 27, 140 (1983).
* [3] K. Adcox et al., _Formation of dense partonic matter in relativistic nucleus nucleus collisions at RHIC: Experimental evaluation by the PHENIX collaboration_ , Nucl. Phys. A 757, 184 (2005) [nucl-ex/0410003].
* [4] J. Adams et al., _Experimental and theoretical challenges in the search for the quark gluon plasma: The STAR collaboration’s critical assessment of the evidence from RHIC collisions_ , Nucl. Phys. A 757, 102 (2005) [nucl-ex/0501009].
* [5] I. Arsene et al., _Quark gluon plasma and color glass condensate at RHIC? The perspective from the BRAHMS experiment_ , Nucl. Phys. A 757, 1 (2005) [nucl-ex/0410020].
* [6] B. B. Back et al., _The PHOBOS perspective on discoveries at RHIC_ , Nucl. Phys. A 757, 28 (2005) [nucl-ex/0410022].
* [7] M. Gyulassy and L. McLerran, _New forms of QCD matter discovered at RHIC_ , Nucl. Phys. A 750, 30 (2005) [nucl-th/0405013]; A. Peshier and W. Cassing, _The hot non-perturbative gluon plasma is an almost ideal colored liquid_ , Phys. Rev. Lett. 94, 172301 (2005) [hep-ph/0502138]; B. Muller and J. L. Nagle, _Results from the Relativistic Heavy Ion Collider_ , Ann. Rev. Nucl. Part. Sci. 56, 93 (2006) [nucl-th/0602029]; S. Mrowczynski and M. H. Thoma, _What do electromagnetic plasmas tell us about quark-gluon plasma?_ , ibid. 57, 61 (2007) [nucl-th/0701002]; E. V. Shuryak, _Physics of Strongly coupled Quark-Gluon Plasma_ , to appear in Prog. Part. Nucl. Phys. [0807.3033].
* [8] Plenary talks at Lattice 2008 by: C. DeTar; S. Ejiri; H. Meyer, _Energy-momentum tensor correlators and viscosity_ [0809.5202]; and M. Teper.
* [9] A. Andronic, P. Braun-Munzinger and J. Stachel, _Hadron production in central nucleus nucleus collisions at chemical freeze-out_ , Nucl. Phys. A 772, 167 (2006) [nucl-th/0511071].
* [10] T. Hirano and Y. Nara, _Hydrodynamic afterburner for the color glass condensate and the parton energy loss_ , Nucl. Phys. A 743, 305 (2004) [nucl-th/0404039].
* [11] M. Luzum and P. Romatschke, _Conformal Relativistic Viscous Hydrodynamics: Applications to RHIC_ [0804.4015].
* [12] R. A. Lacey et al., _Has the QCD critical point been signaled by observations at RHIC?_ , Phys. Rev. Lett. 98, 092301 (2007) [nucl-ex/0609025].
* [13] Y. Hidaka and R. D. Pisarski, _Suppression of the shear viscosity in a “semi” Quark Gluon Plasma_ [0803.0453].
* [14] D. T. Son and A. O. Starinets, _Viscosity, Black Holes, and Quantum Field Theory_ , Ann. Rev. Nucl. Part. Sci. 57, 95 (2007) [0704.0240].
* [15] M. Brigante, H. Liu, R. C. Myers, S. Shenker and S. Yaida, _The Viscosity Bound and Causality Violation_ , Phys. Rev. Lett. 100, 191601 (2008) [0802.3318].
* [16] N. Evans and E. Threlfall, _The thermal phase transition in a QCD-like holographic model_ [0805.0956]; S. S. Gubser and A. Nellore, _Mimicking the QCD equation of state with a dual black hole_ [0804.0434]; U. Gursoy, E. Kiritsis, L. Mazzanti and F. Nitti, _Deconfinement and Gluon Plasma Dynamics in Improved Holographic QCD_ [0804.0899].
* [17] H. Niemi, K. J. Eskola and P. V. Ruuskanen, _Elliptic flow in nuclear collisions at the Large Hadron Collider_ [0806.1116].
* [18] A. Adare et al. _Enhanced production of direct photons in Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV._ [0804.4168]
|
arxiv-papers
| 2008-10-25T08:57:18 |
2024-09-04T02:48:58.450140
|
{
"license": "Public Domain",
"authors": "Robert D. Pisarski",
"submitter": "Robert D. Pisarski",
"url": "https://arxiv.org/abs/0810.4585"
}
|
0810.4667
|
# On total dominating sets in graphs
Maryam Atapour and Nasrin Soltankhah 111Corresponding author: E-mail:
soltan@alzahra.ac.ir, soltankhah.n@gmail.com.
Department of Mathematics
Alzahra University
Vanak Square 19834 Tehran, I.R. Iran
###### Abstract
A set $S$ of vertices in a graph $G(V,E)$ is called a dominating set if every
vertex $v\in V$ is either an element of $S$ or is adjacent to an element of
$S$. A set $S$ of vertices in a graph $G(V,E)$ is called a total dominating
set if every vertex $v\in V$ is adjacent to an element of $S$. The domination
number of a graph $G$ denoted by $\gamma(G)$ is the minimum cardinality of a
dominating set in $G$. Respectively the total domination number of a graph $G$
denoted by $\gamma_{t}(G)$ is the minimum cardinality of a total dominating
set in $G$. An upper bound for $\gamma_{t}(G)$ which has been achieved by
Cockayne and et al. in [1] is: for any graph $G$ with no isolated vertex and
maximum degree $\Delta(G)$ and $n$ vertices, $\gamma_{t}(G)\leq
n-\Delta(G)+1$.
Here we characterize bipartite graphs and trees which achieve this upper
bound. Further we present some another upper and lower bounds for
$\gamma_{t}(G)$. Also, for circular complete graphs, we determine the value of
$\gamma_{t}(G)$.
2000 Mathematics Subject Classification: 05c69
Keywords: total dominating set, total domination number
## 1 Introduction
Let $G(V,E)$ be a graph. For any vertex $x\in V$, we define the neighborhood
of $x$, denoted by $N(x)$, as the set of all vertices adjacent to $x$. The
closed neighborhood of $x$, denoted by $N[x]$, is the set $N(x)\cup\\{x\\}$.
For a set of vertices $S$, we define $N(S)$ as the union of $N(x)$ for all
$x\in S$, and $N[S]=N(S)\cup S$. The degree of a vertex is the size of its
neighborhoods. The maximum degree of a graph $G$ is denoted by $\Delta(G)$ and
the minimum degree is denoted by $\delta(G)$. Here $n$ will denote the number
of vertices of a graph $G$. A set $S$ of vertices in a graph $G(V,E)$ is
called a dominating set if every vertex $v\in V$ is either an element of $S$
or is adjacent to an element of $S$. A set $S$ of vertices in a graph $G(V,E)$
is called a total dominating set if every vertex $v\in V$ is adjacent to an
element of $S$. The domination number of a graph $G$ denoted by $\gamma(G)$ is
the minimum cardinality of a dominating set in $G$. Respectively the total
domination number of a graph $G$ denoted by $\gamma_{t}(G)$ is the minimum
cardinality of a total dominating set in $G$. clearly
$\gamma(G)\leq\gamma_{t}(G)$, also it has been proved that $\gamma_{t}(G)\leq
2\gamma(G)$.
An upper bound for $\gamma_{t}(G)$ has been achieved by Cockayne and et al. in
[1] in the following theorems:
###### THEOREM A
If a graph $G$ has no isolated vertices, then $\gamma_{t}(G)\leq
n-\Delta(G)+1$.
###### THEOREM B
If $G$ is a connected graph and $\Delta(G)<n-1$, then $\gamma_{t}(G)\leq
n-\Delta(G)$
As a result of the above theorems, if $G$ is a graph with
$\gamma_{t}(G)=n-\Delta(G)+1$, then $\Delta(G)\geq n-1$. Hence, if $G$ is a
$k$– regular graph and $\gamma_{t}(G)=n-k+1$, then $G$ is $K_{n}$. As a result
of the above theorems, if $G$ is a graph with $\gamma_{t}(G)=n-\Delta(G)+1$,
then $\Delta(G)\geq n-1$. Hence, if $G$ is a $k$– regular graph and
$\gamma_{t}(G)=n-k+1$, then $G$ is $K_{n}$. Total domination and upper bounds
on the total domination number in graphs were intensively investigated, see e.
g. ( [3], [4]).
Here we characterize bipartite graphs and trees which achieve the upper bound
in Theorem A. Further we present some another upper and lower bounds for
$\gamma_{t}(G)$. Also, for circular complete graphs, we determine the value of
$\gamma_{t}(G)$.
It is easy to prove that for $n\geq 3$,
$\gamma_{t}(C_{n})=\gamma_{t}(P_{n})=\frac{n}{2}$ if $n\equiv 0\pmod{4}$ and
$\gamma_{t}(C_{n})=\gamma_{t}(P_{n})=\lfloor\frac{n}{2}\rfloor+1$ otherwise.
for the definitions and notations not defined here we refer the reader to
texts, such as [2].
## 2 Other bounds for $\gamma_{t}(G)$
In this section we introduce some other upper bounds for $\gamma_{t}(G)$.
###### Theorem 2.1
Let $G$ be a connected graph, then
$\gamma_{t}(G)\geq\lceil\frac{n}{\Delta(G)}\rceil$.
###### Proof:
Let $S\subseteq V(G)$ be a total dominating set in $G$. Every vertex in $S$
dominates at most $\Delta(G)-1$ vertices of $V(G)-S$ and dominate at least one
of the vertices in $S$. Hence, $|S|(\Delta(G)-1)+|S|\geq n$. Since, $S$ is an
arbitrary total dominating set, then
$\gamma_{t}(G)\geq\lceil\frac{n}{\Delta(G)}\rceil$.
If $G=K_{n}$, $G=C_{4n}$, or $G=P_{4n}$ then
$\gamma_{t}(G)=\lceil\frac{n}{\Delta(G)}\rceil$. so the above bound is sharp.
###### Theorem 2.2
Let $G$ be a graph with diam$(G)=2$ then, $\gamma_{t}(G)\leq\delta(G)+1$.
###### Proof:
Let $x\in V(G)$ and deg$(x)=\delta(G)$. Since, diam$(G)=2$, then $N(x)$ is a
dominating set for $G$.
Now $S=N(x)\cup\\{x\\}$ is a total dominating set for $G$ and
$|S|=\delta(G)+1$. Hence, $\gamma_{t}(G)\leq\delta(G)+1$.
As we know, $\gamma_{t}(C_{5})=3$ and also $\delta(C_{5})=2$, $diam(C_{5})=2$
then $\gamma_{t}(C_{5})=\delta(C_{5})+1$. Hence, the above bound is sharp.
###### Theorem 2.3
If $G$ is a connected graph with the girth of length $g(G)\geq 5$ and
$\delta(G)\geq 2$, then $\gamma_{t}(G)\leq n-\lceil\frac{g(G)}{2}\rceil+1$.
###### Proof:
Let $G$ be a connected graph with $g(G)\geq 5$ and let $C$ be a cycle of
length $g(G)$. Remove $C$ from $G$ to form a graph $G^{\prime}$. Suppose an
arbitrary vertex $v\in V(G^{\prime})$, since $\delta(G)\geq 2$, then $v$ has
at least two neighbors say $x$ and $y$. Let $x,y\in C$. If $d(x,y)\geq 3$,
then replacing the path from $x$ to $y$ on $C$ with the path $x,v,y$ reduces
the girth of $G$, a contradiction. If $d(x,y)\leq 2$, then $x,y,v$ are on
either $C_{3}$ or $C_{4}$ in $G$, contradicting the hypothesis that $g(G)\geq
5$. Hence, no vertex in $G^{\prime}$ has two or more neighbors on $C$. Since
$\delta(G)\geq 2$, the graph $G^{\prime}$ has minimum degree at least
$\delta(G)-1\geq 1$. Then $G^{\prime}$ has no isolated vertex. Now let
$S^{\prime}$ be a $\gamma_{t}$–set for $C$. Then $S=S^{\prime}\cup
V(G^{\prime})$ is a total dominating set for $G$. Hence, $\gamma_{t}(G)\leq
n-\lceil\frac{g(G)}{2}\rceil+1$(note that
$\gamma_{t}(C)\leq\lfloor\frac{g(G)}{2}\rfloor+1$) .
## 3 Bipartite graphs with $\gamma_{t}(G)=n-\Delta(G)+1$
In this section we charactrize the bipartite graphs achieving the upper bound
in the theorem A.
###### Theorem 3.4
Let $G$ be a bipartite graph with no isolated vertices. Then
$\gamma_{t}(G)=n-\Delta(G)+1$ if and only if $G$ is a graph in form of
$K_{1,t}\bigcup rK_{2}$ for $r\geq 0$.
###### Proof:
If $G$ is $K_{1,t}\cup rK_{2}(r\geq 0)$, clearly
$\gamma_{t}(G)=n-\Delta(G)+1$. Now let $G$ be a bipartite graph with
partitions $A\bigcup B$ and $x\in A$ where ${\rm deg}(x)=\Delta(G)=t$. We
continue our proof in four stages:
Stage 1: We claim that for every vertex $y\in A-\\{x\\}$,
$N(y)-N(x)\neq\emptyset$. If it is not true, there exists a vertex in
$A-\\{x\\}$, say $y$, such that $N(y)\subseteq N(x)$. So let $u\in N(y)$, the
set $S=V-(N(x)\cup\\{y\\})\bigcup\\{u\\}$ is a total dominating set and
$|S|=n-\Delta(G)$, a contradiction. So we have $n\geq 2|A|+\Delta(G)-1$.
Stage 2: For every vertex $y\in A$, let $u_{y}\in N(y)$. Clearly the set
$S=A\cup(\cup_{y\in A}\\{u_{y}\\})$ is a total dominating set for $G$ and
$|S|\leq 2|A|$, so $\gamma_{t}(G)\leq 2|A|$. Now let $y\in A-\\{x\\}$ such
that $|N(y)-N(x)|\geq 2$. Hence, we have:
$n\geq 2|A|+\Delta(G)$
$\Rightarrow$ $\gamma_{t}(G)+\Delta(G)-1\geq 2|A|+\Delta(G)$
$\Rightarrow$ $\gamma_{t}(G)\geq 2|A|+1$,
a contradiction. Hence, for every vertex $y\in A-\\{x\\}$, $|N(y)-N(x)|=1$.
Stage 3: Let $y\in A-\\{x\\}$ and $N(y)\cap N(x)\neq\emptyset$. Let $u\in
N(y)\cap N(x)$. Now, $S=(V-N(x)\cup\\{y\\})\cup\\{u\\}$ is a total dominating
set and $|S|=n-\Delta(G)$. So, $\gamma_{t}(G)\leq n-\Delta(G)$, a
contradiction.
Stage 4: Let $y,z\in A-\\{x\\}$ and $N(y)\cap N(z)\neq\emptyset$. Now
$S=(V-(\\{z\\}\cup N(x)))\cup\\{u\\}$, where $u\in N(x)$, is a total
dominating set and $|S|=n-\Delta(G)$. So, $\gamma_{t}(G)\leq n-\Delta(G)$, a
contradiction. Hence, $G$ is a graph in form of $K_{1,t}\cup rK_{2}$.
###### COROLLARY 3.1
Let $T$ is a Tree. Then $\gamma_{t}(T)=n-\Delta(T)+1$ if and only if $T$ is a
star.
## 4 Total domination numbers of circular complete graphs
If $n$ and $d$ are positive integers with $n\geq 2d$, then circular complete
graph $K_{n,d}$ is the graph with vertex set
$\\{v_{0},v_{1},\ldots,v_{n-1}\\}$ in which $v_{i}$ is adjacent to $v_{j}$ if
and only if $d\leq|i-j|\leq n-d$. In this section we determine the total
domination of circular complete graphs. It is easy to see that $K_{n,1}$ is
the complete graph $K_{n}$ and $K_{n,2}$ is a circle on $n$ vertices,
therefore we assume that $d\geq 3$.
###### Theorem 4.5
For $n\geq 4d-2$ and $d\geq 3$, $\gamma_{t}(K_{n,d})=2$.
###### Proof:
Clearly, $\gamma_{t}(K_{n,d})\geq 2$. Let $S=\\{v_{0},v_{2d-1}\\}$. We will
show that $S$ is a total dominating set for $K_{n,d}$. Since $n\geq 4d-2$ and
$2d-1\leq 2d$, then $2d-1\leq n-d$. Also $2d-1\geq d$ since $d\geq 3$. Thus
$d\leq 2d-1\leq n-d$ and $v_{0}v_{2d-1}\in E(K_{n,d})$. By definition of
$K_{n,d}$, $v_{0}$ is adjacent to each of the vertices
$v_{d},v_{d+1},\ldots,v_{n-d}$.
Now for each $1\leq i\leq d-1$ we have
$n-d+i-(2d-1)=n-3d+i+1\geq 4d-2-3d+i+1\geq d$
and
$n-d+i-(2d-1)=n-3d+i+1\leq n-3d+d=n-2d<n-d.$
Thus $v_{2d-1}$ is adjacent to each of the vertices
$v_{n-d+1},\ldots,v_{n-1}$. On the other hand, for each $1\leq i\leq d-1$ we
have
$2d-1-i\leq 2d-2\leq 3d-2\leq n-d$
and
$2d-1-i\geq 2d-1-d+1=d.$
Hence $v_{2d-1}$ is adjacent to each of the vertices
$v_{0},v_{1},\ldots,v_{d-1}$ and so $S$ is a total dominating set for
$K_{n,d}$ and $\gamma_{t}(K_{n,d})=2$.
###### Theorem 4.6
For $3d\leq n\leq 4d-3$ and $d\geq 3$, $\gamma_{t}(K_{n,d})=3$.
###### Proof:
Let $S=\\{v_{0},v_{d},v_{2d-1}\\}$. We prove that $S$ is a
$\gamma_{t}(K_{n,d})$\- set. Since $d\leq 2d-2\leq n-d$, $G[S]$ contains no
isolated vertices. Clearly $v_{0}$ and $v_{d}$ are adjacent to each of the
vertices $v_{d},v_{d+1},\ldots,v_{n-d}$ and $v_{2d},v_{2d+1},\ldots,v_{n-d}$
respectively. For $1\leq i\leq d-1$ we have
$2d-1-i\leq 2d-1-d+1=d$
and
$2d-1-i\leq 2d-2\leq 2d\leq n-d$
Thus $v_{2d-1}$ is adjacent to each of the vertices
$v_{1},v_{2},\ldots,v_{d-1}$. Hence $S$ is a total dominating set for
$K_{n,d}$ and so $\gamma_{t}(K_{n,d})\leq 3$. Now we prove that there is no
total dominating set for $K_{n,d}$ of size 2. Let $S^{\prime}=\\{u,v\\}$ be a
$\gamma_{t}(K_{n,d})$\- set. Without loss of generality, let $u=v_{0}$ and
$v=v_{j}$. Clearly $d\leq j\leq n-d$. Since $v_{0}v_{n-d+1}\notin E(K_{n,d})$,
$d\leq n-d+1-j\leq n-d$ and so $1\leq j\leq d+1$. Thus $j=d$ or $j=d+1$. In
both cases, $S^{\prime}$ is not a total dominating set since
$v_{2},v_{3},\ldots,v_{d-1}$ are not dominated by $S^{\prime}$ a
contradiction. This completes the proof.
## References
* [1] E.J. Cockayne, R.M. Dawes, and S.T. Hedetniemi, Total dominations in graphs, Networks, 10 (1980), 211–219.
* [2] T.W. Haynes and S.T. Hedetniemi, Funamentals domination in graphs, Marcel Dekker, New York, 1998.
* [3] M.A. Henning and A. Yeo , A new upper bound on the total domination number of a graph, Electronic J. of Combinatorics, 14 (2007), $\\#$R65.
* [4] P.C.B. Lam and B. Wei , On the total domination number of graphs, Utilitas Math., 72 (2007), 223–240.
|
arxiv-papers
| 2008-10-26T07:08:25 |
2024-09-04T02:48:58.460760
|
{
"license": "Public Domain",
"authors": "Maryam Atapour and Nasrin Soltankhah",
"submitter": "Nasrin Soltankhah",
"url": "https://arxiv.org/abs/0810.4667"
}
|
0810.4681
|
# Dynamical properties of quasiparticles in a gapped graphene sheet
A. Qaiumzadeh Institute for Advanced Studies in Basic Sciences (IASBS),
Zanjan, 45195-1159, Iran School of Physics, Institute for research in
fundamental sciences, IPM 19395-5531 Tehran, Iran F. K. Joibari School of
Physics, Institute for research in fundamental sciences, IPM 19395-5531
Tehran, Iran Reza Asgari School of Physics, Institute for research in
fundamental sciences, IPM 19395-5531 Tehran, Iran
###### Abstract
We present numerical calculations of the impact of charge carriers-carriers
interactions on the dynamical properties of quasiparticles such as
renormalized velocity and quasiparticle inelastic scattering lifetime in a
gapped graphene sheet. Our formalism is based on the many-body
$G_{0}W$-approximation for the self-energy. We present results for the many-
body renormalized velocity suppression and the renormalization constant over a
broad range of energy gap values. We find that the renormalized velocity is
almost independence of the carrier densities at large density regime. We also
show that the quasiparticle inelastic scattering lifetime decreases by
increasing the gap value. Finally, we present results for the mean free path
of charge carriers suppression over the energy gap values.
###### pacs:
71.10.Ay, 81.05.Uw, 71.45.Gm
## I Introduction
The latest rival to succeed silicon’s status is graphene, a single atomic
layer of graphite make a truly tiny transistor to decrease the size and to
improve the operational speed of the electronic devices. Silicon lost it’s
brilliant electronic properties in pieces smaller than about $10$nm and
practically the smallest silicon chips which has been used in silicon-based
electronics is $45$nm. Furthermore, silicon has some limitations in speed of
operations. These restrictions lead to serious challenges for the Moore’s law
which states that the number of transistors can be placed inexpensively on an
integrated circuit has increased exponentially, doubling approximately every
two years. This growth cannot be maintained forever and thus the search is on
to find and use new materials which may be able to produce higher performance
and better functionality.
The recent discovery novoselov of graphene in $2004$, and its fabrication
into a field-effect transistor novoselov , has opened up a new field of
physics and offers exciting prospects for new electronic devices and
apparently possible to come over those aforementioned limitations. Graphene
has instructive and unique physics with special intriguing electronic
properties which has attracted remarkable attentions. Barth First, the
electronic properties of graphene are improved in sizes less than $10$nm .
Second, the massless Dirac-like electrons move through graphene with almost
near-ballistic transport behavior with less resistance because back-scattering
is suppressed. Third, graphene is itself a good thermal conductor such that
graphene’s thermal conductivity is about $\sim 5.3\times 10^{3}$ W/mK at room
temperature which is greater than the thermal conductivity of carbon
nanotubes. thermal Interestingly, the mobility of carriers in graphene is
quite high and it is about $10^{5}$ cm2/Vs at room temperature. morozov It is
important to note that the highest electron mobility recorded on the
semiconductor junction H-Si(111)-vacuum FET is $8\times 10^{3}$cm2/Vs at $4.2$
K or the mobility of electrons in junction Si-SiO2(100) MOSFET systems is
$25\times 10^{3}$cm2/Vs at low temperature eng , make graphene promising for
different applications in devices.
Providing capability to control a type and density of charge carriers by gate
voltage or by the chemical doping dop made graphene instructive for novel
nano-electronic devices. However, a gapped semiconducting behavior would be
more suitable for electronic applications. There have been some proposed in
literature for a gap generation in graphene due to breaking of the sublattice
symmetry by some substrates (such as SiC 3 , graphite 4 and boron nitride5 ),
to adsorbe some molecules (such as water, ammonia 6 and CrO3 cro ), spin-
orbit interaction spin and finite size effect.size In case, we are
interested to carry out the microscopic theory to calculate some physical
quantities of gapped graphene.
Theoretical calculations of quasiparticle properties of electron in
conventional two-dimensional electron liquid are performed within the
framework of Landau’s Fermi liquid theory landau whose key ingredient is the
quasiparticle concept and its interactions. As applied to the electron liquid
model this entails the calculation of effective quasiparticle-quasiparticle
interactions which enter the many-body formalism allowing the calculation of
various physical properties. A number of calculations considered different
variants of the $G_{0}W$-approximation for the self-energy in two-dimensional
electron gasem ; yarlagadda_1994_2 ; em_bohm ; em_dassarma ; zhang ; asgari ;
asgari12 from which density, spin-polarization, and temperature dependence of
quasiparticle properties are obtained.
There is a mechanism for quasiparticle scattering against quasiparticles
because they interact through the Coulomb interaction. This is an inelastic
process and induced a finite lifetime of the quasiparticles. The carrier
lifetime in an epitaxial graphene layers grown on SiC wafers has been recently
measured. tau Since experiments carried out their measurements on graphene
placed on $\rm SiC$, we expect that graphene was gapped. The experimental
measurements are relevant for understanding carrier intraband and interband
scattering mechanisms in graphene and their impact on electronic and optical
devises. williams ; gu
In this paper we focus on the effect of energy gap on the renormalized
velocity, the inelastic scattering lifetime of quasiparticles and the
inelastic mean free path in gapped graphene sheets over the broad range of
energy gap. Our formalism is based on the Landau-Fermi liquid theory
incorporating the $G_{0}W$-approximation for the self-energy. These quantities
are related to some important physical properties of both theoretical and
practical applications such as the band structure of ARPES spectra im , the
energy dissipation rate of injected carriers tau and the width of the
quasiparticle spectral function. martin
The contents of the paper are described briefly as follows. In Section II we
discuss about our theoretical model which contains the effect of gap in the
renormalized velocity of quasiparticles and the inelastic scattering lifetime
$\tau_{in}$, of gapped graphene due to electron-electron interactions by using
$G_{0}W$-approximation. Our numerical results are given in Section III.
Finally, Section V contains the summery and conclusions.
## II Theoretical model
Among the methods designed to deal with the intermediate correlation effects,
of particular interest for its physical appeal and elegance is Landau’s
phenomenological theory landau dealing with low-lying excitations in a Fermi-
liquid. Landau called such single-particle excitations quasiparticles and
postulated a one-to-one correspondence between them and the excited states of
a non-interacting Fermi gas. He wrote the excitation energy of the Fermi-
liquid in terms of the energies of the quasiparticles and of their effective
interaction. The quasiparticle-quasiparticle interaction function can in turn
be used to obtain various physical properties of the system and can be
parameterized in terms of experimentally measurable data. In this paper, we
will compute the energy gap dependence of the renormalized velocity,
renormalization constant and the inelastic scattering lifetime of
quasiparticle in a gapped graphene sheet.
### II.1 Quasiparticle renormalized velocity
The dynamics of quasiparticles in a gapless graphene are described by two-
dimensional (2D) massless Dirac Hamiltonian $\hat{H}=\hbar
v{\bf\sigma}\cdot{\bf k}$, with eigenvalues $\varepsilon_{s\bf k}=s\hbar vk$,
where $s=+(-)$ representing right- and left-handed helicity or chirality for
the electrons and holes, respectively. Note that chirality is the same as
helicity for the massless particles. $v=10^{6}$ m$/$s is the Fermi velocity.
As it has been shown before velocity , contrary to conventional 2D electron
systems, the interactions increase the velocity of quasiparticles in graphene
because of interband exchange interactions and the difference between positive
and negative energy branches due to the chirality.
The dynamics of quasiparticles in a gapped graphene are described by 2D
massive Dirac Hamiltonian given by $\hat{H}=\hbar v_{F}\sigma\cdot{\bf
k}+mv^{2}\sigma_{3}$ with eigenvalues $E_{s\bf k}=s\sqrt{(\hbar
vk)^{2}+\Delta^{2}}$ where $\Delta=mv^{2}$ is the gap energy. Due to massive
term in the Hamiltonian, the chirality differs from the helicity and also the
helicity is conserved but is frame dependence.
From the microscopic point of view, the quasiparticle energy can be calculated
by solving the Dyson equation,
$\displaystyle\delta\varepsilon_{s{\bf k}}^{QP}=\xi_{s{\bf k}}+\Re
e[\delta\Sigma_{s}^{ret}({\bf{\bf
k}},\omega)]|_{\omega=\delta\varepsilon_{s{\bf k}}^{QP}/\hbar},$ (1)
where $\xi_{s{\bf k}}=E_{s{\bf k}}-E_{\rm F}$ is the energy of a quasiparticle
relative to the Fermi energy. The Fermi wave vector in graphene is given by
$k_{\rm F}=(4\pi n/g_{s}g_{v})^{1/2}$ where $g_{s}=g_{v}=2$ are spin and
valley degeneracy, respectively. The Fermi energy of gapless graphene is
$\varepsilon_{\rm F}=\hbar vk_{\rm F}$. The retarded self-energy of gapped
graphene is $\Sigma_{s}^{ret}$ and we define $\delta\Sigma_{s}^{ret}({\bf
k},\omega)=\Sigma_{s}^{ret}({\bf k},\omega)-\Sigma_{s}^{ret}(k_{F},0)$. In the
on-shell approximation Giuliani , on the other hand, the above equation must
be solved by setting $\omega=\xi_{s{\bf k}}/\hbar$.
In the $G_{0}W$\- approximation Giuliani , the self-energy of gapped graphene
at finite temperature ($\beta=1/(k_{B}T)$) is given by
$\displaystyle\Sigma_{s}({\bf{k}},i\omega_{n})$ $\displaystyle=$
$\displaystyle-\frac{1}{\beta}\sum_{s^{\prime}}\int\frac{d^{2}{\bf
q}}{(2\pi)^{2}}F^{ss^{\prime}}({\bf k,k+q})$ $\displaystyle\times$
$\displaystyle\sum_{m=-\infty}^{+\infty}W({\bf
q},i\Omega_{m})G^{(0)}_{s^{\prime}}({\bf k+q},i\omega_{n}+i\Omega_{m}),$
where the dynamic screened effective interaction is $W({\bf
q},i\Omega_{m})=V_{q}/\epsilon(q,i\Omega_{m})$ and $\epsilon(q,i\Omega_{m})$
is the dynamical dielectric function and the bare Coulomb interaction is
$V_{q}=2\pi e^{2}/\kappa q$ where $\kappa$ is the averaged background
dielectric constant of graphene is placed on a substrate.
$G_{s}^{(0)}(q,i\Omega_{m})=1/(i\Omega_{m}-\xi_{s{\bf k}}/\hbar)$ is the
standard noninteracting Green’s function. The overlap function for gapped
graphene $F^{ss^{\prime}}({\bf k,k+q})$, is given by alireza
$\displaystyle F^{ss^{\prime}}({\bf
k,k+q})=\frac{1}{2}(1+ss^{\prime}\frac{\hbar^{2}v^{2}{\bf k}\cdot({\bf k}+{\bf
q})+\Delta^{2}}{E_{\bf k}E_{\bf k+q}})~{}.$ (3)
To evaluate of the zero temperature retarded self-energy, we decompose the
self-energy into the line which is purely a real function and residue
contributions, $\Sigma^{ret}_{s}({\bf k},\omega)=\Sigma^{line}_{s}({\bf
k},\omega)+\Sigma^{res}_{s}({\bf k},\omega)$, where $\Sigma^{line}$ is
obtained by performing the analytic continuation before summing over the
Matsubara frequencies, and $\Sigma^{res}$ is the correction which must be
taken into account in the total self-energy,
$\displaystyle\Sigma^{line}_{s}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle-\sum_{s^{\prime}}\int\frac{d^{2}{\bf
q}}{(2\pi)^{2}}V_{q}F^{ss^{\prime}}({\bf k,k+q})$ $\displaystyle\times$
$\displaystyle\int_{-\infty}^{\infty}\frac{d\Omega}{2\pi}\frac{1}{\epsilon({\bf
q},i\Omega)}\frac{1}{\omega+i\Omega-\xi_{s^{\prime}}({\bf k+q})/\hbar},$
and
$\displaystyle\Sigma^{res}_{s}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle\sum_{s^{\prime}}\int\frac{d^{2}{\bf
q}}{(2\pi)^{2}}\frac{V_{q}}{\epsilon({\bf q},\omega-\xi_{s^{\prime}}({\bf
k+q})/\hbar)}F^{ss^{\prime}}({\bf k,k+q})$ (5) $\displaystyle\times$
$\displaystyle[\Theta(\omega-\xi_{s^{\prime}}({\bf
k+q})/\hbar)-\Theta(-\xi_{s^{\prime}}({\bf k+q})/\hbar)],$
where the dynamic dielectric function is given by $\epsilon({\bf
q},\omega)=1-V_{q}\chi^{(0)}(q,\omega)$ in the random phase approximation
(RPA) and $\chi^{(0)}(q,\omega)$ is the noninteracting polarization function
for gapped graphene. The noninteracting polarization function has been
recently calculated on both along the imaginary and real frequency axis
alireza ; pyatkovskiy . The noninteracting polarization function expressions
along the real frequency axis pyatkovskiy are given in appendix A.
Note that there are two independent parameters in the self-energy. One of them
is the Fermi energy $E_{\rm F}$, and the other is the dimensionless coupling
constant $\alpha_{gr}=g_{s}g_{v}e^{2}/\kappa\hbar v$. The coupling constant in
graphene depends only on the substrate dielectric constant while in the
conventional 2D electron systems the coupling constant is density dependent.
For graphene placed on SiC or graphite substrates, the coupling constant is
about $\alpha_{gr}\simeq 1$.
The quasiparticle energy depends on the magnitude of ${\bf k}$ for the
isotropic systems. Expanding $\delta\varepsilon_{+k}^{QP}$ to first order in
$k-k_{\rm F}$, we obtain $\delta\varepsilon_{+k}^{QP}\simeq\hbar
v^{*}(k-k_{\rm F})$ which effectively defines the renormalized velocity as
$\hbar v^{*}=d\delta\varepsilon_{+k}^{QP}/dk|_{k=k_{\rm F}}$. The renormalized
velocity in the Dyson scheme is thus given by
$\displaystyle\frac{v^{*}}{v}=\frac{(1+\Delta^{2})^{-1/2}+v^{-1}\partial_{k}\Re
e[\delta\Sigma_{+}^{ret}({\bf k},\omega)]|_{\omega=0,k=k_{\rm
F}}}{1-\partial_{\omega}\Re e[\delta\Sigma_{+}^{ret}({\bf
k},\omega)]|_{\omega=0,k=k_{\rm F}}}~{}.$ (6)
In the on-shell approximation, on the other hand, the renormalized velocity is
given by $v^{*}/v=(1+\Delta^{2})^{-1/2}+v^{-1}\partial_{k}\Re
e[\delta\Sigma_{+}^{ret}({\bf k},\omega)]|_{\omega=0,k=k_{\rm
F}}+(1+\Delta^{2})^{-1/2}\partial_{\omega}\Re e[\delta\Sigma_{+}^{ret}({\bf
k},\omega)]|_{\omega=0,k=k_{\rm F}}$. The renormalized velocity in this
approximation demonstrates qualitatively the same behavior obtained by the
Dyson equation, Eq. (6) but its magnitude is larger than the one calculated
within the Dyson scheme. asgari There is an ultraviolet divergence in the
wave vector integrals of the line contribution in a continuum model formulated
as discussed above. velocity We introduce an ultraviolet cutoff for the wave
vector integrals, $k_{c}=\Lambda k_{F}$ which is the order of the inverse
lattice spacing and $\Lambda$ is dimensionless quantity. For definiteness we
take $\Lambda=k_{c}/k_{F}$ to be such that $\pi(\Lambda
k_{F})^{2}=(2\pi)^{2}/{\cal A}_{0}$, where ${\cal A}_{0}=3\sqrt{3}a^{2}_{0}/2$
is the area of the unit cell in the honeycomb lattice, with $a_{0}\simeq 1.42$
Å the carbon-carbon distance. With this choice,
$\Lambda\simeq{(gn^{-1}\sqrt{3}/9.09)^{1/2}}\times 10^{2}$, where $n$ is the
electron density in units of $10^{12}~{}{\rm cm}^{-2}$.
An important quantity in the Fermi-liquid theory is the renormalization
constant $Z$, defined as the square of the overlap between the state of the
system after adding (or removing) of an electron with the Fermi wave vector
and the ground-state of the system. The non-zero renormalization constant
value is always smaller than the one for the normal Fermi-liquid systems and
can be calculated explicitly as followGiuliani
$\displaystyle Z=\frac{1}{1-\partial_{\omega}\Re e[\delta\Sigma_{+}^{ret}({\bf
k},\omega)]|_{\omega=0,k=k_{\rm F}}}.$ (7)
We will show that $Z$ is a finite number for gapped graphene and it confirms
as well that the system is a Fermi-Liquid.
### II.2 Inelastic scattering lifetime
In this subsection, we compute the inelastic scattering lifetime of
quasiparticles due to carriers-carriers interactions at zero temperature and
disorder-free for gapped graphene sheets. This is obtained through the
imaginary part of the self-energy inelastic when the frequency evaluated at
the on-shell energy.
$\displaystyle\tau_{in}^{-1}({\bf k})=\Gamma_{in}({\bf k},{\xi_{+\bf
k}}/\hbar)=-\frac{2}{\hbar}\Im m\Sigma_{+}^{ret}({\bf k},\xi_{+\bf k}/\hbar),$
(8)
where $\Gamma_{in}({\bf k},{\xi_{s\bf k}}/\hbar)$ is the quantum level
broadening of the momentum with eigenstate $|s{\bf k}>$. It is worthwhile to
note that the expression of $\tau_{in}^{-1}$ is identical with a result
obtained by the Fermi’s golden rule summing the scattering rate of electron
and hole contributions at wave vector ${\bf k}$. Giuliani Note again that the
total contribution of the imaginary part of the retarded self-energy comes
from the residue term both intra- and interband contributions, $\Im
m\Sigma^{ret}_{+}({\bf k},\omega)=\Im m\Sigma^{res}_{intra}({\bf
k},\omega)+\Im m\Sigma^{res}_{inter}({\bf k},\omega)$. However, the total
contribution of the imaginary part of the retarded self-energy evaluated at
the on-shell energy comes only from intraband term, $\Im
m\Sigma_{+}^{ret}({\bf k},\xi_{\bf k}/\hbar)=\Im m\Sigma^{res}_{intra}({\bf
k},\xi_{\bf k}/\hbar)$. We will discuss about that with more details in the
appendix B and C.
We turn our attention to investigate the imaginary part of the retarded self-
energy with more details. By starting from Eq. (5), we end up to an expression
for the imaginary part of self-energy which is given by,
$\displaystyle\Im m\Sigma^{ret}_{+}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle\Im m\Sigma^{res}_{intra}({\bf k},\omega)+\Im
m\Sigma^{res}_{inter}({\bf k},\omega)$ (9) $\displaystyle=$
$\displaystyle\int\frac{d^{2}{\bf q}}{(2\pi)^{2}}V_{q}\Im m[\epsilon^{-1}({\bf
q},\omega-\xi_{+}({\bf k+q})/\hbar)]F^{++}({\bf k,k+q})$ $\displaystyle\times$
$\displaystyle[\Theta(\omega-\xi_{+}({\bf k+q})/\hbar)-\Theta(-\xi_{+}({\bf
k+q})/\hbar)]$ $\displaystyle+$ $\displaystyle\int\frac{d^{2}{\bf
q}}{(2\pi)^{2}}V_{q}\Im m[\epsilon^{-1}({\bf q},\omega-\xi_{-}({\bf
k+q})/\hbar)]F^{+-}({\bf k,k+q})$ $\displaystyle\times$
$\displaystyle[\Theta(\omega-\xi_{-}({\bf k+q})/\hbar)-\Theta(-\xi_{-}({\bf
k+q})/\hbar)].$
where the imaginary part of the inverse dielectric function in RPA level is
obtained by
$\Im m[\epsilon^{-1}({\bf q},\omega)]=\frac{V_{q}\Im m\chi^{(0)}({\bf
q},\omega)}{[1-V_{q}\Re e\chi^{(0)}({\bf q},\omega)]^{2}+[V_{q}\Im
m\chi^{(0)}({\bf q},\omega)]^{2}}.$ (10)
It is worth to note that the plasmon contributions in the imaginary part of
self-energy comes from the zero-solutions of denominator in Eq. (10).
## III Numerical results
We turn to a presentation of our main numerical results. We present some
illustrative results for the quasiparticle dynamic properties such
renormalized velocity, renormalization constant and inelastic scattering
lifetime. All numerical data are calculated in the Dyson scheme at $\alpha=1$.
The Fermi liquid phenomenology of Dirac electrons in gapless graphene im ;
velocity and conventional 2D electron liquid asgari have the same structure,
since both systems are isotropic and have a single circular Fermi surface. The
strength of interaction effects in a conventional 2D electron liquid increases
with decreasing carrier density. At low densities, the quasiparticle
renormalization constant $Z$ is small, the renormalized velocity is suppressed
asgari , the charge compressibility changes sign from positive to negative,
and the spin-susceptibility is strongly enhanced asgari2 . These effects
emerge from an interplay between exchange interactions and quantum
fluctuations of charge and spin in the 2D electron liquid.
In the 2D massless electron graphene, on the other hand, it has been shown
yafis ; velocity ; im that interaction effects also become noticeable with
decreasing density, although more slowly, the quasiparticle renormalization
constant, $Z$ tends to larger values, that the renormalized velocity is
enhanced rather than suppressed, and that the influence of interactions on the
compressibility and the spin-susceptibility changes sign. These qualitative
differences are due to exchange interactions between electrons near the Fermi
surface and electrons in the negative energy sea and to interband
contributions to Dirac electrons from charge and spin fluctuations.
In this paper we have shown the results for gapped graphene which are
determined values between the gapless graphene evaluated at $\triangle=0$ and
the conventional 2D electron liquid where $\triangle\rightarrow\infty$.
In Fig. 1, we have plotted the renormalized velocity as a function of carrier
density for the various energy gap. As a result, we see that the impact of
energy gap on quasiparticles velocity which is similar to the effect of
impurity to that on grapheneimpurity . The renormalized velocity is almost
density independent in gapped graphene at large carrier densities. The
renormalized velocity reduces dramatically by increasing the energy gap
especially in the low carrier densities. Importantly, the renormalized
velocity becomes less than the bare velocity at large energy gap and low
density values. It is physically accepted since the system tends to
conventional 2D electron liquid by increasing the energy gap values. Note that
in the conventional 2D electron systems, the renormalized velocity is
suppressed by increasing the coupling constant or reducing the density.
We have shown the renormalization constant $Z$, as a function of the energy
gap in Fig. 2. The renormalization constant enormously reduces by increasing
the energy gap in mild densities, however it decreases quite slowly in high
densities.
Fig. 3(a) is shown the absolute value of $\Im m\Sigma^{\rm\scriptstyle
ret}_{+}({\bf k},\omega)$ as from Eq. (9), evaluated at $\omega=\xi_{\bf
k}/\hbar$. By increasing the gap value, this function takes a finite jump at
the wave number of the plasmon dip and at large $\triangle$ values, a
discontinuity appears. The discontinuity is peculiar to $2D$ electron liquid.
giuliani_quinn It is absent in gapless graphene and starts to arise from the
fact that the oscillator strength of the plasmon pole is non-zero at special
$k$ value for gapped graphene.
Fig. 3(b) is clearly shown the behavior of the energy gap dependence of the
inverse inelastic scattering lifetime. As it is argued in the Appendix B, the
imaginary part of self-energy evaluated at the on-shell energy start from
$\triangle-\sqrt{\varepsilon_{\rm F}^{2}+\triangle^{2}}$ and in case the
results are truncated below that. The quasiparticle lifetime decreases by
increasing the gap value and it is a clear difference between 2D massless
Dirac electron and gapped graphene. Consequently, the inelastic scattering
lifetime in graphene is always larger than the conventional 2D electron
liquid. In the case of gapless graphene, scattering rate is a smooth function
because of the absence of both plasmon emission and interband
processes,inelastic nevertheless with generating a gap and increasing the
amount of it, plasmon emission causes to arise a discontinuity in the
scattering time, similar to conventional 2D electron liquid. asgari We have
thus two mechanisms for scattering of the quasiparticles. The excitation of
electron-hole pairs which is dominant process at low wave vectors and the
excitation of plasmon appears in a specific wave vector. We also see in Fig.
3(b) that the scattering rate is quite sensitive to the gap energy and the
scattering rate increases by increasing the energy gap.
In Fig. 4, we have depicted the inelastic mean free path $l_{in}({\bf
k})=v^{*}\tau_{in}({\bf k})$, as a function of the on-shell energy for various
gap energies. To this purpose we multiplied the results of $\tau_{in}({\bf
k})$ to a proper renormalized velocity. As a result the mean free path of a
gapped graphene is shorter than that obtained for gapless graphene.
Furthermore, the massless graphene has larger $l_{in}$ and it decreases by
increasing the energy gap values. Note that the typical value of energy gap
due to breaking sublattice symmetry is $\triangle=10-100$ meV corresponding
the inelastic mean free path is $l_{in}=20-50$ nm which implies that the
system remains in the semi-ballistic regime. 3 ; 4 ; 5 ; 6 .
## IV Summery and concluding remarks
In summary, we have studied the problem of the microscopic calculation of the
quasiparticle self-energy and many-body renormalized velocity suppression over
the energy gap in a gapped graphene. We have carried out calculations of both
the real and the imaginary part of the quasiparticle self-energy within
$G_{0}W$-approximation. We have also presented results for the renormalized
velocity suppression and for the renormalization constant over a wide range of
energy gap. We have shown that the renormalized velocity for a gapped graphene
is almost independent of the carrier density at high density. We have finally
presented results for the quasiparticle inelastic scattering lifetime
suppression over the energy gap and show that the mean free path of the charge
carriers of a gapless graphene is larger than a gapped graphene one. In case
the mean free path of charge carriers decrease by increasing the energy gap.
A possible role of correlations including the charge-density fluctuations
beyond the Random Phase Approximation, remains to be examined.
###### Acknowledgements.
R. A. would like to thank the International Center for Theoretical Physics,
Trieste for its hospitality during the period when part of this work was
carried out. A. Q is supported by IPM grant.
## Appendix A The dynamic polarization function for a gapped graphene
In this appendix we present the real and imaginary part of the noninteracting
polarization function for a gapped graphene, which is calculated recently by
Pyatkovskiy. pyatkovskiy The dynamic polarization function for gapped
graphene in the imaginary frequency axis is also calculated by us in Ref.
[alireza, ]. Importantly, the noninteracting polarization function along the
imaginary frequency axis can be obtained by performing analytical continuation
from real axis and those results are the same. private
First, by introducing some following notations,
$\displaystyle f(k,\omega)$ $\displaystyle=$
$\displaystyle\frac{g_{s}g_{v}k^{2}}{16\pi\sqrt{|\hbar^{2}v^{2}k^{2}-\hbar^{2}\omega^{2}|}},$
$\displaystyle g_{\pm}$ $\displaystyle=$ $\displaystyle\frac{2E_{\rm
F}\pm\hbar\omega}{\hbar vk},$ $\displaystyle x_{0}$ $\displaystyle=$
$\displaystyle\sqrt{1+\frac{4\Delta^{2}}{\hbar^{2}v^{2}k^{2}-\hbar^{2}\omega^{2}}},$
$\displaystyle G_{<}(x)$ $\displaystyle=$ $\displaystyle
x\sqrt{x_{0}^{2}-x^{2}}-(2-x_{0}^{2})\cos^{-1}(x/x_{0}),$ $\displaystyle
G_{>}(x)$ $\displaystyle=$ $\displaystyle
x\sqrt{x^{2}-x_{0}^{2}}-(2-x_{0}^{2})\cosh^{-1}(x/x_{0}),$ $\displaystyle
G_{0}(x)$ $\displaystyle=$ $\displaystyle
x\sqrt{x^{2}-x_{0}^{2}}-(2-x_{0}^{2})\sinh^{-1}(x/\sqrt{-x_{0}^{2}}),$ (11)
the real part of noninteracting polarization function is given by,
$\displaystyle\Re e\chi^{(0)}(k,\omega)$ $\displaystyle=$
$\displaystyle-\frac{g_{s}g_{v}E_{\rm F}}{2\pi
v_{F}^{2}}+f(k,\omega)\times\left\\{\begin{array}[]{ll}0,&\hbox{1A}\\\
G_{<}(g_{-}),&\hbox{2A}\\\ G_{<}(g_{+})+G_{<}g_{-}),&\hbox{3A}\\\
G_{<}(g_{-})-G_{<}(g_{+}),&\hbox{4A}\\\
G_{>}(g_{+})-G_{>}(g_{-}),&\hbox{1B}\\\ G_{>}(g_{+}),&\hbox{2B}\\\
G_{>}(g_{+})-G_{>}(-g_{-}),&\hbox{3B}\\\
G_{>}(-g_{-})+G_{>}(g_{+}),&\hbox{4B}\\\
G_{0}(g_{+})-G_{0}(g_{-}),&\hbox{5B}\\\ \end{array}\right.$ (21)
and the imaginary part of noninteracting polarization function is given by,
$\displaystyle\Im m\chi^{(0)}(k,\omega)$ $\displaystyle=$ $\displaystyle
f(k,\omega)\times\left\\{\begin{array}[]{ll}G_{>}(g_{+})-G_{>}(g_{-}),&\hbox{1A}\\\
G_{>}(g_{+}),&\hbox{2A}\\\ 0,&\hbox{3A}\\\ 0,&\hbox{4A}\\\ 0,&\hbox{1B}\\\
-G_{<}(g_{-}),&\hbox{2B}\\\ \pi(2-x_{0}^{2}),&\hbox{3B}\\\
\pi(2-x_{0}^{2}),&\hbox{4B}\\\ 0,&\hbox{5B}\\\ \end{array}\right.$ (31)
with the followings regions in the $(k,\omega)$ space,
$\displaystyle\begin{array}[]{cc}1A&\hbar\omega<E_{\rm
F}-\sqrt{\hbar^{2}v^{2}(k-k_{\rm F})^{2}+\Delta^{2}},\\\ 2A&|E_{\rm
F}-\sqrt{\hbar^{2}v_{F}^{2}(k-k_{\rm F})^{2}+\Delta^{2}}|<\hbar\omega<-E_{\rm
F}+\sqrt{\hbar^{2}v^{2}(k+k_{\rm F})^{2}+\Delta^{2}},\\\
3A&\hbar\omega<-E_{\rm F}+\sqrt{\hbar^{2}v^{2}(k-k_{\rm
F})^{2}+\Delta^{2}},\\\ 4A&-E_{\rm F}+\sqrt{\hbar^{2}v^{2}(k+k_{\rm
F})^{2}+\Delta^{2}}<\hbar\omega<\hbar vk,\\\ 1B&k<2k_{\rm
F},~{}~{}~{}\sqrt{\hbar^{2}v^{2}k^{2}+4\Delta^{2}}<\hbar\omega<E_{\rm
F}+\sqrt{\hbar^{2}v^{2}(k-k_{\rm F})^{2}+\Delta^{2}},\\\ 2B&E_{\rm
F}+\sqrt{\hbar^{2}v^{2}(k-k_{\rm F})^{2}+\Delta^{2}}<\hbar\omega<E_{\rm
F}+\sqrt{\hbar^{2}v^{2}(k+k_{\rm F})^{2}+\Delta^{2}},\\\ 3B&\hbar\omega>E_{\rm
F}+\sqrt{\hbar^{2}v^{2}(k+k_{\rm F})^{2}+\Delta^{2}},\\\ 4B&k>2k_{\rm
F},~{}~{}~{}\sqrt{\hbar^{2}v^{2}k^{2}+4\Delta^{2}}<\hbar\omega<E_{\rm
F}+\sqrt{\hbar^{2}v^{2}(k-k_{\rm F})^{2}+\Delta^{2}},\\\ 5B&\hbar
vk<\hbar\omega<\sqrt{\hbar^{2}v^{2}k^{2}+4\Delta^{2}},\\\ \end{array}$ (41)
## Appendix B The intraband contribution of self-energy
Since we are interested in quasiparticle properties, we therefore need only
$s=+$ contribution. Let us focus on the intraband contribution of the retarded
self-energy. The second argument of the dielectric function in Eq. (5) ( by
setting $\hbar=v=1$) is
$\displaystyle\omega-\xi_{+}({\bf k+q})=\omega+E_{\rm
F}-\sqrt{k^{2}+q^{2}+2kq\cos\phi+\Delta^{2}}~{}.$ (42)
In this case, we change the variable $\phi$ and integrate it over
$y=\sqrt{k^{2}+q^{2}+2kq\cos\phi+\Delta^{2}}$. Using the new variable, the
intraband contribution of self-energy changes to
$\displaystyle\Sigma_{intra}^{res}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{2\pi\kappa\sqrt{k^{2}+\Delta^{2}}}\int_{0}^{+\infty}dq\int_{\sqrt{(k-q)^{2}+\Delta^{2}}}^{\sqrt{(k+q)^{2}+\Delta^{2}}}\frac{dy}{\sqrt{4k^{2}q^{2}-(y^{2}-k^{2}-q^{2}-\Delta^{2})^{2}}}\frac{(\sqrt{k^{2}+\Delta^{2}}+y)^{2}-q^{2}}{\epsilon({\bf
q},\omega+E_{\rm F}-y)}$ (43) $\displaystyle\times$
$\displaystyle[\Theta(\omega+E_{\rm F}-y)-\Theta(E_{\rm F}-y)].$
We can now simplify the $\Theta$-functions further in Eq. (43) by considering
the positive and negative regions of $\omega$ as follow
$\displaystyle 1)$ $\displaystyle\omega$ $\displaystyle+E_{\rm
F}-y>0~{}and~{}~{}E_{\rm F}-y<0:~{}It~{}implies~{}that~{}\omega>0~{},$
$\displaystyle 2)$ $\displaystyle\omega$ $\displaystyle+E_{\rm
F}-y<0~{}and~{}~{}E_{\rm F}-y>0:~{}It~{}implies~{}that~{}\omega<0~{}.$
To consider the first case where $\omega>0$, the difference between the two
$\Theta$-functions in Eq. (43) is equal to $+1$ if
$\displaystyle E_{\rm F}<y<\omega+E_{\rm
F}~{}~{}and~{}~{}\sqrt{(k-q)^{2}+\Delta^{2}}<y<\sqrt{(k+q)^{2}+\Delta^{2}}~{}.$
(44)
Now we do need to find the overlap between these two intervals. We simply end
up to inequivalent conditions which are $q>k-\sqrt{\omega^{2}+k_{\rm
F}^{2}+2\omega\sqrt{k_{\rm F}^{2}+\Delta^{2}}}$, $q<k+\sqrt{\omega^{2}+k_{\rm
F}^{2}+2\omega\sqrt{k_{\rm F}^{2}+\Delta^{2}}}$ and $q>k_{\rm F}-k$.
Collecting everything together and using the fact that $q\geq 0$, we finally
find
$\displaystyle\Sigma_{intra}^{res}({\bf k},\omega>0)$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{2\pi\kappa\sqrt{k^{2}+\Delta^{2}}}\int_{max(0,k_{\rm
F}-k,k-\sqrt{\omega^{2}+k_{\rm F}^{2}+2\omega\sqrt{k_{\rm
F}^{2}+\Delta^{2}}})}^{k+\sqrt{\omega^{2}+k_{\rm F}^{2}+2\omega\sqrt{k_{\rm
F}^{2}+\Delta^{2}}}}dq\int^{min(\omega+\sqrt{k_{\rm
F}^{2}+\Delta^{2}},\sqrt{(k+q)^{2}+\Delta^{2}})}_{max(\sqrt{k_{\rm
F}^{2}+\Delta^{2}},\sqrt{(k-q)^{2}+\Delta^{2}})}dy$ (45) $\displaystyle\times$
$\displaystyle\frac{(\sqrt{k^{2}+\Delta^{2}}+y)^{2}-q^{2}}{\epsilon({\bf
q},\omega+E_{\rm F}-y)\sqrt{4k^{2}q^{2}-(y^{2}-k^{2}-q^{2}-\Delta^{2})^{2}}}$
By considering of the second case where $\omega<0$, the difference between the
two $\Theta$-functions in Eq. (5) is equal to $-1$ if
$\displaystyle E_{\rm F}+\omega<y<E_{\rm
F}~{}~{}and~{}~{}\sqrt{(k-q)^{2}+\Delta^{2}}<y<\sqrt{(k+q)^{2}+\Delta^{2}}$
(46)
As what we did before, we calculate overlap between intervals and thus we find
$q>k-k_{\rm F}$ and $q<k+k_{\rm F}$, $q>\sqrt{\omega^{2}+k_{\rm
F}^{2}+2\omega\sqrt{k_{\rm F}^{2}+\Delta^{2}}}-k$. Putting everything together
and using the fact that $q\geq 0$ we finally find
$\displaystyle\Sigma_{intra}^{res}({\bf k},\Delta-E_{\rm F}<\omega<0)$
$\displaystyle=$
$\displaystyle-\frac{e^{2}}{2\pi\kappa\sqrt{k^{2}+\Delta^{2}}}\int_{max(0,k-k_{\rm
F},\sqrt{\omega^{2}+k_{\rm F}^{2}+2\omega\sqrt{k_{\rm
F}^{2}+\Delta^{2}}}-k)}^{k+k_{\rm F}}dq$ (47) $\displaystyle\times$
$\displaystyle\int^{min(\sqrt{k_{\rm
F}^{2}+\Delta^{2}},\sqrt{(k+q)^{2}+\Delta^{2}})}_{max(0,\omega+\sqrt{k_{\rm
F}^{2}+\Delta^{2}},\sqrt{(k-q)^{2}+\Delta^{2}})}dy$ $\displaystyle\times$
$\displaystyle\frac{(\sqrt{k^{2}+\Delta^{2}}+y)^{2}-q^{2}}{\epsilon({\bf
q},\omega+E_{\rm F}-y)\sqrt{4k^{2}q^{2}-(y^{2}-k^{2}-q^{2}-\Delta^{2})^{2}}}$
$\displaystyle\Sigma_{intra}^{res}({\bf k},\omega<-\Delta-E_{\rm F})$
$\displaystyle=$
$\displaystyle-\frac{e^{2}}{2\pi\kappa\sqrt{k^{2}+\Delta^{2}}}\int_{max(0,k-k_{\rm
F})}^{k+k_{\rm F}}dq\int^{min(\sqrt{k_{\rm
F}^{2}+\Delta^{2}},\sqrt{(k+q)^{2}+\Delta^{2}})}_{max(0,\omega+\sqrt{k_{\rm
F}^{2}+\Delta^{2}},\sqrt{(k-q)^{2}+\Delta^{2}})}dy$ (48) $\displaystyle\times$
$\displaystyle\frac{(\sqrt{k^{2}+\Delta^{2}}+y)^{2}-q^{2}}{\epsilon({\bf
q},\omega+E_{\rm F}-y)\sqrt{4k^{2}q^{2}-(y^{2}-k^{2}-q^{2}-\Delta^{2})^{2}}}$
The real and imaginary part of intraband contributions can be computed.
## Appendix C The interband contribution of self-energy
Now, we focus on the interband contribution of the retarded self-energy. The
second argument of the dielectric function in Eq. (5) is
$\displaystyle\omega-\xi_{-}({\bf k+q})=\omega+E_{\rm
F}+\sqrt{k^{2}+q^{2}+2kq\cos\phi+\Delta^{2}}~{}.$ (49)
We change variable $y=\sqrt{k^{2}+q^{2}+2kq\cos\phi+\Delta^{2}}$, then we find
$\displaystyle\Sigma_{inter}^{res}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{2\pi\kappa\sqrt{k^{2}+\Delta^{2}}}\int_{0}^{+\infty}dq\int_{\sqrt{(k-q)^{2}+\Delta^{2}}}^{\sqrt{(k+q)^{2}+\Delta^{2}}}\frac{dy}{\sqrt{4k^{2}q^{2}-(y^{2}-k^{2}-q^{2}-\Delta^{2})^{2}}}\frac{q^{2}-(y-\sqrt{k^{2}+\Delta^{2}})^{2}}{\epsilon({\bf
q},\omega+E_{\rm F}+y)}$ (50) $\displaystyle\times$
$\displaystyle[\Theta(\omega+E_{\rm F}+y)-1].$
Note that $\Sigma_{inter}^{res}$ can be non-zero if $\omega+E_{\rm
F}+y<0~{}~{}and~{}~{}y>0$. It means that $\omega<-E_{\rm F}$. In this case the
difference between the two $\Theta$-functions in Eq. (5) becomes -1 if
$\displaystyle 0<y<-(\omega+E_{\rm
F})~{}~{}and~{}~{}\sqrt{(k-q)^{2}+\Delta^{2}}<y<\sqrt{(k+q)^{2}+\Delta^{2}}~{}.$
(51)
Now we do need to find the overlap between these two intervals. We end up to
inequivalent conditions that $q>k-\sqrt{\omega^{2}+k_{\rm
F}^{2}+2\omega\sqrt{k_{\rm F}^{2}+\Delta^{2}}}$ and
$q<k+\sqrt{\omega^{2}+k_{\rm F}^{2}+2\omega\sqrt{k_{\rm F}^{2}+\Delta^{2}}}$.
Putting everything together and using the fact that $q\geq 0$ we finally get
$\displaystyle\Sigma_{inter}^{res}({\bf k},\omega<-E_{\rm F})$
$\displaystyle=$
$\displaystyle-\frac{e^{2}}{2\pi\kappa\sqrt{k^{2}+\Delta^{2}}}\int_{max(0,k-\sqrt{\omega^{2}+k_{\rm
F}^{2}+2\omega\sqrt{k_{\rm F}^{2}+\Delta^{2}}})}^{k+\sqrt{\omega^{2}+k_{\rm
F}^{2}+2\omega\sqrt{k_{\rm
F}^{2}+\Delta^{2}}}}dq\int_{\sqrt{(k-q)^{2}+\Delta^{2}}}^{min(\sqrt{(k+q)^{2}+\Delta^{2}},-(\omega+\sqrt{k_{\rm
F}^{2}+\Delta^{2}}))}dy$ (52) $\displaystyle\times$
$\displaystyle\frac{q^{2}-(y-\sqrt{k^{2}+\Delta^{2}})^{2}}{\epsilon({\bf
q},\omega+\varepsilon_{\rm
F}+y)\sqrt{4k^{2}q^{2}-(y^{2}-k^{2}-q^{2}-\Delta^{2})^{2}}}.$
If we want to calculate $\Im m\Sigma^{res}_{+}({\bf k},\xi_{+}(\bf k))$ needed
for computing the quasiparticle lifetime, we will only need the intraband
contribution of the self-energy since the interband contribution is zero.
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Figure 1: ( Color online) (a): The renormalized velocity as a function of
density for the various energy gaps at $\alpha_{gr}=1$. (b): The renormalized
velocity as a function of the energy gap for the various densities. Figure 2:
( Color online) The renormalization constant as a function of the energy gap
for the various densities.
Figure 3: ( Color online) (a): The absolute value of the imaginary part of the
retarded self-energy on the energy shell as a function of the wavevector for
the various energy gaps; (b): The inelastic quasiparticle
lifetime($\tau_{in}$) in graphene as a function of the on-shell energy for the
various energy gaps at $n=5\times 10^{12}$cm-2. Figure 4: ( Color online) The
quasiparticle mean free path as a function of the on-shell energy for the
various energy gaps at $n=5\times 10^{12}$cm-2.
|
arxiv-papers
| 2008-10-27T11:59:59 |
2024-09-04T02:48:58.467607
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Qaiumzadeh, F. K. Joibari, Reza Asgari",
"submitter": "Reza Asgari",
"url": "https://arxiv.org/abs/0810.4681"
}
|
0810.4840
|
# The Pursuit For Uniqueness: Extending Valiant-Vazirani Theorem to the
Probabilistic and Quantum Settings
preliminary version
Dorit Aharonov School of Computer Science and Engineering, The Hebrew
University, Jerusalem, Israel. doria@cs.huji.ac.il. Michael Ben-Or School of
Computer Science and Engineering, The Hebrew University, Jerusalem, Israel.
benor@cs.huji.ac.il. Fernando G.S.L. Brandão Institute for Mathematical
Sciences, Imperial College London. fernando.brandao@imperial.ac.uk Or Sattath
School of Computer Science and Engineering, The Hebrew University, Jerusalem,
Israel. sattath@cs.huji.ac.il.
###### Abstract
Valiant-Vazirani showed in 1985 [VV85] that solving NP with the promise that
“yes” instances have only one witness is powerful enough to solve the entire
NP class (under randomized reductions).
We are interested in extending this result to the quantum setting. We prove
extensions to the classes Merlin-Arthur (MA) and Quantum-Classical-Merlin-
Arthur (QCMA) [AN02]. Our results have implications on the complexity of
approximating the ground state energy of a quantum local Hamiltonian with a
unique ground state and an inverse polynomial spectral gap. We show that the
estimation, to within polynomial accuracy, of the ground state energy of poly-
gapped 1-D local Hamiltonians is QCMA-hard, under randomized reductions. This
is in strong contrast to the case of constant gapped 1-D Hamiltonians, which
is in NP [Has07]. Moreover, it shows that unless QCMA can be reduced to NP by
randomized reductions, there is no classical description of the ground state
of every poly-gapped local Hamiltonian which allows the calculation of
expectation values efficiently.
Finally, we discuss a few obstacles towards establishing an analogous result
to the class Quantum-Merlin-Arthur (QMA). In particular, we show that random
projections fails to provide a polynomial gap between two witnesses.
## 1 Introduction and Results
### 1.1 Extending Valiant-Vazirani
One of the properties of the class NP is that the number of witnesses might
vary from zero to exponentially many. How hard is it to distinguish between
“no” instances and “yes” instances that have a unique witness? One might think
that such a problem is easier than solving NP. In a celebrated result, Valiant
and Vazirani [VV85] showed that access to an oracle which can decide between
“no” and “unique yes” instances is enough to solve the NP-complete problem
sat, with high probability, using randomized reductions 111A promise problem A
is reducible to B by a randomized reduction, if there exists a probabilistic
polynomial Turing Machine (TM) $M$ and a polynomial $p$ s.t.: • completeness:
$x\in\textsc{A}_{yes}\Rightarrow Pr_{r}(M(x,r)\in\textsc{B}_{yes})\geq
1/p(|x|)$ • perfect soundness: $x\in\textsc{A}_{no}\Rightarrow\forall r\
M(x,r)\in\textsc{B}_{no}$ where $r$ are the random bits of the TM $M$. We
denote this by $A\preceq_{R}B$. .
The classes MA, QCMA [AN02] and QMA [KVS+02] are probabilistic and quantum
analogues of NP. Informally, we say a problem is in MA if for every “yes”
instance there is a witness which makes the verifier to accept with high
probability (e.g. in the range (2/3, 1)), while for “no” instances he only
accepts with a small probability (e.g. in (0, 1/3)), no matter which witness
is given to him. The class QCMA is defined in a similar manner, but now the
verifier can use a quantum computer to decide whether to accept or not. In
QMA, in turn, not only does the verifier use a quantum computer to check the
proof, but also the proof itself is a quantum state composed of a polynomial
(in the input size) number of qubits.
We can ask a similar question to that of Valiant and Vazirani about each of
these classes: given access to an oracle that can only decide between “no”
instances and “yes” instances which have a unique solution for MA, QCMA, or
QMA, can we solve complete problems for those classes, with high probability?
The quantum related questions are also motivated by physical questions about
ground states of local Hamiltonians. We provide some interesting implications
in this direction, which we will soon describe.
In this paper we partially solve these questions: we present a generalization
of the Valiant-Vazirani result to MA and QCMA. We also discuss some
obstructions towards establishing a similar result to QMA, which is left as an
open problem.
We define UMA and UQCMA as the restrictions of MA and QCMA, respectively, to
instances with a unique witness. Roughly speaking, in a “yes” instance of a
problem in UMA or UQCMA, one proof convinces the verifier with probability
larger than e.g. 2/3, while any other witness makes him accept with
probability of at most $1/3$. In a “no” instance, the verifier accepts any
witness with probability at most 1/3. Our two main results are:
###### Theorem 1
$\textsf{MA}\stackrel{{\scriptstyle R}}{{=}}\textsf{UMA}$ 222We say that the
class $C_{1}$ is included in $C_{2}$ under randomized reduction, and denote it
by $C_{1}\stackrel{{\scriptstyle R}}{{\subseteq}}C_{2}$ if for every $L_{1}\in
C_{1}$ there exists $L_{2}\in C_{2}$ s.t. $C_{1}\preceq_{R}C_{2}$..
###### Theorem 2
$\textsf{UQCMA}\stackrel{{\scriptstyle R}}{{=}}\textsf{QCMA}$.
The proofs of both theorems rely heavily on the Valiant-Vazirani construction
[VV85, AB09], which can be divided into three components:
1. 1.
We could guess the size of the accepting witness set, and use a random
“filter” with a certain degree of screening, which is determined by the set
size. If we guess correctly, then with constant probability, exactly one
witness will pass the filter.
2. 2.
We notice that it is not crucial to guess the exact size of the set - and a
multiplicative approximation is enough. In this way, the possible number of
guesses is reduced from exponentially many in the previous component, to
linear (in the length of the witness).
3. 3.
we replace the random “filter” with a pseudo random “filter” - a universal
hash function - without loosing any of the properties. These pseudo-random
objects have the advantage of an efficient description, unlike truly random
sets.
The probabilistic setting of MA and QCMA raises a new difficulty: on “yes”
instances there might be an exponentially larger number of witnesses in the
gap-interval (e.g. $(1/3,2/3)$) than in the “yes” interval $(2/3,1)$. Thus, a
random choice of one of the witnesses - in the spirit of the Valiant-Vazirani
approach - would, with overwhelming large probability, fail to choose a
witness from the “yes” interval. The main idea in overcoming this obstacle is
to divide the “gap” interval into polynomially many smaller intervals, and
argue that in at least one of them, the number of witnesses inside it is not
much larger than the number of witnesses in the intervals above it.
We can also define the class UQMA \- a unique variant of QMA \- with the hope
of proving the analogous result. It is defined as follows: the conditions for
a “no” instance are the same as in QMA, but for a “yes” instance, we demand
that there exists a $|\psi\rangle$ which is accepted above the
“yes”-threshold, and all states $|\phi\rangle$ orthogonal to it are accepted
with probability below the “no”-threshold. Before we proceed to show that an
analogous result for QMA is probably impossible to achieve using similar
techniques to the ones we employ, we use this definition together with Theorem
2 to derive interesting implications.
### 1.2 Implications to Ground State and Hamiltonian Complexity
We say a Hamiltonian, acting on $n$ $d$-dimensional particles, is $k$-local if
it can be written as a sum of $\textsf{poly}(n)$ terms which act non-trivially
at most on $k$ sites.
###### Definition 3
k-local hamiltonian: We are given a $k$-local Hamiltonian on $n$ qubits
$H=\sum_{j=1}^{r}H_{j}$ with $r=\textsf{poly}(n)$. Each $H_{j}$ has a bounded
operator norm $||H_{j}||\leq\textsf{poly}(n)$. We are also given two constants
$a$ and $b$ with $b-a\geq 1/\textsf{poly}(n)$. In “yes” instances, the
smallest eigenvalue of $H$ is at most $a$. In “no” instances, it is larger
than $b$. We should decide which one is the case.
In a seminal work, Kitaev showed that the 5-local hamiltonian problem is
complete for QMA [Kit99]. Improvements in parameters (dimensionality and
locality) were given in [KR03, KKR06, OT05], leading to the QMA-completeness
of 1-d 2-local hamiltonian [AGIK07], which is the variant of the original
problem to one-dimensional nearest-neighbors Hamiltonians (with $d=12$). The
importance of these results stems not only from the fact that local
hamiltonian is probably the most representative QMA-complete problem, but also
from the key role of local Hamiltonians and their ground-state energy in
physics.
An important parameter when dealing with the complexity of ground states and
local Hamiltonians is the spectral gap of local Hamiltonians, given by the
difference of the ground and the first excited energy levels,
$\Delta:=\lambda_{1}(H)-\lambda_{0}(H)$. When the spectral gap is constant,
the Hamiltonian is said to be gapped. When it is inverse polynomial, we say
the Hamiltonian is poly-gapped.
What are the implications of a gap for the local hamiltonian problem? A
groundbreaking result by Hastings shows that ground states of 1-D gapped
Hamiltonians have an efficient classical description, as a Matrix-Product-
State (MPS) of polynomial bond dimension [Has07]333A state
$|\psi\rangle\in(\mathbb{C}^{d})^{\otimes n}$ has an MPS representation with
bond dimension $D$ if it can be written as
$|\psi\rangle=\sum_{i_{1},...,i_{n}=1}^{d}\text{tr}(A_{i_{1}}^{[1]}...A_{i_{n}}^{[n]})|i_{1},...,i_{n}\rangle,$
(1) with $A_{i}^{[k]}$ $D\times D$ matrices. Note that only $ndD^{2}$ complex
numbers are needed to specify the state.. Since expectation values of local
observables of an MPS can be calculated in polynomial time in the number of
sites and in its bond dimension (see e.g. [PGVWC06]), Hastings’ result implies
that 1-d constant-gap local hamiltonian (the restriction of the original
problem to 1-D gapped Hamiltonians) belongs to NP.
It has been asked whether such efficient descriptions might exist for the
ground state of 1-D poly gapped Hamiltonians. We show that using Theorem 2,
and some more work, one can deduce that the answer to this question is
negative (under some reasonable complexity assumption). The reasoning is as
follows.
We define the unique local hamiltonian problem to be similar to the local
hamiltonian problem, where the conditions for a “no” instance are the same,
but for a “yes” instance we demand that there exists a $|\psi\rangle$ with
energy below the low-threshold, and all other eigenvalues are above the upper-
threshold. We also define the unique 1-d 2-local hamiltonian in a similar
manner.
It is not difficult to show (by observing that the construction used in
[AGIK07] preserves the uniqueness) that:
###### Lemma 4
unique 1-d 2-local hamiltonian is UQMA-Complete.
Together with Theorem 2, which implies that
$\textsf{QCMA}\stackrel{{\scriptstyle
R}}{{\subseteq}}\textsf{UQCMA}\subseteq\textsf{UQMA}$, we have
###### Theorem 5
unique 1-d 2-local hamiltonian is QCMA-hard, under randomized reductions.
From Theorem 5 we can deduce the following “no-go” corollary for the ground
state of poly-gapped Hamiltonians. Consider any set of states which are (i)
described by $\textsf{poly}(n)$ parameters and (ii) from which one can
efficiently compute expectation values of local observables. Matrix-Product-
States are an example of such a set, and several others have recently been
proposed [APD+06, Vid07, HKH+08]. We can show:
###### Theorem 6
Ground states of 1-D poly gapped local Hamiltonians cannot be approximated to
inverse polynomial accuracy by states satisfying properties (i) and (ii),
unless $\textsf{QCMA}\stackrel{{\scriptstyle R}}{{=}}\textsf{NP}$.
The reason is that “yes” instances of the unique 1-d 2-local hamiltonian are
poly-gapped, and therefore such a description would place unique 1-d 2-local
hamiltonian in NP.
To further analyze the complexity of the local Hamiltonian problem for poly-
gapped Hamiltonians, we introduce a variant of the UQMA class, which we call
poly-gapped QMA (PGQMA), as follows: in both “yes” and “no” instances we
require there is a gap (given by a pre-determined quantity larger than an
inverse polynomial in the input size) from the witness which accept with the
largest probability to all the others. We show that the problem 1-d poly-gap
local hamiltonian, in which the Hamiltonians are promised to be poly-gapped,
is complete for the class. We also present a simple randomized reduction from
any UQMA problem to a PGQMA, which implies
###### Theorem 7
1-d poly-gap local hamiltonian is QCMA-hard, under randomized reductions.
We thus see that, unless $\textsf{BQP}=\textsf{QCMA}$ 444BQP is the class of
problems which can be efficiently solved, with high probability, by a quantum
computer, the determination of the ground energy of poly-gapped 1-D local
Hamiltonians is an intractable problem for quantum computation. Note that this
conclusion cannot be drawn from the previous lower bounds on the complexity of
the problem [AGIK07, SCV08]. Indeed, the results of [AGIK07] concerning
adiabatic quantum computation with a 1-D poly-gapped Hamiltonian indirectly
imply that 1-d poly-gap local hamiltonian is BQP-hard555The construction of
[AGIK07] for adiabatic quantum computation with one-dimensional Hamiltonians
provides a way to encode the outcome of any polynomial quantum computation
into the expectation value of a measurement, in the computational basis, of
the first site of the ground state of a 1-D poly-gapped local hamiltonian,
with a zero ground state energy. By adding a small perturbation to the
Hamiltonian, penalizing the first site when it is not in the zero state, and
with a strength much smaller than the spectral gap, but still inverse
polynomial in the number of sites, we can readily conclude that this
construction shows that 1-d poly-gap local hamiltonian is BQP-hard, while in
[SCV08] the problem was shown to be hard for the class
$\textsf{UP}\cap\textsf{Uco-NP}$ (the intersection of unique NP with unique
co-NP), whose relation with BQP is unknown.
### 1.3 Impossibility Results for UQMA
Finally, we examine the UQMA case. We show that attempting to apply the brute
force analogue of the previous proofs in the case of UQMA, we already fail in
the first (inefficient) component. A new idea seems to be required, if an
extension of the Valiant-Vazirani approach is possible at all for QMA.
To show this we construct a simple family of QMA “yes” instances which we
believe captures the difficulty of the problem.
###### Example 1
Let $C$ be a quantum circuit on $l$ qubits, with the property that there
exists a subspace $V$ of dimension 2, s.t. $\forall|\psi\rangle\in V,\ Pr(C\
accepts\ |\psi\rangle)=1$, and $\forall|\psi\rangle\in V^{\bot},\ Pr(C\
accepts\ |\psi\rangle)=0$.
In the classical case, the analogous example of two solutions is easy to deal
with by choosing a “filter” (hash-function) that screens about half of the
witnesses. The natural quantum analogue to try, is to use a random projection
that will reject half of the space. In proposition 1 we prove that such a
transformation (even if it can be implemented efficiently) does not create an
inverse polynomial gap between the two states in the subspace $V$: with
probability exponentially close to 1, regardless of the dimensionality of the
random projection, all states in $V$ will be accepted with probabilities
exponentially close to each other.
The reason for this is that the projection of every $N$-dimensional vector on
a $d$-dimensional random subspace is concentrated around $\frac{d}{N}$, with a
standard deviation of order $\frac{\sqrt{d}}{N}$, for a sufficiently large
$N$. Therefore, regardless of how we choose $d$, we always get that the gap is
less than $\frac{1}{\sqrt{N}}$, which is exponentially small. Hence, the
behavior of random sets - the filters in the classical setting - is very
different from the behavior of random subspaces, the natural quantum analogue.
One might hope that a more refined measurement would help. In fact [Sen06] has
shown that the two distributions resulting from applying a random von Neumann
measurement on two arbitrary orthogonal states have a constant total variation
distance with all but exponentially small probability. This sounds promising;
Moreover, a similar effect can be achieved efficiently by quantum $t$-designs
as shown by [AE07]. Unfortunately, a constant total variation distance between
two distributions does not imply an efficient method to distinguish between
them; this problem is tightly related to complete problems for the complexity
class SZK, which are not known to have a quantum polynomial time algorithm.
Thus, the problem of whether $\textsf{UQMA}\stackrel{{\scriptstyle
R}}{{=}}\textsf{QMA}$ remains wide open.
### 1.4 Organization of the paper
The structure of the rest of the paper is as follows: in Section 2.1 we
present the definitions. Section 3 reviews the proof of the Valiant-Vazirani
Theorem, while Sections 4 and 5 contain the extension of the theorem to the
classes MA and QCMA, respectively. In section 6 we discuss some alternate
definitions of the class UQMA, and complete problems for this class. We also
show that the two classes are equivalent, under randomized reductions.
Finally, in section 7 we prove impossibility results regarding extending our
results to QMA using similar ideas.
## 2 Definitions
We start by defining a few standard complexity classes which we will consider
throughout the paper. Then we turn to the definition of unique versions of MA,
QCMA, and QMA, which to the best of our knowledge, have not been formalized
before.
### 2.1 Background Definitions
###### Definition 8 (Nondeterministic Polynomial (NP))
A language $L\in\textsf{NP}$ if there exists a Turing Machine (TM) $M$ which
runs in polynomial time in its first argument s.t.:
1. 1.
$x\in L\Rightarrow\exists y\ s.t.\ M(x,y)\ accepts$.
2. 2.
$x\notin L\Rightarrow\forall y$ $M(x,y)\ rejects$.
###### Definition 9 (Unique Nondeterministic Polynomial (UP))
A promise problem $L=(L_{yes},L_{no})\in\textsf{UP}$ if there exists a Turing
Machine (TM) $M$ which is polynomial in its first argument s.t.:
1. 1.
$x\in L_{yes}\Rightarrow\exists y\ s.t.\ M(x,y)\ accepts$ and $\forall
y^{\prime}\neq y\ M(x,y^{\prime})\ rejects$.
2. 2.
$x\in L_{no}\Rightarrow\forall y$ $M(x,y)\ rejects$.
###### Definition 10 (Merlin-Arthur (MA))
A promise problem $L=(L_{yes},L_{no})\in\textsf{MA}$ if there exists a
probabilistic polynomial TM $M$ which is polynomial in its first argument, and
its random bits are denoted by the string $r$, s.t.:
1. 1.
$x\in L_{yes}\Rightarrow\exists y\ s.t.\ Pr_{r}(M(x,y,r)\ accepts)\geq 2/3$.
2. 2.
$x\in L_{no}\Rightarrow\forall y$ $Pr_{r}(M(x,y,r)\ accepts)\leq 1/3$.
###### Definition 11 (Quantum Classical Merlin-Arthur (QCMA))
A promise problem $L=(L_{yes},L_{no})\in\textsf{QCMA}$ if there exists a
polynomial quantum circuit $U_{x}$ which can be computed in $poly(|x|)$ time,
having $l(x)$ qubits as input and requiring $m(x)$ ancilla qubits initialized
to $|0^{m}\rangle$, such that
1. 1.
$x\in L_{yes}\Rightarrow\exists y\ s.t.\
\lVert\Pi_{1}U_{x}(|y\rangle\otimes|0^{m}\rangle)\rVert^{2}\geq 2/3$.
2. 2.
$x\in L_{no}\Rightarrow\forall y\
\lVert\Pi_{1}U_{x}(|y\rangle\otimes|0^{m}\rangle)\rVert^{2}\leq 1/3$.
$\Pi_{1}$ is the projection onto $|1\rangle$ in the first qubit, i.e.
$\Pi_{1}:=|1\rangle{\langle 1|}\otimes I_{l+m-1}$. We write $l=l(x)$ and
$m=m(x)$ when $x$ can be understood from the context.
###### Definition 12 (Quantum Merlin-Arthur (QMA))
A promise problem $L=(L_{yes},L_{no})\in\textsf{QMA}$ if there exists a
polynomial quantum circuit $U_{x}$ which can be computed in $poly(|x|)$ time,
having $l(x)$ qubits as input and requiring $m(x)$ ancilla qubits initialized
to $|0^{m}\rangle$, s.t.
1. 1.
$x\in L_{yes}\Rightarrow\exists|\psi\rangle\ s.t.\
\lVert\Pi_{1}U_{x}(|\psi\rangle\otimes|0^{m}\rangle)\rVert^{2}\geq 2/3$.
2. 2.
$x\in L_{no}\Rightarrow\forall|\psi\rangle$
$\lVert\Pi_{1}U_{x}(|\psi\rangle\otimes|0^{m}\rangle)\rVert^{2}\leq 1/3$.
$\Pi_{1}$ is the projection onto $|1\rangle$ in the first qubit.
### 2.2 New Definitions
We now describe the analogue unique versions for the classes MA and QCMA and
QMA.
###### Definition 13 (Unique Merlin-Arthur (UMA))
A promise problem $L=(L_{yes},L_{no})\in\textsf{UMA}$ if there exists a
probabilistic TM $M$ which is polynomial in its first argument s.t.:
1. 1.
$x\in L_{yes}\Rightarrow\exists y\ s.t.\ Pr_{r}(M(x,y,r)\ accepts)\geq 2/3$
and $\forall y^{\prime}\neq y$, $Pr_{r}(M(x,y^{\prime},r)\leq 1/3$.
2. 2.
$x\in L_{no}\Rightarrow\forall y$ $Pr_{r}(M(x,y,r)\ accepts)\leq 1/3$.
###### Definition 14 (Unique Quantum Classical Merlin-Arthur (UQCMA))
A promise problem $L=(L_{yes},L_{no})\in\textsf{UQCMA}$ if there exists a
polynomial quantum circuit $U_{x}$ which can be computed in $poly(|x|)$ time,
having $l(x)$ qubits as input and requiring $m(x)$ ancilla qubits initialized
to $|0^{m}\rangle$, such that
1. 1.
$x\in L_{yes}\Rightarrow\exists y\ s.t.\
\lVert\Pi_{1}U_{x}(|y\rangle\otimes|0^{m}\rangle)\rVert^{2}\geq 2/3$ and
$\forall y^{\prime}\neq y$,
$\lVert\Pi_{1}U_{x}(|y^{\prime}\rangle\otimes|0^{m}\rangle)\rVert^{2}\leq 1/3$
2. 2.
$x\in L_{no}\Rightarrow\forall y\
\lVert\Pi_{1}U_{x}(|y\rangle\otimes|0^{m}\rangle)\rVert^{2}\leq 1/3$.
$\Pi_{1}$ is the projection onto $|1\rangle$ in the first qubit.
###### Definition 15 (Unique Quantum Merlin-Arthur (UQMA))
A promise problem $L=(L_{yes},L_{no})\in\textsf{UQMA}$ if there exists a
polynomial quantum circuit $U_{x}$ which can be computed in $poly(|x|)$ time,
having $l(x)$ qubits as input and requiring $m(x)$ ancilla qubits initialized
to $|0^{m}\rangle$, s.t.
1. 1.
$x\in
L_{yes}\Rightarrow\exists|\psi\rangle\lVert\Pi_{1}U_{x}(|\psi\rangle\otimes|0^{m}\rangle)\rVert^{2}\geq
2/3\hskip 2.84544pt\text{and}\hskip
2.84544pt\forall|\phi\rangle\bot|\psi\rangle,\hskip
1.42271pt\lVert\Pi_{1}U_{x}(|\phi\rangle\otimes|0^{m}\rangle)\rVert^{2}\leq
1/3$
2. 2.
$x\in L_{no}\Rightarrow\forall|\psi\rangle$
$\lVert\Pi_{1}U_{x}(|\psi\rangle\otimes|0^{m}\rangle)\rVert^{2}\leq 1/3$.
## 3 The Valiant-Vazirani Proof Revisited
In this section, we review the results of [VV85]. We divide the proof into
three components, so that we can better understand which components of the
original construction fail in the probabilistic and quantum setting. The main
result proved by Valiant and Vazirani can be stated as follows:
###### Theorem 16
[VV85] If
$\textsf{UP}\subseteq\textsf{RP}\Rightarrow\textsf{NP}\subseteq\textsf{RP}$.
The standard proof of the theorem works with the well known NP-complete
problem sat. We will not use it, as there is no simple variant of sat which is
complete for the classes MA and QCMA.
###### Definition 17 (Trivial NP Problem (tnpp))
The words in $L$ are tuples, $\langle V,x,l,t\rangle$, where V is a
description of a deterministic Turing machine, x is a string of length n, and
$l,t\in\mathbb{N}$, given in unary.
$\langle V,x,l,t\rangle\in L$ if there exists a $y$ with $|y|=l$ s.t. $V(x,y)$
accepts in $t$ steps.
It can easily be seen that tnpp is NP-Complete. The following promise problem
is a “unique” version of tnpp.
###### Definition 18 (Unique-NP Promise Problem (unppp))
The promise problem is $L=(L_{yes},L_{no})$. The words in $L$ are tuples,
$\langle V,x,l,t\rangle$, where V is a description of a deterministic Turing
machine, x is a string of length n, and $l,t\in\mathbb{N}$, given in unary.
$\langle V,x,l,t\rangle\in L_{yes}$ if there exists exactly one string $y$
s.t. $|y|=l$ and $V(x,y)$ accepts in $t$ steps. $\langle V,x,l,t\rangle\in
L_{no}$ if for all strings $y$ s.t. $|y|=t$, $V(x,y)$ does not accept in $t$
steps.
### 3.1 Proof Sketch
We begin with an instance $\hat{I}$ and a language $L\in NP$, and we should
decide if $\hat{I}\in L$. The first step is to use the completeness of $TNPP$
to find an instance $I=\langle V,x,l,t\rangle$ with the property $\hat{I}\in
L\iff I\in TNPP$.
There are three main components in the proof, which we shall, now, explain.
#### Component 1: The right random “filter” for the right size
Let $W$ be the set of accepting witnesses: $W:=\\{y:\ |y|=l\ and\ V(x,y)\
accepts\ in\ t\ steps\\}$, and let $|W|=w$. Notice that $I\in TNPP\iff w\neq
0$.
###### Definition 19 (R-restriction)
Let $R$ be a set of strings, each one of them of size $l$, with the property
that there is an algorithm that answers whether $y\in R$ in exactly $T$ time
steps. Given a Turing machine $V$, we call the following Turing machines the
$R$-restriction of V, and denote it by $V_{R}$:
1. 1.
If $y\notin R$, Reject. Otherwise, Continue.
2. 2.
Run $V$ on $(x,y)$.
We see the $R$-restriction as a filter added to the original problem, because
the new machine accepts only accepting witnesses of the original machine,
which belong to the set $R$.
Let us denote by $I^{\prime}$ the instance $\langle V_{R},x,l,t+T\rangle$.
Component 1 takes the filter $R$ to be a random set, where each string in
$\\{0,1\\}^{l}$ is chosen independently with probability $w^{-1}$. Notice that
the Turing machine $V_{R}$ might not have a short description, because in
order to decide whether $y\in R$, all the elements of $R$ should somehow be
“hard-wired” to the machine. If $|R|$ is exponential in $l$, then by using
Kolmogorov Theory arguments[CTWI06], there is no short description for such a
circuit, therefore the description of $V_{R}$ will not be short. Therefore,
the mapping between $I$ to $I^{\prime}$ is not efficient. This drawback will
be circumvent in component 3.
We claim that $I^{\prime}$ will be in $UNPPP_{yes}$ with probability
$\Omega(1)$. Let $W^{\prime}=\\{y:\ |y|=l\ \text{and}\ V_{R}(x,y)\
\text{accepts in}\ t+T\ \text{steps}\\}$. Defining
$W=\\{w_{1},...,w_{|W|}\\}$,
$\displaystyle Pr(I^{\prime}\in UNPPP_{yes})$ $\displaystyle=$ $\displaystyle
Pr(|W^{\prime}|=1)$ (2) $\displaystyle=$ $\displaystyle Pr(|W\cap R|=1)$
$\displaystyle=$ $\displaystyle Pr\left(\bigcup_{i=1}^{w}(w_{i}\in
R\cap_{j\neq i}w_{j}\notin R)\right)$ $\displaystyle=$ $\displaystyle
w\frac{1}{w}(1-\frac{1}{w})^{w-1}$ $\displaystyle\geq$ $\displaystyle 1/e.$
The first equality follows from $I^{\prime}\in UNPPP\iff w^{\prime}=1$ and the
second from $W^{\prime}=W\cap R$. The third is a direct consequence of the
definition of $w_{i}$. The fourth stems from the facts that the events in the
line above are all disjoint, and using the definition of the set $R$.
Therefore, querying the oracle with $\langle
V^{\prime},x,l,t+t^{\prime}\rangle$ results in a “yes” with probability of at
least $\frac{1}{e}$.
Using this idea, we create $2^{l}$ instances, $I_{1},...,I_{2^{l}}$, one for
every possible value of $w$: $I_{j}=\langle V_{j},x,l,t+t^{\prime}\rangle$. We
claim:
###### Lemma 20
(Completeness) If $I\in TNPP$, then there exists a $j$ for which, with
probability $\Omega(1)$ over the choice of $R$, $I_{j}\in UNPPP_{yes}$.
(Soundness) If $I\notin TNPP$, then all the $I_{j}$ are in $UNPPP_{no}$.
Proof: Completeness: Follows from the previous argument: one of the $I_{j}$’s
is $I_{w}$. $I_{w}\in UNPPP_{yes}$ with probability of at least $1/e$.
Soundness: $I\notin TNPP\Rightarrow W=\emptyset$. As $W_{j}=W\cap R_{j}$,
$W_{j}=\emptyset$, and therefore $I_{j}\in UNPPP_{no}$.
Our algorithm consists of querying $UNPPP$ with $I_{1},...,I_{2^{l}}$. If one
of the results is yes, we accept. The completeness asserts that for a “yes”
instance, we accept with constant probability. The soundness asserts that we
always reject in “no” instances.
#### Component 2: Approximated “filter” also works
The second component concerns the fact that we do not know the value $w$ and,
therefore, in order to use the algorithm given in component 1, we need
exponentially many queries to the $UNPPP$ oracle. The key to the solution is
to realize that being wrong about the size of $w$ by a constant factor, only
changes the probability of having a unique solution by another constant
factor.
More explicitly, we transform our instance $I$ into a polynomial number of
random instances: $I_{1},I_{2},...,I_{l}$. These instances are formed by
choosing random sets $R_{k}$ again; but now, each element is taken with
probability $\frac{1}{2^{k}}$.
A similar statement to Lemma 20 also holds here. To analyze the completeness
of the protocol, we notice that for some $k$, $2^{k}\leq w\leq 2^{k+1}$.
Hence, for such $k$,
$\displaystyle Pr(I_{k}\in UNPPP_{yes})$ $\displaystyle=$ $\displaystyle
Pr(|W_{k}|=1)$ $\displaystyle=$ $\displaystyle Pr(|W\cap R_{k}|=1)$
$\displaystyle=$ $\displaystyle Pr\left(\bigcup_{i=1}^{w}(y_{i}\in
W\cap_{j\neq i}y_{j}\notin W)\right)$ $\displaystyle=$ $\displaystyle
w\frac{1}{2^{k}}(1-\frac{1}{2^{k}})^{w-1}$ $\displaystyle\geq$
$\displaystyle(1-\frac{1}{2^{k}})^{2^{k+1}-1}\geq e^{-2}.$
Therefore, when asking the oracle $l-1$ queries, at least one of the answers
will be “yes”, with probability of at least $1/e^{2}$. The soundness analysis
uses the same argument as in component 1.
#### Component 3: Approximated pseudo random filter is just as good
The third component deals with the inefficiency of randomness: a random and
exponential large set $R$ cannot be determined by a polynomial description.
The solution is to replace the randomness by a suitable notion of pseudo-
randomness. In this case, the pseudo-random objects of interest are pairwise
independent universal hash functions [AB09].
###### Definition 21 (pairwise independent hash functions)
A family of functions $\mathbb{H}_{n,m}$ where each $h\in\mathbb{H}$,
$h:\\{0,1\\}^{n}\rightarrow\\{0,1,\\}^{m}$, is called a pairwise independent
universal family of hash-functions if:
1. 1.
$\forall y_{1}\neq y_{2}\in A,\hskip 2.84544pt\forall a,b\in B,\quad
Pr_{h\sim_{\mathcal{U}}\mathbb{H}}(h(y_{1})=a\hskip 2.84544pt\text{and}\hskip
2.84544pth(y_{2})=b)=\frac{1}{2^{2m}}$
2. 2.
There exists a Turing Machine $PRINT$-$H$ s.t. for every $n,m\in\mathbb{N}$
and $j\in\mathbb{H}_{n,m}$, $PRINT$-$H(n,m,j)$ prints a description of another
Turing machine, which computes $h_{j}\in\mathbb{H}_{n,m}$. By abuse of
notation, we also denote the Turing machine which computes $h_{j}$ by $h_{j}$.
The printing is done in $poly(n,m)$ time.
3. 3.
The running time of each $h\in\mathbb{H}_{n,m}$ is bounded by some $poly(n,m)$
time.
Note that this probability is the same as if the map $h$ was random, although
$h$ has a short description (unlike a random function which has no compact
description).
Instead of choosing $R_{k}$ to be a random set, we pick a random universal
hash function $h_{k}$ from the set $H_{l,k+2}$; The set $R_{k}$ is
$h_{k}^{-1}(0)=\\{y|h_{k}(y)=0\\}$. Evaluating $h_{k}(y)$ is polynomial in
$l$, and therefore, step 1 of $V_{k}$ takes only polynomial time. To conclude,
our algorithm is described in Alg. 1.
Input: The tuple $\langle V,x,l,t\rangle$.
Output: if $x\in TNPP$ accept with some constant probability, if $x\notin
TNPP$ reject (with probability 1)
foreach _ $k\in[l]$_ do Sample a hash-function uniformly at random
$h_{k}\sim_{\mathcal{U}}\mathbb{H}_{l,k+2}$ and let $R_{k}=h_{k}^{-1}(0)$
Denote by $V_{k}$ the $R_{k}$-restriction of $V$.
Query the $UNPPP$ oracle with $I_{k}=\langle V_{k},x,l,t+T_{l,k+2}\rangle$,
and put the result in $r_{k}$. 666We will denote by $T_{a,b}$ the running time
of $h(y)$ where $h\in\mathbb{H}_{a,b}$. We need the reasonable assumption that
the running time is the same for all $h$’s and $y$’s and that it is an easy to
compute function. We changed the time $t$ to be $t+T_{l,i+2}$, because the
machine $V_{k}(x,y)$ needs to do one evaluation of the hash function, compared
to the machine $V$. end if _ $\exists k\ s.t.\ r_{k}=1$ _ then accept else
reject end
Algorithm 1 tnpp solver, which uses polynomially many queries to unppp
2
2
It hence suffices to prove lemma 20 in order to show
$\textsf{UP}\subseteq\textsf{RP}\Rightarrow\textsf{NP}\subseteq\textsf{RP}$,
because then Alg. 1 is in RP. First, we need to show that the algorithm takes
polynomial time. The only suspect is step 1. The preparation of the
description $V_{k}$ takes polynomial time, as in the definition of hash
function (definition 21).
Soundness: In the case that $I\notin TNPP$, then by the soundness of lemma 20,
all the $r_{k}$’s in step 1 are false, and, therefore, in step 1 the condition
does not hold, so we always reject.
Completeness: By combining the assumption that $UNPPP$ is in RP, and the
completeness of lemma 20, we have that if $I\in TNPP$, then with probability
$\Omega(1)$ over the choice of $h_{k}$, $I_{k}\in UNPPP_{yes}$, and therefore
for that $k$ the query in step 1 will return “accept” with probability $2/3$.
Therefore, the overall probability of accepting is at least
$\frac{2}{3}\Omega(1)=\Omega(1)$.
Proof of Lemma 20: Soundness: Same argument as before.
Completeness: We make use of the following lemma:
###### Lemma 22
Let $W\subset\\{0,1\\}^{n}$ of size $w$, such that $2^{k}\leq w\leq 2^{k+1}$,
and let h be a random universal hash function from the set
$\mathbb{H}_{l,k+2}$, which is a set of functions from $\\{0,1\\}^{l}$ to
$\\{0,1\\}^{k+2}$. Then,
$Pr\left(|h^{-1}(0)\cap W|=1\right)\geq 1/8.$
.
We prove this lemma in Appendix A. Note that $I_{k}=\langle
V_{k},x,y,l,t+T_{l,k+2}\rangle\in UNPPP_{yes}$ is equivalent to $|W_{k}|=1$.
We have that $W_{k}=W\cap R_{k}=W\cap h_{k}^{-1}(0)$ and Lemma 22 tells us
that $|h_{k}^{-1}(0)\cap W|=1$ with probability at least $1/8$ over the choice
of $h$.
The fact that the description of $V_{k}$ is efficient makes sure that step 1
of Alg. 1 only takes polynomial time. All the other steps can be easily seen
to take polynomial time as well.
## 4 Valiant-Vazirani Extended to the Class MA
In this section we prove Theorem 1, which can also be formulated as:
###### Theorem 23
$\textsf{UMA}\in\textsf{RP}\Longrightarrow\textsf{MA}\in\textsf{RP}$.
###### Definition 24 (Trivial MA Promise Problem (tmapp))
$\textsc{tmapp}=(L_{yes},L_{no})$. The words in tmapp are tuples, $\langle
V,x,p_{1},p_{2},l,t\rangle$, where V is a description of a probabilistic
Turing machine, x is a string of length n, and $0\leq p_{1}<p_{2}\leq 1$,
where $p_{2}-p_{1}\geq 1/poly(n)$, and $l,t\in\mathbb{N}$, given in unary.
$\langle V,x,p_{1},p_{2},l,t\rangle\in L_{yes}$ if there exists a string $y$
s.t. $|y|=l$ and $Pr(V(x,y)\ accepts\ in\ t\ steps)\in``yes-
interval^{\prime\prime}$.
$\langle V,x,p_{1},p_{2},l,t\rangle\in L_{no}$ if for all strings $y$ of
length $l$, $Pr(V(x,y)\ accepts\ in\ t\ steps)\in``no-
interval^{\prime\prime}$.
It can be easily seen that tmapp is MA-Complete.
We start with a language $L\in\textsf{MA}$ and an instance $I^{\prime}$ and we
should decide whether $I^{\prime}\in L$ or not. The first step, as was done in
the $NP$ case, is to use the completeness of tmapp, and reduce it to the
question whether
$\hat{I}=\langle\hat{V},x,p_{1},p_{2},l,t\rangle\in\textsc{tmapp}_{yes}$ or
$\hat{I}\in\textsc{tmapp}_{no}$.
$1$$p_{2}$$p_{1}$$1$
$1$$p_{2}$$p_{1}$$0$
Figure 1: Typical “no” and “yes” instances
The y-axis is probability. The ellipses are all the $2^{l}$ different
witnesses of a specific instance. The red lines outline the
boundaries,$[p_{1},p_{2}]$ \- the maximal acceptance probability of a MA
instance are promised not to be in that interval. The left one is a “no”
instance, the maximal probability of acceptance is less than $p_{1}$. The
right one is a “yes” instance, because the maximal probability of acceptance
is greater than $p_{2}$.
$1$$p_{2}$$p_{1}$$0$
Figure 2: A “unique yes” instance
There is exactly one witness which is accepted with probability greater than
$p_{2}$, and all others are accepted with probability smaller than $p_{1}$.
Hence, our goal is to create a transformation which takes a
$\textsc{tmapp}_{yes}$ instance (right side of Fig. 1) to a
$\textsc{umapp}_{yes}$ instance (Fig. 2) with constant probability, and a
$\textsc{tmapp}_{no}$ instance to a $\textsc{umapp}_{no}$ instance (left side
of Fig. 1) with probability 1. We divide the potential witnesses into 3
groups, by their probability of acceptance:
$\displaystyle Y_{no}=\\{y|\ |y|=l\ and\ Pr(\hat{V}(x,y)\ accepts\ in\ t\
steps)\in``no-interval^{\prime\prime}\\}$ $\displaystyle Y_{gap}=\\{y|\ |y|=l\
and\ Pr(\hat{V}(x,y)\ accepts\ in\ t\ steps)\in``gap-
interval^{\prime\prime}\\}$ $\displaystyle Y_{yes}=\\{y|\ |y|=l\ and\
Pr(\hat{V}(x,y)\ accepts\ in\ t\ steps)\in``yes-interval^{\prime\prime}\\}$
(3)
Let us look at the $R$-restriction of $V$, $V_{R}$, where $R$ is a random set
and each element in $[2^{l}]$ is taken with some probability $p$. We denote it
by $I^{\prime}=\langle V_{R},x,p_{1},p_{2},l,t+t^{\prime}\rangle$, where
$t^{\prime}$ is the time taken for the machine $V_{R}$ to make its first step.
Define $Y^{\prime}_{yes},Y^{\prime}_{gap},Y^{\prime}_{no}$ for $I^{\prime}$,
as was done for $\hat{I}$ in Equation 3. For every $y$ of length $l$, denote
by $f(y)=Pr(V(x,y)\ accepts\ in\ t\ steps)$, and
$f^{\prime}(y)=Pr(V^{\prime}(x,y)\ accepts\ in\ t+t^{\prime}\ steps)$.
###### Observation 25
$f^{\prime}(y)=\begin{cases}0&\text{if }y\notin R\\\ f(y)&\text{if }y\in
R\end{cases}$
Therefore, $Y^{\prime}_{yes}=Y_{yes}\cap R$ and $Y^{\prime}_{gap}=Y_{gap}\cap
R$.
Using the same method as in the NP case clearly fails, as we explicitly show
in the following section.
### 4.1 Problems with the first component
We present an instance that shows the failure of implementing component 1 in
the probabilistic case. The example is a $I^{problematic}=\langle
V^{problematic},x,p_{1},p_{2},l,t\rangle\in\textsc{tmapp}_{yes}$ instance
which can be seen in Fig.3, with the property that
$|Y^{problematic}_{yes}|=2,|Y^{problematic}_{gap}|=2^{l}-2$ and
$|Y^{problematic}_{no}|=0$.
$0$$1$$p_{2}$$p_{1}$
Figure 3: A problematic ma-instance: it has numerous witnesses with
probability inside the “gap-interval” and very few in the “yes-interval”.
Because the size of the set $Y_{gap}$ is exponentially bigger than $Y_{yes}$,
we cannot “filter” - by using the random set $R$ \- one element from $Y_{yes}$
and none from $Y_{gap}$ with non-negligible probability: Suppose we pick the
size of $R$ by the set $W_{0}$, so each element is chosen with probability
$1/2$. With probability $\Omega(1)$ exactly one element will be chosen from
$W_{0}$, but about half of the elements of $W_{1}$ will also be chosen.
Therefore, it fails to hold the second property of a $\textsc{umapp}_{yes}$
instance. If we pick elements in $R$ by the size $W_{1}$, which means that
each element is picked with probability $\frac{1}{2^{l}-2}$ then with
probability $(1-\frac{1}{2^{l}-2})^{2}$ (which is exponentially close to one),
no element will be picked from $W_{0}$, therefore it fails to hold the first
property of a $\textsc{umapp}_{yes}$ instance.
### 4.2 the fourth component
The missing property in the example of section 4.1 is formalized in the next
definition:
###### Definition 26 (“lightweight-gap” instance)
An instance $I=\langle V,x,p_{1},p_{2},l,t\rangle$ is a “lightweight-gap”
$\textsc{tmapp}_{yes}$ instance if it is a $\textsc{tmapp}_{yes}$ instance,
and $|Y_{gap}|\leq 3|Y_{yes}|$.
Lemma 30 explains how this kind of instances does not have the problem that
was shown in section 4.1. But first we will see how to create a very simple
transformation which takes a general $\textsc{tmapp}_{yes}$ instance to a
“lightweight-gap” $\textsc{tmapp}_{yes}$ instance:
###### Lemma 27
Let $\hat{I}$ be a tmapp instance. There exists an efficient transformation
that maps $\hat{I}$ to several instances $I_{1},...,I_{l-2}$ with the
following properties:
* •
If $\hat{I}\in\textsc{tmapp}_{yes}$ then $\exists k\ s.t.\ I_{k}$ is a
“lightweight-gap” $\textsc{tmapp}_{yes}$ instance.
* •
If $\hat{I}\in\textsc{tmapp}_{no}$ then $\forall k\
I_{k}\in\textsc{tmapp}_{no}$ instance.
Proof: The transformation is the following. We start by applying
amplification: We can reduce the instance
$\hat{I}=\langle\hat{V},x,p_{1},p_{2},l,t\rangle$ to $I=\langle
V,x,\frac{1}{l},1-\frac{1}{l},l,t\rangle$. This is done by using standard
error reduction techniques.
###### Observation 28
Let $I_{1}=\langle V,x,p_{1},p_{2},l,t\rangle$ and let $I_{2}=\langle
V,x,q_{1},q_{2},l,t\rangle$, where $[q_{1},q_{2}]\subset[p_{1},p_{2}]$.
* •
$I_{1}\in\textsc{tmapp}_{yes}\Rightarrow I_{2}\in\textsc{tmapp}_{yes}$.
* •
$I_{1}\in\textsc{tmapp}_{no}\Rightarrow I_{2}\in\textsc{tmapp}_{no}$.
The observation follows immediately from the definitions of tmapp.
The second step of the transformation is the following: we take the instance
$I=\langle V,x,\frac{1}{l},1-\frac{1}{l},l,t\rangle$ and create $l-2$
instance,$I_{1},...,I_{l-2}$, where $I_{j}=\langle
V,x,\frac{j}{l},\frac{j+1}{l},l,t\rangle$. By observation 28, we know that if
$I\in\textsc{tmapp}_{yes}\Rightarrow\forall k\ I_{k}\in\textsc{tmapp}_{yes}$,
and that $I\in\textsc{tmapp}_{no}\Rightarrow\forall k\
I_{k}\in\textsc{tmapp}_{no}$.
But in the case of a “yes” instance, the lemma demands a “lightweight-gap”
$\textsc{tmapp}_{yes}$ instance. This is achieved using the following
observation:
###### Observation 29 (Existence of lightweight range)
We define $l$ ranges: $r_{j}=[\frac{j}{l},\frac{j+1}{l}),\ 1\leq j\leq l-1$.
We define
$Y_{j}=\\{y|\ |y|=l\ and\ Pr(\hat{V}(x,y)\ accepts\ in\ t\ steps)\in r_{j}\\}$
If $I=\langle
V,x,\frac{1}{l},1-\frac{1}{l},l,t\rangle\in\textsc{tmapp}_{yes}$, then there
exists a $j$ s.t. $|Y_{j}|<3|Y_{j+1}|$.
$p_{1}=\frac{1}{l^{2}}$$1$$0$$1-\frac{2}{l}$$1-\frac{2}{l}$$1-\frac{3}{l}$$\frac{2}{l}$a
lightweight-interval$\\}$$p_{2}=1-\frac{1}{l^{2}}$
Figure 4: A yes-instance, with its lightweight range.
Proof: First, notice that $|Y_{l}|\geq 1$, due to the fact that
$I\in\textsc{tmapp}_{yes}$. Now, assume that the inequality does not hold for
every j, i.e. $|Y_{j}|\geq 3|Y_{j+1}|$. Then, $|Y_{1}|\geq 3^{l-1}>2^{l}$. The
total number of the witnesses is $2^{l}$. Contradiction.
All we need to notice to prove lemma 27 is that if $|Y_{j}|<3|Y_{j-1}|$, then
$I_{j}$ is a “lightweight-gap” $\textsc{tmapp}_{yes}$ instance. Observation 29
asserts that such a $j$ indeed exists.
Until now we have shown how to transform the instance to a “lightweight-gap”.
The following lemma proves that component 1 works for this kind of instances:
###### Lemma 30
Suppose $I=\langle V,x,p_{1},p_{2},l,t\rangle$ is a lightweight-gap
$\textsc{tmapp}_{yes}$ instance. Define $I^{\prime}=\langle
V_{R},x,p_{1},p_{2},l,t+t^{\prime}\rangle$, where $V_{R}$ is the R-restriction
of $V$ where each element in $R$ is taken with probability
$p=\frac{1}{|Y_{gap}|+|Y_{yes}|}$. Then, with probability $\Omega(1)$ (over
the choice of R), $I^{\prime}$ is a $\textsc{umapp}_{yes}$ instance.
Proof:
As was shown in component 1, with probability $\Omega(1)$ exactly one witness
will be picked from the set $Y_{yes}\cup Y_{gap}$. The probability that the
instance is from the set $Y_{yes}$ is proportional to its size. Therefore
$Pr(I^{\prime}\in\textsc{umapp}_{yes})=\Omega(1)\frac{|Y_{yes}|}{|Y_{gap}|+|Y_{yes}|}\geq\frac{1}{4}\Omega(1)$.
Component 2 works without any change in the probabilistic setting: a constant
approximation of the size $|Y_{yes}|$ is sufficient. In order to adapt
component 3 to the present case, we need a simple variant of lemma 22:
###### Lemma 31
Let $S\subset\\{0,1\\}^{l}$ of size $b$, such that $2^{k}\leq b\leq 2^{k+1}$,
$S_{1}\subset S$ of size $a$, and $S_{2}=S\setminus S_{1}$. Let h be picked
randomly from the set $\mathbb{H}_{n,k+2}$. Then,
$Pr(|h^{-1}(0)\bigcap S_{1}|=1\wedge|h^{-1}(0)\bigcap
S_{2}|=0]\geq\frac{a}{8b}$
.
The proof is given in Appendix A. We apply lemma 31 to our construction by
setting $S_{1}=Y_{yes},\ S_{2}=Y_{gap},\ S=S_{1}\cap S_{2}$.
### 4.3 Putting It All Together
Assuming $\textsc{umapp}\in\textsf{RP}$, then algorithm 2, which solves tmapp,
is also in RP.
Input: $I=\langle V,x,1-\frac{1}{l},\frac{1}{l},l,t\rangle$, where V is a
description of a probabilistic Turing machine, x is a string of length n, and
$0\leq p_{1}\leq p_{2}\leq 1$, where $p_{2}-p_{1}\geq 1/poly(n)$, and
$l,t\in\mathbb{N}$, given in unary.
Output: if $x\in\textsc{tmapp}_{yes}$ accept with some constant probability,
if $x\in\textsc{tmapp}_{no}$ reject (with probability 1)
foreach _ $k\in[l-2]$_ do Define $I_{k}=\langle
V,x,\frac{k}{l},\frac{k+1}{l},l,t\rangle$.
foreach _ $b\in[l]$_ do Sample a hash-function in random
$h_{b}\in\mathbb{H}_{n,b+2}$. Denote by $R_{b}=h_{b}^{-1}(0)$
Create the $R_{b}$-restriction of $V$, $V_{b}$:
if _$h_{b}(y)\neq 0$_ then return _“no”_ else $result\leftarrow$ Run
(simulate) $V(x,y)$
return _result_
end
Define $I_{k,b}=\langle
V_{b},x,\frac{k}{l},\frac{k+1}{l},l,t+T_{l,b+2}\rangle$.777We will denote by
$T_{a,b}$ the running time of $h(y)$ where $h\in\mathbb{H}_{a,b}$. We need the
reasonable assumption that the running time is the same for all $h$’s and
$y$’s and that it is an easy to compute function. We have changed the time $t$
to be $t+T_{l,i+2}$ because the machine $V_{k}(x,y)$ needs to do one
evaluation of the hash function, compared to the machine $V$, and therefore we
need the additional time.
Query the umapp oracle with $I_{k,b}$ and put the result in $r_{k,b}$ .
end end if _ $\exists k,b\ s.t.\ r_{k,b}=1$ _ then accept else reject end
Algorithm 2 tmapp solver, which uses polynomially many queries to umapp
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That the algorithm takes polynomial time can be seen in the same manner as the
NP case. For the soundness, we have that $\forall k,b\
I\in\textsc{tmapp}_{no}\Rightarrow I_{k,b}\in\textsc{tmapp}_{no}$, by using
observation 28 and observation 25. Because a $\textsc{tmapp}_{no}$ instances
is also a $\textsc{umapp}_{no}$ instance, step 2 will always output 0, and
therefore in step 2 we will always reject. Finally, let us analyze the
completeness of the protocol. We know that $I\in\textsc{tmapp}_{yes}$.
According to lemma 27, for some $k$, $I_{k}$ is a “lightweight-gap”
$\textsc{tmapp}_{yes}$ instance. Define $Y^{k}_{yes},Y^{k}_{gap}$ for $I_{k}$
in similar manner to Equation (3). According to lemma 31, with
$S_{1}=Y^{k}_{yes},\ S_{2}=Y^{k}_{gap},\ S=S_{1}\cap S_{2}$, we have that
$I_{k,b}\in\textsc{umapp}_{yes}$, for a $b$ such that $2^{b}\leq Y_{k}\leq
2^{b+1}$, with probability $\frac{1}{24}$.
## 5 Valiant-Vazirani Extended to the class QCMA
The proof of Theorem 2 is identical to the MA case.
Theorem 2 can also be formulated as:
###### Theorem 32
$\textsf{UQCMA}\in\textsf{RP}\Longrightarrow\textsf{QCMA}\in\textsf{RP}$.
We define the QCMA analogue of tmapp and umapp to be:
###### Definition 33 (tqcmapp)
$\textsc{tqcmapp}=(L_{yes},L_{no})$. The words in tqcmapp are tuples, $\langle
U,p_{1},p_{2}\rangle$ where $U$ is a description of a quantum circuit, with
input of size $l$, s.t.:
1. 1.
$\langle U,p_{1},p_{2}\rangle\in L_{yes}$ if there exists a string $y$ of
length l, s.t. $Pr(U\ accepts\ |y\rangle)\in``yes-interval"$.
2. 2.
$\langle U,p_{1},p_{2}\rangle\in L_{no}$ if for all strings $y$ of length l
$Pr(U\ accepts\ |y\rangle)\in``no-interval"$.
###### Definition 34 (uqcmapp)
$\textsc{uqcmapp}=(L_{yes},L_{no})$. The words in uqcmapp are tuples, $\langle
U,p_{1},p_{2}\rangle$ where $U$ is a description of a quantum circuit, with
input of size $l$, s.t.:
1. 1.
$\langle U,p_{1},p_{2}\rangle\in L_{yes}$ if there exists a string $y$ of
length l, s.t. $Pr(U\ accepts\ |y\rangle)\in``yes-interval"$ and $\forall
y^{\prime}\neq y\ Pr(U\ accepts\ |y\rangle)\in``no-interval"$.
2. 2.
$\langle U,p_{1},p_{2}\rangle\in L_{no}$ if for all strings $y$ of length l
$Pr(U\ accepts\ |y\rangle)\in``no-interval"$.
All the steps realized previously can also be done here: We begin with a
language $L\in\textsf{QCMA}$ and an instance $I^{\prime}$, and we need to
decide whether $I^{\prime}\in L$ or not. We use the completeness of tqcmapp to
reduce it to the question whether
$\hat{I}=\langle\hat{U},p_{1},p_{2}\rangle\in L$ or not. Notice that in order
to use component 4, and apply lemma 27, we need to perform gap amplification,
i.e. to transform $\langle\hat{U},p_{1},p_{2}\rangle$ to $\langle
U,\frac{1}{l},1-\frac{1}{l}\rangle$. This is not a problem, because standard
amplification works also for QCMA: Given $y$ we can create several copies of
it without worrying about the “no cloning theorem”, by measuring $y$ in the
standard basis, without disturbing $|y\rangle$.
The tqcmapp solver appears in Alg. 3.
Input: $I=\langle U,\frac{1}{l},1-\frac{1}{l}\rangle$, where $U$ is a
description of a Quantum Circuit, and $0\leq p_{1}\leq p_{2}\leq 1$, where
$p_{2}-p_{1}\geq 1/poly(n)$
Output: if $x\in\textsc{tmapp}_{yes}$ accept with some constant probability,
if $x\in\textsc{tmapp}_{no}$ reject (with probability 1)
foreach _ $k\in[l-2]$_ do Define $I_{k}=\langle
U,\frac{k}{l},\frac{k+1}{l}\rangle$.
foreach _ $b\in[l]$_ do Sample a hash-function in random
$h_{b}\in\mathbb{H}_{n,b+2}$. Denote by $R_{b}=h_{b}^{-1}(0)$
Create the $R_{b}$-restriction of $U$, $U_{b}$, which is implemented by a
quantum circuit:
if _$h_{b}(y)\neq 0$_ then return _“no”_ else $result\leftarrow$ Run the
circuit $U$ on the state $|y\rangle$,
return _result_
end
Define $I_{k,b}=\langle U_{b},\frac{k}{l},\frac{k+1}{l},\rangle$.
Query the uqcmapp oracle with $I_{k,b}$ and put the result in $r_{k,b}$ .
end end if _ $\exists k,b\ s.t.\ r_{k,b}=1$ _ then accept else reject end
Algorithm 3 tqcmapp solver, which uses polynomially many queries to uqcmapp
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Soundness and Completeness follow from the same arguments used in the MA case.
This ends the proof of Theorem 2.
## 6 The Robustness of UQMA
### 6.1 Discussion about QMA and the Marriott-Watrous Formalism
In this section we discuss the robustness of our definition of unique QMA and
prove Lemma 4.
From Definition 12 we see that for a given QMA verification scheme and a state
$|\psi\rangle$, its probability of acceptance is:
$Pr(\text{verifier accepts
}|\psi\rangle)=\lVert\Pi_{1}U_{x}(I\otimes|0^{m}\rangle)|\psi\rangle\rVert^{2}$
A useful operator in this context, as defined in [MW05], is the following
$Q=(I_{m}\otimes\langle 0^{m}|)U^{\dagger}\Pi_{1}U(I\otimes|0^{m}\rangle).$
(4)
Note that
$Pr(\text{verifier accepts }|\psi\rangle)=\langle\psi|Q|\psi\rangle.$ (5)
As $Q$ is Hermitian, there is a basis of orthonormal eigenvectors
$\\{|\psi_{i}\\}\rangle_{i=1}^{2^{l}}$ for which
$Q=\sum_{i}\lambda_{i}|\psi_{i}\rangle{\langle\psi_{i}|}$, where
$\lambda_{i}(Q)\geq\lambda_{i+1}(Q)$ are the eigenvalues of $Q$. Note that by
knowing the eigenvectors and eigenvalues of $Q$ we can find out the acceptance
probability of every witness in a simple way
$\displaystyle\langle\psi|Q|\psi\rangle$
$\displaystyle=\sum_{i,j}a_{i}^{*}a_{j}\langle\psi_{i}|Q|\psi_{j}\rangle$ (6)
$\displaystyle=\sum_{i,j}a_{i}^{*}a_{j}\lambda_{j}\langle\psi_{i}|\psi_{j}\rangle=\sum_{i}|a_{i}|^{2}\lambda_{i},$
where $a_{i}=\langle\psi_{i}|\psi\rangle$.
Let us consider another possible definition of the class UQMA.
###### Definition 35 (UQMA)
A promise problem $L=(L_{yes},L_{no})\in\textsf{UQMA}$ if there exists a
polynomial quantum circuit $U_{x}$ which can be computed in $poly(|x|)$ time,
having $l(x)$ qubits as input and requiring $m(x)$ ancilla qubits initialized
to $|0^{m}\rangle$, s.t.
1. 1.
$x\in L_{yes}\Rightarrow\lambda_{1}(Q)\geq 2/3\text{ and }\lambda_{2}(Q)\leq
1/3$.
2. 2.
$x\in L_{no}\Rightarrow\lambda_{1}(Q)\leq 1/3$.
Where $\lambda_{1}\geq\lambda_{2}\geq\ldots\lambda_{2^{l}(x)}$ are the
eigenvalues of $Q$.
###### Lemma 36 (Equivalence of Definitions 15 and 35)
A language $L=(L_{yes},L_{no})\in\textsf{UQMA}$ according to Definition 15
$\Longleftrightarrow L\in\textsf{UQMA}$ according to Definition 35
Proof: We start proving that given a $I\in L_{yes}$ according to Definition
15, it is also in $L_{yes}$ according to Definition 35. We now from Definition
15 that there is state $|\psi\rangle$ which is accepted with probability of at
least $2/3$. According to Eq. (5), the acceptance probability of
$|\psi\rangle$ is $\langle\psi|Q|\psi\rangle=p\geq 2/3$. From Eq. (6), in
turn, we see that $p$ can be written as a convex combination of the
$\lambda$’s. Therefore, $\lambda_{1}\geq 2/3$.
We now prove that $\lambda_{2}\leq 1/3$. Denote by $V$ the subspace spanned by
the eigenvectors with eigenvalue greater than $1/3$. Note that
$\forall|\phi\rangle\in V\ \langle\phi|Q|\phi\rangle>1/3$. If $dim(V)\geq 2$,
there must exist a $|\phi\rangle\in V$ orthogonal to $|\psi\rangle$ and,
therefore, the acceptance probability of $|\phi\rangle$ is greater than $1/3$,
which is in contradiction to the properties of an $L_{yes}$ instance according
to definition 15.
The other directions is straightforward.
We now turn to the proof of Lemma 4. Let us start with the precise definition
of the problem unique 1-d 2-local hamiltonian:
###### Definition 37
unique 1-d 2-local hamiltonian: We are given a $2$-local Hamiltonian on $n$
$d$-dimensional sites $H=\sum_{j=1}^{r}H_{j}$ with $r=\textsf{poly}(n)$
arranged in a line. Each $H_{j}$ has a bounded operator norm
$||H_{j}||\leq\textsf{poly}(n)$. We are also given two constants $a$ and $b$
with $b-a\geq 1/\textsf{poly}(n)$. In “yes” instances, the smallest eigenvalue
of $H$ is at most $a$ and all the other eigenvalues are above $b$. In “no”
instances, the smallest eigenvalue is larger than $b$. We should decide which
one is the case.
We now prove Lemma 4. That the problem is in UQMA can be seen by the following
verification procedure. We expect as a proof the unique ground state of $H$.
Given a witness $|\psi\rangle$, we use the phase estimation algorithm (see
e.g. Ref. [WZ06]) to determine, within inverse polynomial accuracy $\delta$
with exponentially high probability, its energy, i.e.
$\langle\psi|H|\psi\rangle$. Case it is smaller than $a+\delta$, we accept;
otherwise we reject. It is clear that in “yes” instances, there is one witness
which is accepted with probability exponentially close to one (the ground
state of $H$), while any state orthogonal to it is accepted only with an
exponentially small probability (which is the probability that the phase
estimation does not give the correct answer).
The hardness of the problem for UQMA is a simple application of the
construction of [AGIK07], which presents a reduction from any problem in QMA
to 1-d 2-local hamiltonian with $d=12$. The details of the construction are
not important here. We only note that the low-lying eigenvectors of the
Hamiltonian considered are well approximated, within an inverse polynomial, to
a class of states parametrized by all possible proofs - called history states
- with the property that two orthogonal proofs give raise to two orthogonal
history states. Moreover, the probability of acceptance of a given proof is
imprinted in the energy of the associated history state - again up to inverse
polynomial accuracy. It is then clear that a problem in UQMA will give raise
to valid instance of unique 1-d 2-local hamiltonian, since in “yes” instances
of the problem (which is the only case we must analyze), the second eigenvalue
of the Hamiltonian, which is well approximated by the energy of the history
state associated to the witness which has the second highest probability of
acceptance, will be separated from the ground state energy by a constant
factor.
### 6.2 Yet Another New Class and its Equivalence To UQMA
One might define a similar class to QMA, with the additional promise of the
gap of its acceptance probability.
###### Definition 38 (Poly-Gapped QMA (PGQMA))
A promise problem $L=(L_{yes},L_{no})\in\textsf{GQMA}$ if there exists a
polynomial $\delta(|x|)$, and a polynomial quantum circuit $U_{x}$ which can
be computed in $poly(|x|)$ time, having $l(x)$ qubits as input and requiring
$m(x)$ ancilla qubits initialized to $|0^{m}\rangle$, s.t.
1. 1.
$x\in L_{yes}\Rightarrow\lambda_{1}(Q)\geq 2/3\text{ and
}(\lambda_{1}(Q)-\lambda_{2}(Q))\geq\delta(|x|)$.
2. 2.
$x\in L_{no}\Rightarrow\lambda_{1}\leq 1/3\text{ and
}(\lambda_{1}(Q)-\lambda_{2}(Q))\geq\delta(|x|)$.
Where $\lambda_{1}\geq\lambda_{2}\geq\ldots\lambda_{2^{l}(x)}$ are the
eigenvalues of the operator $Q$, defined in Eq. (4).
The above definition is motivated by the local hamiltonian problem, with the
additional promise that the spectral gap of the Hamiltonian is inverse
polynomial. Its one dimensional version is defined as follows.
###### Definition 39
1-d poly-gap local hamiltonian: We are given a $2$-local Hamiltonian on $n$
$d$-dimensional sites $H=\sum_{j=1}^{r}H_{j}$ with $r=\textsf{poly}(n)$
arranged in a line. Each $H_{j}$ has a bounded operator norm
$||H_{j}||\leq\textsf{poly}(n)$. We are also given three constants $a$, $b$
and $\Delta$ with $b-a,\Delta\geq 1/\textsf{poly}(n)$. We have the promise
that the spectral gap of $H$ is larger than $\Delta$. In “yes” instances, the
smallest eigenvalue of $H$ is at most $a$. In “no” instances, the smallest
eigenvalue is larger than $b$. We should decide which one is the case.
As in the unique case, we can show
###### Lemma 40
1-d poly-gap local hamiltonianis PGQMA-Complete.
The proof is completely analogous to the reasoning we provided for Lemma 4. In
order to prove Theorem 7, we need the following result.
###### Lemma 41
$\textsf{PGQMA}\stackrel{{\scriptstyle R}}{{=}}\textsf{UQMA}$.
Proof: We first show that $\textsf{UQMA}\subset\textsf{PGQMA}$. This inclusion
is not immediate because of the following reason: If $I\in
L_{no}\in\textsf{UQMA}$, then we know that $\lambda_{1}(Q)\leq 1/3$, but we do
not know whether $(\lambda_{1}(Q)-\lambda_{2}(Q))\geq\delta$.
In order to resolve this issue, we use the amplification property of QMA, and
change the “no”-probability to be $1/3-\delta$ instead of $1/3$: so we have
$\lambda_{1}(Q)\leq 1/3-\delta$. Then, by a simple construction which we shall
explain in the sequel, we add a single state which is accepted with
probability $1/3$, having $\lambda_{1}(Q)=1/3$ and $\lambda_{2}(Q)\leq
1/3-\delta$, which provides the necessary gap.
Adding the $1/3$-eigenvalue is done by changing the circuit: we append another
qubit to the input qubits, and measure it in the beginning of the circuit. If
its state is 0, then we proceed as before. If it is 1, we measure all other
input qubits in the computational basis. If all of them are 1, we accept with
probability 1/3. Otherwise we reject. A simple calculation shows that the
action of such a procedure is exactly as we want: it adds one
$1/3$-eigenvalue, and $2^{l}-1$ 0-eigenvalues, which do not concern us.
We now show that $\textsf{PGQMA}\stackrel{{\scriptstyle
R}}{{\subseteq}}\textsf{UQMA}$. This is again not immediate, as case $I\in
L_{yes}\in\textsf{PGQMA}$, we know that $\lambda_{1}(Q)\geq 2/3$, but we do
not know whether $\lambda_{2}(Q)$ is below the “no”-probability. For this we
use the fact that $\textsf{UQMA}_{1/3,2/3}=\textsf{UQMA}_{a,b}$, where
$(b-a)\geq 1/poly$. We know that for a $I\in L_{yes}$ there exists a for which
$\lambda_{1}(Q)\geq 2/3+(j+1)\frac{\delta}{2}$ and $\lambda_{2}(Q)\leq
2/3+j\frac{\delta}{2}$. So, we give the circuit as a
$\textsf{UQMA}_{2/3+j\frac{\delta}{2},2/3+(j+1)\frac{\delta}{2}}$ problem, for
$j=1,\ldots,\lfloor j\rfloor$, and for at least one $j$, it will be in
$L_{yes}$. Thus by picking $j$ at random, we get the required property. It is
also easy to see that we have soundness in the above construction.
## 7 The QMA Case
### 7.1 Random Projections Fail to Create Inverse Polynomial Gap
As mentioned earlier, we have divided the proof of the Valiant-Vazirani
Theorem into 3 components. Component 1 solves the problem in the simple case
where the number of the accepting witnesses is known; Component 2 improves it
by observing that the size of the set can be only approximated, without a
considerable effect on the probability of acceptance; Finally, Component 3
shows that we may achieve the same results by using a two-universal hash
function instead of a random function, rendering the reduction efficient.
In this section we show that even in the case where the number of solutions is
known, as in component 1, we cannot - at least in the most direct approach -
create a transformation that maps it to a “unique instance”. The main
difficulty in the QMA case is that we do not know in which basis to operate.
Notice that if there exists a description (which Merlin can supply) of how to
efficiently transform a standard basis state to one of the states that is
accepted with probability greater than $2/3$, then the problem is in QCMA.
Let us define a possible quantum analogue of a $R$-restriction. A natural
generalization is - instead of restricting to witnesses which belong to some
set $R$ \- to project onto some subspace $R$; We call this procedure a quantum
$R$-restriction. As we did in the discussion of component 1, we will not
consider the efficiency of implementing the restriction. A diagram of a
general circuit and its $R$-restriction is given in Figure 5.
$\textstyle{U}$$\textstyle{|\psi\rangle}$$\textstyle{\vdots}$
$\textstyle{|0\rangle}$ $\textstyle{\vdots}$$\textstyle{|0\rangle}$
$\textstyle{U}$
$\textstyle{\Pi_{R}}$$\textstyle{|\psi\rangle}$$\textstyle{\vdots}$
$\textstyle{|0\rangle}$ $\textstyle{\vdots}$$\textstyle{|0\rangle}$
Figure 5: A quantum R-restriction. On the left: a general description of a QMA
verification scheme. On the right: its $R$-restriction, where $\Pi_{R}$ is the
projection on the subspace $R$. The state is accepted only if in both
measurements the outcome was 1.
While the relevant operator for the original verification is
$Q=(I_{l}\otimes\langle
0^{m}|)U^{\dagger}\Pi_{1}U(I_{l}\otimes|0^{m}\rangle)$, after the the
$R$-restriction it is given by $Q_{R}=(I_{m}\otimes\langle
0^{m}|)U^{\dagger}\Pi_{1}\Pi_{R}\Pi_{1}A(I_{m}\otimes|0^{k}\rangle)$, where
$\Pi_{R}$ is a projection onto the subspace $R$. The quantum analogue of
component 1 consists of taking the subspace $R$ to be a random subspace of
dimension $d$, chosen accordingly to the Haar measure, for some convenient
$d$. The next proposition shows that this approach, unfortunately, fails.
###### Proposition 1
For every $\epsilon>0$, with probability larger than $1-\epsilon$, applying
the quantum random $R$-restriction, with arbitrary $d$, to example 1 creates
an instance with a gap smaller than $\epsilon^{-1}2^{-l/2+2}$.
Proof: As the verification circuit already rejects any state in the orthogonal
complement of the two-dimensional subspace $V$, it is clear that we only have
to analyze the gap created on states in $V$.
A rank $d$ random projector can be written as $UP_{d}U^{\cal y}$, where $U$ is
a unitary drawn from the Haar measure and
$P_{d}:=\sum_{j=1}^{d}|j\rangle\langle j|$. Let
$m_{V}(U,d):=\max_{|\psi\rangle\in V}\langle\psi|UP_{d}U^{\cal
y}|\psi\rangle-\langle\psi^{\bot}|UP_{d}U^{\cal y}|\psi^{\bot}\rangle$, where
$|\psi^{\bot}\rangle$ is the - up to a phase - unique orthogonal vector to
$|\psi\rangle$ in $V$. We consider the following quantity, which gives the
expectation value of the gap created by applying the random $R$-projection
defined by $UP_{d}U^{\cal y}$:
$\mathbb{E}_{U\sim\text{Haar}}(m_{V}(U,d))=\int_{U(2^{l})}dUm_{V}(U,d),$ (7)
where the integral is taken over the Haar measure of the unitary group
$U(2^{l})$.
Let $\\{|0\rangle,|1\rangle\\}$ be a basis for $V$. Note that $m_{V}(U,d)$ is
given by the difference of the maximum $\lambda_{\max}$ and minimum
$\lambda_{\min}$ eigenvalues of the following matrix
$V_{U,k}:=\left(\begin{array}[]{cc}\langle 0|UP_{d}U^{\cal y}|0\rangle&\langle
0|UP_{d}U^{\cal y}|1\rangle\\\ \langle 1|UP_{d}U^{\cal y}|0\rangle&\langle
1|UP_{d}U^{\cal y}|1\rangle\end{array}\right)$
By Gersgorin disc Theorem ([BB97] p. 244), we find
$|\lambda_{\max}(V_{U,k})-\lambda_{\min}(V_{U,k})|\leq|\langle 0|UP_{d}U^{\cal
y}|0\rangle-\langle 1|UP_{d}U^{\cal y}|1\rangle|+2|\langle 0|UP_{d}U^{\cal
y}|1\rangle|,$
from which follows that
$\int_{U(2^{l})}dUm_{V}(U,d)\leq\int_{U(2^{l})}dU|\langle 0|UP_{d}U^{\cal
y}|0\rangle-\langle 1|UP_{d}U^{\cal y}|1\rangle|+2\int_{U(2^{l})}dU|\langle
0|UP_{d}U^{\cal y}|1\rangle|.$
Applying Lemma 42 to each of the two terms in the R.H.S. of the equation
above,
$\int_{U(2^{l})}dUm_{V}(U,d)\leq\sqrt{\frac{2k(2^{l}-k)}{(2^{l}+1)2^{l}(2^{l}-1)}}+2\sqrt{\frac{k(2^{l}-k)}{(2^{l}+1)2^{l}(2^{l}-1)}}\leq
2^{-l/2+2},$
for any $1\leq k\leq 2^{l}$. To complete the proof, note that by Markov’s
inequality,
$\int_{U:m_{V}(U,d)\geq\lambda}dU\leq 2^{-l/2+2}/\lambda,$
for every $\lambda>0$. Setting $\lambda=2^{-l/2+2}/\epsilon$, we find that
with probability
$\int_{U:m_{V}(U,d)<\lambda}dU=1-\int_{U:m_{V}(U,d)\geq\lambda}dU\geq
1-\epsilon,$
$m_{V}(U,d)$ is smaller than $2^{-l/2+2}/\epsilon$.
###### Lemma 42
For any traceless operator $X\in{\cal B}(\mathbb{C}^{N})$,
$\int_{U(N)}dU|\text{tr}(UP_{k}U^{\cal
y}X)|\leq\sqrt{\frac{k(k-K)\text{tr}(X^{\cal y}X)}{(N+1)N(N-1)}},$ (8)
where $P_{k}:=\sum_{j=1}^{k}|j\rangle\langle j|$.
Proof: From the convexity of the square function,
$\left(\int_{U(N)}dU|\text{tr}(UP_{k}U^{\cal
y}X)|\right)^{2}\leq\int_{U(N)}dU|\text{tr}(UP_{k}U^{\cal y}X)|^{2}.$
To compute the R.H.S. of the equation above, we first note that
$\displaystyle\int_{U(N)}dU|\text{tr}(UP_{k}U^{\cal y}X)|^{2}$
$\displaystyle=$ $\displaystyle\int_{U(N)}dU\text{tr}(U^{\otimes
2}P_{k}^{\otimes 2}(U^{\cal y})^{\otimes 2}X\otimes X^{\cal y})$ (9)
$\displaystyle=$ $\displaystyle\text{tr}(\left(\int_{U(N)}dUU^{\otimes
2}P_{k}^{\otimes 2}(U^{\cal y})^{\otimes 2}\right)X\otimes X^{\cal y}).$
By Schur’s Lemma [FH91],
$\displaystyle\int_{U(N)}dUU^{\otimes 2}P_{k}^{\otimes 2}(U^{\cal y})^{\otimes
2}$ $\displaystyle=$ $\displaystyle\text{tr}\left(P_{k}^{\otimes
2}(\mathbb{I}-\text{SWAP})\right)\frac{\mathbb{I}-\text{SWAP}}{N(N-1)}$
$\displaystyle+$ $\displaystyle\text{tr}\left(P_{k}^{\otimes
2}(\mathbb{I}+\text{SWAP})\right)\frac{\mathbb{I}+\text{SWAP}}{N(N+1)}$
$\displaystyle=$
$\displaystyle\frac{k(k-1)}{N(N-1)}(\mathbb{I}-\text{SWAP})+\frac{k(k+1)}{N(N+1)}(\mathbb{I}+\text{SWAP}),$
where SWAP if the swap operator and we used that
$\text{tr}(\text{SWAP}(P_{k}\otimes
P_{k}))=\text{tr}(P_{k}^{2})=\text{tr}(P_{k})=k$. Then, from Eq. (9),
$\int_{U(D)}dU\text{tr}(UP_{k}U^{\cal y}X)^{2}=\text{tr}(X^{\cal
y}X)\left(\frac{k(k+1)}{N(N+1)}-\frac{k(k-1)}{N(N-1)}\right),$
from which the lemma easily follows.
### 7.2 Using a Many-Outcome Measurement
In the previous section we tried to solve example 1 by applying the most
natural idea that comes to mind: do a random 2-outcome measurement, and see if
one state can “pass” the projection with an amount which is not negligible,
compared to the other state on the subspace. We found out that such a
procedure fails. In this section, we analyze the use a many-outcome
measurement. We begin by applying a measurement in a random basis (or, to put
it differently, by applying a random unitary according to the Haar measure,
and then measuring in the standard basis). This, of course, cannot be done
efficiently, but we will deal with it later.
Radhakrishnan et al. [RRS05] have shown,
###### Theorem 43
[RRS05] Let $|\psi_{1}\rangle,|\psi_{2}\rangle$ be two orthogonal quantum
states in $\mathbb{C}^{N}$. Then,
$\mathbb{E}_{\hat{M}}\left(\left\|\hat{M}(|\psi_{1}\rangle)-\hat{M}(|\psi_{2}\rangle)\right\|_{1}\right)=\Omega(1)$
where $\hat{M}$ is a orthogonal basis chosen uniformly from the Haar measure.
A stronger result was presented in Theorem 1 of [Sen06], which implies the
same kind of result, but instead of the expectation, it asserts that the same
holds with all but an exponentially small probability.
Furthermore, Ambainis and Emerson [AE07] have shown that:
###### Theorem 44
Let $|\psi_{1}\rangle,|\psi_{2}\rangle$ be two orthogonal quantum states in
$\mathbb{C}^{N}$. Then,
$\left\|\hat{M}(|\psi_{1}\rangle)-\hat{M}(|\psi_{2}\rangle)\right\|_{1}=\Omega(1)$
where $\hat{M}$ is a POVM with respect to an $\epsilon$-approximate (4,
4)-design.
For our purpose, there is no need to understand what is an
$\epsilon$-approximate (4, 4)-design, but only that there exists an efficient
construction which enables us to realize the POVM $\hat{M}$ for any constant
$\epsilon$. Notice that this is a constant POVM, and for every 2 states, the
TVD of the distributions is constant. For more details of how one can
implement a 4-design, see Theorem 1 of [AE07]. Although the POVM is constant,
it achieves the same result as a random object (many-outcome measurements) but
in an efficient way, and therefore we see it as a “pseudo-random” object.
So, how can we take advantage of that? Suppose we had the description of the
distribution of $\hat{M}(|\psi_{1}\rangle)$ and $\hat{M}(|\psi_{2}\rangle)$.
Then we could select a unique witness by accepting only when we measure an
outcome $j$ associated to the $j$’s for which
$\hat{M}(|\psi_{1}\rangle)(j)>\hat{M}(|\psi_{2}\rangle)(j)$. In this way we
would get by Theorem 44 that $|\psi_{1}\rangle$ is accepted with a $\Omega(1)$
probability larger than $|\psi_{2}\rangle$. Of course this approach does not
lead to the solution of the problem, as the promise of having a description of
the distributions is too strong.
Indeed, although there is a classical description which would let us
distinguish, with high probability, between the two cases, there is no known
general way to achieve that which is in BQP. We would like to note that there
is a resemblance between this problem and the SZK-Complete given in Ref.
[Vad99], where in both problems, it is required to distinguish between two
probabilities with some total variation distance.
## 8 Acknowledgments
This work is part of the QIP-IRC supported by EPSRC (GR/S82176/0) as well as
the Integrated Project Qubit Applications (QAP) supported by the IST
directorate as Contract Number 015848’ and was supported by an EPSRC
Postdoctoral Fellowship for Theoretical Physics.
## Appendix A Proofs
Proof of Lemma 22: Let $\\{y_{1},y_{2},...,y_{w}\\}$ be the elements of $W$.
$\displaystyle Pr(|h^{-1}(0)\bigcap W|=1)=$ $\displaystyle
Pr(\bigcup_{i=1}^{w}(h(y_{i})=0\bigcap_{j\neq i}h(y_{j})\neq 0))$ (10)
$\displaystyle=\sum_{i=1}^{w}Pr(h(y_{i})=0\bigcap_{j\neq i}h(y_{j})\neq 0)$
(11) $\displaystyle=\sum_{i=1}^{w}Pr(h(y_{i})=0)Pr(\bigcap_{j\neq
i}h(y_{j})\neq 0|h(y_{i})=0)$
$\displaystyle=\sum_{i=1}^{w}Pr(h(y_{i})=0)(1-Pr(\bigcup_{j\neq
i}h(y_{j})=0|h(y_{i})=0)$
$\displaystyle\geq\sum_{i=1}^{w}Pr(h(y_{i})=0)(1-\sum_{j\neq
i}Pr(h(y_{j})=0|h(y_{i})=0)$ (12)
Equation (10) follows from the fact that all the elements in the union of
equation (10) are disjoint. Equation (12) follows from the union bound.
Because $h$ is taken from a universal hash function set, we have that
$Pr(h(y_{i})=0)=1/2^{k+2}$, $Pr(h(y_{j})=0|h(y_{i})=0)=1/2^{k+2}$. It was also
given that $w/2^{k+2}>1/4$ and $w/2^{k+2}\leq 1/2$. So,
$\displaystyle=w/2^{k+2}(1-\frac{w-1}{2^{k+2}})\geq 1/8$ (13)
Proof of Lemma 31:
The proof is almost the same: Let $y_{1},...,y_{a}$ be the elements of
$S_{1}$, and $y_{a+1},...,y_{b}$ the elements of $S_{2}$. So,
$\displaystyle Pr(|h^{-1}(0)\bigcap S_{1}|=1\wedge|h^{-1}(0)\bigcap S_{2}|=0)$
$\displaystyle=Pr(\bigcup_{i=1}^{a}(h(y_{i})=0\bigcap_{1\leq j\leq b,j\neq
i}h(y_{j})\neq 0)).$
The next steps are exactly the same, until we get to:
$\displaystyle\geq\sum_{i=1}^{a}Pr(h(y_{i})=0)(1-\sum_{1\leq j\leq b,j\neq
i}Pr(h(y_{j})=0|h(y_{i})=0))$ $\displaystyle\geq
a/2^{k+2}(1-(b-1)/2^{k+2})\geq 1/8\frac{a}{b}$
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|
arxiv-papers
| 2008-10-27T18:23:31 |
2024-09-04T02:48:58.479397
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Dorit Aharonov, Michael Ben-Or, Fernando G.S.L. Brandao, Or Sattath",
"submitter": "Or Sattath",
"url": "https://arxiv.org/abs/0810.4840"
}
|
0810.4951
|
11institutetext: SOFIA-USRA, NASA Ames Research Center, MS N211-3, Moffett
Field, CA 94035, USA
11email: jdebuizer@sofia.usra.edu 22institutetext: Gemini Observatory, Casilla
603, La Serena, Chile 33institutetext: National Research Council of Canada,
5071 W. Saanich Rd, Victoria, BC V9E 2E7, Canada 44institutetext: School of
Physics, University of New South Wales, Sydney, 2052 NSW, Australia
55institutetext: Australia Telescope National Facility, CSIRO, PO Box 76
Epping, NSW 1710, Australia
# SiO Outflow Signatures Toward Massive Young Stellar Objects with Linearly
Distributed Methanol Masers
J. M. De Buizer 11 2 2 R. O. Redman 33 S. N. Longmore 4455 J. Caswell 55 and
P. A. Feldman 33
(Received ; accepted )
###### Abstract
Context. Methanol masers are often found in linear distributions, and it has
been hypothesized that these masers are tracing circumstellar accretion disks
around young massive stars. However, recent observations in H2 emission have
shown what appear to be outflows at similar angles to the maser distribution
angles, not perpendicular as expected in the maser-disk scenario.
Aims. The main motivation behind the observations presented here is to
determine from the presence and morphology of an independent outflow tracer,
namely SiO, if there are indeed outflows present in these regions and if they
are consistent or inconsistent with the maser-disk hypothesis.
Methods. For ten sources with H2 emission we obtained JCMT single dish SiO
(6–5) observations to search for the presence of this outflow indicator. We
followed up those observations with ATCA interferometric mapping of the SiO
emission in the (2–1) line in six sources.
Results. The JCMT observations yielded a detection in the SiO (6–5) line in
nine of the ten sources. All of the sources with bright SiO lines display
broad line wings indicative of outflow. A subset of the sources observed with
the JCMT have methanol maser velocities significantly offset from their parent
cloud velocities, supporting the idea that the masers in these sources are
likely not associated with circumstellar disks. The ATCA maps of the SiO
emission show five of the six sources do indeed have SiO outflows (the only
non-detection being the same source that was a non-detection in the JCMT
observations). The spatial orientations of the outflows are not consistent
with the methanol masers delineating disk orientations. Overall, the
observations presented here seem to provide further evidence against the
hypothesis that linearly distributed methanol masers generally trace the
orientations of circumstellar disks around massive young stars.
###### Key Words.:
stars: formation – stars: early type – ISM: jets and outflows – circumstellar
matter – molecular data – masers – line: profiles – infrared: ISM – radio
continuum: ISM – radio lines: ISM
## 1 Introduction
Our understanding of star formation, despite decades of research, is still
quite limited. The development of a reasonably detailed model of isolated low
mass star formation through core accretion onto a disk (Shu, Adams, & Lizano
1987), which has recently been reviewed by Arce et al. (2007), has been guided
by extensive observations of nearby protostars. However, there are problems
when modeling the formation of the highest mass stars with core accretion -
most notably the effects of radiation pressure which may inhibit further
accretion once the star has accreted $\ga$10 M⊙. Since stars more massive than
10 M⊙ do exist, and since they tend to form in the middle of dense clusters,
the idea has been proposed that massive stars form through a process of
coalescence of low mass stars or protostars (e.g., Bonnell, Bate & Zinnecker
1998) or through a process of “competitive accretion” (e.g., Bonnell et al.
2001; Bonnell & Bate 2006) within the cluster. However, recent modeling by
several authors (e.g., McKee & Tan 2002, 2003; Krumholz, McKee, & Klein 2005)
has shown that despite all the alleged problems, the highest mass stars may
indeed be formed theoretically in a scaled-up version of low-mass star
formation via core accretion onto disks. However, it is not known with
certainty if massive stars form in this way because direct imaging of the
accretion disks that are hypothesized to be feeding very young B and O type
stars is very difficult.
Several factors complicate the observational problem. First, regions of
massive star formation lie at distances of typically a few to 10 kpc away,
making it harder to resolve spatial detail than for low mass star forming
regions where there are many examples (e.g., Taurus) that are much less than 1
kpc away. Second, the earliest stages of massive star formation are difficult
to observe because they occur extremely rapidly, and furthermore occur in the
most obscured regions of giant molecular clouds. There are still no directly
imaged accretion disks confirmed to exist around a star of spectral type B2 or
earlier.
Despite these difficulties, it is relatively easy to find massive young
stellar objects (YSOs) at a phase just prior to the formation of an UC HII
region, because they often excite methanol maser emission in the surrounding
molecular gas. Surveys of methanol maser emission by Norris et al. (1998) and
Walsh et al. (1998) found numerous massive YSOs scattered along the Galactic
Plane, and subsequent studies (see for example Minier et al. 2003, Walsh et
al. 2003, and Ellingsen 2008) confirm that methanol masers are reliable
indicators of massive star formation. Walsh et al. (1998) note that methanol
maser activity may fade out as the UC HII develops, indicating that this is a
transitory phase tracing mainly the earliest stages of massive star formation
when accretion is thought to be occurring.
Approximately half of the sources with methanol maser emission display this
emission as a grouping of many discrete maser “spots” arranged in a roughly
linear structure as projected on the sky. Norris et al. (1998) hypothesized
that these masers are excited in edge-on accretion disks surrounding the stars
at the center of the massive YSOs. The surprisingly large fraction of edge-on
disks was attributed in large part to the longer path lengths within the disk
for maser amplification compared to the path lengths in more face-on systems.
Velocity gradients that are only occasionally present along the line of maser
spots ($\sim$12% according to Walsh et al. 1998) are thought to be suggestive
of rotating disks. A theory of how masers could be excited in an accretion
disk has been developed by Durisen et al. (2001). There is some observational
evidence from individual sources that methanol masers may indeed be excited in
disks. For example, Bartkiewicz et al. (2005) have found a ring-like structure
of methanol masers around the candidate high-mass YSO G23.657-0.127.
Observations of methanol maser regions at higher resolution with VLBI and the
VLBA (e.g., NGC 7538 IRS 1, Pestalozzi et al. 2004) have shown linear
structures also exist at smaller scales (0.01-0.2$\arcsec$), some of which
have velocity structures consistent with an edge-on Keplerian disk. However,
only a small number of these sources have been studied in detail, and they may
or may not be related to the larger (0.3-1.5$\arcsec$) linear distributions we
are studying here. More recent evidence of the maser-disk connection comes
from Pillai et al. (2006), who interpret the line of methanol masers
associated with a massive YSO in the infrared-dark cloud G11.11$-$0.12 as
evidence for an accretion disk driving an outflow traced by H2O maser
emission.
There are, however, both observational and theoretical reasons to question
whether the methanol masers are actually excited in the accretion disks around
massive YSOs in general, even in the cases where the maser spots have a linear
distribution. Observationally, Walsh et al. (1998), in a major survey of
methanol maser sources, concluded that the maser emission from the majority of
the sources with linear distributions of maser spots was not likely to have
arisen in disks. Alternative models generally invoke shocks. Dodson, Ojha, &
Ellingsen (2004) have shown that externally driven planar shocks moving
through the molecular gas can reproduce many of the observed properties when
viewed edge-on. Even Bartkiewicz et al. (2005) have considered whether the
ring of maser spots in G23.657-0.127 might be better explained by a spherical
shock encountering a planar structure in the molecular gas, rather than a disk
around the massive young stellar object itself.
In this paper we will refer to the hypothesis that the linearly distributed
methanol masers are being excited in an edge-on accretion disk around a
massive YSO as the maser-disk hypothesis, and we explicitly note that our data
only include methanol maser sources that are linearly distributed on the sky
with arcsecond scales (0.3 to 1.5$\arcsec$), due to the numerous studies of
these sources at these spatial scales.
Although accretion disks have been difficult to observe unambiguously, an
active accretion disk should still reveal its presence by driving bipolar
outflows into the surrounding medium, and these are often easier to detect and
characterize. The bipolar outflow emerges along the axis of rotation
perpendicular to the plane of the accretion disk. Lee et al. (2001) were the
first to observe H2 (1$-$0) S(1) emission near three methanol maser sources.
They concluded that the H2 emission most likely arises in terminal shocks at
the tips of high-speed bipolar outflows and that the orientation of the line
of methanol masers for one source (IRAS 16076-5134) was consistent with the
masers being excited in an edge-on accretion disk that could be driving the
outflow responsible for the H2 emission. In a larger study of 28 sources
specifically chosen to have linear distributions of methanol masers, De Buizer
(2003) tried to test the maser-disk hypothesis by searching for outflows
perpendicular to the methanol maser distributions. He obtained wide-field
images of the sites of linearly distributed methanol masers using the 2.12
$\mu$m H2 ($1-0$) S(1) line as the outflow diagnostic. H2 emission from
potential outflows were found to be aligned perpendicular to the maser
distribution (as would have been expected under the maser-disk hypothesis) in
only 2 of the 28 cases. Surprisingly, the emission was distributed within 45∘
of parallel in 12 of the 15 fields where H2 emission was detected and thought
to be outflow related. It was therefore suggested that the methanol masers in
these sources do not delineate circumstellar disks, but may have some
relationship to the outflows, as Moscadelli et al. (2002) found.
The interpretation of the results from De Buizer (2003) remains ambiguous
because 2.12 $\mu$m H2 line emission can be excited both by outflow shocks and
by radiative UV excitation and cascade. In fact, a subset of the sources in
that survey showed clear signs of H2 emission associated with radiative
excitation from nearby dusty star-forming centers. Of the 15 sources in that
survey where the H2 emission was deemed to be not associated with radiative
excitation, the overall morphologies of the emission did not resemble the
simple bipolar outflows seen around young, low-mass stars. Consequently,
without additional evidence, it could not be conclusively ascertained which
mechanism is stimulating the H2 emission near these massive YSOs, nor
definitively link the alleged outflows to the methanol masers.
We therefore undertook a series of observations of these sources in a set of
independent outflow indicators. Our primary outflow tracer was thermal
emission from SiO that is liberated from grains and excited into emission
behind strong shocks. SiO can be a useful shock tracer because its abundance
is enhanced by factors of up to 106 behind strong shocks within a high-speed
outflow or along its immediate boundary (Avery & Chiao 1996; Dutrey,
Guilloteau, & Bachiller 1997; Arce et al. 2007). Emission from more volatile
molecules such as methanol and SO are also enhanced by shocks as the higher
temperatures liberate molecules frozen onto grains, but these molecules are
also commonly present in the molecular gas of the ambient cloud and especially
in the hot cores surrounding massive YSOs. Emission from these molecules
typically arises in a dense shell surrounding the outflow cavity and may
indicate a wider outflow angle than the SiO emission. The wings of these lines
can be useful indicators of high-speed outflows, but the line cores are better
indicators of conditions in the ambient gas. By choosing our molecular
transitions appropriately, we can therefore get a graduated set of probes. H2
spots trace bow shocks on the tips of high-speed outflows. SiO emission will
trace the high-speed outflows responsible for exciting the H2 spots. SO and
thermal methanol emission serve as secondary indicators of the outflows (in
the line wings) and also measure the properties of the hot cores surrounding
the massive YSO.
In the rest of this paper, we will describe the single dish observations we
have obtained using the JCMT in the SiO ($6-5$) line to detect the presence of
SiO in selected sources from the H2 survey of De Buizer (2003). We will also
describe the follow-up interferometric observations with the ATCA in the SiO
($2-1$) line, used to map the SiO outflows from a subsample of those JCMT
targets. We will show that there are indeed outflows from these sources but
that they are not orthogonal to the linear distributions of methanol masers.
That, and other evidence presented, create a serious problem for the
hypothesis that linearly distributed methanol masers generally arise in, and
define the orientation of, circumstellar disks.
## 2 Observations
### 2.1 The JCMT SiO(6–5), SO, and CH3OH Observations
JCMT observations of the SiO $J=6-5$ transition at 260.51802 GHz were made in
service mode during the 2005A (February to July) observing semester, with data
being taken on April 2, 3, 4, 12, May 3, 8, and July 13. The sources observed
with JCMT are listed in Table 1. The observations were taken with the receiver
RxA3. At this frequency the beam size is 18.4$\arcsec$.
Since it was recognized that the SiO lines might be very weak, that the source
elevations would never be very high, and that the velocities of the molecular
clouds were often unknown, the observations were designed to distinguish weak,
wide SiO lines from irregularities in the spectral baseline. We chose to
observe the SiO $J=6$ – $5$ line at 260.51802 GHz, which would allow us to
observe simultaneously the SO ($N_{J}=6_{7}$ – $5_{6}$) at 261.84368 GHz and
CH3OH ($N_{K,v}=2_{1,1}$ – $1_{0,1}$) at 261.80570 GHz. To this end, a custom
observing mode was developed by the JCMT staff that split the correlator into
two sections centered on the SiO and SO lines, the latter spectrum also
covering the CH3OH line, with an offset equivalent to 42 km/s.
All three of these molecules are expected to have enhanced abundance behind
shocks where high temperatures liberate molecules from the icy mantles of
grains. Much higher temperatures are required to create significant quantities
of SiO, so in the cold gas near low-mass protostars SiO is an excellent tracer
of high-speed shocks (Arce et al. 2007).
Calibration of data taken with RxA3 at this frequency is unusually difficult,
with a systematic error of about 25% that varied from night-to-night. With an
appropriate choice of sideband, however, the sensitivity of the receiver was
not impaired. The correct sideband was determined each night from observations
of the sources IRAS 16293-2422 and L1157. In this paper, we only use the
spectra to determine the kinematics of the molecular cloud and outflowing gas,
which does not require accurate calibration of the temperature scale.
The data were reduced using the Starlink SPECX package, rebinning the data to
0.5 km/s resolution. Position-switched observations with a single-detector
receiver like RxA cannot be used to measure the continuum emission from any
but the brightest sources, because the continuum signal is dominated by
fluctuations in the atmospheric transmission between the on-source and off-
source integrations. Also, on physical grounds we expect that continuum
emission from dust will normally be several orders of magnitude weaker than
optically thick line emission in most astrophysical sources. A linear baseline
has therefore been subtracted from all of the spectra shown in Figures 1 and
2, taking care not to fit the baselines to parts of the spectra with apparent
emission in the line wings, and Table 1 does not include any estimate of the
continuum emission at 1.2 mm. It is notable that none of our spectra show
systematically negative signals, indicating that the baseline fits are good in
spite of the challenging observational circumstances, and that even the weak,
extended line wings are believable in the spectra.
There were 15 sources in the survey of De Buizer (2003) where H2 was detected
and not ruled out as unrelated to outflow. Of those, only one source could not
be reached with the JCMT because of its large southern declination
(G305.21+0.21). Therefore our original source list for the JCMT comprised the
remaining 14 sources but required only 12 pointings since there were two pairs
of sources too close to separate with the JCMT beam (G345.01+1.79/G345.01+1.80
and G321.031-0.484/G321.034-0.483). Final observations were for 10 pointings
since weather and time constraints prevented observations for 2 of the 12
pointings (G321.031-0.484/G321.034-0.483 and G313.77-0.86).
Table 1 lists the line properties of all ten sources observed in the JCMT
survey. Plots of the spectra obtained are shown in Figures 1-2. Since most of
the lines have significant wings, the systemic velocity ${\rm V_{LSR}}$
reported in Table 1 is simply the velocity of the peaks of the SO and CH3OH
lines, shown in Figure 1 and 2 as the vertical dashed line. These ${\rm
V_{LSR}}$ values were checked against other molecular species not affected by
outflow (i.e., NH3, CH3CN), and the line-of-sight velocities were very similar
in all cases where data was available (e.g. Longmore et al. 2007, Purcell et
al. 2006). The velocity width $\Delta$$v$ in Table 1 was determined by a
visual inspection of where the line wings reached zero intensity. This number
may be significantly underestimated for the CH3OH line because the blue wing
of that line often blends with the red wing of the SO line, and because the
red wing may be cut off by both the edge of the spectrum and the baseline fit.
The integrated intensity $\int T_{K}$ given in Table 1 is a simple summation
of the measured intensity between the zero-intensity limits multiplied by the
channel spacing $\delta v$. The formal RMS error in each pixel $\sigma$ was
estimated from an empty part of the spectrum and the formal RMS error of $\int
T_{K}$ was calculated from $\sigma\cdot\Delta v\cdot(\delta v/\Delta
v)^{1/2}$. For the weaker lines, this error estimate can be used to determine
the signal-to-noise of the detection, but it does not include the calibration
error that could be as large as 25%.
Of the ten sources observed only G308.918+0.123 yielded a non-detection in the
SiO line. G308.918+0.123 was observed in the wrong sideband, resulting in an
effective system temperature four times worse than normal and yielding the
rather noisy spectra in Figure 1. Even here, integrating the SiO signal over
the velocity range of the SO line yields 0.30$\pm$0.13, i.e. a formal
significance of 2.3 $\sigma$. This is suggestive but not a formal detection,
so we only claim a formal 3 $\sigma$ upper limit of 0.4 K km/s in Table 1. In
addition, the weakest detected source, G11.50-1.49, may be considered marginal
with a formal significance of only 4 $\sigma$ (Figure 2). Since the average
detection rate of SiO in outflows around low-mass protostars is only $\sim$50%
(Codella, Bachiller, & Reipurth 1999), this is a notably high detection rate.
### 2.2 The ATCA SiO (2–1) and 3mm Continuum Observations
The ATCA observations in the SiO $J=2-1$ (86.8469 GHz) line were taken on 2006
September 12-13. The compact, H75 array configuration (baselines of 31 to 86
m) was used with 5 antennae in both east-west and north-south baselines to
allow for snapshot imaging. The primary and synthesized beam sizes (field of
view and angular resolution) were $\sim$33$\arcsec$ and $\sim$7$\arcsec$,
respectively. The $FULL\\_64\\_128\\_2P-1F$ correlator setting was used,
providing 128 channels across 64 MHz (221 km s-1) bandwidth for a velocity
resolution of 1.7 km s-1. Only 6 of the 10 fields observed with the JCMT were
observed with the ATCA because of time constraints. The fields observed with
the ATCA are given in Table 2. Each field was observed for 8-9 10 min cuts
separated over 6 h. A bright ($>$ 1.5 Jy), close, phase calibrator was
observed for 3 min before and after each cut. PKS 1253-055 and Mars were used
as the bandpass and flux calibrator for all the observations.
The data were reduced using the MIRIAD (see Sault, Tueben & Wright 1995)
package. Bad visibilities were flagged, edge channels removed and the gain
solution from the calibrator applied to the source. The visibilities were
Fourier transformed to form image cubes and CLEANed to remove the sidelobes of
the synthesized beam response. Continuum emission was extracted by fitting a
low-order polynomial to the line-free channels and imaged in the same way. The
noise characteristics and detected line and continuum flux densities are
listed in Table 2.
We were awarded enough time to observe 6 of our 10 JCMT sources with the ATCA.
We chose G328.81+0.63, G331.28-0.19, G331.132-0.244 because they were the
three strongest SiO (6–5) detections at the JCMT. G308.918+0.123,
G318.95-0.20, and G320.23-0.28 were chosen because they were the three most
impressive sources of H2 emission from De Buizer (2003).
### 2.3 The Gemini T-ReCS Mid-IR Observations
Sources G318.95-0.20, G328.81+0.63, and G331.28-0.19 were also observed with
the Thermal Region Camera and Spectrograph (T-ReCS) at the Gemini Telescope in
Chile on 2005 April 19. All three sources were observed using the _Si-5_
($\lambda$c=11.7 $\mu$m, $\Delta\lambda$=1.1 $\mu$m) filter and _Qa_ filter
($\lambda$c=18.3 $\mu$m, $\Delta\lambda$=1.6 $\mu$m), with on-source exposure
times of 108s for both filters. T-ReCS utilizes a Raytheon 320$\times$240
pixel Si:As BIB array which is optimized for use in the 7–26 $\mu$m wavelength
range. The pixel scale is 0.089$\arcsec$/pixel, yielding a field of view of
28$\aas@@fstack{\prime\prime}$8$\times$21$\aas@@fstack{\prime\prime}$6\. Sky
and telescope radiative offsets were removed using the standard chop-nod
technique.
Co-added frames were saved every 10 sec, and the telescope was nodded every 30
sec. The co-added frames were examined individually during the data reduction
process and those plagued by clouds (i.e., showing high and/or variable
background or decreased source flux) were discarded. The T-ReCS observations
were made under partly cloudy skies, and most of the images had to have some
frames removed. Final effective exposure times of the images varied from 30 to
80s on-source.
Flux calibration of the final images was difficult given the variable
observing conditions. Standard stars were observed at similar airmasses to the
science targets, and the derived calibration factors varied by 20% among them.
All three maser fields have been previously observed at the CTIO 4-m by De
Buizer, Piña, & Telesco (2000) with a spatial resolution coarser by a factor
of two. However, a comparison of the derived flux densities from the T-ReCS to
the CTIO data show that they agree to within 20%. Therefore we will not quote
here new flux density values for any of the sources already detected by De
Buizer, Piña, &Telesco (2000) since those values will be more accurate. We
will only quote T-ReCS flux densities for any new sources detected in these
same fields.
## 3 Results
### 3.1 General Results from the JCMT Observations
Sufficiently sensitive transitions of outflow tracers like SiO, SO and thermal
methanol should show the lines to have wide wings from gas entrained in the
outflow. As can be seen in Figures 1-2, all sources with strong detections
display line wings in the SiO line. The SO and CH3OH lines show similar line
wings, although they are relatively weaker and narrower than the wings of the
SiO line, as was expected. This behavior is characteristic of emission from
outflows. The green regions plotted in each of the panels in Figures 1-2 show
the velocity ranges of the methanol masers that are linearly distributed and
associated with each source.
It is always possible that a source can have apparent line wings due to
unrelated sources in the beam at different local standard of rest velocities.
In this case one would expect similar structure in all three lines, SiO, SO
and CH3OH. This is apparently the case for G345.01+1.79 (see Figure 2), where
there is a very prominent blue shoulder in all three lines. In this case the
JCMT beam includes G345.01+1.80, another massive young stellar object
$\sim$15$\arcsec$ away from G345.01+1.79. As shown in the figure, the velocity
range of the methanol maser emission is [-14.0, -10.0] km s-1 for G345.01+1.80
and is [-25.0, -16.0] km s-1 for G345.01+1.79 matching the two emission peaks
that are seen in all three lines observed with the JCMT. The primary peak in
SiO in Figure 2 is caused by G345.01+1.80, and the secondary peak by
G345.01+1.79. In addition, there appears to be a very broad blue wing to this
double line, indicating that one or both sources has an outflow.
The near-ubiquity of the wide line wings and especially of SiO emission in
these spectra indicates that powerful outflows driving strong shocks are
present within the JCMT beam for most of these targets. Taken by itself, this
appears to support the interpretation of De Buizer (2003) that the H2 emission
near these sources are being excited in the bow shocks of outflows. We will
revisit this issue in sections 4.1 and 4.2, after considering the rest of the
evidence.
### 3.2 Results from the ATCA and T-ReCS observations
To follow up the JCMT single dish observations we endeavored to map the SiO
emission from these regions to verify the outflow nature of the emission, and
observe the outflow geometries with respect to the methanol maser distribution
angles. Observations with the JCMT of SiO (6–5) emission at 260.5 GHz have
previously been used to detect low-mass YSO outflows in infrared-dark cloud
cores (Feldman et al. 2004). Subsequent BIMA observations of SiO (2–1) at 86.8
GHz from four of these sources verified that the emission arises from bipolar
outflows (R. Redman, priv. communication). Therefore, to map out the outflows
from our southern hemisphere targets, we went to the ATCA to similarly map out
the SiO emission in the (2–1) transition.
Projection effects should be unimportant under the maser disk hypothesis,
because the disks are selected to be edge-on. Thus, the maser-disk hypothesis
makes the clear prediction that outflows traced by SiO emission and H2
emission spots should be on average orthogonal to the disks traced by the
maser emission. Even allowing for precession of the inner accretion disk and
for outflows with significant opening angles, we would expect the majority of
outflow tracers to be found within 45∘ of perpendicular to the orientation of
the disk, and hence perpendicular to the line of maser spots.
Furthermore, if the methanol masers do indeed exist in actively accreting
disks, all such disk sources should display outflow. Unlike low-mass stars,
massive stars are not believed to have a long period after accretion where
their disks are passive. The environment of massive stars is so caustic (i.e.
photo-ionization, radiation pressure, winds), that as soon as accretion halts,
the timescale for dispersal of the disk is very short ($\la$104 yrs; Blum et
al. 2004, Shen & Lou 2006).
Geometrically, we can verify that the outflow traced by SiO emission is driven
by the massive YSO by noting whether the patches of SiO emission are colinear
with the location of the massive YSO. Under the maser-disk hypothesis, the
massive YSO should be spatially coincident with the methanol maser spots
because of the very high MIR intensities required to excite the methanol
masers (Sobolev & Deguchi 1994, Sobolev et al. 1997). In other outflows where
SiO is observed it is excited primarily behind a small number of strong shocks
that are propagating along the outflows, so in our sources we expect that the
morphology of the SiO emission will consist of discrete bright spots along the
outflow marking the locations of the strongest shocks rather than a continuous
line; under such conditions colinearity of the SiO emission with the massive
YSO/masers is the best that can be established by the observations. If the SiO
emission lobes are not colinear with the masers, then the outflow must
originate in a different star.
In the following subsections, we will discuss the ATCA and T-ReCS observations
and results on a source-by-source basis in the context of the maser-disk
hypothesis. Since many of these targeted regions have been observed here for
the first time at these wavelengths and resolutions, there are several results
that are not directly associated with the main theme of this work. These
results are summarized in the Appendix. A summary of observational properties
for each target is listed in Table 3, showing results derived from our new
data and from the literature.
#### 3.2.1 G308.918+0.123 (IRAS 13395-6153)
G308.918+0.123 has the most impressive collection of H2 knots in the entire
survey of De Buizer (2003), spread out over a 120$\arcsec$$\times$80$\arcsec$
area (Figure 3a). The vast majority of H2 spots are contained within regions
45∘ from parallel with respect to the methanol maser distribution. The 8.6 GHz
radio continuum observations of Phillips et al. (1998) show the linear
distribution of four maser spots lie on the northern edge of a 15$\times$15
arcsec2 UC HII region (Figure 3b).
This target was a formal non-detection in the JCMT SiO (6–5) survey, but was
accidentally observed in the wrong sideband, resulting in a much higher noise
level than other observations with comparable integration times. The
observation is still adequate to show that the integrated emission of the SiO
line is weaker for this source than for any of the other targets except
G11.50-1.49. Given the interesting nature of the H2 emission, we observed this
target with the ATCA. However, no SiO (2–1) was detected on the field with the
ATCA, though a bright (49 mJy) unresolved 3mm continuum source was detected
with a peak $\sim$5$\arcsec$ from the methanol maser location and coincident
with the cm UC HII region (Figure 3b).
Since there was no SiO detection we can draw no further conclusions about the
maser-disk hypothesis with respect to this source, other than saying that
further observations will be needed in other outflow indicators to see if
there really is an outflow at this location as indicated by the H2 emission.
However, a confirmed lack of any outflow would also be counter to the maser-
disk hypothesis since all of these sources are likely to be actively
outflowing if they have accretion disks with masers in them.
#### 3.2.2 G318.95-0.20
The methanol maser emission for this target consists of seven maser spots in a
linear pattern spanning $\sim$0.5$\arcsec$. There is a semi-ordered velocity
gradient along the spot distribution (Norris et al. 1993). Seen in Figure 4b,
knots of H2 emission, as well as some diffuse H2 emission was found at this
site (De Buizer 2003). The maser location is coincident with a bright near-IR
continuum source (De Buizer 2003), that was also detected in the mid-IR (De
Buizer, Piña, & Telesco 2000) and re-imaged here with T-ReCS at higher angular
resolution (Figure 4a, $\S$A.2).
The outflow maps obtained with the ATCA show the SiO emission to be coincident
with the H2 emission on this field (Figure 4b) thereby confirming the outflow
nature of the H2 emission suggested by De Buizer (2003). However, this region
may be too complicated to be modeled adequately with just a single outflow
(however, see $\S$A.2). In addition to the red-shifted velocity component to
the northwest of the maser sources, there is a blue-shifted component to the
southwest, on the side dominated by red-shifted emission. These two components
can reasonably be attributed to a second outflow from a second source.
The HCO+ emission contours for this source from Minier et al. (2004) are shown
in Figure 4c. Since the blue-shifted component of the HCO+ emission coincides
with the second blue-shifted component of the SiO emission, the HCO+ emission
may be dominated by the second outflow. If so, it seems to be oriented more
nearly north-south, and is even more closely parallel to the line of methanol
maser spots than the main outflow.
Minier et al. (2004) detect an unresolved source of thermal CH3OH emission
coincident with the HCO+ red-shifted emission peak, which they claim may be
the hot molecular core exciting the HCO+ outflow. The peak of this CH3OH core
is offset $\sim$2$\arcsec$ to the northwest of the methanol maser location and
further offset from the mid-infrared source. Therefore it appears that this
region has two outflows with slightly different position angles and with
opposite senses of outflow direction, one apparently centered on the mid-
infrared emission and the other on a nearby hot molecular core.
The simplest interpretation of this region is that the massive YSO responsible
for exciting the methanol masers is also likely driving a high-speed outflow
traced by the brightest part of the SiO emission, and that this outflow is
responsible for most if not all of the H2 emission spots. A second, smaller
outflow is also likely to be present in the field that may be contributing
significantly to the HCO+ emission. Both of the outflows are aligned within 45
degrees of parallel to the line of methanol maser spots, and their
orientations are both inconsistent with the scenario that the methanol masers
are tracing the orientation of a circumstellar disk.
#### 3.2.3 G320.23-0.28 (IRAS 15061-5814)
There are ten methanol maser spots in the linear distribution associated with
G320.23-0.28, spread out over 0.5$\arcsec$ and at an angle of $\sim$86∘. In De
Buizer (2003), it is explained that the H2 emission from this source most
closely resembles a bi-lobed outflow morphology. Also, the H2 emission is
situated exactly parallel to the position angle of the methanol maser
distribution. In De Buizer (2003) this source was considered to be the best
candidate for further observations in disproving the circumstellar disk
hypothesis for linearly distributed methanol masers.
Our ATCA observations have revealed SiO emission distributed at the same angle
as the H2 emission and the methanol maser position angle (Figure 5a). The
blue-shifted lobe of the SiO outflow is nearly coincident with the western H2
emission region. The red-shifted SiO emission is found to be coincident with
the maser location, and not the eastern H2 emission. However, all of the H2
and SiO emission lie along the same outflow axis and so are presumed to be all
coming from a single outflow parallel to the methanol maser linear
distribution angle.
A 3-color image created from the Spitzer GLIMPSE archival data for this region
is shown in Figure 5b. In this image we can see that the masers are located on
the western edge of a large dusty region, which is also seen in the near-IR
images of De Buizer (2003). However, there is no source seen specifically at
the maser location in these Spitzer images, though there are nearby
($\sim$5$\arcsec$) mid-infrared sources to the east. The wavelength
represented as green in Figure 5b is the IRAC channel 2. This filter is
centered at 4.5 $\mu$m and has been shown to be a tracer of shock in the
outflows of many astrophysical sources (i.e., Noriega-Crespo et al. 2004). In
Figure 5b we see the 4.5 $\mu$m emission (green) is distributed at the same
position angle as all the other outflow indicators in this field, adding
further evidence to the outflow nature of the H2 emission.
Given that all outflow indicators (SiO, H2, Spitzer 4.5 $\mu$m emission) are
all distributed within 15∘ of parallel to the methanol maser distribution,
this source is clearly not compatible with the maser-disk hypothesis.
#### 3.2.4 G328.81+0.63 (IRAS 15520-5234)
The nine linearly distributed maser spots (Norris et al. 1998) at this
location are surrounded by a complex of sources and emission at several
wavelengths.
The new observations presented here add further complexity to the knowledge of
this region. The high spatial resolution mid-infrared images taken with T-ReCS
(Figure 6b) reveal a large (15$\times$15 arcsec2) extended emission region
with 8 peaks (or knots). The two brightest peaks are situated E-W at a
position angle similar to the methanol maser distribution angle of 86∘. The
overall shape of the extended mid-infrared emission is cometary, with the apex
pointing to the north. This emission is coincident with the cometary UC HII
region seen here at cm wavelengths (Ellingsen, Shabala, & Kurtz 2005).
However, the compact cm continuum source to the northeast of (and just
resolved from) the cometary UC HII has no associated mid-infrared emission.
The methanol masers lie between these two cm continuum sources, and at the
edge of the emission from the brightest mid-infrared source (Figure 6b).
Our observations reveal that the SiO emission is distributed on either side of
the cm/mid-infrared/mm continuum emission in a E-W fashion similar to the
methanol maser distribution angle (Figure 6a). However, the velocity structure
of the SiO emission is not like the others, i.e. there is not simply a red-
shifted lobe and a blue-shifted lobe. The velocity structure of the SiO
emission is quite complex, as demonstrated by the velocity channel map in
Figure 6c. There are two likely reasons for this complex velocity structure.
First, there may be multiple outflows present all with similar E-W
orientations. With several mid-infrared and radio sources between the two SiO
lobes, there is a very good possibility that two or more of these could be the
young stellar sources responsible for the outflows. A second possibility is
that it is a single E-W outflow that is oriented close to the plane of the
sky. Therefore no coherent line-of-sight velocity structure would be apparent.
Even with all of the complexity of this region, the SiO emission is clearly
not coming from an outflow (or outflows) centered at the maser location and
perpendicular to the angle of the methanol maser distribution. Consequently,
these observations are inconsistent with the maser-disk hypothesis.
#### 3.2.5 G331.132-0.244 (IRAS 16071-5142)
This source contains nine methanol maser spots oriented E-W with a velocity
gradient along the spot distribution (Phillips et al. 1998). It is coincident
with an extended (7$\times$10 arcsec2) cm continuum emission region (Phillips
et al. 1998) that is also seen in GLIMPSE 8 $\mu$m images. There is no near-
infrared continuum emission at the maser location, however it does appear that
a nearby and extended near-infrared source may overlap spatially with the
southern part of the extended cm continuum emission here (Figure 7b). There is
another round, extended (10$\arcsec$ in diameter) cm continuum source located
$\sim$15$\arcsec$ southwest from the maser location, but is most likely
unrelated to the maser emission itself.
There is a 3 mm continuum source detected at this location, with the masers
situated on its northeast edge. It overlaps spatially with the near-infrared
continuum emission and the cm continuum emission here. Given the offset, it is
not clear what the relationship of the mm continuum source is with respect to
the maser emission, nor is it clear where the source exciting the outflow is.
There is only one knot of H2 emission on the field that is elongated in its
morphology and situated close to the linear maser distribution axis. Our SiO
map (Figure 7a) shows that the SiO emission is centered on the maser location
and distributed at an angle very close to that of the methanol masers. The H2
emission lies at the eastern edge of the red-shifted SiO lobe. The blue-
shifted emission peak lies to the west of the red-shifted emission peak,
however, there is significant overlap of blue-shifted emission at the location
of the red-shifted emission. Again, this may have something to do with
orientation of the outflow being near the plane of the sky or the presence of
multiple outflows, though it is impossible to tell from the data at present.
However, it is certain that the collective SiO emission is distributed along
an axis that is within 10∘ of parallel with the methanol maser distribution
angle for this source, and clearly not what is expected if the masers
delineate a disk orientation.
#### 3.2.6 G331.28-0.19 (IRAS 16076-5134)
Though there is some disagreement of the maser distribution angle (see the
appendix), this source appears to have masers that are linearly distributed at
an angle of $\sim$170∘, and is one of the only two target fields in the survey
of De Buizer (2003) to have H2 emission distributed in the perpendicular
regions of the field with respect to the maser distribution angle.
Near-infrared continuum observations of the field show that the maser location
is bordered to the south and west by diffuse extended emission (De Buizer
2003; Lee et al. 2001). GLIMPSE 8 $\mu$m observations show that this is a
complex region of extended dust emission with the masers at the “corner” of
extended N-S and E-W “walls” of emission (Figure 8c). It is within these
“walls” of dust that the H2 emission seen to the west by De Buizer (2003), as
well as the more sensitive H2 observations of emission to the south and west
seen by Lee et al. (2001) are located. Given the extensive nature of this
dusty star forming region, it is most likely that the H2 emission is
associated with radiative excitation of the star formation ongoing in these
“walls”. This whole region contains diffuse and extended cm continuum emission
(Phillips et al. 1998) testifying to the radiative ionization of the gas in
the region (Figure 8b).
High resolution T-ReCS observations (Figure 8b) show a resolved mid-infrared
source near ($\sim$3$\arcsec$) the brightest H2 emission seen by De Buizer
(2003), which was first seen by De Buizer, Piña, and Telesco (2000). This
source is likely a massive young stellar object responsible for radiative
stimulation of the H2 emission at that location.
Our 3 mm maps reveal an unresolved mm continuum source with an emission peak
$\sim$1.5$\arcsec$ south of the maser location. SiO emission is also found in
the field, with blue-shifted and red-shifted lobes nearly coincident with the
3 mm continuum location. There appears to be a $\sim$2$\arcsec$ offset between
the two lobes at a position angle of $\sim$206∘. This outflow axis is within
45∘ of parallel with the maser position angle of 170∘, and therefore would be
inconsistent with the maser-disk hypothesis. However, the offset between the
lobes is smaller than the synthesized beam of
6.0$\arcsec$$\times$4.8$\arcsec$. The relative position uncertainty,
$\Delta\theta$, between two unresolved components in a well calibrated image
such as this can be estimated through
$\Delta\theta$$\sim$$\theta_{beam}/(2\times SNR)$, where $\theta_{beam}$ is
the synthesized beam and $SNR$ is the signal-to-noise ratio (Fomalont 1999).
With a signal-to-noise of $>$6$\sigma$ for both lobes, the 2$\arcsec$
separation is $>$4 times the relative position uncertainty, giving us
confidence that the offset between the lobes is real. We can use a similar
argument to test the robustness of the measured orientation angle between the
two peaks. Even if each of the lobes was offset by the relative positional
uncertainty given above, in opposite directions orthogonal to the position
angle between the peaks, this would lead to a maximum change in measured
positional angle of $\sim$25∘. We therefore argue it is statistically unlikely
that calibration errors could alter the measured lobe position angle enough to
change the outflow direction to be perpendicular to the maser position angle.
The H2 emission in this case is likely to not be outflow related at all, and
instead radiatively excited by the nearby mid-infrared source at the H2
location. Further SiO images (or other outflow indicator) at higher spatial
resolution are needed to unambiguously determine an outflow axis.
## 4 Discussion
Table 3 summarizes the observational properties of the sources in the ATCA
survey. The general result from the observations is that these sources of
linear methanol maser emission do indeed have SiO outflows, and that these
outflows are not oriented perpendicular to the linear methanol maser
distributions. In four of the six cases the H2 and SiO emission have
approximately the same position angle (G318.95-0.20, G320.23-0.28,
G328.81+0.63, G331.132-0.244), indicating that H2 is indeed a good tracer of
outflows from these objects. The two exceptions are G308.918+0.123, where no
SiO emission was found in either the JCMT or the ATCA observations, and
G331.28-0.19, where the outflow in SiO is at a different position angle to the
H2 emission on the field. In both cases it is likely that the H2 emission that
was detected is radiatively excited by other massive YSOs and not outflow
related. However, observations in other outflow tracers may still be warranted
considering that massive YSOs likely have a wide variety of chemistries,
energetics, and environmental factors that may favor the excitation of certain
outflow tracers over others.
For the five ATCA sources where SiO emission was detected and mapped, all five
have their linear maser distributions at approximately the same angle as the
overall SiO distributions on the field. The largest deviation from parallel is
G331.28-0.19, where the SiO emission is distributed at an angle $\sim$35∘ from
the methanol maser distribution angle. In the case of G318.95-0.20, there are
possibly two outflows, though both are contained within the quadrants parallel
to the maser distribution angle in Figure 4. The other three sources,
G320.23-0.28, G328.81+0.63, and G331.132-0.244 all have their SiO emission
distributed within 15∘ of parallel to their methanol maser distribution
angles, though G328.81+0.63 is likely to also contain multiple outflows at
similar angles. Because in no case presented here is there evidence for SiO
emission perpendicular to the methanol maser distributions, we conclude that
these observations are incompatible with the hypothesis that linearly
distributed methanol masers are generally delineating the orientations of
circumstellar accretion disks around massive stars.
### 4.1 Linking the outflow emission to the sources exciting the maser
emission
Massive stars do not form in isolation, and therefore the large scale H2
emission and the small-scale maser emission for a source can not be implicitly
linked through the same outflow a priori. However, using our observations we
can clarify two main points. First, the H2 emission does generally seem to be
excited by shocks associated with outflows, and not by fluorescence. And
second, the outflows that excite the H2 emission are driven by the massive
YSOs that excite the methanol masers, and not by nearby low-mass protostars.
We undertook the ATCA observations to address this first point by checking
that the outflows traced by the SiO emission are colinear with the H2 emission
spots. Figures 4 through 7 show that in the four sources imaged with ATCA
which showed significantly extended SiO emission, the dominant features in the
SiO emission are colinear with the H2 emission spots and the massive YSO whose
location is marked by the methanol maser emission. For the three sources,
G318.95-0.20, G320.23-0.28, and G331.132-0.244, this alignment is very clear.
For G328.81+0.63, the complexity of the SiO emission makes the case less
compelling, but the brightest SiO emission spot to the ESE of the massive YSO
is almost perfectly aligned opposite the H2 emission spots to the WNW of the
massive YSO, and a weaker SiO feature appears midway between the massive YSO
and the H2 emission spots.
In only one case, G331.28-0.19, is there evidence that the outflow traced by
SiO emission does not excite the H2 emission spots. As discussed in Section
3.2.6, the H2 emission spots are probably excited by fluorescence in an
unrelated source, and it is surely significant that in this one case the H2
emission spots are not aligned with the axis of the outflow traced by the SiO
emission (which is admittedly very poorly determined in this barely resolved
source).
Now we address our second point. If we had been studying isolated low-mass
protostars, there would have been no question that the outflows observed at
each source are driven by the target YSO and are the same as the outflows
directed towards the H2 emission spots. However, each of the massive YSOs in
our sample is probably surrounded by a crowd of low-mass protostars, some of
which may be driving their own outflows. This makes it important to establish
that the massive YSO responsible for the methanol maser excitation is also
driving the outflow that excites the H2 spots.
A significant result of De Buizer (2003) was that the H2 emission near these
sources is normally confined to a small range of position angles on either
side of each massive YSO. Since the outflows from unrelated YSOs are unlikely
to be aligned, this by itself indicates that there is normally a single
dominant outflow in each target region, and it is the source of that outflow
that is of interest.
Our observations show significant SiO emission in nine of the ten sources,
where observations of low-mass protostars (Codella et al. 1999, Gibb et al.
2004) would have suggested only half of the sources should have exhibited
significant SiO emission. We draw the conclusion that outflows from massive
YSOs may be more energetic than those from lower mass protostars and hence
more likely to excite SiO emission.
Geometrically, we can verify that the outflow traced by SiO emission is driven
by the massive YSO by noting whether the SiO emission regions excited by the
outflow are colinear with the location of the massive YSO deduced from the
maser spots. If unrelated YSOs were responsible for the strongest outflow, we
would expect significant misalignments between the massive YSO and the axis
defined by the SiO emission and the H2 emission spots in most sources. The
possible presence of weaker outflows that are not aligned across the massive
YSO in the fields of G318.95-0.20 and G328.81+0.63 confirms this expectation,
and again emphasizes that outflows capable of exciting SiO emission are quite
rare, even in regions like these that should be crowded with young YSOs and
protostars undergoing active accretion. In fact, in the five sources where SiO
emission was detected, there is a dominant outflow in the field and the
massive YSO is coincident with the SiO emission and/or lies on the axis
defined by the brightest SiO emission and H2 spots.
With the ATCA observations we have presented here, where no SiO emission is
found in any of the cases to be perpendicular to the maser alignment, linking
the outflow emission to the sources exciting the masers is not strictly
necessary for the testing of the maser-disk hypothesis. If these sources have
accretion disks delineated by methanol masers, we should see at least some
fields with confirmed outflows perpendicular to the maser distribution in our
observations, and this is not the case. While this presents a major problem
for the maser-disk hypothesis, the establishment of the direct relationship of
the massive YSO exciting the masers to the outflow is important for testing
other possible hypotheses, which we will address in the next section.
### 4.2 Association with shocks or outflows?
One consequence of the linking of the masers to the outflows in the last
section is that the relative position angles of the lines of maser spots and
H2 emission spots can validly be compared to test other suggested mechanisms
of methanol maser emission near high-mass YSOs.
Dodson, Ojha, & Ellingsen (2004) have developed a model in which the methanol
masers arise in an edge-on shock propagating through the hot core around the
massive YSO. This model has many attractive properties, especially in its
natural explanation of velocity gradients across the line of maser spots.
However, it is perhaps surprising that so many massive YSOs would be
associated with externally driven, edge-on shocks during the very brief period
in which the massive YSO is sufficiently bright to excite methanol maser
emission but before the growing H II region overwhelms the hot core. It is
unclear what other energy sources could be driving all these shocks when the
massive YSOs are the most powerful sources in the region. The results of this
paper are even more difficult to accommodate within this model, since we have
demonstrated that in most cases massive YSOs with linearly distributed
methanol masers are driving bipolar outflows traced by SiO emission that are
aligned with both the lines of methanol maser spots and shock excited H2
emission. It is not plausible that an externally driven shock should routinely
align itself with an outflow jet.
The fact that 12 of the 15 the fields in De Buizer (2003) showed H2 emission
organized within 45∘ of parallel to their maser distribution angles led him to
hypothesize that most linearly distributed methanol masers may be directly
associated with outflows, an idea we will refer to as the maser-outflow
hypothesis. Nor was he the first to reach this conclusion; Minier, Booth, &
Conway (2000) also considered that shocks associated with bipolar outflows
provided a more generally satisfactory paradigm than locating the methanol
masers in disks.
The first evidence of this may come from our JCMT data. Given that for each
JCMT target the SiO, SO, and thermal methanol lines all peak at the same
velocity, and that in general massive stars are found closest to the cluster
centers, it is reasonable to assume the massive YSOs in our sample are at the
local velocity of their parent molecular clouds. Furthermore it is reasonable
to assume that any outflows present in the region will be dominated by the
massive young stellar source. It is therefore striking that in four of the ten
sources (G308.918+0.123, G328.81+0.63, G331.28-0.19, and G339.88-1.26), the
methanol maser velocities lie mostly or entirely on one side of the SO line
core (Figures 1-2). If the maser emission is associated with outflows, this
behavior would not be unexpected because in some lower-mass sources, like
L1157, the thermal methanol emission can come almost entirely from one side of
the bipolar outflow (Avery & Chiao 1996). In L1157 the thermal methanol and
SiO emission each come from opposite sides of the bipolar outflow, probably
because the speed of the outflow and the temperature and density of the
ambient gas is different on the two sides. Likewise, masers in bipolar
outflows from YSOs should also generally have both red- and blue-shifted
velocities with respect to the YSO velocity, but there will be some instances
where chemistry or geometry is only appropriate on one side of the bipolar
outflow to generate maser emission observable from the Earth.
It is also possible, but much less likely, to get such asymmetric maser
emission from methanol masers excited in circumstellar disks. Although we
would expect a high degree of circular symmetry in a disk, clumping and
turbulence would ensure that only a random selection of lines of sight through
the disk would have strong maser amplification. Recent modeling by Krumholtz,
Klein, & McKee (2007) has shown that massive disks have strong gravitational
instabilities and may have significant sub-structure and non-axisymmetry in
the disks. They claim that such star-disk velocity offsets could be of the
order of a few km s-1. However, the largest offset in maser velocities in
G339.88-1.26 and G331.28-0.19 are 9 km s-1 and 12 km s-1, respectively, which
are much larger than expected from disk asymmetry. Therefore, in our sample
there may be kinematic evidence for the association of the methanol masers
with outflows rather than circumstellar disks.
Although an analytical model of a bipolar outflow with structures on scales
ranging from the size of the maser spots out to the bow shocks that excite the
H2 emission would be far too complex to develop in an observational paper such
as this, we do have enough evidence in hand to qualitatively guide what such a
model might look like.
Our JCMT observations demonstrate that the high-speed jet that forms the core
of the outflow has speeds much higher than the methanol masers. For most
sources in Figures 1-2, the maser lines have a velocity distribution slightly
wider than the line cores of the SO and CH3OH lines, but not nearly as wide as
the wings of the SiO lines when these are observed. If these masers are indeed
associated with outflows, this suggests that the masers are excited in gas
that has been shocked and partially entrained by the outflows, but not in the
high-speed outflows themselves. The methanol masers would be excited behind
shocks propagating away from the axis of the outflow, which might account for
gradients in the maser velocities across the line of the emission spots.
This suggestion is very similar to the conclusions of Moscadelli et al. (2002)
who observed the 12 GHz methanol masers in W3(OH) and found that the maser
proper motions were consistent with outflow away from the central source, with
plane-of-the-sky speeds around 4 km/s. The gas involved would be moving
supersonically but substantially more slowly than the gas on-axis in the
bipolar outflows. They suggest a model in which the masers originate in gas
entrained on a conical boundary delimiting the outflow cavity. In such cases,
with an appropriate allowance for the opening angle of the outflow, the maser
spots would align closely, but likely not exactly parallel, with the outflow
axis, even for distributions of maser spots that do not form well-collimated
lines. This conical opening angle of the outflow may explain why in our data
the maser distribution angles are often not exactly colinear with the SiO (or
H2) outflow axes.
To conclude, there may be evidence in our observations, as well as those in
the literature, that linearly distributed methanol masers are directly
associated with outflow and/or outflow cavities. However, to date, the
evidence is only suggestive and further observations as well as a detailed
physical model will be needed to confirm this possible relationship.
### 4.3 Association of continuum emission to the masers
Unrelated to the main goals of our observations, we can make some comment on
the association of the methanol maser emission to the mid-IR and mm continuum
emission sources that we have detected in our fields. For the six ATCA sources
we have continuum information at cm, mm, mid-infrared, and near-infrared
wavelengths. Table 3 shows that there are fewer detections in the near-
infrared than any other wavelength, and that the mid-infrared has the highest
detection rate. This is not to be unexpected given that methanol masers are
radiatively pumped by mid-infrared photons (Sobolev & Deguchi 1994, Sobolev et
al. 1997). The canonical size of a massive YSO is about 0.1 pc, which at the
distance of our farthest source (5 kpc) would subtend about 5$\arcsec$. Given
this, Table 3 lists the detection of unresolved sources with emission peaks
$<$5$\arcsec$ from the maser reference location, or if the masers lie within
an extended continuum source. However, it is clear that several of the
continuum sources that are at the edge of this cut-off are far enough away
that they might not be directly associated with the masers themselves.
For instance, in the mid-infrared, sources with definite spatial coincidence
(to within the FWHM of the mid-infrared source) with the masers are
G308.918+0.123, G318.95-0.20, and G331.132-0.244, even though mid-infrared
emission is detected on all six fields. For the other sources the masers lie
either outside the FHWM of the mid-infrared emission or are completely
spatially separate from the mid-infrared source on the field. Likewise in the
mm we detect sources on five of the six fields, however, even at the coarse
resolution of our mm maps ($\sim$7$\arcsec$) we can see that only for three
targets do the methanol masers lie within the FWHM of the mm sources:
G328.81+0.63, G331.132-0.244, and G331.28-0.19.
According to the survey of Beuther et al. (2002) there is a 100% detection
rate of 1.2 mm continuum emission towards methanol maser sites at a spatial
resolution of $\sim$11$\arcsec$. Perhaps this means higher angular resolution
is needed in order to say for sure if there is real physical connection
between the mm emission and the masers, or perhaps the sources seen at 1 mm
are completely different sources than what we could detect at 3 mm. Indeed,
when we look to the 1.2 mm continuum survey of Hill et al. (2005), we find two
of our sources included in their sample. Both sources, G318.95-0.20 and
G331.28-0.19, were measured to have peak intensities approximately 1.9
Jy/beam. Extrapolating this to 3 mm assuming a $\nu^{-4}$ dependence leads to
an estimate of 2.8 mJy/beam peak intensity, which means given our
sensitivities (Table 2) we would be unlikely to have detected dust emission
from those sources at 3 mm. Therefore, because of the clustered nature of
massive star formation, the presence of emission at different wavelengths may
come from several different nearby sources unrelated to the maser emission,
and may explain why in some cases in this survey the 3 mm continuum emission
is not associated with the massive YSO exciting the methanol masers.
## 5 Conclusions
The main motivation behind the observations presented in this article was to
determine from the presence and morphology of the SiO emission if there are
indeed outflows present in these regions, and if they are consistent or
inconsistent with the hypothesis that linearly distributed methanol masers
generally trace circumstellar accretion disks around young massive stars. We
obtained JCMT single dish observations of ten sources from the H2 survey of De
Buizer (2003) and all but one yielded a detection in the SiO (6–5) line. All
of the sources with bright SiO lines displayed broad line wings indicative of
outflow.
It also appears from comparisons between the JCMT thermal line velocities and
the maser velocities that there may be kinematic evidence that the masers are
not associated with disks, and that perhaps there indeed is an association
between outflows and methanol masers. Four of the ten sources observed with
the JCMT have methanol maser velocities significantly offset from the thermal
line velocities of their parent clouds, which may support the suggestion of De
Buizer (2003) that the masers in these sources are participating in the
outflows themselves.
We followed up the JCMT single dish SiO (6–5) observations with ATCA
interferometric mapping in the SiO (2–1) line of 6 sources. None of these
fields had outflows oriented within 45∘ of perpendicular to the position
angles to the linear methanol maser distributions. However, five of these six
sources of linear methanol maser emission do indeed have SiO outflows, the
only non-detection being the same source that was a non-detection in the JCMT
observations.
G331.28-0.19 was one of only two sources in the De Buizer (2003) survey where
the H2 emission was actually found to be perpendicular to the linear methanol
maser distribution, and therefore could be an example of a case where the
linear methanol masers are tracing an accretion disk. However, the ATCA SiO
maps for this source reveal an outflow that is not perpendicular to the maser
distribution. Mid-infrared observations appear to show that the H2 emission is
likely associated with a nearby star-forming region and not an outflow from
the maser location.
From the ATCA SiO observations of G318.95-0.20 and G328.81+0.63 it is obvious
that some of these sources are highly complex in their SiO emission, and
higher spatial resolution observations are needed to understand the velocity
patterns in the outflow lobes. It is likely that multiple outflows are
present, but they cannot be distinguished in our coarse resolution
observations. Follow-up studies at higher spatial resolution and with other
outflow tracers may reveal further important clues regarding the relationship
between the SiO, H2, and methanol maser emission.
Unrelated to our main goals, we have also found that there are fewer
detections of continuum emission towards the maser locations in the near-
infrared than any other wavelength, and that the mid-infrared has the highest
detection rate. There also appears to be a higher detection rate of continuum
emission at 1 mm ( 100% according to Beuther et al. 2002) than at 3 mm towards
methanol maser sites. We find that only half of our methanol maser locations
are co-spatial with 3 mm continuum emission.
To conclude, overall the new SiO observations presented here seem to provide
further evidence against the hypothesis that, in general, linearly distributed
methanol masers are tracing circumstellar disks around massive young stars.
This is not to say that there cannot exist cases where there are methanol
masers in disks. However, as a population, these 0.2–1.5$\arcsec$-scale
linearly distributed methanol masers are not disk indicators. 80% of the
fields in De Buizer (2003) showed H2 emission organized parallel to their
maser distribution angles, and the probability of this occurring simply by
chance is low. Consequently, that data led to the hypothesis advanced by De
Buizer (2003) that, in general, linearly distributed methanol masers may be
associated with outflows. Kinematic evidence from the JCMT observations
presented here, as well as the overall geometries of the SiO outflows we have
mapped out with the ATCA, are compatible with (but do not yet provide
sufficient proof to confirm) this maser-outflow scenario, but are clearly
inconsistent with the maser-disk hypothesis. However, without a detailed
analytic model, and given the complexity of some of the sources presented
here, there is likely a lot of detail “swept under the rug” that needs to be
explored before one can definitely say that, as a population, linearly
distributed methanol masers are general tracers of outflows from young massive
stars.
###### Acknowledgements.
We would like to thank the referee, Andrew Walsh, for coherent and
constructive comments, which helped to improve the paper.This research was
based partially on observations obtained at the James Clerk Maxwell Telescope
under program M05AC16. The JCMT is operated by the Joint Astronomy Centre on
behalf of the Science and Technology Facilities Council of the United Kingdom,
the Netherlands Organisation for Scientific Research, and the National
Research Council of Canada. The Australia Telescope Compact Array is part of
the Australia Telescope which is funded by the Commonwealth of Australia for
operation as a National Facility managed by CSIRO. JMDB was partially
supported by Gemini Observatory, which is operated by the AURA, Inc., under a
cooperative agreement with the NSF on behalf of the Gemini partnership: NSF
(US), PPARC (UK), NRC (Canada), CONICYT (Chile), ARC (Australia), CNPq
(Brazil) and CONICET (Argentina). Gemini program ID associated with the
results in this paper is GS-2005A-DD-5. This research has made use of the
NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion
Laboratory, California Institute of Technology, under contract with the
National Aeronautics and Space Administration.
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## Appendix A Further details of the individual sources in the survey
We summarize here for each source in our survey any further observational
results determined from our data.
### A.1 G308.918+0.123 (IRAS 13395-6153)
In the near infrared observations of De Buizer (2003), there is a bright
continuum source at 2 $\mu$m at the location of the masers. This source is
also observed at 11.5 $\mu$m by Phillips et al. (in prep) as an unresolved but
very bright source (266 Jy). Given the spatial coincidence, this near-IR/mid-
IR object is likely to be the massive stellar source within the UC HII region
with a direct association to the masers.
Though this field has an impressive collection of H2 knots, there is very
little extended H2 emission on the field. Phillips et al. (1998) point out
that the size and shape of the UC HII region indicates that there may be
substructure, possibly due to a cluster of massive stars. Along the lines of
the discussion in $\S$4.5, a 3-color image created from the Spitzer GLIMPSE
archival data for this region shows no enhanced emission from IRAC channel 2
(4.5 $\mu$m), which is believed to be a tracer of shock in the outflows of
many astrophysical sources. This fact, combined with no clear sign of SiO
outflow, means that another possibility is that the cm/mm continuum source (or
possibly unresolved group of sources) and H2 knots are all YSOs in a large
cluster centered on the methanol maser location. H2 emission could be
radiatively excited in the near vicinity of hot massive stars. Outflow
observations in another indicator will be needed to completely understand the
nature of the observed H2 emission in this region.
### A.2 G318.95-0.20
Other than the near- and mid-infrared emission source, there are no other
detected continuum sources in the area around G318.95-0.20; no compact cm
continuum emission (Ellingsen, Shabala, & Kurtz 2005), and we detect no 3 mm
continuum in the field.
This infrared source was imaged with Gemini and T-ReCS (Figure 4a), and was
found to be elongated in its mid-IR continuum emission at a similar position
angle ($\sim$145∘) to the methanol maser distribution angle (151∘). Astrometry
tests were performed during these T-ReCS observations, and the absolute
astrometric accuracy of the mid-infrared data is good to 0.6$\arcsec$. The
mid-infrared source peak is offset approximately 0.80$\arcsec$ southeast of
the maser reference spot (see Figure 4a). The coincidence of the mid-infrared
emission and the masers, the velocity gradient along the maser spots, and the
fact that the maser distribution is at the same position angle as the
elongated thermal dust emission would appear to favor the disk interpretation
for this source. However, the ATCA outflow observations show that this is not
the case.
The ATCA SiO observations shows that the outflow direction is not
perpendicular to the maser distribution angle (Figure 4). Given that the SiO
outflow is only $\sim$28∘ from the mid-infrared source elongation angle, it is
possible that the mid-infrared emission from this source is elongated because
we are seeing emission from the dusty outflow or outflow cavities. High-
resolution observations are showing massive young stellar objects appear to
often have strong, extended, mid-infrared continuum emission from their
outflows (e.g., De Buizer & Minier 2005; De Buizer 2006; De Buizer 2007). In
this case we may be seeing one or both outflow cavities from this source, but
lack the resolution required to confirm this.
While we discuss in $\S$4.2 the idea that there are likely two outflows in
this region, a plausible alternate scenario could be that there is emission
from the red-shifted wing at the location of the brightest part of the blue-
shifted wing of the SiO emission because the outflow is nearly in the plane of
the sky. Turbulence or an expansion of the shock away from the axis of the
high-speed outflow would easily account for both line wings being visible at
the same location. As discussed in the introduction, emission from volatile
molecules like methanol and HCO+ will most likely arise from a dense, shocked
shell of gas surrounding the cavity excavated by the high-speed outflow. These
molecules can indicate a wider opening angle for the outflow than suggested by
the high-speed jet, either because the jet wanders with time or because it is
surrounded by a lower-speed outflow that clears out the cavity (Arce et al.
2007). This would be consistent with the interpretation of the HCO+ emission
contours from Minier et al. (2004) that are shown in Figure 4c.
This source, along with G331.132-0.244, are the only two sources mapped with
the ATCA that have a clear velocity gradient in their methanol maser linear
distributions. If linearly distributed methanol masers do indeed trace some
area or material near the root of an outflow, as has been alternatively
proposed (De Buizer 2003), one may expect that the red- and blue- shifted
masers should be oriented toward the red- and blue-shifted SiO outflow
components, respectively. In G318.95-0.20 the masers are more blue-shifted to
the northwest, and red-shifted more to the southeast. Figure 4 shows that the
majority of the blue-shifted SiO emission is indeed to the northwest and the
majority of the red-shifted emission is to the southeast. This velocity
correspondence between masers and SiO emission is also seen in G331.132-0.244
(see $\S$A.5). However, we remind the reader that velocity gradients in the
methanol maser distributions are not common (Walsh et al. 1998), and there are
scenarios where the masers can be linearly distributed and associated with the
outflow, but need not have any velocity gradient (i.e., if the masers are
located in the material of the outflow cavity walls).
### A.3 G320.23-0.28 (IRAS 15061-5814)
This source was not observed with T-ReCS, and there exist no high spatial
resolution mid-infrared images of this site. There is no source detected in
near-infrared continuum emission directly at the maser location (De Buizer
2003), and the cm continuum source observed by Walsh et al. (1998) is located
$\sim$8$\arcsec$ northeast of the maser location.
We detect a 3 mm continuum source at the same location as the cm continuum
source of Walsh et al. (1998). This continuum source does not appear to have
any direct relationship to the masers or the outflow (Figure 5a).
### A.4 G328.81+0.63 (IRAS 15520-5234)
While there appears to be no near-infrared source at the maser location at 2
$\mu$m (De Buizer 2003; Goedhart et al. 2002) there is some emission seen in
the L band (3.3 $\mu$m, Walsh et al. 2001). There are two UC HII regions near
the maser location (Figure 6b), one compact and round, and the other large and
cometary shaped (Ellingsen, Shabala, & Kurtz 2005).
### A.5 G331.132-0.244 (IRAS 16071-5142)
As mentioned in $\S$A.2, this source, along with G318.95-0.20, are the only
two sources in our SiO ATCA sample that have a clear velocity gradient in
their methanol maser linear distributions. G331.132-0.244 has blue-shifted
masers more to the west of the linear maser distribution and red-shifted
masers more to the east. Figure 7a shows that this matches the velocity
pattern of the SiO emission. Therefore if linearly distributed methanol masers
do indeed trace some area or material near the root of an outflow, both
sources have maser velocities consistent with this idea.
### A.6 G331.28-0.19 (IRAS 16076-5134)
It has been pointed out since the De Buizer (2003) paper was published that
there is some disagreement in the literature as to whether the masers
associated with this location are in a linear distribution or not. According
to Norris et al. (1993, 1998) this field contains nine 6.7 GHz methanol masers
and four 12.2 GHz methanol masers, all distributed in an elongated fashion at
an angle of 166∘. More sensitive follow-up observations by Phillips et al.
(1998) discovered eleven 6.7 GHz methanol maser spots distributed in a similar
pattern in an elongated structure at a similar position angle of 170∘.
Interestingly, the 6.7 GHz methanol maser observations of Walsh et al. (1998)
show a grouping of ten maser spots with no discernible linear distribution to
within the relative astrometric precision of the individual maser spots
(0.3$\arcsec$). It is unclear why the maser distribution from Walsh et al.
(1998) is so different to that of Norris et al. (1993, 1998) and Phillips et
al. (1998) given the fact that the maser components in their spectra have
velocities that are very similar. Norris et al. (1998) quote a relative
astrometric precision of the individual maser spot locations of 0.2$\arcsec$,
however even if one adopts the 0.3$\arcsec$ accuracy of Walsh et al. (1998),
the masers shown in Norris et al. (1993, 1998) and Phillips et al. (1998)
would still appear to be in an elongated distribution.
Unlike previous mid-infrared images of this region, the T-ReCS images detect a
small source southeast of the maser location near the 3 mm peak, which also
has an associated cm continuum peak (Figure 8b). We derive a flux density for
this source of 23 mJy at 11.7 $\mu$m, from the T-ReCS images. No Qa flux
density can be quoted because of the very poor images of this site at that
wavelength. In the T-ReCS images, diffuse extended emission can also be seen
corresponding to the “walls” of dust seen in the Spitzer images. A knot of cm
continuum emission is also seen at the maser location, however no near-
infrared continuum source is detected there, and there is no mid-infrared
continuum source at the precise maser location in the T-ReCS images.
Table 1: Properties of Sources Observed with the JCMT | | | | | | SiO (6–5) | | SO (67–56) | | CH3OH (21,1–10,1)
---|---|---|---|---|---|---|---|---|---|---
Source | RA | Dec | Dist. | VLSR | | $\int$TK | $\Delta$$v$ | | $\int$TK | $\Delta$$v$ | | $\int$TK | $\Delta$$v$
| (J2000.0) | (J2000.0) | kpc | km/s | | K km s-1 | km s-1 | | K km s-1 | km s-1 | | K km s-1 | km s-1
G308.918+0.123 | 13 43 01.75 | –62 08 51.3 | 5.2 | –51.0 | | $<$0.4 | – | | 3.1 $\pm$ 0.28 | 8 | | 1.2 $\pm$ 0.28 | 8
G318.95–0.20 | 15 00 55.40 | –58 58 53.0 | 2.4 | –34.0 | | 2.3$\pm$2 0.11 | 45 | | 5.6 $\pm$ 0.14 | 30 | | 4.2 $\pm$ 0.11 | 20
G320.23–0.28 | 15 09 51.95 | –58 25 38.1 | 4.7 | –66.0 | | 0.9$\pm$2 0.15 | 20 | | 2.6 $\pm$ 0.15 | 20 | | 2.8 $\pm$ 0.13 | 15
G328.81+0.63 | 15 55 48.61 | –52 43 06.2 | 3.0 | –41.5 | | 10.0$\pm$2 0.08 | 60 | | 55.3 $\pm$ 0.10 | 30 | | 22.1 $\pm$ 0.08 | 22
G331.132–0.244 | 16 10 59.74 | –51 50 22.7 | 5.2 | –86.0 | | 7.4$\pm$2 0.09 | 65 | | 18.9 $\pm$ 0.10 | 35 | | 10.8 $\pm$ 0.09 | 28
G331.28–0.19 | 16 11 26.60 | –51 41 56.6 | 4.8 | –87.5 | | 10.4$\pm$2 0.05 | 65 | | 24.7 $\pm$ 0.13 | 50 | | 4.1 $\pm$ 0.08 | 20
G335.789+0.174 | 16 29 47.33 | –48 15 52.4 | 3.4 | –50.0 | | 1.2$\pm$2 0.05 | 30 | | 3.9 $\pm$ 0.11 | 30 | | 5.5 $\pm$ 0.10 | 25
G339.88–1.26 | 16 52 04.66 | –46 08 34.2 | 3.1 | –33.0 | | 1.7$\pm$2 0.07 | 50 | | 11.8 $\pm$ 0.09 | 30 | | 4.5 $\pm$ 0.06 | 15
G345.01+1.79 | 16 56 47.56 | –40 14 26.2 | 2.1 | –12.8 | | 4.5$\pm$2 0.06 | 30 | | 17.9 $\pm$ 0.10 | 42 | | 7.7 $\pm$ 0.08 | 24
G11.50–1.49 | 18 16 22.13 | –19 41 27.3 | 1.7 | –10.5 | | 0.12 $\pm$ 0.03 | 16 | | 3.0 $\pm$ 0.07 | 16 | | 1.2 $\pm$ 0.07 | 16
Table 2: Continuum and Line Properties for Sources Observed with the ATCA
Target | F3mm | F3mm RMS | SiO (2-1) Peak | SiO (2-1) RMS | Vred Range | Vblue Range
---|---|---|---|---|---|---
| (mJy) | (mJy/beam) | (mJy/beam/chan) | (mJy/beam/chan) | (km/s) | (km/s)
G308.918+0.123 | 49 | 7 | - | 25 | - | -
G318.95-0.20 | - | 8 | 178 | 16 | -24.0/-32.6 | -37.8/-39.5
G320.23-0.28 | 87 | 5 | 90 | 14 | -63.7/-65.5 | -70.6/-72.4
G328.81+0.63 | 500 | 26 | 326 | 12 | -23.9/-36.0 | -51.5/-60.2
G331.132-0.244 | 56 | 3 | 266 | 11 | -63.5/-84.7 | -91.1/-110.1
G331.28-0.19 | 16 | 2 | 546 | 14 | -61.8/-79.0 | -92.8/-117.0
Note – Absolute flux calibration errors are estimated to be $\sim$20%.
Table 3: Summary of Observational Properties for Sources Observed with the
ATCA
Target | 3 cm | 3 mm | MIR | NIR | H2 Angle w.r.t. | SiO (6-5) | SiO (2-1) | Maser | SiO | SiO Angle w.r.t.
---|---|---|---|---|---|---|---|---|---|---
| Cont? | Cont? | Cont? | Cont? | Maser P.A. | Line? | Line? | P.A. | P.A. | Maser P.A.
G308.918+0.123 | Yesa | Yes | Yesb | Yes | Parallel | No | No | 137 | n/a | n/a
G318.95-0.20 | Noc | No | Yes | Yes | Parallel | Yes | Yes | 151 | 117 (148)d | Parallel
G320.23-0.28 | Noe | No | Yesf | No | Parallel | Yes | Yes | 86 | 75 | Parallel
G328.81+0.63 | Yesc | Yes | Yes | Yesg | Parallel | Yes | Yes | 86 | $\sim$90h | Parallel
G331.132-0.244 | Yesa | Yes | Yesf | No | Parallel | Yes | Yes | 90 | 81 | Parallel
G331.28-0.19 | Yesa | Yes | Yes | No | Perpendiculari | Yes | Yes | 170i | 206 | Paralleli
Note – Near-infrared continuum (2 $\mu$m) and H2 and maser position angle
information is from De Buizer (2003). Mid-infrared continuum (12 and 18
$\mu$m) information unless otherwise indicated is from the T-ReCS data
presented in this work. Columns 2, 3, and 4 address the presence of continuum
emission at the maser location only. For unresolved continuum sources this
means the emission peak is within 5$\arcsec$ of the reference feature. See
$\S$3 for details on individual sources.
aFrom Phillips et al. (1998).
bFrom Phillips et al. (in prep).
cFrom Ellingsen, Shabala, & Kurtz (2005).
dThere appears to be two outflows present in G318.95-0.20. See $\S$4.2.
eFrom Walsh et al. (1998).
fFrom GLIMPSE 8$\mu$m data.
gFrom Walsh et al. (2001). These are L-band (3.3 $\mu$m) observations.
hG328.81+0.63 has a complex velocity behavior in SiO emission; however the
overall distribution angle is parallel to the methanol maser distribution
angle.
iThere is some inconsistency in the maser distribution orientation in the
literature. See $\S$4.6.
Figure 1: Spectral line profiles for sources observed with the JCMT. The black
spectrum is of the SiO (6–5) line, the red spectrum shows the SO (67–56) line,
and the blue spectrum is of the CH3OH (21,1–10,1) line. The antenna
temperature ${\rm T_{A}^{*}}$ for the SiO line is shown on the left side of
each plot. Note that the SO and CH3OH lines have been offset and in may cases
scaled to fit onto the same scale, with the scale factor shown as the divisor
at the end of the label on the right side of each plot. For example, the SO
and CH3OH lines for G328.81+0.63 have been scaled down by a factor of 8. The
conversion factor from temperature to flux density per beam is about 7
Jy/beam/K with a systematic calibration uncertainty of about 25%, although the
coupling of different components into the beam is another major uncertainty in
such complicated sources. All line profiles have been adjusted to local
standard of rest velocities (vlsr), with the adopted systemic velocity of the
YSO (based on the velocity of the peak of the thermal emission) shown by a
vertical dashed line. The green areas correspond to the range of vlsr of the
6.7 GHz methanol maser spots associated with each source. These maser
velocities are taken from either Norris et al. (1993), Phillips et al. (1998),
or Walsh et al. (1998).
Figure 2: Same as the caption for Figure 1. For G345.01+1.79 there are two
vlsr vertical lines. Both G345.01+1.79 and G345.01+1.80 are in the JCMT beam.
The short-dashed vertical line on the right corresponds to the vlsr of
G345.01+1.80, and the green area on the right shows the range of vlsr for the
methanol masers associated with this source only. The long-dashed vertical
line on the left corresponds to the vlsr of G345.01+1.79, and the green area
on the left shows the range of vlsr for the methanol masers associated with
that source only.
Figure 3: Observations of G308.918+0.123. a) A grayscale image of the H2
emission (De Buizer 2003) with the mm continuum contours (thick brown)
overlaid. The cross represents the maser group location, and the elongated
axis shows the position angle of the linear maser distribution. Dashed
ellipses encompass areas of positively identified H2 emission (other
“emission” in the field is likely due to improper continuum subtraction). The
dashed lines divide the frame into quadrants parallel (marked with a “$\|$”)
and perpendicular (marked with a “$\bot$”) to the linearly distributed
methanol maser position angle. The ellipse in the corner of the image shows
the mm beamsize. The mm continuum contours shown are 2 and 4-$\sigma$ the rms
noise given in Table 2. b) A close-up of the cm continuum from Phillips et al.
1998 (thin black contours) and mm continuum (thick brown contours), and the
location of the masers (cross). The mm beam size is shown by the ellipse in
the corner. The mm continuum contours shown are 2, 3, 4, and 5-$\sigma$ the
rms noise given in Table 2.
Figure 4: Observations of G318.95-0.20. a) The 11.7 $\mu$m T-ReCS image of the
elongated mid-IR emission, with the methanol masers overplotted (triangles).
The cross represents the absolute astrometric error of the mid-IR image with
respect to the maser positions. b) H2 emission (De Buizer 2003) is shown in
grayscale, with the SiO red and blue-shifted outflow contours overlaid. The
SiO contours shown are 3, 5, 7, and 9-$\sigma$ the rms noise given in Table 2.
Mid-IR contours (green) are also shown. Other symbols are same as in Figure
3a. c) The SiO (solid contours) with the HCO+ (dashed contours) of Minier
(2004) overplotted. The green contours are the mid-IR emission.
Figure 5: Observations of G320.23-0.28. a) The H2 emission (De Buizer 2003) in
grayscale, with the SiO red and blue-shifted outflow contours overlaid. Dashed
ellipses encompass areas of positively identified H2 emission (other
“emission” in the field is likely due to improper continuum subtraction). SiO
contours are 3, 4, and 5-$\sigma$ for red and 3 and 4-$\sigma$ for blue the
rms given in Table 2. Millimeter continuum contours (brown) are also shown,
with contours of 8, 12, and 16-$\sigma$ the rms in Table 2. The cm continuum
(black contours) of Walsh et al. (1998) are also overplotted. All other
symbols are the same as in Figure 3a. b) A 3-color Spitzer IRAC image of the
region centered on the maser location (white cross). Red is 8.0 $\mu$m, green
is 4.5 $\mu$m, and blue is 3.6 $\mu$m. The 4.5 $\mu$m filter encompasses many
outflow lines and shows enhanced emission above the other filters in the
outflow at this location. Angular scale and image orientation are also given.
Figure 6: Observations of G328.81+0.63. a) The H2 emission (De Buizer 2003) in
grayscale, with the SiO integrated emission contours overlaid (gray). Dashed
ellipses encompass areas of positively identified H2 emission (other
“emission” in the field is likely due to improper continuum subtraction and
filter ghosts). SiO contours are 2, 3, 4, 5, 6, 7, and 8-$\sigma$ the rms
value of 0.5 Jy/beam km/s. Millimeter continuum contours (brown) are also
shown, with contours of 8, 12, and 16-$\sigma$ the rms in Table 2. All other
symbols are the same as in Figure 3a. b) Overlays of the mid-infrared T-ReCS
image (green contours) and the cm continuum emission (thin black contours) of
Ellingsen et al. (2005). c) Velocity channel maps showing the complex velocity
structure of the SiO emission.
Figure 7: Observations of G331.132-0.244. a) The H2 emission (De Buizer 2003)
is shown as grayscale. The SiO emission red-shifted and blue-shifted contours
are overlaid with values of 2, 3, 4, and 5-$\sigma$ the rms given in Table 2.
Also shown are the mm continuum emission (brown contours) with contour levels
of 3, 6, 9, and 12-$\sigma$ the rms given in Table 2. All other symbols are
the same as in Figure 3a. b) The near-IR continuum emission from De Buizer
(2003) is shown in grayscale, and the cm continuum emission from Phillips et
al. (1998) is overplotted as black contours. Also shown are the mm continuum
contours (brown).
Figure 8: Observations of G331.28-0.19. a) The SiO red and blue-shifted
outflow contours. The contours shown are 2, 3, 4 and 5-$\sigma$ the rms value
given in Table 2. Overlaid are the millimeter continuum emission contours
(brown), with values of 2, 4, and 6-$\sigma$ the rms value given in Table 2.
b) The same region surrounding the mm continuum contours (brown), with the cm
continuum (black contours) of Phillips et al. (1998), 11.7 $\mu$m image from
T-ReCS (green contours), and the H2 emission of De Buizer (2003) shown in
grayscale. All other symbols are the same as in Figure 3a. c) A 3-color
Spitzer IRAC image of the region. Red is 8.0 $\mu$m, green is 4.5 $\mu$m, and
blue is 3.6 $\mu$m. The white cross marks the maser location. Angular scale
and image orientation are also given.
|
arxiv-papers
| 2008-10-27T23:10:31 |
2024-09-04T02:48:58.493905
|
{
"license": "Public Domain",
"authors": "J.M. De Buizer (1,2), R.O. Redman (3), S.N. Longmore (4,5), J. Caswell\n (5), and P.A. Feldman (3) ((1) SOFIA-USRA, (2) Gemini Observatory, (3) NRCC,\n (4) U. New South Wales, (5) ATNF)",
"submitter": "James M. De Buizer",
"url": "https://arxiv.org/abs/0810.4951"
}
|
0810.5036
|
# Dehn twists have roots
Dan Margalit and Saul Schleimer Dan Margalit
Department of Mathematics
503 Boston Ave
Tufts University
Medford, MA 02155 dan.margalit@tufts.edu Saul Schleimer
Department of Mathematics
University of Warwick
Coventry, CV4 7AL, UK s.schleimer@warwick.ac.uk
This work is in the public domain.
Let $S_{g}$ denote a closed, connected, orientable surface of genus $g$, and
let $\operatorname{Mod}(S_{g})$ denote its mapping class group, that is, the
group of homotopy classes of orientation preserving homeomorphisms of $S_{g}$.
Fact. If $g\geq 2$, then every Dehn twist in $\operatorname{Mod}(S_{g})$ has a
nontrivial root.
It follows from the classification of elements in
$\operatorname{Mod}(S_{1})\cong\operatorname{SL}(2,\mathbb{Z})$ that Dehn
twists are primitive in the mapping class group of the torus.
For Dehn twists about separating curves, the fact is well-known: if $c$ is a
separating curve then a square root of the Dehn twist $T_{c}$ is obtained by
rotating the subsurface of $S_{g}$ on one side of $c$ through an angle of
$\pi$. In the case of nonseparating curves, the issue is more subtle. We give
two (equivalent) constructions of roots below.
## Geometric construction.
Fix $g\geq 2$. Let $P$ be a regular $(4g-2)$-gon. Glue opposite sides to
obtain a surface $T\cong S_{g-1}$. The rotation of $P$ about its center
through angle $2\pi g/(2g-1)$ induces a periodic map $f$ of $T$. Notice that
$f$ fixes the points $x,y\in T$ that are the images of the vertices of $P$.
Let $T^{\prime}$ be the surface obtained from $T$ by removing small open disks
centered at $x$ and $y$. Define $f^{\prime}=f|T^{\prime}$.
Let $A$ and $B$ be annular neighborhoods of the boundary components of
$T^{\prime}$. Modify $f^{\prime}$ by an isotopy supported in $A\cup B$ so that
* $\cdot$
$f^{\prime}|\partial T^{\prime}$ is the identity,
* $\cdot$
$f^{\prime}|A$ is a $g/(2g-1)$–left Dehn twist, and
* $\cdot$
$f^{\prime}|B$ is a $(g-1)/(2g-1)$–right Dehn twist.
Identify the two components of $\partial T^{\prime}$ to obtain a surface
$S\cong S_{g}$ and let $h:S\to S$ be the induced map. Then $h^{2g-1}$ is a
left Dehn twist along the gluing curve, which is nonseparating.
## Algebraic construction.
Let $c_{1},\dots,c_{k}$ be curves in $S_{g}$ where $c_{i}$ intersects
$c_{i+1}$ once for each $i$, and all other pairs of curves are disjoint. If
$k$ is odd, then a regular neighborhood of $\cup c_{i}$ has two boundary
components, say, $d_{1}$ and $d_{2}$, and we have a relation in
$\operatorname{Mod}(S_{g})$ as follows:
$(T_{c_{1}}^{2}T_{c_{2}}\cdots T_{c_{k}})^{k}=T_{d_{1}}T_{d_{2}}.$
This relation comes from the Artin group of type $B_{n}$, in particular, the
factorization of the central element in terms of standard generators [2]. In
the case $k=2g-1$, the curves $d_{1}$ and $d_{2}$ are isotopic nonseparating
curves; call this isotopy class $d$. Using the fact that $T_{d}$ commutes with
each $T_{c_{i}}$, we see that
$[(T_{c_{1}}^{2}T_{c_{2}}\cdots T_{c_{2g-1}})^{1-g}T_{d}]^{2g-1}=T_{d}.$
## Other roots.
All roots of Dehn twists are obtained in a similar way. That is, if $f$ is a
root of a Dehn twist $T_{d}$ then the canonical reduction system for $f$ is
$d$ [1]. By the Nielsen–Thurston classification for surface homeomorphisms
[3], if we cut the surface along $d$, then $f$ restricts to a finite order
element.
## Roots of half-twists.
Let $S_{0,2g+2}$ be the sphere with $2g+2$ punctures (or cone points) and let
$d$ be a curve in $S_{0,2g+2}$ with 2 punctures on one side and $2g$ on the
other. On the side of $d$ with 2 punctures, we perform a left half-twist, and
on the other side we perform a $(g-1)/(2g-1)$–right Dehn twist by arranging
the punctures so that one puncture is in the middle, and the other punctures
rotate around this central puncture. The $(2g-1)^{\textrm{\tiny st}}$ power of
the composition is a left half-twist about $d$. Thus, we have roots of half-
twists in $\operatorname{Mod}(S_{0,2g+2})$ for $g\geq 2$. There is a 2-fold
orbifold covering $S_{g}\to S_{0,2g+2}$ where the relation from our algebraic
construction above descends to this relation in
$\operatorname{Mod}(S_{0,2g+2})$. A slight generalization of this construction
gives roots of half-twists in any $\operatorname{Mod}(S_{0,n})$ with $n\geq
5$.
## Roots of elementary matrices.
If we consider the map
$\operatorname{Mod}(S_{g})\to\operatorname{Sp}(2g,\mathbb{Z})$ given by the
action of $\operatorname{Mod}(S_{g})$ on $H_{1}(S_{g},\mathbb{Z})$, we also
see that elementary matrices in $\operatorname{Sp}(2g,\mathbb{Z})$ have roots;
for instance, we have
$\left(\begin{array}[]{rrrr}1&0&0&1\\\ 0&1&0&0\\\ 0&1&-1&1\\\ 0&1&-1&0\\\
\end{array}\right)^{3}=\left(\begin{array}[]{rrrr}1&1&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&0&0&1\\\ \end{array}\right)_{.}$
By stabilizing, we obtain cube roots of elementary matrices in
$\operatorname{Sp}(2g,\mathbb{Z})$ for $g\geq 2$.
## Roots of Nielsen transformations.
Let $F_{n}$ denote the free group generated by $x_{1},\dots,x_{n}$, let
$\operatorname{Aut}(F_{n})$ denote the group of automorphisms of $F_{n}$, and
assume $n\geq 2$. A Nielsen transformation in $\operatorname{Aut}(F_{n})$ is
an element conjugate to the one given by $x_{1}\mapsto x_{1}x_{2}$ and
$x_{k}\mapsto x_{k}$ for $2\leq k\leq n$. The following automorphism is the
square root of a Nielsen transformation in $\operatorname{Aut}(F_{n})$ for
$n\geq 3$.
$\begin{array}[]{rcl}x_{1}&\mapsto&x_{1}x_{3}\\\
x_{2}&\mapsto&x_{3}^{-1}x_{2}x_{3}\\\ x_{3}&\mapsto&x_{3}^{-1}x_{2}\\\
\end{array}$
Taking quotients, this gives a square root of a Nielsen transformation in
$\operatorname{Out}(F_{n})$ and, multiplying by $-\textrm{Id}$, a square root
of an elementary matrix in $\operatorname{SL}(n,\mathbb{Z})$, $n\geq 3$.
Finally, our roots of Dehn twists in $\operatorname{Mod}(S)$ can be modified
to work for punctured surfaces, thus giving “geometric” roots of Nielsen
transformations in $\operatorname{Out}(F_{n})$.
## References
* [1] Joan S. Birman, Alex Lubotzky, and John McCarthy. Abelian and solvable subgroups of the mapping class groups. Duke Math. J., 50(4):1107–1120, 1983.
* [2] Egbert Brieskorn and Kyoji Saito. Artin-Gruppen und Coxeter-Gruppen. Invent. Math., 17:245–271, 1972.
* [3] William P. Thurston. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.), 19(2):417–431, 1988.
|
arxiv-papers
| 2008-10-28T14:27:04 |
2024-09-04T02:48:58.509822
|
{
"license": "Public Domain",
"authors": "Dan Margalit and Saul Schleimer",
"submitter": "Dan Margalit",
"url": "https://arxiv.org/abs/0810.5036"
}
|
0810.5039
|
# Deformed spaces and loop cosmology
Marco Valerio Battisti ICRA and Phys. Dept. University of Rome “Sapienza”,
P.le A. Moro 5 00185 Rome, Italy battisti@icra.it
###### Abstract
The non-singular bouncing solution of loop quantum cosmology is reproduced by
a deformed minisuperspace Heisenberg algebra. This algebra is a realization of
the Snyder space, is almost unique and is related to the $\kappa$-Poincaré
one. Since the sign of the deformation parameter it is not fixed, the
Friedmann equation of braneworlds theory can also be obtained. Moreover, the
sign is the only freedom in the picture and these frameworks are the only ones
which can be reproduced by our deformed scheme. A generalized uncertainty
principle for loop quantum cosmology is also proposed.
## Introduction
In the last years a wide interest in the analysis of the non-commutative
framework is increased (for a review see [1]). In particular, this approach
can be considered a plausible candidate for describing physics at the Planck
scale [2] and can be related to the intuitions of doubly special relativity
(DSR) [3] which arises as a semi-classical limit of quantum gravity (see [4]
and the references therein). On the other hand, a prerequisite of any quantum
theory of gravity is to solve the space-time singularities predicted by the
Einstein theory of general relativity. One of the most important example is
the big-bang singularity appearing in the standard model of cosmology, which
Friedmann dynamics is expected to be modified by quantum effects in the regime
of small scale factor. Loop quantum cosmology (LQC) [5] leads to a resolution
of the singularity replacing the big-bang by a big-bounce as soon as the
matter energy density reaches the Planck scale, i.e. in this extreme region
(quantum) gravity behaves repulsively [6].
From this perspective it is natural to investigate the fate of the
cosmological singularity in a deformed (minisuper)-space framework [7]. Here,
we consider generalized commutation relations which leave undeformed the
translation group and preserve the rotational invariance, and apply this
framework to the Friedmann-Robertson-Walker (FRW) Universes (for more details
see [8]). As a result the effective Friedmann equation of LQC [9] naturally
arises from the deformed scheme. Interesting, since the deformed Heisenberg
algebra is fixed up to a sign, also the braneworlds Friedmann equation [10] is
predicted by our model. In other words, this deformed phase space can be
regarded, from a phenomenological point of view, as an effective framework
which is able to describe the results obtained in both LQC and Randall-Sundrum
braneworlds scenario. In this deformed scheme, the different predictions of
such theories can be easily understood considering the opposite sign in the
deformation term. It is worth noting that, a generalized uncertainty principle
naturally arises in our framework and, in the braneworlds-like case, it
resembles the one predicted by string theory. At the same level, a generalized
uncertainty relation can also be proposed for LQC.
The paper is organized as follows. Section 1 is devoted to discuss a non-
commutative space and its commutations relations. In Section 2 this framework
is applied to the isotropic cosmological models analyzing the modifications
induced on the Friedmann equation. Finally, in Section 3, implications of the
picture are analyzed. Concluding remarks follow.
## 1 Deformed Heisenberg algebra
Let us start by considering a $n$-dimensional deformed (non-commutative)
Euclidean space such that commutator between the coordinates has the non-
trivial structure
$[\tilde{x}_{i},\tilde{x}_{j}]=\mp i\alpha M_{ij},$ (1)
where with $\tilde{x}$ we refer to the non-commutative coordinates and
$\alpha>0$ is the deformation parameter with dimension of a length square. We
then demand that the rotation generators $M_{ij}=x_{i}p_{j}-x_{j}p_{i}$
satisfy the ordinary $SO(n)$ algebra and that the translation group is not
deformed, i.e. $[p_{i},p_{j}]=0$. Is then natural to assume the commutators
between $M_{ij}$ and $\tilde{x}_{k}$, as well as between $M_{ij}$ and $p_{k}$,
as undeformed. This way we deal with the (Euclidean) Snyder space [11].
We now consider a general rescaling of the non-commutative coordinates
$\tilde{x}_{i}$ in terms of a momentum-dependent function $\varphi\left(\alpha
p^{2}\right)$ as
$\tilde{x}_{i}=x_{i}\varphi\left(\alpha p^{2}\right),$ (2)
i.e. we consider a realization of the algebra (1) in terms of ordinary phase
space variables [12]. It is not difficult to show that the function $\varphi$
is uniquely fixed (up to a sign) by our natural assumptions and reads
$\varphi\left(\alpha p^{2}\right)=\sqrt{1\pm\alpha p^{2}}$. This way the
commutator between non-commutative coordinates and momenta is given by
$[\tilde{x}_{i},p_{j}]=i\delta_{ij}\sqrt{1\pm\alpha p^{2}}$ (3)
and the ordinary Heisenberg algebra is recovered as soon as $\alpha\rightarrow
0$. This deformed algebra closes in sense that all the Jacobi identities are
satisfied and can be related to the $\kappa$-Poincaré one [13]. Interest in
non-commutative (or deformed) phase space arises in order to mathematically
describe DSR and, in particular, some of DSR models can be formulated as
generalizations of Snyder model [14]. As last point, it is worth noting that
in the case of minus sign, the momentum is limited from above since $p\in
I\equiv\left(-1/\sqrt{\alpha},1/\sqrt{\alpha}\right)$. Therefore, in this
case, we have a truncation of the phase space at $p^{2}\geq\alpha$.
## 2 Friedmann dynamics in the deformed framework
The FRW cosmological models are characterized by imposing the isotropy on the
Cauchy surfaces which fill the space-time manifold. Isotropy reduces the phase
space of GR to be two dimensional with coordinates $(a,p_{a})$, where the
scale factor $a$ is the only degree of freedom of the system. It describe the
expansion of the Universe and the standard model of cosmology is based on this
models [15]. The dynamics of these Universes can be obtained from the extended
Hamiltonian
$\mathcal{H}_{E}=\frac{2\pi G}{3}N\frac{p_{a}^{2}}{a}+\frac{3}{8\pi G}Nak-
Na^{3}\rho+\lambda\pi,$ (4)
where $\lambda$ is a Lagrange multiplier and the parameter $k$ can be zero or
$\pm 1$ leading to the flat, open or closed Universe respectively. The term
$\lambda\pi$ is introduced since $\pi$, the momentum conjugate to the lapse
function $N$, vanishes. In the expression above $\rho=\rho(a)$ denotes a
generic energy density we have introduced into the dynamics.
Let us now consider the modifications induced on the dynamics of the FRW
models by the deformed Heisenberg algebra discussed above [8]. In particular,
we assume the symplectic structure of the minisuperspace as deformed and thus
the Poisson bracket (for any two-dimensional phase space function) appears to
be
$\\{F,G\\}_{\alpha}=\left(\frac{\partial F}{\partial a}\frac{\partial
G}{\partial p_{a}}-\frac{\partial F}{\partial p_{a}}\frac{\partial G}{\partial
a}\right)\sqrt{1\pm\alpha p_{a}^{2}}.$ (5)
We deal with a one-dimensional mechanical system and thus the only non-trivial
commutators is given by
$\\{a,p_{a}\\}=\sqrt{1\pm\alpha p_{a}^{2}}.$ (6)
Since the Poisson bracket $\\{N,\pi\\}=1$ is not affected by the deformations
induced by the $\alpha$ parameter on the system, the equations of motion
$\dot{N}=\\{N,\mathcal{H}_{E}\\}=\lambda$ and
$\dot{\pi}=\\{\pi,\mathcal{H}_{E}\\}=\mathcal{H}=0$ remain unchanged111As
usually, $\mathcal{H}=0$ is the scalar constraint and is obtained by requiring
the primary constraint $\pi=0$ will be satisfied at all times.. On the other
hand, the equations of motion of the scale factor $a$ and its conjugate
momentum $p_{a}$ become modified in such an approach via the deformed
symplectic geometry (5) and read
$\displaystyle\dot{a}$ $\displaystyle=$
$\displaystyle\\{a,\mathcal{H}_{E}\\}_{\alpha}=\frac{4\pi
G}{3}N\frac{p_{a}}{a}\sqrt{1\pm\alpha p_{a}^{2}},$ (7)
$\displaystyle\dot{p}_{a}$ $\displaystyle=$
$\displaystyle\\{p_{a},\mathcal{H}_{E}\\}_{\alpha}=N\left(\frac{2\pi
G}{3}\frac{p_{a}^{2}}{a^{2}}-\frac{3}{8\pi
G}k+3a^{2}\rho+a^{3}\frac{d\rho}{da}\right)\sqrt{1\pm\alpha p_{a}^{2}}.$
The equation of motion for the Hubble rate $(\dot{a}/a)$ can be obtained
solving the scalar constraint $\mathcal{H}=0$ with respect to $p_{a}$ and then
considering the first equation of (7). Explicitly it becomes
$\left(\frac{\dot{a}}{a}\right)^{2}=\left(\frac{8\pi
G}{3}\rho-\frac{k}{a^{2}}\right)\left[1\pm\frac{3\alpha}{2\pi
G}a^{2}\left(a^{2}\rho-\frac{3}{8\pi G}k\right)\right].$ (8)
We refer to this equation as the deformed Friedmann equation as it entails the
modification arising from the deformed Heisenberg algebra previously analyzed.
Of course, for $\alpha\rightarrow 0$ the ordinary Friedmann equation is
recovered. Let us consider the flat FRW Universe, i.e. the $k=0$ model. In
this case the deformed equation (8) appears to be
$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi
G}{3}\rho\left(1\pm\frac{\rho}{\rho_{c}}\right),$ (9)
where $\rho_{c}=(2\pi G/3\alpha)\rho_{P}$ is the critical energy density, and
$\rho_{P}$ denotes the Planck one. When the limit $\alpha\rightarrow 0$ is
taken into account, the critical energy density diverges leading to the
ordinary dynamics. It is worth stressing that we have assumed the existence of
a fundamental minimal length. In fact, as widely accepted, one of the most
peculiar consequences of all promising quantum gravity theories is the
existence of a fundamental cut-off length, which should be related to the
Planck one [16]. Therefore, although this minimal length appears differently
in distinct contexts, it is reasonable that the scale factor (the energy
density) has a minimum (maximum) at the Planck scale.
The modified Friedmann equations (9) are known in literature. The one with the
$(-)$ deformation term is the effective equation incorporating the LQC effects
on the FRW dynamics [9]. It can be obtained using the geometric formulation of
quantum mechanics [17]. On the other hand, the string inspired Randall-Sundrum
braneworlds scenario leads to a modified Friedmenn equation as in (9) with the
$(+)$ sign [10]. The opposite sign of the $\rho^{2}$-term in such an equation,
is the well-known key difference between the effective LQC and the Randall-
Sundrum framework. In fact, the former approach leads to a non-singular
bouncing cosmology while in the latter, because of the positive sign, the
Hubble rate can not vanish and the Universe can not experiences a bounce (or
more generally a turn-around) in the scale factor.
## 3 Quantum mechanical implications of the deformed picture
As we have seen, the deformed algebra (3) leads to effective dynamics of loop
and braneworlds cosmologies in the minus or plus case respectively. Let us now
investigate some implications of the deformed framework in the quantum theory.
Firstly, we have to stress that the ($\pm$)-frameworks are, of course,
physically different. More precisely, the deformed Hilbert spaces
$\mathcal{F}_{\pm}$ underling the algebras (6) can be written as [8]
$\mathcal{F}_{\pm}=L^{2}\left(\mathbb{R}(I),dp_{a}/\sqrt{1\pm\alpha
p_{a}^{2}}\right),$ (10)
for the $\pm\alpha$ deformation of the ordinary Heisenberg theory,
respectively. It is worth noting that these Hilbert spaces are unitarily
inequivalent each other and also with respect to the ordinary one
$L^{2}(\mathbb{R},dp_{a})$. This is not surprising since the deformation of
the canonical commutation relations can be viewed, from the realization (2),
as an algebra homomorphism which is a non-canonical transformation and thus it
can not be implemented as an unitary transformation. New features are then
introduced at both classical and quantum level. (For an application of the
formalism to the harmonic oscillator problem see [8].)
A peculiar feature which deserves to be analyzed is the uncertainty principle
underlying the deformed symplectic structure (5). The generalized uncertainty
relation can be immediately obtained from the commutator (6) and reads
$\Delta a\Delta p_{a}\geq\frac{1}{2}|\langle\left(1\pm\alpha
p_{a}^{2}\right)^{1/2}\rangle|,$ (11)
by which, using the basic proprieties $\langle p_{a}^{2n}\rangle\geq\langle
p_{a}^{2}\rangle$ and $\langle p_{a}^{2}\rangle=(\Delta p_{a})^{2}+\langle
p_{a}\rangle^{2}$, the boundary of the allowed region begin
$\Delta a=\frac{1}{2}\left|\left(\frac{1\pm\alpha\langle
p_{a}\rangle^{2}}{(\Delta p_{a})^{2}}\pm\alpha\right)^{1/2}\right|.$ (12)
As we can easily see, for an infinite uncertainty in momentum (or better when
the relation $\Delta p_{a}\gg(\Delta
p_{a})^{\star}\equiv\sqrt{(1\pm\alpha\langle p_{a}\rangle)/\alpha}$ holds),
the uncertainty in the scale factor $a$ no longer vanishes but approaches the
minimal value of $\Delta a_{\text{min}}=\sqrt{\alpha}/2$. It is interesting to
note that, $\Delta a_{\text{min}}$ is the global minimum in the $(+)$-sector,
while $\Delta a_{\text{min}}=0$ is allowed in the $(-)$-one. This appears as
soon as the dispersion on its conjugate momentum $\Delta p_{a}$ reaches the
critical value of $(\Delta p_{a})^{\star}$.
Summarizing, in the $(+)$-sector a nonzero minimal uncertainty in the particle
(Universe) position (the scale factor) appears. The resulting implications are
quite profound. In fact, it is no longer possible to spatially localize a wave
function with arbitrary precision and then no physical states which are
position eigenstates exist at all since they were only formal ones [18]. As we
have seen, this framework leads to the same Friedamnn equation of the Randall-
Sundrum braneworlds scenario and thus, it is not unexpected that the plus
deformed uncertainty relation (11) contains, at the leading order in $\alpha$,
the string theory result $\Delta x\sim(1/\Delta p+l_{s}^{2}\Delta p)$ [19], in
which the string length $l_{s}$ can be identify with $\sqrt{\alpha/2}$.
On the other hand, to obtain the Friedmann equation found in the effective
loop cosmology, the $(-)$-deformed Heisenberg algebra is required. In this
case a vanishing uncertainty in position is allowed and therefore the position
eigenstates are true physical states (in the same sense of the ordinary
quantum mechanics). However, differently from the Heisenberg framework, an
infinite uncertainty in momentum is no longer required and $\Delta
x_{\text{min}}=0$ appears as soon as the finite value $(\Delta
p)^{\star}\propto 1/\sqrt{\alpha}$ is considered (we also remember that in
this scheme there is a cut-off on the momentum, i.e. $|p|\leq
1/\sqrt{\alpha}$). This way, considering the minus relation (11) at the first
order in $\alpha$, a generalized uncertainty principle for LQC can be proposed
to be $\Delta x\sim|1/\Delta p-l_{L}^{2}\Delta p|$, where
$l_{L}=\sqrt{\alpha/2}$ can be regarded as the loop cut-off length scale.
## Conclusions
The equations of motion of the FRW models obtained in LQC and in the
braneworlds scenario can be reproduced by a deformed Heisenberg algebra. This
algebra in the unique one which is consistent, in the sense of the Jacobi
identities, with the assumptions that both the translation and rotation groups
are undeformed and that the commutator between the non-commutative coordinates
is as in (1). Notably, it is also related to the $\kappa$-Poincaré algebra and
the only freedom in (2) lies in the $\pm$ sign. The $(+)$-framework leads to
the effective Friedmann dynamics of Randall-Sundrum braneworlds scenario,
while the opposite one to that of LQC. From this perspective, the former
framework is such that a vanishing uncertainty in position (the scale factor
of the Universe) is not longer allowed. On the other hand, the $(-)$-scheme
implies that the zero uncertainty in position appears for a finite uncertainty
in the momentum, proportional to the natural cut-off of the framework.
Summarizing, a non-commutative (deformed) picture which leads, at
phenomenological level, to the prediction of more general theories can be
formulated. The validity and applicability of this model to more complicated
(and physically interesting) arenas will deserve future investigations.
## References
## References
* [1] M.R.Douglas and N.A.Nekrasov, Rev.Mod.Phys. 73 (2001) 977.
* [2] S.Doplicher, K.Fredenhagen and J.E.Roberts, Phys.Lett.B 331 (1994) 39.
* [3] G.Amelino-Camelia, Int.J.Mod.Phys.D 11 (2002) 35; Phys.Lett.B 510 (2001) 255; J.Magueijo and L.Smolin, Phys.Rev.Lett. 88 (2002) 190403.
* [4] C.Rovelli, arXiv:0808.3505; L.Smolin, arXiv:0808.3765.
* [5] M.Bojowald, Living Rev.Rel. 8 (2005) 11.
* [6] A.Ashtekar, T.Pawlowski and P.Singh, Phys.Rev.Lett. 96 (2006) 141301; Phys.Rev.D 73 (2006) 124038.
* [7] M.V.Battisti and G.Montani, Phys.Lett.B 656 (2007) 96; Phys.Rev.D 77 (2008) 023518; B.Vakili and H.R.Sepangi, Phys.Lett.B 651 (2007) 79; A.Bina, K.Atazadeh and S.Jalalzadeh, Int.J.Theor.Phys. 47 (2008) 1354; H.Garcia-Compean, O.Obregon and C.Ramirez, Phys.Rev.Lett. 88 (2002) 161301.
* [8] M.V.Battisti, arXiv:0805.1178.
* [9] P.Singh, Phys.Rev.D 73 (2006) 063508; P.Singh, K.Vandersloot and G.V.Vereshchagin Phys.Rev.D 74 (2006) 043510.
* [10] R.Maartens, Living Rev.Rel. 7 (2004) 7.
* [11] H.S.Snyder, Phys.Rev. 71 (1947) 38.
* [12] S.Meljanac, M.Milekovic and S.Pallua, Phys.Lett.B 328 (1994) 55; L.Jonke and S.Meljanac, Phys.Lett.B 526 (2002) 149.
* [13] M.Maggiore, Phys.Lett.B 304 (1993) 65; Phys.Rev.D 49 (1994) 5182.
* [14] J.Kowalski-Glikman, Phys.Lett.B 547 (2002) 291; J.Kowalski-Glikman and S.Nowak, Class.Quant.Grav. 20 (2003) 4799.
* [15] E.W.Kolb and M.S.Turner, The Early Universe (Adison-Wesley Reading, 1990).
* [16] L.J.Garay, Int.J.Mod.Phys.A 10 (1995) 145.
* [17] V.Taveras, arXiv:0807.3325.
* [18] A.Kempf, G.Mangano and R.B.Mann, Phys.Rev.D 52 (1995) 1108,; A.Kempf, J.Math.Phys. 38 (1997) 1347.
* [19] D.J.Gross and P.F.Mendle, Nucl.Phys.B 303 (1988) 407; K.Konishi, G.Paffuti and P.Provero, Phys.Lett.B 234 (1990) 276.
|
arxiv-papers
| 2008-10-28T14:35:20 |
2024-09-04T02:48:58.514640
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marco Valerio Battisti",
"submitter": "Marco Valerio Battisti",
"url": "https://arxiv.org/abs/0810.5039"
}
|
0810.5173
|
# Discrete Coordinate Scattering Approach to Nano-optical Resonance
Jun Yang Department of Physics, Peking University, 100871, Beijing, P.R.
China H. Dong dhui@itp.ac.cn Institute of Theoretical Physics, Chinese
Academy of Sciences, 100190, Beijing, P.R. China C.P. Sun suncp@itp.ac.cn
Institute of Theoretical Physics, Chinese Academy of Sciences, 100190,
Beijing, P.R. China
###### Abstract
In this letter, we investigate the coherent tunneling process of photons
between a defected circular resonator and a waveguide based on the recently
developed discrete coordinate scattering methods (L. Zhou et al., Phys. Rev.
Lett. 101, 100501 (2008)). We show the detailed microscopic mechanism of the
tunneling and present a simple model for defect coupling in the resonator. The
Finite-Difference Time-Domain(FDTD) numerical results is explored to
illustrate the analysis results.
Recently, resonant optical cavities has attracted lots of interesting from
both experimental and theoretical aspect vahla03review . Devices based on
these micro-cavities has been widely used for processing optical signals
joannopoulos . In most hardware of these signal processing, the elementary
unit consists of a micro-cavity and a side coupling waveguide vahla03review .
This unit has been experimentally realized based on photonic crystal, most
recently ring made with GeN nanowire pdyang06prl . With such kind photonic
setups, photon blockade was experimentally observed in this unit kimble08 .
For this unit, it is well known that the incident wave is totally reflected by
the ring resonator when it is resonant with the modes of the ring. Thus, the
resonator takes the role of frequency-selection little98opt in this unit.
Such transfer process in the unit is crucial to design more complicated
devices. To investigate this phenomena, one way is to directly solve the
Maxwell equations. However, it is impossible to get an analytic result to
reveal the key physical mechanism for even a simple configuration. Thus, this
is usually done by numerical simulation, such as Finite-Differential-Time-
Domain(FDTD) method taflove , which is used as a validation approach.
Theoretically, coupled mode theory(CMT) is introduced to study the resonator
scattering properties in this unit little98lightware ; little98opt . However,
this is done in the mode space, and therefore not intuitive in real space.
In this letter, we generalize the most recently developed discrete coordinate
scattering methods lan08 to investigate the detailed scattering properties
for the optical signal processing. Here we model the resonator and the
waveguide as array of coupled cavities. And the coupling between resonator and
waveguide is modeled as hopping of the photon among cavities. We present the
method by a qualitatively analysis of an example of a notched-ring resonator.
Rigorous FDTD numerical simulations is also performed to verify the obtained
analytical result.
Figure 1: (Color online) Frequency-selective filter. (a) A notched ring is
evanescently coupled to the waveguide. The notch couples the CW and CCW
traveling modes; (b) Model of the unit by discrete coupled resonator array.
The notch is characterized as point inhomogeneous of the tunneling strength
along the ring.
The elementary units under investigation is schematically illustrated in Fig.
1(a). The defected ring resonator is evanescently coupled to the signal
waveguide. For perfect ring, it supports the clockwise(CW) and counter
clockwise(CCW) traveling wave and there is no interaction between the two
traveling wave modes. However, the scattering of the defected notch causes the
recombination of the CW and CCW modes. In the context of CMT little98opt , the
notched ring is mapped to the coupled double-ring resonator and the notch in
single ring plays the role of the gap, which couples the double rings. Here,
we treat the defected circular resonator as the discrete coupled cavity array
with coupling constant different at the defected area, which is illustrated in
Fig. 1(b). The array consists of $N$ resonators with tight-binding coupling.
The notch in the ring is marked with a dashed line. The Hamiltonian here reads
$H_{c}=\sum_{j}\omega
a_{j}^{\dagger}a_{j}-J\left(a_{j}^{\dagger}a_{j+1}+\text{h.c.}\right)+\delta
J\left(a_{0}^{\dagger}a_{1}+\text{h.c.}\right),$ (1)
where $a_{j}$ is the creation operator for photon with energy $\omega$ the
$j-th$ cavity and $J$ is the tunneling strength between nearby cavities with
defected induced inhomogeneous coupling $\delta J$. By the Fourier
transformation
$a_{j}^{\dagger}=\sum\exp\left(-ijk\right)c_{k}^{\dagger}/\sqrt{N}$, the
Hamiltonian in the momentum space reads $H_{c}=\sum_{k}\left(\omega-2J\cos
k\right)c_{k}^{\dagger}c_{k}+\delta
J\sum_{k,p}\left[\exp\left(ip\right)c_{k}^{\dagger}c_{p}+h.c.\right]/N$. In
the rotation wave approximation, we keep only $p=\pm k$ term and obtain the
Hamiltonian as
$H_{c}=\sum_{k>0}\varepsilon_{k}\left(c_{k}^{\dagger}c_{k}+c_{-k}^{\dagger}c_{-k}\right)+\sum_{k>0}2\delta
J\left(\exp\left(-ik\right)c_{k}^{\dagger}c_{-k}+\text{h.c.}\right)/N$, where
$\varepsilon_{k}=\omega-2\left(J-\delta J/N\right)\cos k$. Physically,
$c_{k}$and $c_{-k}$ stand for the CW and CCW traveling wave, which has been
introduced phenomenally in CMT. The coupling between the two modes is induced
by the defect of the inhomogeneous of tunneling $\delta J$. Diagonalizing the
Hamiltonian with unitary transformation
$c_{k}^{\dagger}=\exp\left(ik/2\right)(\alpha_{k}^{\dagger}+\beta_{k}^{\dagger})\sqrt{2}$
and
$c_{-k}^{\dagger}=\exp\left(-ik/2\right)(\alpha_{k}^{\dagger}-\beta_{k}^{\dagger})\sqrt{2}$
for $k>0$, we obtain
$H_{c}=\sum_{k>0}\varepsilon_{k}^{+}\alpha_{k}^{\dagger}\alpha_{k}+\varepsilon_{k}^{-}\beta_{k}^{\dagger}\beta_{k},$
(2)
where $\varepsilon_{k}^{\pm}=\left(\varepsilon_{k}\pm 2\delta J/N\right)$ and
$\displaystyle\alpha_{k}^{\dagger}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2N}}\sum_{j}2a_{j}^{\dagger}\cos\left[(j-1/2)k\right]$
(3) $\displaystyle\beta_{k}^{\dagger}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2N}}\sum_{j}2ia_{j}^{\dagger}\sin\left[(j-1/2)k\right].$
(4)
are the eigen-modes of the notched ring. And they were related to photonic
molecule in Ref. bayer98prl ; pdyang06prl . Actually, this single defected
notch induced the split of the degenerate resonant peak of CW and CCW
traveling wave. The splitting can be probed by checking the transmission rate
in the waveguide, which will be investigated in the following parts. Actually,
those modes correspond to a symmetric and antisymmetric state with respect to
the notch. The amplitude for the modes $\alpha_{k}$ and $\beta_{k}$ is
illustrated in Fig. 2. In this discrete model of the ring resonator, the
position $j_{\mathrm{notch}}$ of notch should between the resonator $a_{0}$
and $a_{1}$, namely $j_{\mathrm{notch}}=1/2$. Thus for the $\alpha$ mode, the
amplitude of the field at the notch is at maximum. However, there is no field
in the notch for the $\beta_{k}$ mode.
Figure 2: Ez-field amplitude of $\alpha_{k}$(Blue line) and $\beta_{k}$(Red
dashed line) mode on the notched ring. The position of notch is marked with
dot line. For $\alpha_{k}$ mode, the amplitude of Ez-field at the notch is at
maximum. However, the amplitude is null for $\beta_{k}$ mode.
Figure 3: Plot of the reflection rate with the parameters
$\omega=\omega_{1}=8.772\times 10^{14}Hz$, $J=J^{\prime}=3.3\times 10^{14}Hz$,
$g=3\times 10^{13}Hz$ and $\delta J=2.98\times 10^{13}Hz$. (a) Theoretical
calculation of the reflection of the filter; (b) FDTD simulation of the
reflection and transition rate. Two resonator peaks appear at the cavity mode
$\varepsilon_{k}^{\pm}$.
To probe the optical property of the resonator, a transmission line waveguide
is introduced to couple the signal in the microcavity. Physically, the
waveguide can be optical fiber, nanowire and defected lines on photonic
crystal for different systems. Here, we also model the transmission line
waveguide as the discrete coupled resonator, which is illustrated in Fig.
1(b). Each resonator couples to the near-by resonators by means of photon
hopping. Those coupled-resonator array has been proposed to be a new type of
optical waveguideyariv99opl . The Hamiltonian for the transmission line
waveguide here reads
$H_{l}=-J^{\prime}\sum\left(b_{j}^{\dagger}b_{j+1}+\text{h.c.}\right)+\omega^{\prime}b_{j}^{\dagger}b_{j},$
(5)
where $b_{j}^{\dagger}$ is the bosonic creation operator for photon on the
$j-th$ resonator. The tight-binding Hamiltonian here is a good approximation
for modeling the coupled resonator array, which is proved in Ref. yariv99opl .
And in Ref. lan08 , it is showed the model is equivalent to the simple
dielectric waveguide under the linear dispersion area with right going wave
$\phi_{R}(x)$ and left going wave $\phi_{L}(x)$ as showed in Ref. fan98 . The
waveguide couples to the cavity at the $0$-th resonator as the tunneling
between 0-th resonator and the $j_{0}$-th resonator of the ring resonator
$H_{I}=g\left(a_{j_{0}}^{\dagger}b_{0}+a_{j_{0}}b_{0}^{\dagger}\right)$. The
total Hamiltonian for this elementary unit reads $H=H_{c}+H_{l}+H_{I}$. For
the single photon case, the eigenwave stationary scattering function can be
written as
$\left|\Phi\right\rangle=\sum_{j}u_{j}b_{j}^{\dagger}\left|0\right\rangle+\sum_{k>0}\left(v_{k}\alpha_{k}^{\dagger}+w_{k}\beta_{k}^{\dagger}\right)\left|0\right\rangle,$
(6)
where $u_{j}$ is the amplitude of photon in the $j-th$ resonator on the
waveguide and $v_{j}$($w_{j}$) is the amplitude of photon on the resonator
with mode $\alpha^{\dagger}$($\beta^{\dagger}$). Substituting the stationary
scattering function into the time-independent Schrödinger equation
$H\left|\Phi\right\rangle=E\left|\Phi\right\rangle$, one can obtain equations
for the amplitude of the site as
$\displaystyle-J^{\prime}\left(u_{j+1}+u_{j-1}\right)+\omega^{\prime}u_{j}+\frac{g\delta_{j,0}}{\sqrt{2N}}\sum_{k>0}\left(v_{k}+w_{k}\right)e^{-i\left(1/2-j_{0}\right)k}+\left(v_{k}-w_{k}\right)e^{-ik(j_{0}-1/2)}$
$\displaystyle=$ $\displaystyle Eu_{j},$ (7)
$\displaystyle\varepsilon_{k}^{+}v_{k}+\frac{gu_{0}}{\sqrt{2N}}\left(e^{i\left(1/2-j_{0}\right)k}+e^{i(j_{0}-1/2)k}\right)$
$\displaystyle=$ $\displaystyle Ev_{k},$ (8)
$\displaystyle\varepsilon_{k}^{-}w_{k}+\frac{gu_{0}}{\sqrt{2N}}\left(e^{i\left(1/2-j_{0}\right)k}-e^{i(j_{0}-1/2)k}\right)$
$\displaystyle=$ $\displaystyle Ew_{k}.$ (9)
When the incident energy coincides with the energy of $\alpha$ mode, namely
$E=\varepsilon_{k}^{+}$, the amplitude $u_{0}=0$ by Eq. (8) and $w_{k}=0$ by
Eq. (9). Thus only $\alpha$ mode is excited here and the Ez-field at the notch
is at maximum. When the incident energy is resonant with the energy of $\beta$
mode, namely $E=\varepsilon_{k}^{-}$, the amplitude $u_{0}=0$ by Equation(9)
and $v_{k}=0$ by Equation(8). Here, only $\beta$ mode is excited and the
amplitude of the field is zero. This phenomena has been numerically predicted
in Ref. little98opt .
For the scattering process, the photon incidents from the left side of the
line waveguide and is partially reflected by the notched ring resonator array.
The amplitude in the waveguide reads
$u_{j}=\left[\exp\left(iqj\right)+r\exp\left(-iqj\right)\right]\theta\left(-j\right)+s\exp\left(iqj\right)\theta\left(j\right)$
, where $r$($s$) is the reflection(transmission) coefficient of the photon.
The coefficient of the transition here is obtained
$s=\left(1-\frac{Q}{2iJ^{\prime}\sin q}\right)^{-1},$ (10)
where
$Q=2g^{2}/N\sum_{k>0}\cos^{2}(k(j_{0}-1/2))/\left(E-\varepsilon_{k}^{+}\right)+\sin^{2}(k(j_{0}-1/2))/\left(E-\varepsilon_{k}^{-}\right)$.
The reflection rate $1-|s|^{2}$ peaks at the resonance with the internal ring
mode $\varepsilon_{k}^{\pm}$, where the optical signal is totally reflected.
To see this resonant filtering effect, we plot the transmission rate $|s|^{2}$
verse the the incident energy of photon in Fig. 3(a). For comparison, a
rigorous FDTD taflove simulation is performed with free available code F2P
qiu for Transverse-Magnetic mode. The simulation is performed with the
following parameter: $n_{0}=1$, $n_{w}=3.2$, $R_{1}=5a$, $R_{2}=4.4a$,
$\theta=54^{\mathrm{o}}$ and $\delta\theta=1.5^{\mathrm{o}}$, where $a=0.25\mu
m$. The transmission and reflection rate is plotted in Fig. 3(b) around the
resonant frequency $\varepsilon_{k}=0.331\times 2\pi c/a$. In Fig. 3(a), the
parameters for this defect ring resonator unit are chosen to fit the numerical
simulation: $\omega=\omega_{1}=8.772\times 10^{14}Hz$, $J=J^{\prime}=3.3\times
10^{14}Hz$, $g=3\times 10^{13}Hz$ and $\delta J=2.98\times 10^{13}Hz$.
Therefore, our model confirms the function of the filter and its mechanism.
However, we should notice a drawback to this approach: the quantitative
analysis will relies on the numerical simulation. At present, the coupling
constant at the notch and between the transmission line and the resonator
should be fitted by numeric simulation.
In conclusion, we have presented a theoretical investigation of the notched
ring filter with discrete coordinate scattering methods. The detailed
mechanism of the coupling between the clockwise and count-clockwise is
revealed to be the inhomogeneous tunneling in the resonator array system. We
have predicted some experimentally accessible results for the coherent
transmission and reflection property and present detailed explanation to field
amplitude distribution at the notch.
This work is supported by NSFC No.10474104, No. 60433050, and No. 10704023,
NFRPCNo. 2006CB921205 and 2005CB724508.
## References
* (1) K. J. Vahala, Nature 424, 839 (2003).
* (2) J. D. Joannoupoulos, R. D. Meade and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press), 1995\.
* (3) P. J. Pauzauskie, D. J. Sirbuly, and P. Yang, Phys. Rev. Lett. 96, 143903 (2006).
* (4) B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, H. J. Kimble, Science 319, 1062 (2008).
* (5) B. E. Little, S. T. Chu and H. A. Haus, Opt. Lett. 23, 1570 (1998).
* (6) A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Methods (Artech House, Boston,2000)
* (7) B. E. Little, S. T. Chu, H. A. Haus, J. S. Foresi and J.-P. Laine, J. Lightware Technol. 15, 998 (1997).
* (8) Lan Zhou, Z. R. Gong, Yu-xi Liu, C. P. Sun and F. Nori, Phys. Rev. Lett. 101, 100501 (2008).
* (9) A. Yariv, Y. Xu, R. K. Lee and A. Scherer, Opt. Lett. 24, 711 (1999).
* (10) J. T. Shen, S. Fan, Phys. Rev. Lett. 95, 213001 (2005); ibid. 98, 153003 (2007); Opt. Lett. 30, 2001 (2005)..
* (11) M. Qiu, F2P: Finite-difference time-domain 2D simulator for photonic devices, http://www.imit.kth.se/info/FOFU/PC/F2P/.
* (12) M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, Phys. Rev. Lett. 81, 2582 (1998).
|
arxiv-papers
| 2008-10-29T00:32:55 |
2024-09-04T02:48:58.523193
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jun Yang, H. Dong and C.P. Sun",
"submitter": "H. Dong",
"url": "https://arxiv.org/abs/0810.5173"
}
|
0810.5233
|
# LOCV calculations for polarized liquid ${}^{3}\mathrm{He}$ with the spin-
dependent correlation
G.H. Bordbar111Corresponding author 222E-Mail: bordbar@physics.susc.ac.ir and
M.J. Karimi Department of Physics, Shiraz University, Shiraz 71454, Iran
###### Abstract
We have used the lowest order constrained variational (LOCV) method to
calculate some ground state properties of polarized liquid ${}^{3}He$ at zero
temperature with the spin-dependent correlation function employing the
Lennard-Jones and Aziz pair potentials. We have seen that the total energy of
polarized liquid ${}^{3}He$ increases by increasing polarization. For all
polarizations, it is shown that the total energy in the spin-dependent case is
lower than the spin-independent case. We have seen that the difference between
the energies of spin-dependent and spin-independent cases decreases by
increasing polarization. We have shown that the main contribution of the
potential energy comes from the spin-triplet state.
Keywords: Liquid ${}^{3}He$, Spin polarized, Correlation, Spin-dependent
## I Introduction
Helium has two stable isotopes: that of mass 4 is readily available as helium
gas or liquid from the atmosphere or gas wells, while that of mass 3 is
extremely rare in nature and only became available commercially in the 1950s
Dob . Liquid ${}^{3}He$ is particularly suited to study correlation among the
strongly interacting many-body fermionic systems. Several approaches have been
used for investigating the properties of normal liquid ${}^{3}He$. These are
mainly based on the STLS scheme Nd , mott localization Gl , spin fluctuation
theory Ms , Green’s function Monte Carlo (GFMC) GFMCa , FN-DMC, DMC, VMC and
EMC simulations DVEF , CBPT formalism FPS , nonperturbative renormalization
group equation KW , nonlocal density functional formalism PT , correlated
basis functions (CBF) CBFa and Fermi hyper-netted chain (FHNC) FHNCa . The
spin polarized liquid ${}^{3}He$ as an interesting many-body system has been
investigated using different approaches such as FHNC FHNCb , GFMC GFMCb , CBF
CBFb and transport theory Trans .
In recent years, we have studied both normal and polarized liquid ${}^{3}He$
at zero and finite temperature Gha1 ; Gha2 ; Gha3 ; Gha4 . In these
calculations, the lowest order constrained variational (LOCV) method based on
the cluster expansion of the energy functional has been used. This method is
fully self-consistent, since it does not introduce any free parameter to the
calculations. We have also used the LOCV method in many-body calculations of
dense matter Ghb . Recently, we have used this method to calculate some
properties of the polarized neutron matter and the polarized symmetrical and
asymmetrical nuclear matters Ghc . In these works, a comparison of our results
and those of other many-body techniques indicates that the LOCV method is a
powerful microscopic technique to calculate the properties of the polarized
matter.
In this work, we use the LOCV method to compute the ground state energy of the
polarized liquid ${}^{3}He$ at zero temperature by employing the spin-
dependent correlation function with the Lennard-Jones LJ and Aziz Aziz1 ;
Aziz2 pair potentials.
## II Lowest Order Constrained Variational Method
We consider a system of $N$ interacting ${}^{3}He$ atoms with $N^{+}$ spin up
and $N^{-}$ spin down atoms. The total number density ($\rho$) and spin
asymmetry parameter ($\xi$) are defined as
$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\rho^{+}+\rho^{-},$
$\displaystyle\xi$ $\displaystyle=$ $\displaystyle\frac{N^{+}-N^{-}}{N}.$ (1)
$\xi$ shows the spin ordering of matter which can have a value in the range of
$\xi=0.0$ (unpolarized matter) to $\xi=1.0$ (fully polarized matter). For this
system, we consider the energy per particle up to the two-body term in the
cluster expansion,
$E=E_{1}+E_{2},$ (2)
where
$\displaystyle E_{1}$ $\displaystyle=$
$\displaystyle\frac{3}{10}\frac{\hbar^{2}}{m}(3\pi^{2}\rho)^{\frac{2}{3}}[(1+\xi)^{\frac{5}{3}}+(1-\xi)^{\frac{5}{3}}],$
$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2N}\sum_{i,j}\langle ij\mid w(12)\mid ij-ji\rangle.$
(3)
In the above equation, $w(12)$ is the effective pair potential,
$w(12)=-\frac{\hbar^{2}}{2m}[F(12),[\nabla_{12}^{2},F(12)]]+F(12)V(12)F(12),$
(4)
where $F(12)$ is the two-body correlation operator and $V(12)$ is the pair
potential between the helium atoms. In our calculations, we use the Lennard-
Jones LJ and Aziz Aziz1 ; Aziz2 pair potentials. The Lennard-Jones pair
potential is as follows LJ ,
$V(r)=4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right],$
(5)
where
$\epsilon=10.22K,\hskip 56.9055pt\sigma=2.556A\cdot$ (6)
The Aziz pair potential has the following form Aziz1 ; Aziz2 ,
$V(r)=\epsilon\left\\{Ae^{-\alpha
r/r_{m}}-\left[C_{6}\left(\frac{r_{m}}{r}\right)^{6}+C_{8}\left(\frac{r_{m}}{r}\right)^{8}+C_{10}\left(\frac{r_{m}}{r}\right)^{10}\right]F(r)\right\\},$
(7)
where
$\displaystyle F(r)$ $\displaystyle=$
$\displaystyle\left\\{\begin{tabular}[]{lll}$e^{-(\frac{Dr_{m}}{r}-1)^{2}}$&;&$\frac{r}{r_{m}}\leq
D$\\\ 1&;&$\frac{r}{r_{m}}>D$,\end{tabular}\right.$ (10)
and
(15)
Now, we consider a spin-dependent correlation function as follows
$F(12)=f_{0}(r_{12})P_{0}+f_{1}(r_{12})P_{1},$ (16)
where
$\displaystyle P_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(1-\sigma_{1}.\sigma_{2}),$ $\displaystyle P_{1}$
$\displaystyle=$ $\displaystyle\frac{1}{4}(3+\sigma_{1}.\sigma_{2}).$ (17)
$f_{0}$ and $f_{1}$ indicate the spin-singlet and spin-triplet two-body
correlation functions, respectively. With the above two-body correlation
function, we have derived the following relation for the effective pair
potential,
$w_{s}(r)=\frac{\hbar^{2}}{m}({f_{s}{{}^{\prime}}}(r))^{2}+f_{s}^{2}(r)V(r),$
(18)
and then the two-body energy $E_{2}$ is found by
$E_{2}=2\pi\rho\sum_{s=0,1}\int_{0}^{\infty}drr^{2}w_{s}(r)a_{s}.$ (19)
In Eq. (19)
$\displaystyle a_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(1-\xi^{2})[1+l(k_{F^{+}}r)l(k_{F^{-}}r)],$
$\displaystyle a_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{4}[(1+\xi)^{2}(1-l^{2}(k_{F^{+}}r))+(1-\xi)^{2}(1-l^{2}(k_{F^{-}}r))$
(20) $\displaystyle+(1-\xi^{2})(1-l(k_{F^{+}}r)l(k_{F^{-}}r))].$
$k_{F^{\pm}}=(6\pi^{2}\rho^{\pm})^{\frac{1}{3}}$ is the Fermi momentum and
$l(x)$ is given by
$l(x)=\frac{3}{x^{3}}[\sin(x)-x\cos(x)].$ (21)
Now, we minimize the two-body energy Eq. (19) with respect to the variations
in the two-body correlation function subject to the normalization constraint
Clark ,
$\frac{1}{N}\sum_{i,j}\langle ij|h^{2}(12)-F^{2}(12)|ij-ji\rangle=1.$ (22)
The normalization constraint is conveniently re-written in the integral form
as
$4\pi\rho\sum_{s=0,1}\int_{0}^{\infty}drr^{2}[h^{2}(r)-f_{s}^{2}(r)]a_{s}=1,$
(23)
where the Pauli function $h(r)$ is
$h(r)=\\{1-\frac{1}{4}[(1+\xi)^{2}l^{2}(k_{F^{+}}r)+(1-\xi)^{2}l^{2}(k_{F^{-}}r)]\\}^{-\frac{1}{2}}.$
(24)
The minimization of the two body energy $E_{2}$ gives the following Euler-
Lagrange differential equation for the two-body correlation function
$f_{s}(r)$,
$f_{s}^{{}^{\prime\prime}}(r)+(\frac{2}{r}+\frac{a_{s}^{{}^{\prime}}}{a_{s}})f_{s}^{{}^{\prime}}(r)-\frac{m}{\hbar^{2}}\left(V(r)-2\lambda\right)=0.$
(25)
The Lagrange multiplier $\lambda$ imposes by normalization constraint. For
$s=0$ and $s=1$ states, the two-body correlation function $f_{s}(r)$ is
obtained by numerically integrating Eq. (25). Using this two-body correlation
function we can determine the effective pair potential $w_{s}(r)$ as a
function of interatomic distance from Eq. (18). Finally, the two-body energy
$E_{2}$ and the total energy of system can be calculated.
## III Results and Discussion
We have calculated some ground state properties of the polarized liquid
${}^{3}He$ at zero temperature with the Lennard-Jones LJ and Aziz Aziz1 ;
Aziz2 pair potentials using the spin-dependent correlation function. Our
results are as follows.
The spin-dependent two-body correlation functions at $s=0$ state and $s=1$
states for different values of spin asymmetry parameter ($\xi$) are shown in
Fig. 1. These figures show that the correlation function at $s=1$ state heals
to pauli function, $h(r)$, more rapidly than $s=0$ state. Therefore, the $s=1$
state has a shorter correlation length with respect to $s=0$ state. For large
values of $r$, $f_{0}(r)$ and $f_{1}(r)$ have the same values and therefore,
the spin-dependent part of correlation operator (Eq. 16) is vanished. In these
figures, the spin-independent two-body correlation function are also plotted
for comparison. It is seen that the spin-dependent correlation function
differs from spin-independent correlation function, except for the fully
polarized matter ($\xi=1.0$). This is due to the fact that for fully polarized
matter, there is only $s=1$ state. From Fig. 1, we can see that the
correlation functions with the Lennard-Jones and Aziz potentials are nearly
identical.
In Fig. 2, we have shown the total energy of polarized liquid ${}^{3}He$
versus number density calculated both with the spin-dependent correlation and
the spin-independent correlation at different values of spin asymmetry
parameter $\xi$. We can see that the total energy increases by increasing
$\xi$. Fig. 2 indicates that in the spin-dependent case, the total energy of
the liquid ${}^{3}He$ is lower than the spin-independent case. It is also seen
that for all values of $\xi$, the energy curve has a minimum which shows the
existence of a bound state for this system. It is shown that the difference
between the energies of spin-dependent case and spin-independent case
decreases by increasing $\xi$ and it becomes zero as $\xi$ approaches to one.
It is seen that for all values of the density and spin asymmetry parameter,
the total energy with the Aziz pair potential is grater than that of the
Lennard-Jones pair potential.
The potential energy of the polarized liquid ${}^{3}He$ for different values
of $\xi$ are presented in Fig. 3, for spin-dependent and spin-independent
cases. This figure indicates that the potential energy decreases by increasing
the polarization. According to the above results, we can conclude that the
increasing of kinetic energy dominates and this leads to the increasing of
total energy by increasing $\xi$. Fig. 3 shows that the potential energy in
the spin-dependent case has lower values with respect to the spin-independent
case. It is also seen that the difference between the potential energies of
spin-dependent and spin-independent cases decrease by increasing $\xi$. We see
that the potential energies with the Aziz and the Lennard-Jones pair
potentials are different. This difference increases by increasing the density.
In Fig 4, the potential energies of $s=0$ and $s=1$ states for different
values of $\xi$ are compared. We have seen that the potential energy at $s=1$
state is lower than at $s=0$ state. It can be concluded that the spin-triplet
state has the main contribution in the potential energy of polarized liquid
${}^{3}He$. It is also seen that the potential energy of $s=0$ ($s=1$) state
increases (decreases) by increasing $\xi$. For $s=0$ state, we can see that at
low densities, the potential energies with the Lennard-Jones and Aziz pair
potentials are nearly identical. However, for $s=1$ state and high densities,
the difference between these potential energies becomes appreciable.
The equation of state of polarized liquid ${}^{3}He$, $P(\rho,\xi$), can be
obtained using
$\displaystyle P(\rho,\xi)=\rho^{2}\frac{\partial E(\rho,\xi)}{\partial\rho}$
(26)
In Fig. 5, we have presented the pressure of liquid ${}^{3}He$ as a function
of the density ($\rho$) for fully polarized ($\xi=1.0$) and unpolarized
($\xi=0.0$) cases. This figure shows that for different values of the
polarization, the equations of state of liquid ${}^{3}He$ are nearly
identical. From Fig. 5, it is seen that for both $\xi=1.0$ and $\xi=0.0$, the
equation of state with the Aziz pair potential is stiffer than that of the
Lennard-Jones pair potential.
## IV Summary and Conclusion
We have considered a system consisting of Helium atoms $(^{3}He)$ with
asymmetrical spin configuration and derived the two-body term in the cluster
expansion of the energy functional by employing spin-dependent correlation
function. Then, we have minimized the two-body energy term under the
normalization constraint and obtained the Euler-Lagrange differential
equation. By numerically solving this differential equation, we have computed
the correlation function and then calculated the other properties of this
system with the Lennard-Jones and Aziz pair potentials. It is shown that for
the two different spin-singlet and spin-triplet states, the correlation
functions are different from each other. Our results show that the
introduction of the spin-dependent term in the correlation operator reduces
the total energy of system by about $10\%$. It is also shown that the total
energy increases by increasing the polarization. The difference between the
energies of the spin-dependent and spin-independent cases decreases by
increasing the polarization. We have seen that, the potential energy of these
states have a remarkable difference. It is shown that the main contribution of
the potential energy comes from $s=1$ state. Our calculations show that there
is a difference between the results with the Lennard-Jones and Aziz pair
potentials, especially at high densities.
###### Acknowledgements.
Financial support from the Shiraz University research council is gratefully
acknowledged.
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* (16) G. H. Bordbar, M.J. Karimi and J. Vahedi, _Int. J. Mod. Phys._ B27 (2008) in press.
* (17) G. H. Bordbar, S. M. Zebarjad, M. R. Vahdani and M. Bigdeli, _Int. J. Mod. Phys._ B19, 3379 (2005).
* (18) G.H. Bordbar, S. M. Zebarjad and F. Shojaei, _Int. J. Theor. Phys._ 43, 1863 (2004).
* (19) G. H. Bordbar and M. Hashemi, _Int. J. Theor. Phys., Group Theory and Nonlinear Optics_ 8, 251 (2002).
* (20) G. H. Bordbar, _Int. J. Theor. Phys._ 43, 399 (2004); ibid. 41, 309 (2002); 41, 1135 (2002); G. H. Bordbar and N. Riazi, ibid. 40, 1671 (2001); G. H. Bordbar, _Int. J. Mod. Phys._ A18, 2629 (2003). G. H. Bordbar and M. Modarres, _Phys. Rev_ C57, 714 (1998); _J. Phys. G: Nucl. Part. Phys._ 23, 1631 (1997).
* (21) G. H. Bordbar and M. Bigdeli, _Phys. Rev_ C77, 015805 (2008); ibid. C76, 035803 (2007); C75, 045804 (2007).
* (22) J. de Boer and A. Michels, Physica 6, 409 (1939).
* (23) R. A. Aziz et al., J. Chem. Phys. 70, 4330 (1979).
* (24) R. A. Aziz, F. R. W. McCourt and C. C. K. Wong, Mol. Phys. 61, 1487 (1987).
* (25) J. W. Clark, _Prog. Part. Nucl. Phys._ 2, 89 (1979).
Figure 1: The correlation function with the Aziz (dashed curves) and Lennard-
Jones (full curves) pair potentials in the case of spin-dependent at $s=0$ and
$s=1$ states for $\xi=0.0$ (a), $\xi=0.33$ (b), $\xi=0.66$ (c) and $\xi=1.0$
(d). Our results for the spin-independent correlation function are also
presented for comparison.
Figure 2: Our results for the total energy of the polarized liquid ${}^{3}He$
with the Aziz and Lennard-Jones (LJ) pair potentials in the case of spin-
dependent (full curve) and spin-independent (dotted curve) correlation
functions for $\xi=0.0$ (a), $\xi=0.33$ (b), $\xi=0.66$ (c) and $\xi=1.0$ (d).
Figure 3: As Fig. 2, but for the potential energy of the polarized liquid
${}^{3}He$.
Figure 4: Our results for the potential energy of the polarized liquid
${}^{3}He$ with the Aziz and Lennard-Jones (LJ) pair potentials at $s=0$
(dotted curve) and $s=1$ (full curve) states for $\xi=0.0$ (a), $\xi=0.33$
(b), $\xi=0.66$ (c) and $\xi=1.0$ (d). Figure 5: The equation of state of the
fully polarized (dotted curve) and unpolarized (full curve) liquid ${}^{3}He$
with the Aziz and Lennard-Jones (LJ) pair potentials.
|
arxiv-papers
| 2008-10-29T10:24:49 |
2024-09-04T02:48:58.532253
|
{
"license": "Public Domain",
"authors": "G.H. Bordbar and M.J. Karimi",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/0810.5233"
}
|
0810.5237
|
# Lowest Order Constrained Variational Calculation of Structure Properties of
Protoneutron Star
G.H. Bordbar111Corresponding author, S.M. Zebarjad and R. Zahedinia
Department of Physics, Shiraz University, Shiraz 71454, Iran222Permanent
address
and
Research Institute for Astronomy and Astrophysics of Maragha,
P.O. Box 55134-441, Maragha, Iran
###### Abstract
We calculate the structure properties of protoneutron star such as equation of
state, maximum mass, radius and temperature profile using the lowest order
constrained variational method. We show that the mass and radius of
protoneutron star decrease by decreasing both entropy and temperature. For the
protoneutron star, it is shown that the temperature is nearly constant in the
core and drops rapidly near the crust.
## I Introduction
Neutron star which is highly compact stellar objects is a result of supernova
explosion. The structure properties of this object, especially its maximum
mass, is of a great interest for astrophysicists. Since the compactness
parameter for a neutron star is about $0.2-0.4$, its structure should be
studied using the general theory of relativity. In fact, the computation of
the structure properties of a neutron star can be derived using the Tolman-
Oppenheimer-Volkoff (TOV) equation 1 .
Just after the supernova collapse, a newly-born neutron star called
protoneutron star is formed. At these stages, the neutron stars are rich in
leptons. This is due to the fact that the neutrinos are trapped in the
protoneutron star matter. The temperature of protoneutron star is greater than
$10MeV$ 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 . Therefore, the high temperature of
these stages cannot be neglected with respect to the Fermi temperature
throughout the calculation of its structure. Depending on the total number of
nucleons, a protoneutron star evolves either to black hole or to stable
neutron star 3 ; 7 ; 8 ; 9 . Therefore, computation of the maximum mass of
protoneutron star is of crucial importance.
In recent years, we have investigated the properties of neutron star matter 10
; 11 ; 12 ; 13 ; 14 . In the present work, we intend to compute the structure
properties of protoneutron star at different stages employing the lowest order
constrained variational approach using the modern $AV_{18}$ potential wiringa
.
## II Formalism
In this section, we give the formalism of calculation for the protoneutron
star matter equation of state.
We write the total energy per baryon ($E$) as the sum of contributions from
leptons and nucleons,
$E=E_{lep}+E_{nucl}\cdot$ (1)
The contribution from the energy of leptons is
$E_{lep}=\frac{m^{4}c^{5}}{\pi^{2}\rho\hbar^{3}}\int_{0}^{\infty}n(x)\sqrt{1+x^{2}}x^{2}dx,$
(2)
where $\rho$ is the total number density of nucleons and $n(x)$ is the Fermi-
Dirac distribution function, where $x$ defined,
$x=\frac{\hbar k}{mc}\cdot$ (3)
We calculate the contribution from the energy of nucleons using the lowest
order constrained variational method. We consider up to the two-body term in
the cluster expansion for the energy functional,
$E_{nucl}=E_{1}+E_{2}\cdot$ (4)
The one-body energy $E_{1}$ is
$E_{1}=\sum_{i=n,p}\frac{\hbar^{2}}{2m_{i}\rho\pi^{2}}\int_{0}^{\infty}n_{i}(k)k^{4}dk\cdot$
(5)
The two-body energy $E_{2}$ is
$E_{2}=(2A)^{-1}\sum_{ij}<ij\mid\\{-\frac{\hbar^{2}}{2m}\left[f(12),[\nabla_{12},f(12)]\right]+f(12)V(12)f(12)\\}\mid
ij>_{a},$ (6)
where f(12) and V(12) are the two-body correlation function and nucleon-
nucleon potential, respectively. V(12) has the following general form wiringa
,
$V(12)=\sum_{p=1}^{18}V^{p}(r_{12})O^{p}_{12}\cdot$ (7)
Finally, we minimize the two-body energy, $E_{2}$, with respect to the
variation in the two-body correlation function subjected to the normalization
constraint. From this minimization, we get a set of Euler-Lagrange
differential equations. The correlation functions are calculated by solving
these differential equations and then, the two-body energy, $E_{2}$, is
computed 17 ; 18 . This leads to the equations of state used in our present
work.
## III Calculations of Protoneutron Star Structure Properties
In this section, we calculate the structure properties of neutron star in the
different stages just after its formation by numerically integrating the TOV
equation. The structure properties of neutron star in these stages such as
maximum mass, radius and temperature profile are calculated using the modern
microscopic equations of state derived from constrained variational method
employing the Argonne $V_{18}$ potential wiringa as the inter-nucleonic
interaction. At low densities, we also consider a hot dense matter model and
use the equations of state obtained by Gondek et al. 15 and Strobel et al. 16
. Here is our results for different stages.
### III.1 Lepton rich protoneutron star
Since a neutron star, at the beginning of its lifetime (protoneutron star), is
opaque with respect to neutrinos therefore, it contains a high lepton fraction
($Y$), $0.3-0.4$. Furthermore, at this stage of neutron star, the entropy per
baryon ($s$) is nearly constant throughout the star, $1-2k_{B}$. Since, we are
also dealing with uncharged neutron star matter, the lepton and proton
fractions should be equal 2 ; 3 ; 4 ; 5 .
Our results for the pressure of protoneutron star matter as a function of mass
density are presented in Fig. 1 at entropies $s=1k_{B}$ and $s=2k_{B}$ with
$Y=0.4$ and $Y=0.3$. It is seen that for a fixed value of entropy, the
pressure of protoneutron star matter increases by decreasing the lepton
fraction. We see that for a given value of lepton fraction, the equation of
state for $s=2.0k_{B}$ is stiffer than for $s=1.0k_{B}$. In Fig. 1, We have
compared our results with those of Gondek et al. 15 and Strobel et al. 16 .
We have seen that the results of Gondek et al. 15 are nearly in agreement
with our results for $s=2.0k_{B}$ and $Y=0.3$. But, the equations of state
calculated by Strobel et al. 16 are stiffer than those of ours.
In Fig. 2, we have presented the gravitational mass of protoneutron star as a
function of central mass density for different values of entropy and lepton
fraction. From Fig. 2, we can see that the gravitational mass increases by
increasing the central mass density. It is shown that for different cases of
entropy and lepton fraction, the gravitational mass exhibits different mass
limits (maximum mass) as given in Table 1. From this table, the decreasing of
maximum mass by decreasing the entropy and increasing the lepton fraction is
seen. In Fig. 2 and Table 1, the results of Gondek et al. 15 and Strobel et
al. 16 are also given for comparison. It is seen that these results are
different from our calculations.
The radius of protoneutron star as a function of central mass density is shown
in Fig. 3 for different entropies and lepton fractions. It is seen that as the
central mass density increases, the radius decreases very rapidly and then
reaches a nearly constant value. The radius of protoneutron star corresponding
to the maximum mass for different entropies and lepton fractions is given in
Table 2. It can be seen that the stiffer equation of state leads to the
relatively higher radius.
The variations of temperature throughout the protoneutron star matter with
respect to the radial coordinate ($r$) are presented in Figs. 4 and 5 for
$s=1k_{B}$ and $s=2k_{B}$ with $Y=0.4$ and $Y=0.3$. We can see from the Figs.
4 and 5 that the temperature is decreasing slowly by increasing $r$, but it
drops rapidly near the protoneutron star crust.
### III.2 Beta-stable protoneutron star
After complete deleptonization, neutrino trapped within the hot interior
matter of neutron star do not affect the beta stability condition and
therefore the lepton fraction is determined from the beta-equilibrium criteria
2 ; 3 ; 4 ; 5 . In this stage, we have calculated the structure properties of
protoneutron star for both isentropic and isothermal paths.
#### III.2.1 Isentropic paths
For different values of entropy, our calculated equations of state of beta-
stable protoneutron star are shown in Fig. 6. It is seen that the pressure of
beta-stable protoneutron star matter increases by increasing entropy. In Fig.
6, the results of Strobel et al. 16 are also plotted for comparison. There is
a compatible difference between our results and those of Strobel et al. 16 .
Our calculated gravitational mass of beta-stable protoneutron star as a
function of central mass density for $s=1.0k_{B}$ and $s=2.0k_{B}$ is
presented in Fig. 7. From this figure, the increasing of gravitational mass by
increasing both central mass density and entropy is seen. As it is shown in
Fig. 7, for higher values of central mass density, the gravitational mass
shows a limiting value (maximum mass). The maximum mass of beta-stable
protoneutron star is given in Table 3 for different entropies. It is seen that
the maximum mass decreases by decreasing the entropy. In Fig. 7 and Table 3,
we have compared our results with the results of Strobel et al. 16 . It is
seen that our results are different from those of Strobel et al. 16 .
In Fig. 8, our results for the radius of beta-stable protoneutron star as a
function of central mass density are plotted at different values of entropy.
We see that the radius decreases very rapidly by increasing the central mass
density and then exhibits a nearly constant value. The radius corresponding to
the maximum mass of beta stable protoneutron star for $s=1.0k_{B}$ and
$s=2.0k_{B}$ are also given in Table 3. This shows the decreasing of radius
with respect to decreasing entropy.
The temperature of beta-stable protoneutron star matter as a function of
radial coordinate ($r$) is shown in Fig. 9 for different entropies. The same
as Fig.s 4 and 5, we see that the temperature decreases slowly by increasing
$r$, but drops rapidly near the beta-stable protoneutron star crust.
#### III.2.2 Isothermal paths
We are now able to do the above calculations for the protoneutron star in the
beta-equilibrium case at different isothermal paths.
Our results for the equation of state, gravitational mass and radius of beta-
stable protoneutron star at different values of temperature are presented in
Figs. 10-12, respectively. At different temperatures, we have extracted the
maximum mass and corresponding radius from these figures. Our results are
given in Table 4. From this table, we can see that the maximum mass and radius
of beta-stable protoneutron star decrease by decreasing the temperature.
## IV Summary and Conclusion
The protoneutron stars which are lepton rich and hot objects are formed just
after the supernova explosion. In this paper, we have integrated TOV equation
to compute the maximum mass, radius and temperature profile for this object.
As we have shown at a fixed value of lepton fraction, the mass and radius of
protoneutron star decreases by decreasing entropy. The temperature is nearly
constant in the core of protoneutron star and drops rapidly near its crust.
The same properties of protoneutron star at beta equilibrium condition have
been also calculated along the isentropic and isothermal paths. We have shown
that the maximum mass and corresponding radius increase by increasing
temperature.
###### Acknowledgements.
This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha. We wish to thanks Shiraz University Research Council.
## References
* (1) S. Shapiro and S. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars (Wiley, New York, 1983).
* (2) J.A. Pons et al., Astrophys. J. 513, 780 (1999).
* (3) A. Burrows and J.M. Lattimer, Astrophys. J. 307, 178 (1986).
* (4) H.A. Bethe, Rev. Mod. Phys. 62, 801 (1990).
* (5) A. Burrows, J. Hayes and B.A. Fryxell, Astrophys. J. 450, 830 (1995).
* (6) W. Keil and H. Th. Janka, Astron. Astrophys. 296, 145 (1995).
* (7) H.Th. Janka, astro-ph/0402200.
* (8) K. Strobel and M.K. Weigel, Astron. Astrophys. 367, 582 (2001).
* (9) D. Gondek, P. Haensel and J.L. Zdunik, astro-ph/0012541.
* (10) G.H. Bordbar, Int. J. Theor. Phys. 41, 309 (2002).
* (11) G.H. Bordbar and N. Riazi, Int. J. Theor. Phys. 40, 1671 (2001).
* (12) G.H. Bordbar and N. Riazi, Int. J. Theor. Phys., GTNO, 7, 73 (2001).
* (13) G.H. Bordbar and N. Riazi, Astrophys. Space Sci. 282, 563 (2002).
* (14) G.H. Bordbar and M. Hayati, Int. J. Mod. Phys. A21, 1555 (2006).
* (15) R. B. Wiringa, V. Stoks and R. Schiavilla, Phys. Rev. C 51, 38 (1995).
* (16) G. H. Bordbar, M. Modarres, J. Phys. G: Nucl. Part. Phys. 23, 1631 (1997).
* (17) G. H. Bordbar and M. Modarres, Phys. Rev. C 57, 714 (1998).
* (18) D. Gondek, P. Haensel and J.L. Zdunik, Astron. Astrophys. 325, 217 (1997).
* (19) K. Strobel, Ch. Schaab and M.K. Weigel, Astron. Astrophys. 350, 497 (1999).
Table 1: Our results for the maximum mass ($M_{\odot}$) of protoneutron star at different values of entropy ($s$) and lepton fraction ($Y$). The results of Gondek et al. [18] (GHZ) and Strobel et al. [19] (SSW) are also given for comparison. | $s=1.0k_{B}$ | $s=2.0k_{B}$
---|---|---
$Y=0.4$ | 1.55 | 1.56
$Y=0.3$ | 1.63 | 1.70
SSW, $Y=0.4$ | 2.01 | 2.03
GHZ, $Y=0.4$ | | 1.91
Table 2: Our results for the radius ($km$) of protoneutron star corresponding to the maximum mass at different entropies and lepton fractions. | $s=1.0k_{B}$ | $s=2.0k_{B}$
---|---|---
$Y=0.4$ | 7.62 | 7.68
$Y=0.3$ | 8.13 | 8.35
Table 3: Our results for the maximum mass ($M_{\odot}$ ) of beta-stable protoneutron star and corresponding radius ($km$) at different entropies. The results of Strobel et al. [19] (SSW) for the maximum mass of beta-stable protoneutron star are also given for comparison. | $s=1.0k_{B}$ | $s=2.0k_{B}$
---|---|---
Mass | 1.68 | 1.76
Radius | 8.05 | 8.15
SSW | 1.98 | 2.03
Table 4: Our results for the maximum mass ($M_{\odot}$ ) of beta-stable protoneutron star and corresponding radius ($km$) at different temperatures. | $T=5.0MeV$ | $T=10.0MeV$ | $T=20.0MeV$
---|---|---|---
Mass | 2.19 | 2.20 | 2.23
Radius | 7.85 | 7.88 | 8.01
Figure 1: Our results for the pressure ($10^{34}dyn/cm^{2}$) of the
protoneutron star matter versus mass density ($10^{14}gr/cm^{3}$) at entropies
$s=1k_{B}$ and $s=2k_{B}$ with lepton fractions $Y=0.4$ and $0.3$. The results
of Gondek et al. [18] (GHZ) and Strobel et al. [19] (SSW) are also given for
comparison.
Figure 2: As Fig. 1, but for the gravitational mass ($M_{\odot}$) of
protoneutron star versus central mass density ($10^{14}gr/cm^{3}$).
Figure 3: Our results for the radius ($km$) of protoneutron star versus
central mass density ($10^{14}gr/cm^{3}$) at entropies $s=1k_{B}$ and
$s=2k_{B}$ with lepton fractions $Y=0.4$ and $0.3$.
Figure 4: Our results for the temperature profile of protoneutron star at
different entropies for $Y=0.4$.
Figure 5: As Fig. 4, but for $Y=0.3$.
Figure 6: Our results for the pressure ($10^{34}dyn/cm^{2}$) of beta-stable
protoneutron star matter as a function of mass density ($10^{14}gr/cm^{3}$) at
different entropies. The results of Strobel et al. [19] (SSW) are also given
for comparison.
Figure 7: As Fig. 6, but for the gravitational mass ($M_{\odot}$) of beta-
stable protoneutron star versus central mass density ($10^{14}gr/cm^{3}$).
Figure 8: Our results for the radius ($km$) of beta-stable protoneutron star
versus central mass density ($10^{14}gr/cm^{3}$) at different entropies.
Figure 9: Our results for the temperature profile of beta-stable protoneutron
star at different entropies.
Figure 10: Our results for the pressure ($10^{34}dyn/cm^{2}$) of beta-stable
protoneutron star matter as a function of mass density ($10^{14}gr/cm^{3}$) at
different temperatures.
Figure 11: As Fig. 10, but for the gravitational mass ($M_{\odot}$) of beta-
stable protoneutron star versus central mass density ($10^{14}gr/cm^{3}$).
Figure 12: As Fig. 10, but for the radius ($km$) of beta-stable protoneutron
star versus central mass density ($10^{14}gr/cm^{3}$).
|
arxiv-papers
| 2008-10-29T10:37:16 |
2024-09-04T02:48:58.537510
|
{
"license": "Public Domain",
"authors": "G.H. Bordbar, S.M. Zebarjad and R. Zahedinia",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/0810.5237"
}
|
0810.5405
|
# Berry Phase in a Single Quantum Dot with Spin-Orbit Interaction
Huan Wang111Email: wanghuan2626@sjtu.edu.cn, Ka-Di Zhu222Email:
zhukadi@sjtu.edu.cn Department of Physics, Shanghai Jiao Tong University,
Shanghai 200240, People’s Republic of China
###### Abstract
Berry phase in a single quantum dot with Rashba spin-orbit coupling is
investigated theoretically. Berry phases as functions of magnetic field
strength, dot size, spin-orbit coupling and photon-spin coupling constants are
evaluated. It is shown that the Berry phase will alter dramatically from $0$
to $2\pi$ as the magnetic field strength increases. The threshold of magnetic
field depends on the dot size and the spin-orbit coupling constant.
###### pacs:
71.70.Ej, 03.65.Vf
††preprint: APS/123-QED
## I Introduction
Due to its important role in encoding information, the phase of wavefunction
attracts a lot of interest in information science. Those properties can also
be used in future quantum information and quantum computer. Thus selecting one
kind of phases which can be manipulated by quantum effect is very important.
Berry phase is believed to be a promising candidate. As a quantum mechanical
system evolves cyclically in time such that it return to its initial physical
state, its wavefunction can acquire a geometric phase factor in addition to
the familiar dynamic phaseSegao ; Anandan . If the cyclic change of the system
is adiabatic, this additional factor is known as Berry’s phaseBerry , and is,
in contrast to dynamic phase, independent of energy and time. Fuentes-Guridi
et al.Fuentes calculated the Berry phase of a particle in a magnetic field in
consideration of the quantum nature of the light field. Yi et al.Yi studied
the Berry phase in a composite system and showed how the Berry phases depend
on the coupling between the two subsystems. In a recent paper, San-Jose et
al.San have described the effect of geometric phases induced by either
classical or quantum electric fields acting on single electron spins in
quantum dots. Wang and Zhu Wang H. have investigated the voltage-controlled
Berry phases in two vertically coupled InGaAs/GaAs quantum dots. Most
recently, observations of Berry phases in solid state materials are
reportedYuanbo ; Vartiainen ; Leek . Leek et al.Leek demonstrated the
controlled Berry phase in a superconducting qubit which manipulates the qubit
geometrically using microwave radiation and observes the phase in an
interference experiment.
Spin-related effects have potential applications in semiconductor devices and
in quantum computation. Rashba et al.Rashba have described the orbital
mechanisms of electron-spin manipulation by an electric field. SoninSonin has
demonstrated that an equilibrium spin current in a 2D electron gas with Rashba
spin-orbit interaction can result in a mechanical torque on a substrate near
an edge of the medium. SerebrennikovSerebrennikov considered that the
coherent transport properties of a charge carrier. The transportation will
cause a spin precession in zero magnetic fields and can be described in purely
geometric terms as a consequence of the corresponding holonomy.
The spin-orbit interaction in semiconductor heterstructures is increasingly
coming to be seen as a tool which can manipulate electronic spin states Egues
; Efros . Two basic mechanisms of the spin-orbit coupling of 2D electrons are
directly related to the symmetry properties of QDs. They stem from the
structure inversion asymmetry mechanism described by the Rashba termRashba ;
Bychkov and the bulk inversion asymmetry mechanism described by the
Dresselhaus termDresselhaus . Recently, Debald and Emary Debald have
investigated a spin-orbit driven Rabi oscillation in a single quantum dot with
Rashba spin-orbit coupling. However, the influence of spin-orbit interaction
on Berry phase in a single quantum dot is still lacking. In the present paper
we will give a detail study on the Berry phase evolution of a single quantum
dot with spin-orbit interaction in a time-dependent quantized electromagnetic
environment. We will borrow quantum optics method to investigate the impact of
the spin-orbit interaction and spin-photon interaction on Berry phase.
The paper is organized as follows. In Sec.II, we give the model Hamiltonian
including both spin-orbit interaction and spin-photon interaction and
calculated Berry phases as functions of magnetic field strength, dot size,
spin-orbit coupling and photon-spin coupling constants. In Sec.III, we draw
the figures of the Berry phase as a function of magnetic field strength and
some discussions are given. The final conclusion is presented in Sec.IV.
## II Theory
We consider a simple two-dimensional quantum dot with parabolic lateral
confinement potential in a perpendicular magnetic field $\bf{B}$ which points
along $z$ direction. Then the electron system can be described by the
Hamiltonian Debald ,
$\displaystyle{H_{s}}=\frac{(\textbf{p}+\frac{e}{c}\textbf{A})^{2}}{2m^{*}}+\frac{m^{*}}{2}\omega^{2}_{0}(x^{2}+y^{2})+\frac{1}{2}g\mu_{B}B\sigma_{z},$
(1)
where p is the linear momentum operator of the electron,
$\textbf{A}(\textbf{r})=\frac{\textbf{B}}{2}(-y,x,0)$ is the vector potential
in the symmetric gauge, $\omega_{0}$ is the characteristic confinement
frequency, and $\sigma=(\sigma_{x},\sigma_{y},\sigma_{z})$ is the vector Pauli
matrices. $m^{*}$ is the effective mass of the electron and $g$ its
gyromagnetic factor. $\mu_{B}$ is the Bohr magneton. In the second quantized
notation, Eq.(1) becomes
$\displaystyle{H_{s}}=(a_{x}^{+}a_{x}+a_{y}^{+}a_{y}+1)\hbar\widetilde{\omega}+\frac{\hbar\omega_{c}}{2i}(a_{x}^{+}a_{y}-a_{x}a_{y}^{+})$
$\displaystyle+\frac{1}{2}g\mu_{B}B\sigma_{z},$ (2)
where $\omega_{c}=\frac{eB}{m^{*}c}$ and
$\widetilde{\omega}^{2}=\omega_{0}^{2}+\frac{\omega_{c}^{2}}{4}$. If we set
$\displaystyle
a_{+}=\frac{1}{\sqrt{2}}(a_{x}-ia_{y}),a_{-}=\frac{1}{\sqrt{2}}(a_{x}+ia_{y}),$
(3)
Then, the Hamiltonian (2) can be written as
$\displaystyle{H_{s}}=n_{+}\hbar\omega_{+}+n_{-}\hbar\omega_{-}+\frac{1}{2}g\mu_{B}B\sigma_{z},$
(4)
where $\omega_{\pm}=\widetilde{\omega}\pm\omega_{c}/2$, $n_{+}=a_{+}^{+}a_{+}$
and $n_{-}=a_{-}^{+}a_{-}$. In what follows we include the spin-orbit
interaction which is described as Rashba Hamiltonian in this system Rashba
$\displaystyle{H_{so}}=-\frac{\alpha}{\hbar}[(\textbf{p}+\frac{e}{c}\textbf{A})\times\sigma]_{\it
z},$ (5)
where $\alpha$ is the spin-orbit coupling constant which can be controlled by
gate voltage in experiment. On substituting Eq.(3) into Eq.(5) and then
$\displaystyle{H_{so}}=\frac{\alpha}{\widetilde{l}}[\gamma_{+}(\sigma_{+}a_{+}+\sigma_{-}a_{+}^{+})-\gamma_{-}(\sigma_{-}a_{-}+\sigma_{+}a_{-}^{+})],$
(6)
where $\gamma_{\pm}=1\pm\frac{1}{2}(\widetilde{l}/l_{B})^{2}$,
$\widetilde{l}=({\hbar}/{m^{*}\widetilde{\omega}})^{\frac{1}{2}}$ and
$l_{B}=({\hbar}/{m^{*}\omega_{c}})^{\frac{1}{2}}$.
The Hamiltonians of photons and the coupling to the electron spin can be
written as follows:
$\displaystyle H_{p}=\hbar\omega_{p}b^{+}b,$ (7) $\displaystyle
H_{p-s}=g_{c}(\sigma_{+}+\sigma_{-})(b^{\dagger}+b),$ (8)
where $b^{+}$ ($b$) and $\omega_{p}$ are the creation (annihilation) operator
and energy of the photons, respectively. $g_{c}$ is the spin-photon coupling
constant. Hence we obtain the total Hamiltonian of the electron and photons:
$\displaystyle{H}=H_{s}+H_{so}+H_{p}+H_{p-s}$
$\displaystyle=\hbar\omega_{+}a_{+}^{+}a_{+}+\hbar\omega_{-}a_{-}^{+}a_{-}+\frac{1}{2}g\mu_{B}B\sigma_{z}$
$\displaystyle+\frac{\alpha}{\widetilde{l}}[\gamma_{+}(\sigma_{+}a_{+}+\sigma_{-}a_{+}^{+})-\gamma_{-}(\sigma_{-}a_{-}+\sigma_{+}a_{-}^{+})]$
$\displaystyle+\hbar\omega_{p}b^{+}b+g_{c}(\sigma_{+}+\sigma_{-})(b^{\dagger}+b).$
(9)
Performing a unitary rotation of the spin such that
$\sigma_{z}\rightarrow-\sigma_{z}$ and $\sigma_{\pm}\rightarrow-\sigma_{\mp}$,
we arrive at the Hamiltonian
$\displaystyle{H}=\hbar\omega_{+}a_{+}^{+}a_{+}+\hbar\omega_{-}a_{-}^{+}a_{-}-\frac{1}{2}g\mu_{B}B\sigma_{z}$
$\displaystyle+\frac{\alpha}{\widetilde{l}}[\gamma_{-}(\sigma_{+}a_{-}+\sigma_{-}a_{-}^{+})-\gamma_{+}(\sigma_{-}a_{+}+\sigma_{+}a_{+}^{+})]$
$\displaystyle+\hbar\omega_{p}b^{+}b-g_{c}(\sigma_{+}+\sigma_{-})(b^{\dagger}+b).$
(10)
We now derive an approximation form of this Hamiltonian by borrowing the
observation from quantum optics that the terms preceded by $\gamma_{+}$ in
Eq.(10) are counterrotating, and thus negligible under the rotating-wave
approximation when the spin-orbit coupling is small compared to the
confinement Debald . The last term in Eq.(10) treats in the conventional
rotaing-wave approximation of quantum optics.
$\displaystyle{H}=\hbar\omega_{+}a_{+}^{+}a_{+}+\hbar\omega_{-}a_{-}^{+}a_{-}+\frac{1}{2}|g|\mu_{B}B\sigma_{z}$
$\displaystyle+\lambda(\sigma_{+}a_{-}+\sigma_{-}a_{-}^{+})+\hbar\omega_{p}b^{+}b-g_{c}(\sigma_{+}b+\sigma_{-}b^{+}),$
(11)
where $\lambda={\alpha\gamma_{-}}/{\widetilde{l}}$. Since $g$ is negative in
InGaAs, we choose the absolute value $|g|$ of $g$. It is obvious that the
$\omega_{+}$ mode is decoupled from the rest of the system, giving
$H=\hbar\omega_{+}n_{+}+H_{JC}$ where
$\displaystyle
H_{JC}=\hbar\omega_{-}a_{-}^{+}a_{-}+\frac{1}{2}|g|\mu_{B}B\sigma_{z}+\hbar\omega_{p}b^{+}b$
$\displaystyle+\lambda(\sigma_{+}a_{-}+\sigma_{-}a_{-}^{+})-g_{c}(\sigma_{+}b+\sigma_{-}b^{+}).$
(12)
This is the well-known two mode Jaynes-Cummings model of quantum optics. In
general this Hamiltonian can not be solved exactly except
$\omega_{p}=\omega_{-}$. In what follows, for the sake of analytical
simplicity, we consider $\omega_{p}=\omega_{-}$ which we can use a frequency-
controllable laser and a special circuit to satisfy this condition in real
experiments.
In order to solve the above Hamiltonian, we define the normal-mode operators:
$A=e_{1}a_{-}+e_{2}b,$ (13) $K=e_{2}a_{-}-e_{1}b,$ (14)
where
$e_{1}=\frac{\lambda}{\sqrt{\lambda^{2}+g_{c}^{2}}},e_{2}=\frac{-g_{c}}{\sqrt{\lambda^{2}+g_{c}^{2}}},$
(15)
with $e_{1}^{2}+e_{2}^{2}=1$. The new operators satisfy the commutation
relationsMarchiolli
$\begin{split}[A,A^{\dagger}]=1,[N_{A},A]=-A,[N_{A},A^{\dagger}]=A^{\dagger}&,\\\
[K,K^{\dagger}]=1,[N_{K},K]=-K,[N_{K},K^{\dagger}]=K^{\dagger}&,\\\
[A,K]=0,[A,K^{\dagger}]=0,[N_{A},N_{K}]=0&,\end{split}$ (16)
where $N_{A}=A^{\dagger}A(N_{K}=K^{\dagger}K)$ is the number operator related
to the normal-mode operator $A(K)$. Introducing the number-sum operator
$S=N_{A}+N_{K}$ and the number-difference $D=N_{A}-N_{K}$, we can verify that
the Hamiltonian (12) transforms into the following
Hamiltonian:(i)$S=n_{a}+n_{b}$ is a conserved quantity
($n_{a}=a_{-}^{\dagger}a_{-}$ and $n_{b}=b^{\dagger}b$); (ii) the operator D
can be written in terms of the generators$\\{Q_{+},Q_{-},Q\\}$ of the SU(2)
Lie algebra,
$D=2(e_{1}^{2}-e_{2}^{2})Q_{0}+2e_{1}e_{2}(Q_{+}+Q_{-}),$ (17)
where $Q_{-}=a_{-}b^{\dagger},Q_{+}=a_{-}^{\dagger}b$, and
$Q_{0}=\frac{1}{2}(a_{-}^{\dagger}a_{-}-b^{\dagger}b)$, with
$[Q_{-},Q_{+}]=-2Q_{0}$ and $[Q_{0},Q_{\pm}]=\pm Q_{\pm}$;(iii) the
commutation relation between the operators S and D is null, i.e., $[S,D]=0$;
and consequently, (iv) the Hamiltonian $H_{JC}$ simplifies to
$H_{JC}=H_{0}+V$, where
$\begin{split}H_{0}=\hbar\omega_{p}(S+\frac{1}{2}\sigma_{z})&,\\\
V=\frac{1}{2}\delta\sigma_{z}+\lambda_{A}(\sigma_{-}A^{+}+\sigma_{+}A)&,\end{split}$
(18)
with $[H_{0},V]=0$. $\lambda_{A}=\sqrt{\lambda^{2}+g_{c}^{2}}$ is an effective
coupling constant and $\delta=\omega_{p}-|g|\mu_{B}B/\hbar$. The above
Hamiltonian can be solved exactly. The eigenstates of this Hamiltonian are
given by
$|\Psi^{(n,\pm)}\rangle=cos\theta^{(n,\pm)}|n,\uparrow\rangle+sin\theta^{(n,\pm)}|n+1,\downarrow\rangle,$
(19) $tan\theta^{(n,\pm)}=(\delta\pm\Delta_{n})/2\lambda_{A}\sqrt{(n+1)},$
(20)
where $\Delta_{n}=\sqrt{\delta^{2}+4\lambda_{A}^{2}(n+1)}$ and $|\uparrow>$
($|\downarrow>$) is the spin-up (down) state.
According to Ref.Fuentes , since only the quasi-mode $A$ is coupled with the
spin of the electron, so the phase shift operator $U(\varphi)=e^{-i\varphi
A^{\dagger}A}$ is introduced. Applied adiabatically to the Hamiltonian (18),
the phase shift operator alters the state of the field and gives rise to the
following eigenstates:
$\begin{split}|\psi^{(n,\pm)}>=e^{-in\varphi}cos\theta^{(n,\pm)}|n,\uparrow>+&\\\
e^{-i(n+1)\varphi}sin\theta^{(n,\pm)}|n+1,\downarrow>&.\end{split}$ (21)
Changing $\varphi$ slowly from 0 to $2\pi$, the Berry phase is calculated as
$\Gamma_{l}=i\int_{0}^{2\pi}{{}^{l}}\langle\psi|\frac{\partial}{\partial\varphi}|\psi\rangle^{l}d\varphi$
which is given by
$\Gamma_{l}=2\pi[sin\theta^{(n,l)}]^{2}.$ (22)
This Berry phase is composed of two parts. One is induced by spin-orbit
interaction, the other is induced by quantized light. Therefore if we can
measure the total Berry phase and either part of two Berry phase, we will
measure the other part of Berry phase.
## III Numerical Results
Figure 1: The Berry phase $\Gamma_{+}$ as a function of magnetic field
strength $B$ with three spin-orbit coupling constants ( $\alpha=0.4\times
10^{-12}eVm$, $0.8\times 10^{-12}eVm$ and $1.2\times 10^{-12}eVm$). The other
parameters used are $g=-4$, $m^{*}/m_{e}=0.05$, $g_{c}=0.01meV$, $l_{0}=80nm$,
and $n=0$. Figure 2: The Berry phases of $\Gamma_{+}$ and $\Gamma_{-}$ as a
function of magnetic field strength $B$. The parameters used are
$\alpha=0.4\times 10^{-12}eVm$, $g=-4$, $m^{*}/m_{e}=0.05$, $g_{c}=0.01meV$,
$l_{0}=80nm$, and $n=0$.
For the illustration of the numerical results, we choose the typical
parameters of the InGaAs: $g=-4$, $m^{*}/m_{e}=0.05$ ($m_{e}$ is the mass of
free electron). The dot size is defined by
$l_{0}=\sqrt{\hbar/m^{*}\omega_{0}}$. Figure 1 depicts the Berry phases
$\Gamma_{+}$ as a function of the magnetic field strength for three spin-orbit
couplings. In Figure 1, we can find that all the Berry phases change almost
from 0 to $2\pi$ as the magnetic field strength varies from $20mT$ to $50mT$.
When other parameters are fixed, the spin-orbit coupling constant changes as
$\alpha=0.4\times 10^{-12}eVm$, $0.8\times 10^{-12}eVm$ and $1.2\times
10^{-12}eVm$, the Berry phases $\Gamma_{+}$ will have a slight movement in the
figure. When $B<20mT$ and $B>50mT$, the Berry phase changes gradually, while
when $20mT<B<50mT$, the Berry phase changes dramatically. As the coupling
constant increases, the Berry phase changes from sharply to slowly. The
Shördinger equation has two different eigenenergies when $n=0$. The two
eigenenergies will give two different Berry phases. Figure 2 illustrates these
two Berry phases. In Figure 2, when the others parameter are fixed, one of the
Berry phase changes from 0 to $2\pi$ , while the other changes from $2\pi$ to
0 as the magnetic field strength varies from $20mT$ to $50mT$. Two Berry
phases have an intersecting point at approximatively $B=33mT$, which is
corresponding to the resonant point.
Figure 3: The Berry phases $\Gamma_{+}$ as a function of $B$ with three three
different dot sizes ($l_{0}=70nm,80nm,90nm$). The parameters used are
$\alpha=0.4\times 10^{-12}eVm$, $g=-4$, $m/m_{e}=0.05$, $g_{c}=0.01meV$, and
$n=0$. Figure 4: The Berry phases $\Gamma_{+}$ as a function of $B$ with
three different light coupling constants ($g_{c}=0.01meV,0.02meV,0.03meV$).
The parameters used are $\alpha=0.4\times 10^{-12}eVm$, $g=-4$,
$m/m_{e}=0.05$, $l_{0}=80nm$, and $n=0$.
Figure 3 shows the effect of the dot size on the Berry phase. When dot size
becomes large from 70nm to 90nm, although all three Berry phases change from 0
to $2\pi$, the threshold points of the magnetic field have a large movement.
When the dot size is 70nm, the Berry phase will change dramatically at
approximately 40mT, while the dot sizes are 80nm and 90nm, the turning points
are approximately at 30mT and 20mT, respectively. This implies that the bigger
the dot, the smaller the threshold of the magnetic field strength. Figure 4
illustrates the influence of spin-photon coupling constant on Berry phase. As
the coupling constant becomes large, the Berry phase becomes less drastic as
shown in Figure 4.
In a recent paper, Giuliano et al.Giuliano have designed an experimental
arrangement, which is capacitively coupled the dot to one arm of a double-path
electron interferometer. The phase carried by the transported electrons may be
influenced by the dot. The dot’s phase gives raise to an interference term in
the total conductance across the ring. More recently, Leek et al.Leek have
measured Berry phase in a Ramsey fringe interference experiment. Our
experimental setup proposed here is analogous with these two arrangements as
shown in Figure 5. A beam light is split into two beams, one of the beams
passes through the dot, and interferes with the other one. Accurate control of
the light field for dot is achieved through phase and amplitude modulation of
laser radiation coupled to the dot. We choose a special designed electric
circuit to ensure the magnetic and laser vary synchronistically. Through
detecting the interfered light, we can measure the Berry phase.
Figure 5: A sketch of a possible experimental setup to detect the Berry phase.
a, b and c are three mirror, d is a beam splitter.
## IV Conclusions
In conclusion, we have theoretically investigated the Berry phase in a single
quantum dot in the presence of Rashba spin-orbit interaction. Berry phases as
functions of magnetic field strength, dot size, spin-orbit coupling and
photon-spin coupling constants are evaluated. It is shown that for a given
quantum dot, the spin-orbit coupling constant and photon-spin coupling
constant the Berry phase will alter dramatically from $0$ to $2\pi$ as the
magnetic field strength increases. The threshold of magnetic field is
dependent on the Rashba spin-orbit coupling constant, spin-photon coupling
constant and the dot size. We also propose a practicable method to detect the
Berry phase in such a quantum dot system. Finally, we hope that our
predictions in the present work can be testified by experiments in the near
future.
###### Acknowledgements.
This work has been supported in part by National Natural Science Foundation of
China (No.10774101) and the National Ministry of Education Program for
Training PhD.
## References
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* (6) P.San-Jose, B.Scharfenberger, G.Shon, A.Shnirman, and G.Zarand, arXiv:cond-mat/0710.3931(2007).
* (7) H.Wang and K.D.Zhu, _Euro.Phys.Lett_ _82_ , 60006(2008).
* (8) Y.Zhang, Y.W.Tan, H.L.Stormer and P.Kim, _Nature (London)__438_ ,201(2005).
* (9) M.Möttönen, J.J.Vartiainen, and J.P.Pekola, _Phys. Rev. Lett.__100_ , 177201(2008).
* (10) P.J.Leek, J.M.Fink, A.Blais, _Science_ _318_ ,1889(2007).
* (11) E.I.Rashba, _Sov.Phys.Solid State_ _2_ , 1109(1960)
* (12) E.B.Sonin, _Phys. Rev. Lett._ _99_ ,266602(2007).
* (13) Y.A.Serebrennikov, _Phys.Rev.B_ _73_ ,195317(2006).
* (14) J.C.Egues, G.Burkard, and D.Loss, _Phys.Rev.Lett.__89_ , 176401(2002).
* (15) E.I.Rashba, and Al.L.Efros, _Phys. Rev. Lett.__91_ , 126405 (2003).
* (16) Y.A.Bychkov and E.I.Rashba, _JEPT Lett.__39_ , 78(1984).
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|
arxiv-papers
| 2008-10-30T02:43:15 |
2024-09-04T02:48:58.545720
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Huan Wang, Ka-Di Zhu",
"submitter": "Huan Wang",
"url": "https://arxiv.org/abs/0810.5405"
}
|
0810.5573
|
# A branch-and-bound feature selection algorithm for U-shaped cost functions
Marcelo Ris, Junior Barrera, David C. Martins Jr M. Ris, J. Barrera and D. C.
Martins Jr are with the Instituto de Matemática e Estatística, Universidade de
São Paulo, São Paulo, Brazil
###### Abstract
This paper presents the formulation of a combinatorial optimization problem
with the following characteristics: $i$.the search space is the power set of a
finite set structured as a Boolean lattice; $ii$.the cost function forms a
U-shaped curve when applied to any lattice chain. This formulation applies for
feature selection in the context of pattern recognition. The known approaches
for this problem are branch-and-bound algorithms and heuristics, that explore
partially the search space. Branch-and-bound algorithms are equivalent to the
full search, while heuristics are not. This paper presents a branch-and-bound
algorithm that differs from the others known by exploring the lattice
structure and the U-shaped chain curves of the search space. The main
contribution of this paper is the architecture of this algorithm that is based
on the representation and exploration of the search space by new lattice
properties proven here. Several experiments, with well known public data,
indicate the superiority of the proposed method to SFFS, which is a popular
heuristic that gives good results in very short computational time. In all
experiments, the proposed method got better or equal results in similar or
even smaller computational time.
###### Index Terms:
Boolean lattice; branch-and-bound algorithm; U-shaped curve; classifiers;
W-operators; feature selection; subset search; optimal search.
## I Introduction
A combinatorial optimization algorithm chooses the object of minimum cost over
a finite collection of objects, called search space, according to a given cost
function. The simplest architecture for this algorithm, called full search,
access each object of the search space, but it does not work for huge spaces.
In this case, what is possible is to access some objects and choose the one of
minimum cost, based on the observed measures. Heuristics and branch-and-bound
are two families of algorithms of this kind. An heuristic algorithm does not
have formal guaranty of finding the minimum cost object, while a branch-and-
bound algorithm has mathematical properties that guarantee to find it.
Here, it is studied a combinatorial optimization problem such that the search
space is composed of all subsets of a finite set with $n$ points (i.e., a
search space with $2^{n}$ objects), organized as a Boolean lattice, and the
cost function has a U-shape in any chain of the search space or, equivalently,
the cost function has a U-shape in any maximal chain of the search space.
This structure is found in some applied problems such as feature selection in
pattern recognition [dudahart2000chap1, jaduma:2000] and W-operator window
design in mathematical morphology [dmartins2006]. In these problems, a minimum
subset of features, that is sufficient to represent the objects, should be
chosen from a set of $n$ features. In W-operator design, the features are
points of a finite rectangle of $Z^{2}$ called window. The U-shaped functions
are formed by error estimation of the classifiers or of the operators designed
or by some measures, as the entropy, on the corresponding estimated join
distribution. This is a well known phenomenon in pattern recognition: for a
fixed amount of training data, the increasing number of features considered in
the classifier design induces the reduction of the classifier error by
increasing the separation between classes until the available data becomes too
small to cover the classifier domain and the consequent increase of the
estimation error induces the increase of the classifier error. Some known
approaches for this problem are heuristics. A relatively well succeeded
heuristic algorithm is SFFS [pudil94], which gives good results in relatively
small computational time.
There is a myriad of branch-and-bound algorithms in the literature that are
based on monotonicity of the cost-function [frank, nakariyakul, wang, yang].
For a detailed review of branch-and-bound algorithms, refer to [somol]. If the
real distribution of the joint probability between the patterns and their
classes were known, larger dimensionality would imply in smaller
classification errors. However, in practice, these distributions are unknown
and should be estimated. A problem with the adoption of monotonic cost-
functions is that they do not take into account the estimation errors
committed when many features are considered (“curse of dimensionality” also
known as “U-curve problem” or “peaking phenomena” [jaduma:2000]).
This paper presents a branch-and-bound algorithm that differs from the others
known by exploring the lattice structure and the U-shaped chain curves of the
search space.
Some experiments were performed to compare the SFFS to the U-curve approach.
Results obtained from applications such as W-operator window design, genetic
network architecture identification and eight UCI repository data sets show
encouraging results, since the U-curve algorithm beats (i.e., finds a node
with smaller cost than the one found by SFFS) the SFFS results in smaller
computational time for 27 out of 38 data sets tested. For all data sets, the
U-curve algorithm gives a result equal or better than SFFS, since the first
covers the complete search space.
Though the results obtained with the application of the method developed to
pattern recognition problems are exciting, the great contribution of this
paper is the discovery of some lattice algebra properties that lead to a new
data structure for the search space representation, that is particularly
adequate for updates after up-down lattice interval cuts (i.e., cuts by
couples of intervals [0,X] and [X,W]). Classical tree based search space
representations does not have this property. For example, if the Depth First
Search were adopted to represent the Boolean lattice only cuts in one
direction could be performed.
Following this introduction, Section 2 presents the formalization of the
problem studied. Section 3 describes structurally the branch-and-bound
algorithm designed. Section 4 presents the mathematical properties that
support the algorithm steps. Section 5 presents some experimental results
comparing U-curve to SFFS. Finally, Conclusion discusses the contributions of
this paper and proposes some next steps of this research.
## II The Boolean U-curve optimization problem
Let $W$ be a finite subset, $\mathscr{P}(W)$ be the collection of all subsets
of $W$, $\subseteq$ be the usual inclusion relation on sets and, $|W|$ denote
the cardinality of $W$. The search space is composed by $2^{|W|}$ objects
organized in a Boolean lattice.
The partially ordered set $(\mathscr{P}(W),\subseteq)$ is a complete Boolean
lattice of degree $|W|$ such that: the smallest and largest elements are,
respectively, $\emptyset$ and $W$; the sum and product are, respectively, the
usual union and intersection on sets and the complement of a set $X$ in
$\mathscr{P}(W)$ is its complement in relation to $W$, denoted by $X^{c}$.
Subsets of $W$ will be represented by strings of zeros and ones, with $0$
meaning that the point does not belong to the subset and $1$ meaning that it
does. For example, if $W=\\{(-1,0),(0,0),$ $(+1,0)\\}$, the subset
$\\{(-1,0),(0,0)\\}$ will be represented by $110$. In an abuse of language,
$X=110$ means that $X$ is the set represented by $110$.
A chain $\mathcal{A}$ is a collection
$\\{A_{1},A_{2},\ldots,A_{k}\\}\subseteq\mathcal{X}\subseteq\mathscr{P}(W)$
such that $A_{1}\subseteq A_{2}\subseteq\ldots\subseteq A_{k}$. A chain
$\mathcal{M}\subseteq\mathcal{X}$ is maximal in $\mathcal{X}$ if there is no
other chain $\mathcal{C}\subseteq\mathcal{X}$ such that $\mathcal{C}$ contains
properly $\mathcal{M}$.
Let $c$ be a cost function defined from $\mathscr{P}(W)$ to $\mathbb{R}$. We
say that $c$ is decomposable in U-shaped curves if, for every maximal chain
$\mathcal{M}\subseteq\mathscr{P}(W)$, the restriction of $c$ to $\mathcal{M}$
is a U-shaped curve, i. e., for every $A,X,B\in\mathcal{M}$, $A\subseteq
X\subseteq B\Rightarrow\max(c(A),c(B))\geq c(X)$.
Figure 1 shows a complete Boolean lattice $\mathcal{L}$ of degree $4$ with a
cost function $c$ decomposable in U-shaped curves. In this figure, it is
emphasized a maximal chain in $\mathcal{L}$ and its cost function. Figure 2
presents the curve of the same cost function restricted to some maximal chains
in $\mathcal{L}$ and in $\mathcal{X}\subseteq\mathcal{L}$. Note the U-shape of
the curves in Figure 2.
Figure 1: A complete Boolean lattice $\mathcal{L}$ of degree $4$ and the cost
function decomposable in U-shaped curves.
$\mathcal{X}=\mathcal{L}-\\{0000,0010,0001,1110,1111\\}$ is a poset obtained
from $\mathcal{L}$. A maximal chain in $\mathcal{L}$ is emphasized. The
element $0111$ is the global minimum element and $0101$ is the local minimum
element in the maximal chain. Figure 2: The four possible representaion of the
cost function $c$ restricted to some maximal chains in $\mathcal{L}$ (a) and
in $\mathcal{X}\subseteq\mathcal{L}$ (b-d) of Figure 1.
Our problem is to find the element (or elements) of minimum cost in a Boolean
lattice of degree $|W|$. The full search in this space is an exponential
problem, since this space is composed by $2^{|W|}$ elements. Thus, for
moderately large $|W|$, the full search becomes unfeasible.
## III The U-curve algorithm
The U-shaped format of the restriction of the cost function to any maximal
chain is the key to develop a branch-and-bound algorithm, the U-curve
algorithm, to deal with the hard combinatorial problem of finding subsets of
minimum cost.
Let $A$ and $B$ be elements of the Boolean lattice $\mathcal{L}$. An interval
$[A,B]$ of $\mathcal{L}$ is the subset of $\mathcal{L}$ given by
$[A,B]=\\{X\in\mathcal{L}:A\subseteq X\subseteq B\\}$. The elements $A$ and
$B$ are called, respectively, the left and right extremities of $[A,B]$.
Intervals are very important for characterizing decompositions in Boolean
lattices [Banon, Salas].
Let $R$ be an element of $\mathcal{L}$. In this paper, intervals of the type
$[\emptyset,R]$ and $[R,W]$ are called, respectively, lower and upper
intervals. The right extremity of a lower interval and the left extremity of
an upper interval are called, respectively, lower and upper restrictions. Let
$\mathcal{R}_{L}$ and $\mathcal{R}_{U}$ denote, respectively, collections of
lower and upper intervals. The search space will be the poset
$\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$ obtained by eliminating the
collections of lower and upper restrictions from $\mathcal{L}$, i. e.,
$\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})=\mathcal{L}-\bigcup\\{[\emptyset,R]:R\in\mathcal{R}_{L}\\}-\bigcup\\{[R,W]:R\in\mathcal{R}_{U}\\}$.
In cases in which only the lower or the upper intervals are eliminated, the
resulting search space is denoted, respectively, by
$\mathcal{X}(\mathcal{R}_{L})$ and $\mathcal{X}(\mathcal{R}_{U})$ and given,
respectively, by
$\mathcal{X}(\mathcal{R}_{L})=\mathcal{L}-\bigcup\\{[\emptyset,R]:R\in\mathcal{R}_{L}\\}$
and
$\mathcal{X}(\mathcal{R}_{U})=\mathcal{L}-\bigcup\\{[R,W]:R\in\mathcal{R}_{U}\\}$.
The search space is explored by an iterative algorithm that, at each
iteration, explores a small subset of
$\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$, computes a local minimum,
updates the list of minimum elements found and extends both restriction sets,
eliminating the region just explored. The algorithm is initiated with three
empty lists: minimum elements, lower and upper restrictions. It is executed
until the whole space is explored, i. e., until
$\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$ becomes empty. The subset of
$\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$ eliminated at each iteration is
defined from the exploration of a chain, which may be done in down-up or up-
down direction. Algorithm 1 describes this process. The direction selection
procedure (line 5) can use a random or an adaptative method. The random method
states a static probability to select the down-up or up-down direction. The
adaptative method calculates a new probability to each direction giving more
probability to down-up direction if most of the local minima is closest to the
bottom of the lattice and up-down otherwise.
Algorithm 1 U-curve-algorithm()
1: $\mathcal{M}$ $\Leftarrow\emptyset$
2: $\mathcal{R}_{L}\Leftarrow\emptyset$
3: $\mathcal{R}_{U}\Leftarrow\emptyset$
4: while $\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})\neq\emptyset$ do
5: _direction_ $\Leftarrow$ Select-Direction()
6: if _direction_ is _UP_ then
7: Down-Up-Direction($\mathcal{R}_{L}$, $\mathcal{R}_{U}$)
8: else
9: Up-Down-Direction($\mathcal{R}_{L}$, $\mathcal{R}_{U}$)
10: end if
11: end while
An element $C$ of the poset $\mathcal{X}\subseteq\mathcal{L}$ is called a
minimal element of $\mathcal{X}$, if there is no other element $C^{\prime}$ of
$\mathcal{X}$ with $C^{\prime}\subset C$. In Figure 1, the minimal elements of
$\mathcal{X}(\mathcal{R}_{L})$ are: $1000$, $0100$ and $0011$. If the down-up
direction is chosen, the Down-Up-Direction procedure is performed (algorithm
2):
* •
Minimal-Element procedure calculates a minimal element $B$ of the poset
$\mathcal{X}(\mathcal{R}_{L})$. Only the lower restriction set is used to
calculate the minimal element $B$. An element $B$ is said to be covered by the
lower restriction set $\mathcal{R}_{L}$, if $\exists
R\in\mathcal{R}_{L}:B\subseteq R$, and $B$ is said to be covered by the upper
restriction set $\mathcal{R}_{U}$, if $\exists R\in\mathcal{R}_{U}:R\subseteq
B$. When the calculated $B$ is covered by an upper restriction, it is
discarded, i.e., the lower restriction set is updated with $B$ and a new
iteration begins (lines 1-5).
* •
The down-up direction chain exploration procedure begins with a minimal
element $B$ and flows by random selection of upper adjacent elements from the
current poset $\mathcal{X}(\mathcal{R}_{L},$ $\mathcal{R}_{U})$ until it finds
the U-curve condition, i. e., the last element selected ($B$) has cost bigger
than the previous one ($M$) (lines 7-11).
* •
At this point, the element $M$ is the minimum element of the chain explored,
$A$ and $B$ are, respectively, the lower and upper adjacent elements of $M$
(i.e., $A\subset M\subset B$ and $\\{X\in\mathscr{P}(W):A\subset
M\\}=\\{X\in\mathscr{P}(W):M\subset B=\emptyset$) by construction, $c(A)\leq
c(M)\leq C(B)$. It can be proved that any element $C$ of
$\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$, with $C\subset A$, has cost
bigger than $A$ and, any element $D$ of
$\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$, with $B\subset D$, has cost
bigger than $B$. By using this property, the lower and upper restrictions can
be updated, respectively, by $A$ and $B$ (lines 12-17). Figure 3 shows a
schematic representation of the first iteration of the algorithm and the
elements contained in the intervals $[\emptyset,A=1\ldots 1010\ldots 0]$ and
$[B=1\ldots 11110\ldots 0,W]$.
* •
The result list can be updated with $M$ (line 18) , i. e., $M$ will be
included in the result list if it has cost lower than the elements already
saved in the list. The result list can save a pre-defined number of elements
with low costs or only elements with the overall minimum cost.
* •
In order to prevent visiting the element $M$ more than once, a recursive
procedure called minimum exhausting procedure is performed (line 19)
Algorithm 2 Down-Up-Direction(ElementSet $\mathcal{R}_{L}$, ElementSet
$\mathcal{R}_{U}$)
1: $B\Leftarrow$ Minimal-Element($\mathcal{R}_{L}$)
2: if $B$ is covered by $\mathcal{R}_{U}$ then
3: Update-Lower-Restriction($B$, $\mathcal{R}_{L}$)
4: return
5: end if
6: $M\Leftarrow$ null
7: repeat
8: $A\Leftarrow M$
9: $M\Leftarrow B$
10: $B\Leftarrow$ Select-Upper-Adjacent($M$, $\mathcal{R}_{L}$,
$\mathcal{R}_{U}$)
11: until $c(B)>c(M)$ or $B=$ null
12: if $A\neq$ null then
13: Update-Lower-Restriction($A$, $\mathcal{R}_{L}$)
14: end if
15: if $B\neq$ null then
16: Update-Upper-Restriction($B$, $\mathcal{R}_{U}$)
17: end if
18: Update-Results($M$)
19: Minimum-Exhausting($M$, $\mathcal{R}_{L}$, $\mathcal{R}_{U}$)
Figure 3: A schematic representation of a step of the algorithm, the detached
areas represents the elements contained in a lower and upper restrictions.
An element is called a minimum exhausted element in $\mathcal{L}$ if all its
adjacents elements (upper and lower) have cost bigger than it. This definition
can be extended to the poset $\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$,
i. e., all its adjacent elements (upper and lower) in
$\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$ have cost bigger than it. In
Figure 1 we can see that the elements $1010$, $1001$ and $0111$ are minimum
exhauted elements in $\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$, but
$1001$ is not a minimum exhauted element in $\mathcal{L}$. In this paper, the
term minimum exhausted will be applied always refering to a poset
$\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$.
Algorithm 3 Minimum-Exhausting(Element $M$, ElementSet $\mathcal{R}_{L}$,
ElementSet $\mathcal{R}_{U}$)
1: Push $M$ to $\mathcal{S}$
2: while $\mathcal{S}$ is not empty do
3: $T\Leftarrow$ Top($\mathcal{S}$)
4: MinimumExhausted $\Leftarrow$ true
5: for all $A$ adjacent of $T$ in
$\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$ and $A\not\in\mathcal{S}$ do
6: if c($A$) $\leq$ c($T$) then
7: Push $A$ to $\mathcal{S}$
8: MinimumExhausted $\Leftarrow$ false
9: else
10: if $A$ is upper adjacent of $T$ then
11: Update-Upper-Restriction($A$, $\mathcal{R}_{U}$)
12: else
13: Update-Lower-Restriction($A$, $\mathcal{R}_{L}$)
14: end if
15: end if
16: end for
17: if MinimumExhausted then
18: Pop $T$ from $\mathcal{S}$
19: Update-Results($T$)
20: Update-Lower-Restriction($T$, $\mathcal{R}_{L}$)
21: Update-Upper-Restriction($T$, $\mathcal{R}_{U}$)
22: end if
23: end while
24: return
The minimum exhausting procedure (Algorithm 3) is a recursive process that
visit all the adjacent elements of a given element $M$ and turn all of them
into minimum exhausted elements in the resulting poset
$\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$. It uses a stack $\mathcal{S}$
to perform the recursive process. $\mathcal{S}$ is initialized by pushing $M$
to it and the process is performed while $\mathcal{S}$ is not empty (lines
2-22). At each iteration, the algorithm processes the top element $T$ of
$\mathcal{S}$: all the adjacent elements (upper and down) of $T$ in
$\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$ and not in $\mathcal{S}$ are
checked. If the cost of an adjacent element $A$ is lower (or equal) than the
cost of $T$ then $A$ is pushed to $\mathcal{S}$. If the cost of $A$ is bigger
than the cost of $T$ then one of the restriction sets can be updated with $A$,
lower restriction set if $A$ is lower adjacent of $T$ and upper restriction
set if $A$ is upper adjacent of $T$ (lines 5-16). If $T$ is a minimum
exhausted element in $\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$, i. e.,
there is no adjacent element $A$ in
$\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$ with cost lower than $T$, then
$T$ is removed from $\mathcal{S}$ and, also, the restriction sets and the
result list are updated with $T$ (lines 19-21). At the end of this procedure
all the elements processed are minimum-exhausted elements in
$\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$.
Figure 4 shows a graphical representation of the minimum exhausting process.
4-A shows a chain construction process in up direction, the chain has its
edges emphasized. The element $M=010101$ (orange-colored) has the minimum cost
over the chain. The elements in black are the elements eliminated from the
search space by the restrictions obtained by the lower and upper adjacent
elements of the local minimum $M$. The stack begins with the element $M$.
Figure 4-B shows the first iteration of the minimum exhausting process. The
arrows in red and the elements in red indicates the adjacents elements of $M$
(top of the stack) that have cost lower (or equal) than it. These elements
$010001$ and $010111$ are pushed to the stack. The adjacent elements of $M$
with cost bigger than it can update the restriction sets, i. e., the lower
adjacent element $000101$ updates the lower restriction set and the upper
adjacent element $000101$ updates the upper restriction set. Figure 4-C shows
the second iteration: the adjacent elements $010011$ and $000111$ with cost
lower (or equal) than the new top element $010111$ are pushed to the stack and
the other adjacent elements $010110$ and $011111$ with cost bigger than
$010111$ update, respectively, the lower and upper restriction sets. In Figure
4-D the element $000111$ is a minimum exhausted element (grey color) in
$\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$ and it is is removed from
stack. In Figure 4-E the elements eliminated by the new interval
$[\emptyset,000111]$ and $[000111,W]$ are turned into black color. At this
point, $010011$ is a minimum exhausted (grey color) in
$\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$ and it is removed from stack.
From Figure 4-F to Figure 4-H all the elements are removed from stack and the
elements removed by the new restrictions are turned into black color. Figure
4-H shows all the elements removed from a single minimum exhausted process.
Figure 4: Representation of the minimum exhausting process.
The procedures to calculate minimal and maximal elements and the procedure to
update lower and upper restriction sets will be discussed in the next section.
## IV Mathematical foundations
This section introduces mathematical foundations of some modules of the
algorithm.
### IV-A Minimal and Maximal Construction Procedure
Each iteration of the algorithm requires the calculation of a minimal element
in $\mathcal{X}(\mathcal{R}_{L})$ or a maximal element in
$\mathcal{X}(\mathcal{R}_{U})$. It is presented here a simple solution for
that. The next theorem is the key for this solution.
Theorem 1. For every $A\in\mathcal{X}(\mathcal{R}_{L})$,
$A\in\mathcal{X}(\mathcal{R}_{L})\Leftrightarrow A\cap
R^{c}\neq\emptyset,\forall R\in\mathcal{R}_{L}$.
###### Proof:
(in Appendix Section)
Algorithm 4 implements the minimal construction procedure. It builds a minimal
element $C$ of the poset $\mathcal{X}(\mathcal{R}_{L})$. The process begins
with $C=(\underbrace{1\ldots 1}_{n})$ and $S=(\underbrace{1\ldots 1}_{n})$ and
executes a $n$-loop (lines 3-16) trying to remove components from $C$. At each
step, a component $k,k\in\\{1,\ldots,n\\}$ is chosen exclusively from $S$ ($S$
prevents multi-selecting). If the element $C^{\prime}$ resulted from $C$ by
removing the component $k$ is contained in $\mathcal{X}(\mathcal{R}_{L})$ then
$C$ is updated with $C^{\prime}$ (lines 7-15).
Algorithm 4 Minimal-Element(ElementSet $\mathcal{R}_{L}$)
1: $C\Leftarrow\underbrace{1\ldots 1}_{n}$
2: $S\Leftarrow\underbrace{1\ldots 1}_{n}$
3: while $S\neq\underbrace{0\ldots 0}_{n}$ do
4: $k\Leftarrow$ random index in $\\{1,\ldots,n\\}$ where $S[k]=1$
5: $S[k]\Leftarrow 0$
6: $C^{\prime}\Leftarrow C\setminus{k}$
7: RemoveElement $\Leftarrow$ true
8: for all $R$ in $\mathcal{R}_{L}$ do
9: if $R^{c}\cap C^{\prime}=\emptyset$ then
10: RemoveElement $\Leftarrow$ false
11: end if
12: end for
13: if RemoveElement then
14: $C\Leftarrow C^{\prime}$
15: end if
16: end while
17: return $C$
The minimal element calculated is equal to $\underbrace{1\ldots 1}_{n}$ when
$\mathcal{R}_{L}=\\{\underbrace{1\ldots 1}_{n}\\}$. At this point, the poset
$\mathcal{X}(\mathcal{R}_{L},\mathcal{R}_{U})$ is empty and the algorithm
stops in the next iteration.
The next theorem proves the correctness of Algorithm 4 .
Theorem 2. The element $C$ of $\mathcal{X}(\mathcal{R}_{L})$ returned by the
minimal construction process (Algorithm 4) is a minimal element in
$\mathcal{X}(\mathcal{R}_{L})$.
|
arxiv-papers
| 2008-10-30T20:24:28 |
2024-09-04T02:48:58.553988
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marcelo Ris, Junior Barrera, David C. Martins Jr",
"submitter": "David Correa Martins Jr",
"url": "https://arxiv.org/abs/0810.5573"
}
|
0810.5608
|
# Measurements of the observed cross sections for $e^{+}e^{-}\to$ exclusive
light hadrons containing $K_{S}^{0}$ meson at $\sqrt{s}=3.773$ and 3.650 GeV
M. Ablikim1, J. Z. Bai1, Y. Bai1, Y. Ban11, X. Cai1, H. F. Chen15, H. S.
Chen1, H. X. Chen1, J. C. Chen1, Jin Chen1, X. D. Chen5, Y. B. Chen1, Y. P.
Chu1, Y. S. Dai17, Z. Y. Deng1, S. X. Du1, J. Fang1, C. D. Fu14, C. S. Gao1,
Y. N. Gao14, S. D. Gu1, Y. T. Gu4, Y. N. Guo1, K. L. He1, M. He12, Y. K.
Heng1, J. Hou10, H. M. Hu1, T. Hu1, G. S. Huang1a, X. T. Huang12, Y. P.
Huang1, X. B. Ji1, X. S. Jiang1, J. B. Jiao12, D. P. Jin1, S. Jin1, Y. F.
Lai1, H. B. Li1, J. Li1, L. Li1, R. Y. Li1, W. D. Li1, W. G. Li1, X. L. Li1,
X. N. Li1, X. Q. Li10, Y. F. Liang13, H. B. Liao1b, B. J. Liu1, C. X. Liu1,
Fang Liu1, Feng Liu6, H. H. Liu1c, H. M. Liu1, J. B. Liu1d, J. P. Liu16, H. B.
Liu4, J. Liu1, R. G. Liu1, S. Liu8, Z. A. Liu1, F. Lu1, G. R. Lu5, J. G. Lu1,
C. L. Luo9, F. C. Ma8, H. L. Ma1, L. L. Ma1e, Q. M. Ma1, M. Q. A. Malik1, Z.
P. Mao1, X. H. Mo1, J. Nie1, R. G. Ping1, N. D. Qi1, H. Qin1, J. F. Qiu1, G.
Rong1, X. D. Ruan4, L. Y. Shan1, L. Shang1, D. L. Shen1, X. Y. Shen1, H. Y.
Sheng1, H. S. Sun1, S. S. Sun1, Y. Z. Sun1, Z. J. Sun1, X. Tang1, J. P.
Tian14, G. L. Tong1, X. Wan1, L. Wang1, L. L. Wang1, L. S. Wang1, P. Wang1, P.
L. Wang1, W. F. Wang1f, Y. F. Wang1, Z. Wang1, Z. Y. Wang1, C. L. Wei1, D. H.
Wei3, Y. Weng1, N. Wu1, X. M. Xia1, X. X. Xie1, G. F. Xu1, X. P. Xu6, Y. Xu10,
M. L. Yan15, H. X. Yang1, M. Yang1, Y. X. Yang3, M. H. Ye2, Y. X. Ye15, C. X.
Yu10, G. W. Yu1, C. Z. Yuan1, Y. Yuan1, S. L. Zang1g, Y. Zeng7, B. X. Zhang1,
B. Y. Zhang1, C. C. Zhang1, D. H. Zhang1, H. Q. Zhang1, H. Y. Zhang1, J. W.
Zhang1, J. Y. Zhang1, X. Y. Zhang12, Y. Y. Zhang13, Z. X. Zhang11, Z. P.
Zhang15, D. X. Zhao1, J. W. Zhao1, M. G. Zhao1, P. P. Zhao1, B. Zheng1, H. Q.
Zheng11, J. P. Zheng1, Z. P. Zheng1, B. Zhong9 L. Zhou1, K. J. Zhu1, Q. M.
Zhu1, X. W. Zhu1, Y. C. Zhu1, Y. S. Zhu1, Z. A. Zhu1, Z. L. Zhu3, B. A.
Zhuang1, B. S. Zou1
(BES Collaboration)
1 Institute of High Energy Physics, Beijing 100049, People’s Republic of China
2 China Center for Advanced Science and Technology(CCAST), Beijing 100080,
People’s Republic of China
3 Guangxi Normal University, Guilin 541004, People’s Republic of China
4 Guangxi University, Nanning 530004, People’s Republic of China
5 Henan Normal University, Xinxiang 453002, People’s Republic of China
6 Huazhong Normal University, Wuhan 430079, People’s Republic of China
7 Hunan University, Changsha 410082, People’s Republic of China
8 Liaoning University, Shenyang 110036, People’s Republic of China
9 Nanjing Normal University, Nanjing 210097, People’s Republic of China
10 Nankai University, Tianjin 300071, People’s Republic of China
11 Peking University, Beijing 100871, People’s Republic of China
12 Shandong University, Jinan 250100, People’s Republic of China
13 Sichuan University, Chengdu 610064, People’s Republic of China
14 Tsinghua University, Beijing 100084, People’s Republic of China
15 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
16 Wuhan University, Wuhan 430072, People’s Republic of China
17 Zhejiang University, Hangzhou 310028, People’s Republic of China
a Current address: University of Oklahoma, Norman, Oklahoma 73019, USA
b Current address: DAPNIA/SPP Batiment 141, CEA Saclay, 91191, Gif sur Yvette
Cedex, France
c Current address: Henan University of Science and Technology, Luoyang 471003,
People’s Republic of China
d Current address: CERN, CH-1211 Geneva 23, Switzerland
e Current address: University of Toronto, Toronto M5S 1A7, Canada
f Current address: Laboratoire de l’Accélérateur Linéaire, Orsay, F-91898,
France
g Current address: University of Colorado, Boulder, CO 80309, USA
###### Abstract
By analyzing the data sets of 17.3 pb-1 taken at $\sqrt{s}=3.773$ GeV and of
6.5 pb-1 taken at $\sqrt{s}=3.650$ GeV with the BES-II detector at the BEPC
collider, we measure the observed cross sections for the exclusive light
hadron final states of $K_{S}^{0}K^{-}\pi^{+}$,
$K_{S}^{0}K^{-}\pi^{+}\pi^{0}$, $K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{-}$,
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{-}\pi^{0}$,
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{+}\pi^{-}\pi^{-}$ and
$K_{S}^{0}K^{-}\pi^{+}\pi^{0}\pi^{0}$ produced in $e^{+}e^{-}$ annihilation at
the two energy points. We set the upper limits on the observed cross sections
and the branching fractions for $\psi(3770)$ decay to these final states at
$90\%$ C.L..
## I INTRODUCTION
The $\psi(3770)$ resonance is expected to decay almost entirely into
$D\bar{D}$ meson pairs since its width is almost two orders of magnitude
larger than that of $\psi(3686)$ prl39_526 . In recent years, the study of the
$\psi(3770)$ non-$D\bar{D}$ decays becomes an attractive study field in the
charmonium energy region due to the existing puzzle that about $38\%$ of
$\psi(3770)$ does not decay into $D\bar{D}$ meson pairs hepex_0506051 . To
understand the possible excess of the $\psi(3770)$ cross section relative to
the $D\bar{D}$ cross section, BES and CLEO Collaborations made many efforts to
study the $\psi(3770)$ non-$D\bar{D}$ decays. The CLEO Collaboration measured
the $e^{+}e^{-}\to\psi(3770)\to$ non-$D\bar{D}$ cross section to be $(-0.01\pm
0.08^{+0.41}_{-0.30})$ nb prl96_092002 . While the BES Collaboration measured
the branching fraction for $\psi(3770)\to$ non$-D\bar{D}$ decay to be $(15\pm
5)\%$ plb641_145 ; prl97_121801 ; plb659_74 ; prd76_000000 ; pdg07 , which
indicates that, contrary to what is generally expected, the $\psi(3770)$ might
substantially decay into non$-D\bar{D}$ final states or there are some new
structure or physics effects which may partially be responsible for the
largely measured non-$D\bar{D}$ branching fraction of the $\psi(3770)$ decays
prl101_102004 ; plb668_263 . BES Collaboration observed the first
non$-D\bar{D}$ decay mode for $\psi(3770)\to J/\psi\pi^{+}\pi^{-}$, and
measured its decay branching fraction to be ${\mathcal{B}}[\psi(3770)\to
J/\psi\pi^{+}\pi^{-}]=(0.34\pm 0.14\pm 0.09)\%$ hepnp28_325 ; plb605_63 . This
was confirmed by CLEO Collaboration prl96_082004 . Latter, CLEO Collaboration
observed more $\psi(3770)$ exclusive non-$D\bar{D}$ decays, $\psi(3770)\to
J/\psi\pi^{0}\pi^{0}$, $J/\psi\pi^{0}$, $J/\psi\eta$ prl96_082004 ,
$\gamma\chi_{cJ}(J=0,1,2)$ prl96_182002 ; prd74_031106 and $\phi\eta$
prd74_012005 , etc. Summing over these measured branching fractions yields the
sum of the branching fractions for the $\psi(3770)$ exclusive non-$D\bar{D}$
decays not more than 2%. In addition, BES and CLEO Collaborations also
attempted to search for other $\psi(3770)$ exclusive charmless decays
prd70_077101 ; prd72_072007 ; plb650_111 ; plb656_30 ; epjc52_805
prd74_012005 ; prl96_032003 ; prd73_012002 . However, the existing results can
not clarify the possible excess. For better understanding the origin of the
possible excess, search for more $\psi(3770)$ exclusive charmless decays will
be helpful.
In this Letter, we report measurements of the observed cross sections for the
exclusive light hadron final states of $K_{S}^{0}K^{-}\pi^{+}$ (Throughout the
Letter, charge conjugation is implied), $K_{S}^{0}K^{-}\pi^{+}\pi^{0}$,
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{-}$,
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{-}\pi^{0}$,
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{+}\pi^{-}\pi^{-}$ and
$K_{S}^{0}K^{-}\pi^{+}\pi^{0}\pi^{0}$ at the center-of-mass energies of 3.773
and 3.650 GeV with the same method as the one used in our previous works
plb650_111 ; plb656_30 ; epjc52_805 . With the measured cross sections at the
two energy points, we set the upper limits on the observed cross sections and
the branching fractions for $\psi(3770)$ decay to these final states. The
measurements are made by analyzing the data set of 17.3 pb-1 collected at
$\sqrt{s}=3.773$ GeV [called as the $\psi(3770)$ resonance data] and the data
set of 6.5 pb-1 collected at $\sqrt{s}=3.650$ GeV (called as the continuum
data) with the BESII detector at the BEPC collider.
## II BESII detector
The BES-II is a conventional cylindrical magnetic detector that is described
in detail in Refs. nima344_319 ; nima458_627 . A 12-layer vertex chamber (VC)
surrounding the beryllium beam pipe provides input to the event trigger, as
well as coordinate information. A forty-layer main drift chamber (MDC) located
just outside the VC yields precise measurements of charged particle
trajectories with a solid angle coverage of $85\%$ of 4$\pi$; it also provides
ionization energy loss ($dE/dx$) measurements which are used for particle
identification. Momentum resolution of $1.7\%\sqrt{1+p^{2}}$ ($p$ in GeV/$c$)
and $dE/dx$ resolution of $8.5\%$ for Bhabha scattering electrons are obtained
for the data taken at $\sqrt{s}=3.773$ GeV. An array of 48 scintillation
counters surrounding the MDC measures the time of flight (TOF) of charged
particles with a resolution of about 180 ps for electrons. Outside the TOF, a
12 radiation length, lead-gas barrel shower counter (BSC), operating in
limited streamer mode, measures the energies of electrons and photons over
$80\%$ of the total solid angle with an energy resolution of
$\sigma_{E}/E=0.22/\sqrt{E}$ ($E$ in GeV) and spatial resolutions of
$\sigma_{\phi}=7.9$ mrad and $\sigma_{z}=2.3$ cm for electrons. A solenoidal
magnet outside the BSC provides a 0.4 T magnetic field in the central tracking
region of the detector. Three double-layer muon counters instrument the magnet
flux return and serve to identify muons with momentum greater than 500
MeV/$c$. They cover $68\%$ of the total solid angle.
## III EVENT SELECTION
In the reconstruction of the $K_{S}^{0}K^{-}\pi^{+}$,
$K_{S}^{0}K^{-}\pi^{+}\pi^{0}$, $K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{-}$,
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{-}\pi^{0}$,
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{+}\pi^{-}\pi^{-}$ and
$K_{S}^{0}K^{-}\pi^{+}\pi^{0}\pi^{0}$ final states, the $K^{0}_{S}$ and
$\pi^{0}$ mesons are reconstructed through the decays of
$K^{0}_{S}\to\pi^{+}\pi^{-}$ and $\pi^{0}\to\gamma\gamma$.
For each candidate event, we require that at least four charged tracks are
well reconstructed in the MDC with good helix fits, and the polar angle of
each charged track satisfies $|\rm cos\theta|<0.85$. The charged tracks
(except for the $K_{S}^{0}$ meson reconstruction) are required to originate
from the interaction region $V_{xy}<2.0$ cm ($V_{xy}<8.0$ cm) and
$|V_{z}|<20.0$ cm, where $V_{xy}$ and $|V_{z}|$ are the closest approaches in
the $xy$-plane and the $z$ direction, respectively.
The charged particles are identified by using the $dE/dx$ and TOF
measurements, with which the combined confidence levels $CL_{\pi}$ and
$CL_{K}$ for pion and kaon hypotheses are calculated. The pion and kaon
candidates are required to satisfy $CL_{\pi}>0.001$ and $CL_{K}>CL_{\pi}$,
respectively. To reconstruct $K^{0}_{S}$ mesons, we require that the
$\pi^{+}\pi^{-}$ meson pairs must originate from a secondary vertex which is
displaced from the event vertex at least by 4 mm in the $xy$-plane.
The photons are selected with the BSC measurements. The good photon candidates
are required to satisfy the following criteria: the energy deposited in the
BSC is greater than 50 MeV, the electromagnetic shower starts in the first 5
readout layers, the angle between the photon and the nearest charged track is
greater than $22^{\circ}$ plb_597_39 ; plb_608_24 , and the opening angle
between the cluster development direction and the photon emission direction is
less than $37^{\circ}$ plb_597_39 ; plb_608_24 .
For each candidate event, there may be several different charged and/or
neutral track combinations satisfying the above selection criteria for
exclusive light hadron final states. Each combination is subjected to an
energy-momentum conservation kinematic fit. For the processes containing
$\pi^{0}$ meson in the final states, an additional constraint kinematic fit is
imposed on $\pi^{0}\to\gamma\gamma$. Candidates with a fit probability larger
than 1$\%$ are accepted. If more than one combination satisfies the selection
criteria in an event, only the combination with the longest decay distance of
the reconstructed $K_{S}^{0}$ mesons is retained.
To suppress the background from the $D\bar{D}$ decays, we use the double tag
method npb727_395 to remove the $D\bar{D}$ events. For example, for the
$K_{S}^{0}K^{-}\pi^{+}\pi^{0}$ final state, we exclude the all possible events
from $D\bar{D}$ decays by rejecting those in which the $D$ and $\bar{D}$
mesons can be reconstructed in the decay modes of $D^{-}\to K_{S}^{0}K^{-}$
and $D^{+}\to\pi^{+}\pi^{0}$, $D^{-}\to K^{-}\pi^{0}$ and $D^{+}\to
K_{S}^{0}\pi^{+}$, $\bar{D}^{0}\to K_{S}^{0}\pi^{0}$ and $D^{0}\to
K^{-}\pi^{+}$ npb727_395 . For the other final states, the events from
$D\bar{D}$ decays are suppressed similarly. The remaining contaminations from
$D\bar{D}$ decays due to particle misidentification or missing photon(s) are
accounted by using Monte Carlo simulation, as discussed in Section V.
## IV DATA ANALYSIS
In the data analysis, these processes containing $K^{0}_{S}$ meson in the
final state are studied by examining the invariant mass spectra of the
$\pi^{+}\pi^{-}$ combinations satisfying the above selection criteria for the
$K^{0}_{S}$ meson reconstruction. The invariant masses of the $\pi^{+}\pi^{-}$
combinations are calculated with the momentum vectors from the $K^{0}_{S}$
reconstruction. Figure 1 shows the resulting distribution of the invariant
masses of the $\pi^{+}\pi^{-}$ combinations from the selected candidates for
the $K_{S}^{0}K^{-}\pi^{+}$, $K_{S}^{0}K^{-}\pi^{+}\pi^{0}$,
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{-}$,
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{-}\pi^{0}$,
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{+}\pi^{-}\pi^{-}$ and
$K_{S}^{0}K^{-}\pi^{+}\pi^{0}\pi^{0}$ final states. In each figure, the peak
around the $K^{0}_{S}$ nominal mass indicates the production of
$e^{+}e^{-}\to$ exclusive light hadrons containing $K_{S}^{0}$ meson. Fitting
the $\pi^{+}\pi^{-}$ invariant mass spectra with a Gaussian function for the
$K^{0}_{S}$ signal and a flat background yields the number of the events for
each process observed from the $\psi(3770)$ resonance data and the continuum
data. In the fit, the $K^{0}_{S}$ mass and its mass resolution are fixed at
the values obtained by analyzing Monte Carlo samples.
(a)(a’)(b)(b’)(c)(c’)(d)(d’)(e)(e’)(f)(f’)Invariant mass (GeV/c${}^{2})$
Events/0.005GeV/c2
Fig. 1: The $\pi^{+}\pi^{-}$ invariant mass spectra of the candidates for the
(a) $K_{S}^{0}K^{-}\pi^{+}$, (b) $K_{S}^{0}K^{-}\pi^{+}\pi^{0}$, (c)
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{-}$, (d)
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{-}\pi^{0}$, (e)
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{+}\pi^{-}\pi^{-}$ and (f)
$K_{S}^{0}K^{-}\pi^{+}\pi^{0}\pi^{0}$ final states selected from the
$\psi(3770)$ resonance data (left) and the continuum data (right).
## V BACKGROUND SUBTRACTION
Some other events may contribute to the selected candidate events for
$e^{+}e^{-}\to f$ ($f$ represents exclusive light hadron final state). These
include the events from $J/\psi$ and $\psi(3686)$ decays due to ISR returns,
the events from the other final states due to misidentifying a pion as a kaon
or reverse, and the events from $D\bar{D}$ decays. The number $N^{\rm b}$ of
these contaminations should be subtracted from the number $N^{\rm obs}$ of the
candidates for $e^{+}e^{-}\to f$. The estimation of them can be done based on
Monte Carlo simulation. The details about the background subtraction have been
described in Ref. plb650_111 . For each background channel except $D\bar{D}$
decays, 50,000 or 100,000 Monte Carlo events are used in the background
estimation. The Monte Carlo sample of each different background channel is
from ten to several thousands times larger than the data in size.
Monte Carlo study shows that the contaminations from $\psi(3770)\to
J/\psi\pi^{+}\pi^{-}$, $\psi(3770)\to J/\psi\pi^{0}\pi^{0}$, $\psi(3770)\to
J/\psi\pi^{0}$ and $\psi(3770)\to\gamma\chi_{cJ}\hskip 2.84544pt(J=0,1,2)$ can
be neglected.
Even though we have removed the main contaminations from $D\bar{D}$ decays in
the previous event selection (see section III), there are still some events
from $D\bar{D}$ decays satisfying the selection criteria for the light hadron
final states due to particle misidentification or missing photon(s). The
number of these contaminations from $D\bar{D}$ decays are further removed by
analyzing a Monte Carlo sample which is about forty times larger than the
$\psi(3770)$ resonance data. The Monte Carlo events are generated as
$e^{+}e^{-}\to D\bar{D}$ at $\sqrt{s}=$ 3.773 GeV, where the $D$ and $\bar{D}$
mesons are set to decay into all possible final states with the branching
fractions quoted from PDG pdg07 .
Subtracting the number $N^{\rm b}$ of these contaminations from the number
$N^{\rm obs}$ of the candidate events, we obtain the net number $N^{\rm net}$
of the signal events for each process. For the $K_{S}^{0}K^{-}\pi^{+}$ and
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{+}\pi^{-}\pi^{-}$ final states, for which
only a few signal events are observed from the continuum data, we set the
upper limits $N^{\rm up}$ on the number of the signal events at 90% C.L..
Here, we use the Feldman-Cousins method prd57_3873 and assume that the
background is absent. The numbers of $N^{\rm obs}$, $N^{\rm b}$ and $N^{\rm
net}$ (or $N^{\rm up}$) are summarized in the second, third and fourth columns
of Tabs. 1 and 2. For each process, the background events in the $\psi(3770)$
resonance data are dominant by $D\bar{D}$ decays and $\psi(3686)$ decays.
While, there is no $D\bar{D}$ decay in the continuum data, and the
$\psi(3686)$ production cross section at $\sqrt{s}=$ 3.650 GeV is much less
than that at $\sqrt{s}=$ 3.773 GeV. So, the number of the background events in
the continnum data can almost be negligible.
## VI RESULTS
### VI.1 Monte Carlo efficiency
To estimate the detection efficiency $\epsilon$ for $e^{+}e^{-}\to f$, we use
a phase space generator including initial state radiation and vacuum
polarization corrections yf41_377 with $1/s$ energy dependence in cross
section. Final state radiation cpc79_291 decreases the detection efficiency
not more than 0.5%. Detailed analysis based on Monte Carlo simulation for the
BES-II detector nima552_344 gives the detection efficiencies for each process
at $\sqrt{s}=3.773$ and 3.650 GeV, which are summarized in the fifth columns
of Tabs. 1 and 2, where the detection efficiencies do not include the
branching fractions for $K_{S}^{0}\to\pi^{+}\pi^{-}$ and
$\pi^{0}\to\gamma\gamma$, ${\mathcal{B}}(K_{S}^{0}\to\pi^{+}\pi^{-})$ and
${\mathcal{B}}(\pi^{0}\to\gamma\gamma)$.
### VI.2 Observed cross sections
Let ${\mathcal{B}}_{\pi^{0}}$ = ${\mathcal{B}}(\pi^{0}\to\gamma\gamma)$ for
the modes of $K_{S}^{0}K^{-}\pi^{+}\pi^{0}$ and
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{-}\pi^{0}$, ${\mathcal{B}}_{\pi^{0}}$ =
${\mathcal{B}}^{2}(\pi^{0}\to\gamma\gamma)$ for the mode of
$K_{S}^{0}K^{-}\pi^{+}\pi^{0}\pi^{0}$ and ${\mathcal{B}}_{\pi^{0}}$ = 1 for
$K_{S}^{0}K^{-}\pi^{+}$, $K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{-}$ and
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{+}\pi^{-}\pi^{-}$, where
${\mathcal{B}}(\pi^{0}\to\gamma\gamma)$ is the branching fraction for the
decay of $\pi^{0}\to\gamma\gamma$, then the observed cross section for
$e^{+}e^{-}\to f$ can be determined by
$\sigma_{e^{+}e^{-}\to f}=\frac{N^{\rm
net}}{{\mathcal{L}}\times\epsilon\times{\mathcal{B}}(K_{S}^{0}\to\pi^{+}\pi^{-})\times{\mathcal{B}}_{\pi^{0}}},$
(1)
where ${\mathcal{L}}$ is the integrated luminosity of the data set, $N^{\rm
net}$ is the number of the signal events, $\epsilon$ is the detection
efficiency and ${\mathcal{B}}(K_{S}^{0}\to\pi^{+}\pi^{-})$ is the branching
fraction for the decay of $K_{S}^{0}\to\pi^{+}\pi^{-}$. Inserting these
numbers in Eq. (1), we obtain the observed cross sections for each process at
$\sqrt{s}=3.773$ and 3.650 GeV. They are summarized in Tabs. 1 and 2, where
the first error is statistical and the second systematic. In the measurements
of the observed cross sections, the systematic errors arise from the
uncertainties in integrated luminosity of the data set ($2.1\%$ plb641_145 ;
prl97_121801 ), photon selection ($2.0\%$ per photon), tracking efficiency
($2.0\%$ per track), particle identification ($0.5\%$ per pion or kaon),
kinematic fit ($1.5\%$), $K_{S}^{0}$ reconstruction ($1.1\%$ plb_608_24 ),
branching fractions quoted from PDG jpg33_1 ($0.03\%$ for
${\mathcal{B}}(\pi^{0}\to\gamma\gamma)$ and $0.07\%$ for
${\mathcal{B}}(K_{S}^{0}\to\pi^{+}\pi^{-})$), Monte Carlo modeling ($6.0\%$
plb650_111 ; plb656_30 ; epjc52_805 ), Monte Carlo statistics ($1.4\%\sim
4.4\%$), background subtraction ($0.0\%\sim 3.0\%$) and fit to mass spectrum
($0.4\%\sim 8.5\%$). Adding these uncertainties in quadrature yields the total
systematic error $\Delta_{\rm sys}$ for each mode at $\sqrt{s}=3.773$ and
3.650 GeV.
The upper limit $\sigma_{e^{+}e^{-}\to f}^{\rm up}$ on the observed cross
sections for the $K_{S}^{0}K^{-}\pi^{+}$ and
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{+}\pi^{-}\pi^{-}$ final states at
$\sqrt{s}=3.650$ GeV are set with Eq. (1) by substituting $N^{\rm net}$ with
$N^{\rm up}/(1-\Delta_{\rm sys})$, where $N^{\rm up}$ is the upper limit on
the number of the signal event, and $\Delta_{\rm sys}$ is the systematic error
in the cross section measurement. Inserting the corresponding numbers in the
equation, we obtain the upper limits on the observed cross sections for
$e^{+}e^{-}\to K_{S}^{0}K^{-}\pi^{+}$ and $e^{+}e^{-}\to
K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{+}\pi^{-}\pi^{-}$ at $\sqrt{s}=3.650$ GeV,
which are also listed in Tab. 2.
### VI.3 Upper limits on the observed cross sections and the branching
fractions for $\psi(3770)\to f$
If we ignore the possible interference effects between the continuum and
resonance amplitudes, and the difference of the vacuum polarization
corrections at $\sqrt{s}=3.773$ and 3.650 GeV, we can determine the observed
cross section $\sigma_{\psi(3770)\to f}$ for $\psi(3770)\to f$ at
$\sqrt{s}=3.773$ GeV by comparing the observed cross sections
$\sigma_{e^{+}e^{-}\to f}^{3.773\hskip 1.42271pt\rm GeV}$ and
$\sigma_{e^{+}e^{-}\to f}^{3.650\hskip 1.42271pt\rm GeV}$ for $e^{+}e^{-}\to
f$ measured at $\sqrt{s}=3.773$ and 3.650 GeV, respectively. It can be given
by
$\sigma_{\psi(3770)\to f}=\sigma_{e^{+}e^{-}\to f}^{3.773\hskip 1.42271pt\rm
GeV}-f_{\rm co}\times\sigma_{e^{+}e^{-}\to f}^{3.650\hskip 1.42271pt\rm GeV},$
(2)
where $f_{\rm co}=3.650^{2}/3.773^{2}$ is the normalization factor to consider
the $1/s$ cross section dependence. The results are summarized in the second
column of Tab. 3, where the first error is the statistical, the second is the
independent systematic arising from the uncertainties in the Monte Carlo
statistics, in the fit to the mass spectrum and in the background subtraction,
and the third is the common systematic error arising from the other
uncertainties as discussed in the subsection B.
The upper limit on the observed cross section $\sigma^{\rm up}_{\psi(3770)\to
f}$ for $\psi(3770)\to f$ at $\sqrt{s}=3.773$ GeV is set by shifting the cross
section by 1.64$\sigma$, where $\sigma$ is the total error of the measured
cross section. The results on $\sigma^{\rm up}_{\psi(3770)\to f}$ are
summarized in the third column of Tab. 3.
The upper limit on the branching fraction ${\mathcal{B}}^{\rm
up}_{\psi(3770)\to f}$ for $\psi(3770)\to f$ is set by dividing its upper
limit on the observed cross section $\sigma^{\rm up}_{\psi(3770)\to f}$ by the
observed cross section $\sigma^{\rm obs}_{\psi(3770)}=(7.15\pm 0.27\pm 0.27)$
nb plb650_111 for the $\psi(3770)$ production at $\sqrt{s}=3.773$ GeV and a
factor $(1-\Delta\sigma^{\rm obs}_{\psi(3770)})$, where $\Delta\sigma^{\rm
obs}_{\psi(3770)}$ is the relative error of the $\sigma^{\rm
obs}_{\psi(3770)}$. The results on ${\mathcal{B}}^{\rm up}_{\psi(3770)\to f}$
are summarized in the last column of Tab. 3.
## VII SUMMARY
In this Letter, we present the measurements of the observed cross sections for
$K_{S}^{0}K^{-}\pi^{+}$, $K_{S}^{0}K^{-}\pi^{+}\pi^{0}$,
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{-}$,
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{-}\pi^{0}$,
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{+}\pi^{-}\pi^{-}$ and
$K_{S}^{0}K^{-}\pi^{+}\pi^{0}\pi^{0}$ produced in $e^{+}e^{-}$ annihilation at
$\sqrt{s}=3.773$ and 3.650 GeV. These cross sections are obtained by analyzing
the data sets of 17.3 pb-1 taken at $\sqrt{s}=3.773$ GeV and of 6.5 pb-1 at
$\sqrt{s}=3.650$ GeV with the BES-II detector at the BEPC collider. By
comparing the observed cross sections for each process measured at
$\sqrt{s}=3.773$ and 3.650 GeV, we set the upper limits on the observed cross
sections and the branching fractions for $\psi(3770)$ decay to these final
states at $90\%$ C.L.. These measurements provide helpful information to
understand the mechanism of the continuum light hadron production and the
discrepancy between the observed cross sections for $D\bar{D}$ and
$\psi(3770)$ production.
## VIII Acknowledgments
The BES collaboration thanks the staff of BEPC for their hard efforts. This
work is supported in part by the National Natural Science Foundation of China
under contracts Nos. 10491300, 10225524, 10225525, 10425523, the Chinese
Academy of Sciences under contract No. KJ 95T-03, the 100 Talents Program of
CAS under Contract Nos. U-11, U-24, U-25, the Knowledge Innovation Project of
CAS under Contract Nos. U-602, U-34 (IHEP), the National Natural Science
Foundation of China under Contract No. 10225522 (Tsinghua University).
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Tab. 1: The observed cross sections for $e^{+}e^{-}\to$ exclusive light hadrons at $\sqrt{s}=3.773$ GeV, where $N^{\rm obs}$ is the number of events observed from the $\psi(3770)$ resonance data, $N^{\rm b}$ is the total number of background events, $N^{\rm net}$ is the number of the signal events, $\epsilon$ is the detection efficiency, $\Delta_{\rm sys}$ is the relative systematic error of the observed cross section, and $\sigma$ is the observed cross section. $e^{+}e^{-}\to$ | $N^{\rm obs}$ | $N^{\rm b}$ | $N^{\rm net}$ | $\epsilon$($\%$) | $\Delta_{\rm sys}$($\%$) | $\sigma^{\rm obs}[{\rm pb}]$
---|---|---|---|---|---|---
$K^{0}_{S}K^{-}\pi^{+}$ | $18.4\pm 4.6$ | $0.1\pm 0.0$ | $18.3\pm 4.6$ | $10.02\pm 0.14$ | 10.7 | $15.2\pm 3.8\pm 1.6$
$K^{0}_{S}K^{-}\pi^{+}\pi^{0}$ | $41.2\pm 6.6$ | $1.1\pm 0.2$ | $40.1\pm 6.6$ | $3.52\pm 0.08$ | 11.6 | $96.2\pm 15.9\pm 11.1$
$K^{0}_{S}K^{-}\pi^{+}\pi^{+}\pi^{-}$ | $40.0\pm 6.5$ | $1.0\pm 0.2$ | $38.9\pm 6.5$ | $3.56\pm 0.06$ | 14.2 | $91.5\pm 15.3\pm 13.0$
$K^{0}_{S}K^{-}\pi^{+}\pi^{+}\pi^{-}\pi^{0}$ | $24.5\pm 5.2$ | $1.5\pm 0.3$ | $23.0\pm 5.2$ | $0.77\pm 0.03$ | 15.2 | $253.0\pm 57.1\pm 38.4$
$K^{0}_{S}K^{-}\pi^{+}\pi^{+}\pi^{+}\pi^{-}\pi^{-}$ | $4.8\pm 2.2$ | $0.3\pm 0.1$ | $4.5\pm 2.2$ | $0.84\pm 0.03$ | 18.4 | $44.4\pm 21.9\pm 8.2$
$K^{0}_{S}K^{-}\pi^{+}\pi^{0}\pi^{0}$ | $19.8\pm 4.9$ | $2.8\pm 0.5$ | $17.0\pm 4.9$ | $0.99\pm 0.04$ | 14.3 | $147.0\pm 42.4\pm 21.0$
Tab. 2: The observed cross sections for $e^{+}e^{-}\to$ exclusive light hadrons at $\sqrt{s}=3.650$ GeV, where $N^{\rm obs}$ is the number of events observed from the continuum data, $N^{\rm up}$ is the upper limit on the number of the signal events, $\sigma^{\rm up}$ is the upper limit on the observed cross section set at 90% C.L., and the definitions of the other symbols are the same as those in Tab. 1. $e^{+}e^{-}\to$ | $N^{\rm obs}$ | $N^{\rm b}$ | $N^{\rm net}({\rm or}\hskip 4.5ptN^{\rm up})$ | $\epsilon$($\%$) | $\Delta_{\rm sys}$($\%$) | $\sigma^{\rm obs}(\sigma^{\rm up})[\rm pb]$
---|---|---|---|---|---|---
$K^{0}_{S}K^{-}\pi^{+}$ | $2$ | 0.0 | $<5.91$ | $10.55\pm 0.15$ | 12.9 | $<14.3$
$K^{0}_{S}K^{-}\pi^{+}\pi^{0}$ | $7.7\pm 2.9$ | 0.0 | $7.7\pm 2.9$ | $3.62\pm 0.09$ | 11.6 | $47.9\pm 18.0\pm 5.6$
$K^{0}_{S}K^{-}\pi^{+}\pi^{+}\pi^{-}$ | $13.4\pm 3.8$ | 0.0 | $13.4\pm 3.8$ | $3.66\pm 0.06$ | 14.1 | $81.4\pm 23.1\pm 11.5$
$K^{0}_{S}K^{-}\pi^{+}\pi^{+}\pi^{-}\pi^{0}$ | $4.6\pm 2.5$ | 0.0 | $4.6\pm 2.5$ | $0.87\pm 0.03$ | 17.2 | $119.0\pm 64.7\pm 20.5$
$K^{0}_{S}K^{-}\pi^{+}\pi^{+}\pi^{+}\pi^{-}\pi^{-}$ | $0$ | 0.0 | $<2.44$ | $0.95\pm 0.03$ | 18.1 | $<69.7$
$K^{0}_{S}K^{-}\pi^{+}\pi^{0}\pi^{0}$ | $3.3\pm 2.0$ | 0.0 | $3.3\pm 2.0$ | $1.12\pm 0.05$ | 14.1 | $67.1\pm 40.7\pm 9.5$
Tab. 3: The upper limits on the observed cross section $\sigma^{\rm up}_{\psi(3770)\to f}$ at $\sqrt{s}=3.773$ GeV and the branching fraction ${\mathcal{B}}^{\rm up}_{\psi(3770)\to f}$ for $\psi(3770)\to f$ are set at 90% C.L.. The $\sigma_{\psi(3770)\to f}$ is calculated with Eq. (2), where the first error is the statistical, the second is the independent systematic, and the third is the common systematic error. The upper ∗ denotes that we neglect the contributions from the continuum production. Decay Mode | $\sigma_{\psi(3770)\to f}$ (pb) | $\sigma^{\rm up}_{\psi(3770)\to f}$(pb) | ${\mathcal{B}}^{\rm up}_{\psi(3770)\to f}$
---|---|---|---
$K_{S}^{0}K^{-}\pi^{+}$ | $15.2\pm 3.8\pm 0.2\pm 1.6^{*}$ | $<22.0$ | $3.2\times 10^{-3}$
$K_{S}^{0}K^{-}\pi^{+}\pi^{0}$ | $51.4\pm 23.2\pm 2.6\pm 5.8$ | $<90.7$ | $13.3\times 10^{-3}$
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{-}$ | $15.3\pm 26.5\pm 2.2\pm 2.1$ | $<59.0$ | $8.7\times 10^{-3}$
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{-}\pi^{0}$ | $141.6\pm 83.2\pm 14.7\pm 20.7$ | $<284.3$ | $41.8\times 10^{-3}$
$K_{S}^{0}K^{-}\pi^{+}\pi^{+}\pi^{+}\pi^{-}\pi^{-}$ | $44.4\pm 21.9\pm 2.1\pm 7.9^{*}$ | $<82.7$ | $12.2\times 10^{-3}$
$K_{S}^{0}K^{-}\pi^{+}\pi^{0}\pi^{0}$ | $84.2\pm 57.2\pm 8.3\pm 11.2$ | $<180.4$ | $26.5\times 10^{-3}$
|
arxiv-papers
| 2008-10-31T02:45:54 |
2024-09-04T02:48:58.561462
|
{
"license": "Public Domain",
"authors": "BES Collaboration",
"submitter": "Bo Zheng",
"url": "https://arxiv.org/abs/0810.5608"
}
|
0810.5609
|
# Charmed baryon in a string model
Wang Qing-Wu1,2 111qw.wang@impcas.ac.cn and Zhang Peng-Ming1 1Institute of
Modern Physics, Chinese Academy of Science, P.O. Box 31, Lanzhou 730000,
P.R.China 2Graduate School, Chinese Academy of Science, Beijing 100049,
P.R.China
###### Abstract
Charm spectroscopy has studied under a string model. Charmed baryons are
composed of diquark and charm quark which are connected by a constant tension.
In a diquark picture, the quantum numbers $J^{P}$ of confirmed baryons are
well assigned. We give energy predictions for the first and second orbital
excitations. We see some correspondences with the experimental data.
Meanwhile, we have obtained diquark masses in the background of charm quark
which satisfy a splitting relation based on spin-spin interaction.
PACS: 12.39.-x, 14.20.-c,14.20.Lq
## I Introduction
Charm spectroscopy has revived since 2000. Many new excited charmed baryon
states have been discovered by CLEO, BaBar, Belle and Fermilab. Masses of
ground states as well as many of their excitations are known experimentally
with rather good precision. However charmed baryons have narrow widths and non
of their spin or parity are measured except the $\Lambda_{c}(2880)$Mizuk .
The assignments listed in the PDG book almost are based on quark modelpdg .
Theoretically, the study of heavy baryons has a long storyCopley ; Capstick ;
Glozman . Heavy baryons provide a laboratory to study the dynamics of the
light quarks in the environment of heavy quark, such as their chiral
symmetrycheng . The studies of heavy baryons also help us to understand the
nonperturbative QCDRoberts . Furthermore, it really is an ideal place for
studying the dynamics of diquark.
The concept of diquark appeared soon after the original papers on
quarksGellmann ; Ida ; Lich . It was used to calculate the hadron properties.
In heavy quark effective theory, two light quarks often refer to as diquark,
which is treated as particle in parallel with quark itself. There are several
phenomenal manifestations of diquark: the $\Sigma-\Lambda$ mass difference,
the isospin $\Delta I=1/2$ rule, the structure function ratio of neutron to
proton, _et al._ Anselmino ; Jaffe ; Selem ; Wilczek . In this diquark
picture, charmed baryons are composed of one diquark and one charm quark.
Selem and WilczekSelem ; Wilczek have generalized the famous Chew-Frautschi
formula by considering diquark and quark connected by a relativistic string
with constant tension $T$ and rotating with angular momentum $L$. The string
is responsible for the color confinement and is also called as loaded flux
tube if the two ends get masses. In the limit of zero diquark and quark
masses, the usual Chew-Frautschi relationship $E^{2}\sim L$ appears. They have
investigated the $N-\Delta$ spectrum and concluded that “large $L$
spectroscopy would give convincing evidence for energetically significant
diquark correlations” Selem . In hadrons containing one heavy quark, diquark
ought to be a better approximation than in hadrons containing only light
quarks. However Selem and Wilczek have given only a short discussion on the
$\Lambda_{c}$ spectrum. In this paper, we accept the diquark concept and use
their relativistic string model to study the charmed baryon spectroscopy.
In the following, we introduce firstly the diquarks and the string model in
section II. We give our analysis of the doublets and give the quantum number
assignments based on diquark assumption. Numerical results are showed in
section III. In the end summary and discussion are given.
## II Diquark and String Model
### II.1 Diquarks and mass splitting
The single-charmed baryons composed of one diquark and one charm quark. In
literatures, there are two kinds of diquarks: the good diquark with spin zero
and the bad diquark with spin one. The good diquark is more favorable
energetically than the bad one, which is indicated by both the one-gluon
exchange and instanton calculations. In $SU(3)_{f}$, the good diquark has
flavor-spin symmetry $\bar{\textbf{3}}_{F}\bar{\textbf{3}}_{S}$ while the bad
diquark $\textbf{6}_{F}\textbf{6}_{S}$. To give a color singlet state, both
kinds of diquark have the same color symmetry $\bar{\textbf{3}}_{C}$. In the
following we use the $[qq^{\prime}]$ to denote a good diquark, while
$(qq^{\prime})$ the bad and $l$ to denote either $u$ quark or $d$ quark.
In this diquark picture, $\Lambda_{c}$ has a $[ud]$ component, while
$\Sigma_{c}$, $(ll^{\prime})$ and $\Omega_{c}$, $(ss)$. For $\Xi_{c}$, either
kind of diquarks, good or bad, can be formed. Heavy baryons always have been
obtained by continuum productionRoberts . So, the lowest baryons discovered
are more likely to be ground states. These are $\Lambda_{c}(2285)$ and
$\Xi_{c}(2470)$. Three doublets $\Sigma_{c}(2455,2520)$, $\Xi_{c}(2578,2645)$
and $\Omega_{c}(2768,2698)$ would also be states with $L=0$, if they are
composed of bad diquark and charm quark. And we assign the doublets
$\Lambda_{c}(2595,2628)$ and $\Xi_{c}(2790,2815)$ with $L=1$.
The good diquark with spin zero has no spin interaction with the charm quark.
So, the lowest energy is a singlet. Only the $L-S$ coupling may make the
energy splitJaffe :
$\mathcal{H}(q_{c},[qq^{\prime}])=\mathcal{K}_{[qq^{\prime}]}2\vec{L}\cdot\vec{S_{c}},$
(1)
where the coefficient $\mathcal{K}_{L,(qq^{\prime})}$ depends on the diquark
and charm quark masses. This interaction splits baryon with orbital angular
momentum L to baryons with $J=L+1/2$ and $L-1/2$. And the parity is
$P=(-1)^{L}$. For the bad diquark, the spin-spin interaction is:
$\mathcal{H}(q_{c},(qq^{\prime}))=\mathcal{G}_{(qq^{\prime})}2\vec{S}_{(qq^{\prime})}\cdot\vec{S}_{c},$
(2)
where $\vec{S}_{(qq^{\prime})}$ is the spin of the bad diquark, and the
coefficient $\mathcal{G}_{(qq^{\prime})}$ depends on the diquark and charm
quark masses. This spin-spin interaction also lead to a doublet in the
spectrum. Since in our assignments, there is no $L>0$ multiplet of bad diquark
and for simplicity, we will not discuss the splitting caused by $L-S$ coupling
for baryons containing bad diquark.
We have relation
$<2\vec{j}_{1}\cdot\vec{j}_{2}>=J(J+1)-j_{1}(j_{1}+1)-j_{2}(j_{2}+1)$, with
$\vec{J}=\vec{j}_{1}+\vec{j}_{2}$. It is easy to deduce the mass difference of
a doublet. For example, when $j_{1}=1$, $j_{2}=1/2$, they are $M_{0}+\Delta$
and $M_{0}-2\Delta$, with $\Delta$ being $\mathcal{G}$ or $\mathcal{K}$.
Taking a doublet as input, we can obtain the $\Delta$ and $M_{0}$. And it is
not the masses of the doublet, but this $M_{0}$ which enter into the string
model.
### II.2 The string model
In 1960 Chew and Frautschi conjectured that the strongly interacting particles
fall into families where the Regge trajectory functions were straight lines:
$E^{2}=\sigma+kL$ with the same constant $k$ for all the trajectories. The
straight-line Regge trajectories with $\sigma$ zero were later understood as
arising from massless endpoints on rotating relativistic strings at speed of
light transversely. A non-zero values of $\sigma$ may include zero-point
energy for string vibrations and loaded endpoints.
In Selem and Wilczek’s modelSelem ; Wilczek , the two ends of the string have
masses $m_{1}$ and $m_{2}$ respectively, with constant string tension $T$. The
rotaing angular momentum is $L$ with angular velocity $\omega$. If the diquark
and charm quark are away from the center of rotation at distances $R_{1}$ and
$R_{2}$, the energy of the system is:
$E=\sum_{i=1,2}(m_{i}\gamma_{i}+\frac{T}{\omega}\int_{0}^{\omega
R_{i}}{\frac{1}{\sqrt{1-u^{2}}}}du),$ (3)
where $\gamma_{i}$ is the usual Lorentz factor:
$\gamma_{i}=\frac{1}{\sqrt{1-(\omega R_{i})^{2}}}.$ (4)
The angular momentum can be written as:
$L=\sum_{i=1,2}(m_{i}\omega
R_{i}^{2}\gamma_{i}+\frac{T}{\omega^{2}}\int_{0}^{\omega
R_{i}}{\frac{u^{2}}{\sqrt{1-u^{2}}}}du).$ (5)
Furthermore, we have formula relating the tension and the angular velocity:
$m_{i}\omega^{2}R_{i}=\frac{T}{\gamma_{i}^{2}}.$ (6)
From equation (6), we see that $\omega R_{i}$ can be replaced by $m_{i}$,
$\gamma_{i}$ and $T/\omega$. Solving the equations (4) and (6), we can express
the $\gamma_{i}$ by $T/\omega$ and $m_{i}$:
$\gamma_{i}=\sqrt{\frac{1}{2}+\frac{\sqrt{1+4(T/m_{i}\omega)^{2}}}{2}}$ (7)
Now, we have two equations, (3) and(5), and 6 parameters, $E$, $L$, $m_{1}$,
$m_{2}$, $T$ and $\omega$. These equations are more useful than the Chew-
Frautschi formula for they make us able to extract the diquark masses.
For very light mass, it appears that $\omega\to\infty$ as $L\to 0$ and the
Chew-Frautschi relationship is recovered as $E^{2}=(2\pi T)L$. For other
cases, such as the first corrections at small masses see Ref.Selem ; Wilczek .
A reduced formula has being used to study the charmed meson spectroscopyShan
.In this paper we will solve the equations numerically.
## III Numerical Results
### III.1 $\Lambda_{c}$ and $\Xi_{c}$ with good diquark
We chose $L=1$ doublet as input to solve the equations (3) and (5). Firstly,
we take the $m_{c}$ and $T$ as free parameters to get the diquark mass. The
string tension $T$ is universal for baryons with the same components. Then we
use these three parameters to give energy predictions for $L=2$. And we find
that the $T$ is almost equal for $\Lambda_{c}$ and $\Xi_{c}$ if we choose the
value which give a linear Regge trajectory $E^{2}\sim L$ or linear
$(E-M)^{2}\sim L$. The last relation was given by Selem and Wilczek, which can
be obtained by expanding the right hands of equations (3) and (5) in
$m\omega/T$ for terms of light diquark and in $T/(m\omega)$ for terms of charm
quark. The results are listed in Table 1 and 2. The two kinds trajectories are
both linear since the energies for $L=0,1,2$ at $T=0.1$ form a arithmetic
progression and with small common difference. We plot the two kinds
trajectories with $M_{c}=1.7$ $GeV$ for example, on Figure 1 and 2.
Figure 1: Plot of $E^{2}\sim L$ for $\Lambda_{c}$ and $\Xi_{c}$ with
$m_{c}=1.7$ $GeV$ and $T=0.05\sim 0.20$. Figure 2: Plot of $(E-M)^{2}\sim L$
for $\Lambda_{c}$ and $\Xi_{c}$ with $m_{c}=1.7$ $GeV$ and $T=0.05\sim 0.20$.
$\Lambda_{c}$: $m_{[ll\prime]}$, $M_{0}^{L=2}$ | T=0.05 | 0.1
---|---|---
Mc=1.5 | 0.873, 2.781 | 0.737, 2.878
1.6 | 0.770 , 2.781 | 0.630, 2.878
1.7 | 0.665, 2.781 | 0.518, 2.879
1.8 | 0.559, 2.782 | 0.403, 2.881
$\Lambda_{c}$: $m_{[ll\prime]}$, $M_{0}^{L=2}$ | T= 0.15 | 0.2
---|---|---
Mc=1.5 | 0.624, 2.959 | 0.520, 3.032
1.6 | 0.509, 2.960 | 0.395, 3.034
1.7 | 0.387, 2.962 | 0.255, 3.038
1.8 | 0.253, 2.967 | 0.080, 3.043
Table 1: Good diquark mass of $[ll^{\prime}]$ and energy for
$\Lambda_{c}(L=2)$ neglecting the spin-spin interaction. Mass unit is in GeV.
$\Xi_{c}$: $m_{[ls]}$, $M_{0}^{L=2}$ | T=0.05 | 0.1
---|---|---
Mc=1.5 | 1.070, 2.968 | 0.940, 3.063
1.6 | 0.969, 2.967 | 0.835, 3.062
1.7 | 0.866, 2.966 | 0.730, 3.061
1.8 | 0.762, 2.967 | 0.620, 3.063
$\Xi_{c}$: $m_{[ls]}$, $M_{0}^{L=2}$ | T=0.15 | 0.2
---|---|---
Mc=1.5 | 0.835, 3.142 | 0.738, 3.214
1.6 | 0.726, 3.141 | 0.625, 3.213
1.7 | 0.614, 3.141 | 0.505, 3.214
1.8 | 0.498, 3.143 | 0.380, 3.216
Table 2: Good diquark mass of $[ls]$ and energy for $\Xi_{c}(L=2)$ neglecting
the spin-spin interaction. Mass unit is in GeV.
The energies $M_{0}^{L=1}$ is $2.617$ $GeV$ for doublet
$\Lambda_{c}(2595,2628)$ with $\mathcal{K}_{[ll]}=11$ $MeV$. The numerical
result for $L=2$ are $M_{0}=2.879$ $GeV$ which gives doublets
$\Lambda_{c}^{\prime}(2846,2901)$ with the splitting mass formulas
$M_{0}+2\mathcal{K}$ and $M_{0}-3\mathcal{K}$. Here, we use a prime to
indicate our theoretic prediction. We think the $\Lambda_{c}(2880)$ and
$\Lambda_{c}(2940)$ to be a doublet with $L=2$, since the splitting gives
$\mathcal{K}_{[ll]}=12$ $MeV$ which is near equal to the result from mass
difference of $\Lambda_{c}(L=1)$. Then their mass difference gives
$M_{0}^{L=2}=2917$ $MeV$ which is 38 $Mev$ large than our prediction However,
it can be wiped out given $T=1.2$, see Table 3.
The linear fit is:
$\Lambda_{c}:E^{2}=1.632L+5.220,$ (8)
with $\chi/Dof$ almost being zero. So we predict that the quantum numbers
$J^{P}$ of doublet $\Lambda_{c}(2882,2940)$ are $3/2^{+}$ and $5/2^{+}$.
For $\Xi_{c}$, $M_{0}^{L=1}$ is $2.807$ $GeV$ with $\mathcal{K}_{[ls]}=8.3$
$MeV$. If $T=0.12$, we get $M_{0}^{L=2}=3.094$ $GeV$ and the doublet is
$\Xi_{c}^{\prime}(3069,3111)$. And the Regge trajectory is
$\Xi_{c}:E^{2}=1.736L+6.115.$ (9)
The nearest experimental data are $\Xi_{c}(3080)$ and $\Xi_{c}(3123)$ only
about $10MeV$ larger than our predictions. So, we take $\Xi_{c}(3080,3123)$ as
a doublet with $J^{P}=$ $3/2^{+}$ and $5/2^{+}$.
$B_{(qq\prime)}$ | $\Lambda_{c}$ | $\Xi_{c}$
---|---|---
$\mathcal{K}$/MeV | 11 | 8.3
$m_{[qq\prime]}$ /GeV | 0.465 | 0.682
$M_{0}^{L=1}/GeV$ | 2.617 | 2.807
$M_{0}^{L=2}/GeV$ | 2.913 | 3.094
mass splitting | L=1: $M_{0}+\mathcal{K}$ and $M_{0}-2\mathcal{K}$ |
| L=2: $M_{0}+2\mathcal{K}$ and $M_{0}-3\mathcal{K}$ |
Table 3: Good diquark masses and predictions for masses at $L=2$ with $T=0.12$
and $m_{c}=1.7$ $GeV$. By using the mass splitting formula at $L=1$,
$M_{c}^{L=1}$ and $\mathcal{K}$ are easy to be derived.
### III.2 $\Sigma_{c}$, $\Xi_{c}$ and $\Omega_{c}$ with bad diquark
For $\Sigma_{c}$, $\Omega_{c}$ and $\Xi_{c}$ with bad diquark, we have to take
the $L=0$ doublets as input for the lack of data. When $L\to 0$, we have
$\omega\to 0$, $R\to 0$ and $E\to m_{1}+m_{2}$ from which we can deduce the
bad diquark masses. We see from Table 1 and Table 2 that energies are more
depending on $T$ not on quark masses. So we take charm quark mass to be $1.7$
$GeV$. The numerical results with $T=0.12$ are listed in Table 4. Linear fits
of the three groups of baryon masses are:
$\displaystyle\Sigma_{c}:E^{2}=1.987L+6.326,$
$\displaystyle\Xi_{c}:E^{2}=2.035L+6.967,$
$\displaystyle\Omega_{c}:E^{2}=2.084L+7.624,$
with $\chi/Dof$ being about $0.04$ for each fit. The slopes are almost equal
but a little larger than 1.632 and 1.736, the slopes for fitting the spectra
of $\Lambda_{c}$ and $\Xi_{c}$ containing good diquark. However, the diquark
masses are so heavy. And it is unreasonable for a string with zero length. So,
when $L\to 0$ and $R\to 0$, the string model would not be a good
approximation.
In the end, we give the mass predictions for these baryons using linear Regge
trajectory, though there are arguments that hadronic Regge trajectories are
nonlinearnonlinear . We take the slope to be the average of the slopes for
good diquark baryons, that is 1.684. Then use equations (3) and (5) with L=2
to extract the diquark masses. Results are showed in Table 5. In PDG book,
there is $\Sigma_{c}(2800)$ with question mark which is a little lower than
our prediction for $\Sigma_{c}(2815,L=1)$. And note that we have neglected
here all the angular momentum interactions.
$B_{(qq\prime)}$ | $\Sigma_{c}$ | $\Xi_{c}$ | $\Omega_{c}$
---|---|---|---
$\mathcal{G}$/MeV | 21.7 | 22.3 | 23.3
$m_{(qq\prime)}$ /GeV | 0.798 | 0.923 | 1.045
$M_{0}/GeV$ | 2.498 | 2.623 | 2.745
$M_{0}^{L=1}/GeV$ | 2.913 | 3.029 | 3.144
$M_{0}^{L=2}/GeV$ | 3.196 | 3.309 | 3.421
Table 4: Results of the string model using the ground states as input and with $T=0.12$. $B_{(qq\prime)}$ | $\Sigma_{c}$ | $\Xi_{c}$ | $\Omega_{c}$
---|---|---|---
$\mathcal{G}$/MeV | 21.7 | 22.3 | 23.3
$m_{(qq\prime)}$ /GeV | 0.739 | 0.858 | 0.975
$M_{0}/GeV$ | 2.498 | 2.623 | 2.745
$M_{0}^{L=1}/GeV$ | 2.815 | 2.926 | 3.036
$M_{0}^{L=2}/GeV$ | 3.100 | 3.201 | 3.302
Table 5: Results of using Regge trajectory with slope being 1.684. And the
diquark masses are derived by taking the $M_{0}^{L=1}$ as input.
### III.3 Diquark masses
The good and bad diquark masses are listed in Table 3 and Table 5. Bad
diquarks are heavier than good diqarks and diquark with heavier flavor quark
is heavier than the light one. These diquark masses are sensitive to the
background, i.e. the charm quark mass. However, they still satisfy the
relation $(ud)-[ud]$ $>$ $(us)-[us]$ which was expected from spin-spin
interaction that the mass difference would be strongest for lightest
quarksWilczek ; Selem .
We can adopt the string model to the charmed mesons. The non-strange mesons
$D(2400,2420,2430,2460)$ with positive parity would be a multiplet of $L=1$.
The meson $D(2460,J^{P}=2^{+})$ thus has total spin $S=1$ and $<2L\cdot S>=0$.
The same is for charmed and strange meson $D_{s}(2573)$. Using this two states
as input, we have derived the quark masses which are $0.332$ $GeV$ for up and
down quark and $0.468$ $GeV$ for strange quark. The diquark masses can be
defined by $M_{D}=M_{q1}+M_{q2}+E_{12}$, with $E_{12}$ being the binding
energy. We see that the good diquarks have negative binding energies while the
bad positive. This is consistent with result that comes from spin-dependent
colormagnetic interactions of two quarks. The interactions are attractive in a
spin-0 state and repulsive in a spin-1 stateLichtenberg .
## IV Summary and Discussion
We have used a diquark picture and a string model to study the charmed baryon
spectroscopy. The many doublets in the spectroscopy are the results of S-S or
L-S interactions. With string tension $T=0.12$ we have given predictions for
the good diquark baryons with L=2 which have some experimental
correspondences. The possible state $\Sigma_{c}(2800)$ would be the first
orbital excitation of $\Sigma_{c}$. The quantum number $J^{P}$ assignments for
L=0 and L=1 baryons from a diquark picture are the same as PDG book. By using
the string model, we have extracted the diquark masses which satisfy the
expected relation $(ud)-[ud]$ $>$ $(us)-[us]$.
However, there is one problem. Our prediction for $\Lambda_{c}(2880)$
$J^{P}=3/2^{+}$ is contradicted with the experimental result and Selem’s
assignment with $J^{P}=5/2^{+}$Mizuk ; Selem . If it is confirmed by later
experiments, we must reconsider our diquark picture or mass splitting formula
based on angular momentum interactions.
## V Acknowledgments
This work was supported by Chinese Academy of Sciences Knowledge Innovation
Project (KJCX2-SW-No16;KJCX2-SW-No2), National Natural Science Foundation of
China(10435080;10575123), West Light Foundation of The Chinese Academy of
Sciences and Scientific Research Foundation for Returned Scholars, Ministry of
Education of China.
## References
* (1) R. Mizuk et al, Phy. Rev. Lett98 (2007) 262001
* (2) C. Amsler et al, (Particle Data Group), Phy. Lett. B667 (2008) 1
* (3) L. A. Copley, N. Isgur and G. Carl, Phy. Rev. D20 (1979) 768
* (4) S. Capstick and N. Isgur, Phy. Rev. D34 (1986) 2809
* (5) L. Ya Glozman and D. O. Riska, Nucl. Phy. A603 (1996) 326; L. Y. Glozman and D. O. Riska, Nucl. Phy. A620 (1997) 510
* (6) H. Y. Cheng, hep-ph/0709.0958; H. Y. Cheng and C. K. Chua, Phy Rev D75 (2007) 014006
* (7) W .Roberts and M. Pervin, nucl-th/0711.2492
* (8) M. Gell-Mann, Phys. Lett.8 (1964) 214
* (9) M. Ida and R. Kobayashi, Prog. Theor. Phys.36 (1966) 846
* (10) A. Lichtenberg and L. Tassie, Phys. Rev.155 (1967) 1601
* (11) M. Anselmino et al, Rev. Mod. Phy.65 (1993) 1199
* (12) R. L. Jaffe, Phys. Rep.409 (2005) 1
* (13) F. Wilczek, hep-ph/0409168
* (14) A. Selem and F. Wilczek, hep-ph /0602128
* (15) H. Y. Shan and A. L. Zhang, hep-ph /0805.4764
* (16) A. Tang and J. W. Norbury, Phy. Rev. D62(2000) 016006; M. M. Brisudova, L. Burakovsky and T. Goldman, Phy. Rev. D61 (2000) 054013
* (17) D. B. Lichtenberg, J. Phys. G19 (1993) 1257
|
arxiv-papers
| 2008-10-31T02:52:19 |
2024-09-04T02:48:58.567998
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Q.W.Wang and P. M. Zhang",
"submitter": "QingWu Wang",
"url": "https://arxiv.org/abs/0810.5609"
}
|
0811.0298
|
# About a peculiar extra $U(1):Z^{\prime}$ discovery limit
Muon anomalous magnetic moment
Electron electric dipole moment
Jae Ho Heo jheo1@uic.edu Physics Department, University of Illinois at
Chicago, Chicago, Illinois 60607, USA
###### Abstract
The model (Lagrangian) with a peculiar extra $U(1)$ Bar05 is clearly
presented. The assigned extra $U(1)$ gauge charges give a strong constraint to
build Lagrangians. The $Z^{\prime}$ discovery limits are estimated and
predicted at the Tevatron and the LHC. The new contributions of the muon
anomalous magnetic moment are investigated at one and two loops, and we
predict that the deviation from the standard model may be explained. The
electron electric dipole moment could also be generated because of the
explicit CP violation effect in the Higgs sector, and a sizable contribution
is expected for a moderately sized CP phase (argument of the CP-odd Higgs),
$0.1\leq\sin\delta\leq 1$ $(6^{\circ}\leq\arg(A)\leq 90^{\circ})$.
###### pacs:
12.15.-y, 12.60.Cn, 12.60.Fr
††preprint:
## I Introduction
An extra $U(1)$ (or a few extra $U(1)^{\prime}$s) may arise in the context of
grand unified theories Agu90 , superstring theories Ros88 or generically
emerge as simple extensions of the standard model (SM). Therefore, the models
with an extra $U(1)$ (or a few extra $U(1)^{\prime}$s) have been extensively
considered. Recently, Barr and Dorsner Bar05 suggested another possibility
for an extra $U(1)$ gauge, which satisfies all anomaly constraints in a
maximally economical way, whatever its origin111The origin is close to the
Pati-Salam model with an extra $U(1)$Bar05 . is. In the standard model, all
the possible anomalies from triangle diagrams of three gauge bosons must be
canceled if the Ward identities of the gauge theory are to be satisfied. The
existence of an extra $U(1)$ brings six additional anomaly cancellation
conditions, $U(1)_{Y}^{2}\times U(1)_{X}$, $U(1)_{Y}\times U(1)_{X}^{2}$,
$U(1)_{X}^{3}$, $SU(2)^{2}\times U(1)_{X}$, $SU(3)^{2}\times U(1)_{X}$ ,
gravity $\times U(1)_{X}$. These anomaly cancellations are nontrivial222The
general analyses about these cancellations can be found in Ref.Ema02 ., but
Barr and Dorsner showed a remarkably trivial solution Bar05 with a single
extra lepton triplet per family. These gauge anomalies are exactly canceled
for the fermion gauge charges listed at Table I.
In this letter the model (Lagrangian) is clearly presented. With an extra
lepton triplet, an additional Higgs singlet is necessary to provide masses of
the exotic leptons and the extra gauge boson $Z^{\prime}$. The Higgs singlet
would involve the extra $U(1)$ gauge symmetry breaking, and we assume that the
symmetry is broken near the weak scale. A Higgs triplet333The Higgs triplet is
introduced because of the need to induce the CP violating interaction in this
work. We can add additional scalars, such as nongauged Higgs singlets or
doublets, in other ways, but adding the gauged scalars is more generic. If we
only consider neutrino mass generation without electric dipole and dark matter
(we need to impose the discrete $Z_{2}$ symmetry ; see the next section)
phenomenology, the additional Higgs triplet is unnecessary. The interactions
induced by the Higgs triplet could also give a significant contribution to the
anomalous magnetic moment of the muon (see Sec. IV). with the required gauge
charges is added to explain our interesting phenomenology. The new gauge boson
$Z^{\prime}$ that generically emerges as gauging an extra $U(1)$ is the
intrinsic particle that explains the existence of an extra $U(1)$, so its
discovery limits at the Tevatron and LHC are estimated and predicted. The muon
anomalous magnetic moment, $a_{\mu}\equiv(g_{\mu}-2)/2$, has been a powerful
tool to account for new physics because of its importance. We investigate the
contributions involving the new particles at the one- and two-loop levels. One
can also see the explicit CP violation that generates the electric dipole
moment (EDM) of the electron $d_{e}$, and a sizable contribution is expected
via Barr-Zee two-loop mechanism for the moderate size of the CP-phase
(argument of the CP-odd Higgs), $0.1\leq\sin\delta\leq 1$
$(6^{\circ}\leq\arg(A)\leq 90^{\circ})$.
Table 1: Fermion gauge charges. $T_{3}$ is the weak isospin, $Y$ is the hypercharge, $X$ is the extra $U(1)_{X}$ charge, and $Q=T_{3}+Y$ is the electric charge. The charges for the right handed fermions can also be assigned in the identical way. ($f_{L}^{c}\equiv(f_{L})^{c}$ in this letter, so $(f^{c})_{L}$ implies the antiparticle of $f_{R}$). | $u_{L}$ | $d_{L}$ | $(u^{c})_{L}$ | $(d^{c})_{L}$ | $\nu_{L}$ | ${\ell}_{L}$ | $({\ell}^{c})_{L}$ | $E^{+}_{L}$ | $E^{0}_{L}$ | $E^{-}_{L}$
---|---|---|---|---|---|---|---|---|---|---
T3 | ${1\over 2}$ | $-{1\over 2}$ | 0 | 0 | ${1\over 2}$ | $-{1\over 2}$ | 0 | 1 | 0 | $-1$
Y | ${1\over 6}$ | ${1\over 6}$ | $-{2\over 3}$ | ${1\over 3}$ | $-{1\over 2}$ | $-{1\over 2}$ | 1 | 0 | 0 | 0
X | 1 | 1 | $-1$ | $-1$ | 1 | 1 | 1 | $-1$ | $-1$ | $-1$
## II Model description (Lagrangian)
With the new particle content, the Yukawa potential for the lepton sector can
have the following enlarged form without the ad hoc imposition of lepton
number conservation.
$y_{1}\mathrm{{Tr}}(\overline{E_{L}^{c}}E_{L})\eta+y_{2}\overline{L_{L}}\phi\ell_{R}+y_{3}\overline{L_{L}^{c}}i\sigma_{2}\chi
L_{L}+y_{4}\overline{L_{L}^{c}}i\sigma_{2}E_{L}\phi+y_{5}\mathrm{{Tr}}(\overline{E_{L}}\chi)\ell_{R}+h.c.,$
(1)
where $\eta$ and $\chi$ denote a Higgs singlet and a Higgs triplet,
$\phi=(\phi^{+},\phi^{0})^{T}$ is a Higgs doublet, and
$L=\left(\nu_{\ell},\ell\right)^{T}$ is the lepton doublet. The bi doublet
representation is taken for the additional lepton triplet and the Higgs
triplet is also taken in the form of a $2\times 2$ matrix transforming under
$SU(2)$ as $\chi\rightarrow U\chi U^{{\dagger}}$.
$E_{L}=\begin{pmatrix}\frac{1}{\sqrt{2}}E^{0}&E^{+}\\\
E^{-}&-\frac{1}{\sqrt{2}}E^{0}\end{pmatrix}_{L},\text{ \
}\chi=\begin{pmatrix}\frac{1}{\sqrt{2}}\chi^{+}&\chi^{++}\\\
\chi^{0}&-\frac{1}{\sqrt{2}}\chi^{+}\end{pmatrix}.$ (2)
The lepton triplet must be a Majorana combination. It should be noted that the
antisymmetric tensor $i\sigma_{2}$ follows from the antisymmetric property of
the charge conjugation.
Since the gauge charges of the leptons are already assigned by anomaly
constraints, the gauge charges of the Higgses are assigned by the combinations
with leptons in the Yukawa potential under the $SU(2)_{L}\times U(1)_{Y}\times
U(1)_{X}$ gauge invariance. If we introduce the assignment of $U(1)_{X}$
charges for the Higgses, the singlet $\eta$ must have a $U(1)_{X}$ charge of
$2$ from the $y_{1}$-term since $E$ has $-1$; the doublet $\phi$ may have a
charge of $2$ from the $y_{2}$-term and $0$ from the $y_{4}$-term ; and the
triplet $\chi$ may have a charge of $-2$ from the $y_{3}$-term and $0$ from
the $y_{5}$-term. The other gauge charges may be assigned in an analogous way,
and the assigned charges of the Higgses are listed at Table II. Notice that
the Higgs doublet and triplet may have two distinctive $U(1)_{X}$ charges.
Table 2: Higgs gauge charges. $T_{3}$ is the weak isospin, Y is the hypercharge, $X$ is the $U(1)_{X}$ charge, and $Q=T_{3}+Y$ is the electric charge. | $\phi^{+}$ | $\phi^{0}$ | $\eta$ | $\chi^{++}$ | $\chi^{+}$ | $\chi^{0}$
---|---|---|---|---|---|---
$T_{3}$ | ${1\over 2}$ | $-{1\over 2}$ | 0 | 1 | 0 | $-1$
$Y$ | ${1\over 2}$ | ${1\over 2}$ | 0 | 1 | 1 | 1
$X$ | $(0,2)$ | $(0,2)$ | 2 | $(0,-2)$ | $(0,-2)$ | $(0,-2)$
The Yukawa potential with distinct charges takes the form.
$y_{1}\mathrm{{Tr}}\left(\overline{E_{L}^{c}}E_{L}\right)\eta_{(2)}+y_{2}\overline{L_{L}}\phi_{(2)}\ell_{R}+y_{3}\overline{L_{L}^{c}}i\sigma_{2}\chi_{(-2)}L_{L}+y_{4}\overline{L_{L}^{c}}i\sigma_{2}E_{L}\phi_{(0)}+y_{5}\mathrm{{Tr}}\left(\overline{E_{L}}\chi_{(0)}\right)\ell_{R}+h.c.,$
(3)
where the indices in the lower brackets of the Higgses denote $U(1)_{X}$
charges of the Higgses.
A discrete $Z_{2}$ symmetry could be imposed to explain a certain
phenomenology, dark matter. If $E$ is odd and all other particles are even
under $Z_{2}$ symmetry, this would prevent the exotic leptons from coupling
with the ordinary leptons and the neutral lepton $E^{0}$ becomes stable, and
thus could be a dark matter candidate. The Yukawa potential with $Z_{2}$
symmetry is given by
$y_{1}\mathrm{{Tr}}\left(\overline{E_{L}^{c}}E_{L}\right)\eta_{(2)}+y_{2}\overline{L_{L}}\phi_{(2)}\ell_{R}+y_{3}\overline{L_{L}^{c}}i\sigma_{2}\chi_{(-2)}L_{L}+h.c..$
(4)
The Yukawa potential for the quark sector may be built in an analogous way.
$y_{6}\overline{Q_{L}}\widetilde{\phi_{(0)}}u_{R}+y_{7}\overline{Q_{L}}\phi_{(0)}d_{R}+h.c.,$
(5)
where $Q=\left(u,d\right)^{T}$ is the quark doublet and
$\widetilde{\phi_{(0)}}=i\sigma_{2}\phi_{(0)}^{\ast}$. The hypercharge
combinations in the potential prohibits the couplings between quarks and Higgs
triplets. Since quarks receive masses only from $\phi_{(0)}$, there is no tree
level flavor-changing neutral currents. Note that leptons and quarks interact
with two distinct Higgs doublets, which is different from the standard two
Higgs doublet model (2HDM) where one Higgs couples to the up-type quarks and
the other couples to the charged leptons and down-type quarks.
The size of the couplings and vacuum expectation values (VEVs) may be
approximately constrained with the known experimental measurements. The VEV of
$\phi_{(0)}$, $\left\langle\phi_{(0)}\right\rangle$, must be of the order of
$100$ GeV to meet the top quark mass, and $\
\left\langle\phi_{(2)}\right\rangle$ must be $1\sim 100$ GeV to satisfy the
$\tau$-lepton mass and the known SM
VEV444$\left\langle\phi\right\rangle=\sqrt{\left\langle\phi_{(0)}\right\rangle^{2}+\left\langle\phi_{(2)}\right\rangle^{2}}\simeq
174$ GeV.. The Higgs triplet VEV $\left\langle\chi\right\rangle$ must be very
small compared to the Higgs doublet VEV, since the $\rho$ parameter predicted
by the SM is consistent with the experimental measurement in high precision
Cer07 . The neutrino mass may be generated at the tree level in this model.
The mass matrix of neutral leptons is
$\mathcal{M}_{\nu
E}=\begin{pmatrix}m_{\nu}&y_{4}\left\langle\phi_{(0)}\right\rangle\\\
y_{4}\left\langle\phi_{(0)}\right\rangle&M_{E}\end{pmatrix},$
where $m_{\nu}\equiv y_{3}\left\langle\chi_{(-2)}\right\rangle$ and
$M_{E}\equiv y_{1}\left\langle\eta\right\rangle$. The nature of the neutrino
is not known; however, we have approximately predicted the size of the
neutrino mass. We take the exotic lepton $E$ at the weak scale, so the
coupling $y_{4}$ must be very small555According to the famous canonical seesaw
mechanism, the order unity coupling is assumed, with the scale of new physics
of 1013GeV. However, we relax the constraint, as the coupling $y_{4}$ could
approximately be of the order of the electron Yukawa coupling ($\sim
10^{-6}$). since $\left\langle\phi_{(0)}\right\rangle$ is of the order of
$100$ GeV. The seesawlike mechanism is applicable to generate the neutrino
mass. If $Z_{2}$ symmetry is imposed, $y_{4}=0$. The neutrino mass may be
taken as $m_{\nu}$, where $y_{3}$ and/or
$\left\langle\chi_{(-2)}\right\rangle$ be sized for the neutrino mass. For
either case, we predict Majorana-type neutrinos in this model. Since
$\left\langle\chi\right\rangle$ and the coupling $y_{4}$ are small, we can
consider that the massive leptons are in the mass eigenstates for the Yukawa
potential of (3).
The Higgs potential is also amenable to the gauge invariance with the extra
$U(1)$.
$\displaystyle V$ $\displaystyle\supset
V_{2HDM}+\left\\{\mu_{1}\phi_{(0)}^{T}\chi_{(0)}^{{\dagger}}\phi_{(0)}+\mu_{2}\phi_{(2)}^{{\dagger}}\chi_{(-2)}^{{\dagger}}\phi_{(0)}+h.c.\right\\}$
$\displaystyle+\left\\{\lambda_{1}\phi_{(2)}^{{\dagger}}\phi_{(0)}\mathrm{{Tr}}\left(\chi_{(-2)}^{{\dagger}}\chi_{(0)}\right)+\lambda_{2}\phi_{(2)}^{{\dagger}}\sigma^{a}\phi_{(0)}\mathrm{{Tr}}\left(\chi_{(-2)}^{{\dagger}}\sigma^{a}\chi_{(0)}\right)+h.c.\right\\},$
(6)
where $V_{2HDM}$ stands for the Higgs potential involving only Higgs doublets,
and the functional form is the same as the 2HDM with $Z_{2}$ symmetry. In
addition to the two complex trilinear couplings, the two complex quartic
couplings are possible, those involving the CP violation phenomenology. The
phenomenology with two complex trilinear coupings can be found in Ref.Ema98
666They assigned the Higgs triplets of the order of $10^{13}$GeV to explain
neutrino masses and Baryogenesis via Leptogenesis. However the Higgs triplets
are assumed to have masses of the order of weak scale to explain the
interesting phenomenology in our scenario., and the complex quartic couplings
are related to the electric dipole moment of fermions which will be discussed
as a part of this letter. The other interaction terms are trivial and almost
irrelevant to the phenomenology.
## III $Z^{\prime}$ discovery limit
The interactions of the $Z^{\prime}$ boson with the fermions are described by
${\displaystyle\sum\limits_{f}}z_{f}^{\prime}g_{Z^{\prime}}Z_{\mu}^{\prime}\overline{f}\gamma^{\mu}f,$
(7)
where $f=E_{L},Q_{L},L_{L},u_{R},d_{R},e_{R}$ are the lepton and quark fields
and $z_{f}^{\prime}$ is the gauge charge corresponding to the fermion.
The leptonic decays $Z^{\prime}\rightarrow\ell^{+}\ell^{-}(e^{+}e^{-}$ and
$\mu^{+}\mu^{-})$ provide the most distinctive signature for observing the
$Z^{\prime}$ signal at the hadron colliders. The cross section of the
$p{\bar{p}}$ collision in the $\ell^{+}\ell^{-}$ channel can be calculated at
the narrow width $Z^{\prime}$ pole in the center-of-momentum (CM) frame. The
hadronic cross section is given by
$\sigma(Z^{\prime})=K\sum_{q,{\bar{q}}}\int_{0}^{1}dx_{1}dx_{2}(f_{q}^{p}(x_{1})f_{\overline{q}}^{\overline{p}}{(x_{2})}+f{{}_{\overline{q}}^{p}(x_{1})}f_{q}^{\overline{p}}(x_{2})){\hat{\sigma}(Z^{\prime}),}$
(8)
where ${\hat{s}=x}_{1}x_{2}s$ is the partonic fraction of $s$, $f(x)$’s are
the partonic distribution functions (PDFs) and the sum is performed over all
the light quarks. $K$ is the QCD correction factor ($\sim$ 1.3) Mcar04 , which
accounts for higher order QCD corrections. The partonic cross secion
${\hat{\sigma}(Z^{\prime})}$ is calculated in a sum over the spins of the
final states and an average over the spins and colors of the initial states.
${\hat{\sigma}}(Z^{\prime})={\frac{\pi z_{f}^{\prime
2}g_{Z^{\prime}}^{2}}{48}}\delta({\hat{s}}-M_{Z^{\prime}}^{2}).$ (9)
Eq.(8) and (9) lead to the hadronic cross section in the $\ell^{+}\ell^{-}$
channel.
$\sigma(Z^{\prime})\cdot Br_{\ell^{+}\ell^{-}}=K{\frac{\pi z_{f}^{\prime
2}g_{Z^{\prime}}^{2}}{48s}}\sum_{q,{\bar{q}}}\int_{\frac{m_{Z^{\prime}}^{2}}{s}}^{1}{\frac{dx}{x}}\left(f_{q}^{p}(x)f_{\overline{q}}^{\overline{p}}\left({\frac{M_{Z^{\prime}}^{2}}{xs}}\right)+f{{}_{\overline{q}}^{p}(x)}f_{q}^{\overline{p}}\left({\frac{M_{Z^{\prime}}^{2}}{xs}}\right)\right)\cdot
Br_{\ell^{+}\ell^{-}},$ (10)
where $Br_{\ell^{+}\ell^{-}}$ is the branching ratio of $Z^{\prime}$ to
$\ell^{+}\ell^{-}$. We may take $z_{f}^{\prime 2}\simeq 1,$ since precision
measurements of $Z$-pole observables predict the small $Z-Z^{\prime}$ mixing
angle$(\leq 10^{-3})$ Cer07 .
Figure 1: The $Z^{\prime}$ discovery limit at the Tevatron ($\sqrt{s}=1.96$
TeV and $L=1.3$ $fb^{-1}$) and the LHC($\sqrt{s}=14$ TeV and $L=100$
$fb^{-1}$). The horizontal lines indicate the experimental sensitivities, and
the bold lines are predictions of the cross section. The predictions are for
the coupling, $g_{Z^{\prime}}=0.1$ and $0.7$ (SM coupling). MRST LO PDFs Mar02
are used. The intersections of the curves determine the lower mass limits.
For $pp$ collision at the LHC, the proton PDF takes the place of the
antiproton PDF. Fig.1 shows the predicted cross sections with the present
experimental sensitivity at the Tevatron Run II777The CDF Collaboration CDF07
has set the better luminosity for $\sigma(Z^{\prime})\cdot
Br_{\ell^{+}\ell^{-}}$ than the DØ D007 in some reason, so it is considered
for the CDF collider. and the projected experimental sensitvity at the LHC
LHC05 . The actual experimental analysis shows an experimental line with a
more complicated structure than the horizontal line in the figure. For a
nonzero background888The non-zero background is roughly taken from Ref.CDF07 ,
which all the expected backgrounds are considered. The most significant source
of background in this channel is the SM Drell-Yan process via
$Z/\gamma^{\ast}$ as reported in Ref. CDF07 ., $N_{Z^{\prime}}=3$ events are
excluded at the Tevatron. The $Z^{\prime}$ discovery limits are $300$ GeV,
$870$ GeV for $g_{Z^{\prime}}=0.1,0.7$ at the Tevatron, and the LHC may probe
$Z^{\prime}$ upto $3.1$ TeV, $5.7$ TeV for $g_{Z^{\prime}}=0.1,0.7$. Since the
$U(1)_{X}$ gauge charge of the Higgs singlet $\eta$ is $2$,
$M_{Z^{\prime}}\simeq 2g_{Z^{\prime}}\left\langle\eta\right\rangle$. We
predict the lower limit of the extra $U(1)$ symmetry breaking to be around
$200\sim 800$ GeV at the Tevatron (CDF detector).
## IV muon anomalous magnetic moment
The deviation of the current experimental value from the SM prediction is
approximately $3.0\sigma$ and the numerical deviation is $\Delta
a_{\mu}=27.5(8.4)\times 10^{-10}$ Mda07 or $27.7(9.3)\times 10^{-10}$ Fdo08 .
The experimental value is the measurement of the BNL experiment Muo06 . We
investigate one- and two-loop contributions.
Figure 2: The one-loop contributions to $a_{\mu}$ involving the extra
particles, $E,\chi,$ and $Z^{\prime}$.
The diagrams of Fig.2 display one-loop contributions involving the new
particles, $E$, $\chi,$ and $Z^{\prime}$. The relevant interaction Lagrangian
for diagrams (a) and (b) of Fig. 2 comes from the $y_{5}$-term of the Yukawa
potential of (3). The states $\chi_{(0)},\chi_{(-2)}$ may be rotated into the
mass eigenstates $\chi_{\ell},\chi_{h}$, where $\chi_{\ell}$ and $\chi_{h}$
are the light and heavy mass eigenstates. The rotational angle is determined
in the Higgs potential. However, the couplings with the Higgs triplet are free
parameters, so we redefine the new couplings in the mass eigenstates. The
relevant Lagrangian for the light scalar state $\chi_{\ell}$ is given by
$-y\left(\overline{E_{L}^{0}}\chi_{\ell}^{+}\mu_{R}+\overline{E_{L}^{-}}\chi_{\ell}^{0}\mu_{R}-\overline{E_{L}^{+}}\chi_{\ell}^{++}\mu_{R}+h.c.\right),$
(11)
where $y$ is the Yukawa coupling in the mass eigenstates of $\chi$. The
Lagrangian for the heavy mass eigenstates can be given in the same fashion.
The $y_{4}$-term with which the Higgs doublet is involved is neglected due to
the small coupling constrained by the neutrino mass.
The contribution of Fig.2(a) is negligible, since the particles
($E^{+}$,$E^{-}$) on the line which are hooked up by the photon have opposite
electric charges. We calculated the contribution of Fig.2(b), and it is given
by
$\Delta
a_{\mu}^{(\text{one})}=\frac{3y^{2}}{8\pi^{2}}\left(\frac{m_{\mu}}{M_{E}}\right)f\left(\frac{M_{\chi}^{2}}{M_{E}^{2}}\right)\simeq
4.03\times 10^{-6}\cdot
y^{2}\left(\frac{1\text{TeV}}{M_{E}}\right)f\left(\frac{M_{\chi}^{2}}{M_{E}^{2}}\right),$
(12)
where the prefactor of 3 comes from the electric charges of
$\chi^{\pm},\chi^{\pm\pm}$.
The corresponding one-loop function is
$f(z)=\int_{0}^{1}dx\frac{(1-x)x}{zx+1-x}=\frac{1-z^{2}+2z\ln z}{2(1-z)^{3}}$
(13)
which has asymptotic behaviors,
$f(z)\longrightarrow\left\\{\begin{array}[c]{cc}\frac{1}{6}&\text{as }z=1\
,\\\ {\frac{1}{2z}-}\frac{\ln z}{z^{2}}&\text{ for }z\gg 1\ ,\\\
\frac{1}{2}+z\ln z&\text{for }z\ll 1\ .\end{array}\right..$ (14)
We neglect the contribution from the other scalars (called the heavy scalars),
since those scalars are split into light and heavy mass eigenstates, in
general, and the one-loop function behaves $f(z)\rightarrow 0$ as
$z\rightarrow\infty$. Furthermore, the large splitting is necessary to
generate the sizable electric dipole moment, that will be discussed in the
next section. The $M_{\chi}$ or $M_{\chi_{\ell}}$ implies the mass of the
light scalar in this letter.
Figure 3: $\Delta a_{\mu}$ as a function of the exotic lepton mass $M_{E}$ for
various values of $M_{\chi}$ at the one-loop level.
Fig.3 shows the predictions of the anomalous magnetic moment for
$0.1$TeV$<M_{E},M_{\chi}<1$TeV. The range of deviations from the SM is
presented in the dark ”allowed” band Mda07 . The predictions are in the
allowed band around the Yukawa coupling $y=0.05$. Since $\Delta
a_{\mu}^{(\text{one})}\sim y^{2}/M_{E},$ the Yukawa coupling $y$ is very
sensitive to the deviation $\Delta a_{\mu}$. Besides the above region, a
possible scenario is $M_{E}\approx M_{\chi}>1$TeV for the Yukawa coupling
$y>0.06$. The contribution by the $Z^{\prime}$ gauge boson of Fig.2(c) is
negligible, since $\Delta a_{\mu}\sim m_{\mu}^{2}/M_{Z^{\prime}}^{2}$.
Figure 4: Two-loop contributions to $a_{\mu}$ $(d_{e})$ (mirror graphs are not
displayed.).
If $Z_{2}$ symmetry is imposed, there is no one-loop contribution to explain
the deviation. We consider the two-loop contribution via the Barr-Zee type of
mechanism, which is depicted in Fig.4. The relevant Lagrangian to induce the
Barr-Zee two-loop contribution is given by
$-\frac{\sqrt{2}m_{\mu}r_{\mathcal{H}}}{v}\overline{\mu}\mathcal{H}\mu-\frac{\lambda_{+}v}{\sqrt{2}}\mathcal{H}\left(\chi_{\ell}\chi_{\ell}+\chi_{h}\chi_{h}\right),$
where $\mathcal{H}=h$ or $H$, $v=\sqrt{2}\left\langle\phi\right\rangle,$ and
$\lambda_{+}$ is the coupling in the mass eigenstates of $\chi$ for
$\mathcal{H}$. The rotational angles999Coventionally, the rotational angle
between the neutral Higgses in 2HDM is denoted by the symbol $\alpha$. But in
this letter, the symbol $\alpha$ is used for the electric fine structure
constant, so we use the symbol $\beta_{h}$ for the rotational angle.
$r_{h}=-\sin\beta_{h}/\cos\beta$ and $r_{H}=\cos\beta_{h}/\cos\beta$ to the
muon are the same as in the standard 2HDM, since the scalar $\phi_{(2)}$,
which is consistent with the scalar to couple to the charged leptons in the
2HDM, couples to the muon. There is no contribution from the CP-odd Higgs
(pseudoscalar) $A$ because the interaction with the CP-odd Higgs violates CP
symmetry, so the effect of the CP-odd Higgs involves the electric dipole
moment.
The contribution of two loops is given by
$\displaystyle\Delta a_{\mu}^{(\text{two})}$
$\displaystyle\simeq-\sum_{\mathcal{H},\chi}\frac{\alpha
m_{\mu}^{2}}{16\pi^{3}}\frac{Q_{\chi}^{2}r_{\mathcal{H}}\lambda_{+}}{m_{\mathcal{H}}^{2}}\left[F\left(\frac{M_{\chi_{\ell}}^{2}}{m_{\mathcal{H}}^{2}}\right)+F\left(\frac{M_{\chi_{h}}^{2}}{m_{\mathcal{H}}^{2}}\right)\right]$
$\displaystyle=-2.07\times 10^{-11}\cdot\sum_{\mathcal{H}=h,H}\lambda
r_{\mathcal{H}}\left(\frac{200\text{GeV}}{m_{\mathcal{H}}}\right)^{2}\left[F\left(\frac{M_{\chi_{\ell}}^{2}}{m_{\mathcal{H}}^{2}}\right)+F\left(\frac{M_{\chi_{h}}^{2}}{m_{\mathcal{H}}^{2}}\right)\right].$
(15)
Note that $\sum Q_{\chi}^{2}=5$ due to singly and doubly charged scalars in
the inner loop. The two-loop function is
$F\left(z\right)=\int_{0}^{1}dx\frac{x(1-x)}{z-x(1-x)}\ln\left[\frac{x(1-x)}{z}\right]$
(16)
which has asymptotic behaviors,
$F(z)\longrightarrow\left\\{\begin{array}[c]{cc}-{0.344}&\text{as }z=1\ ,\\\
-{\frac{1}{6z}}\ln z-{\frac{5}{18z}}&\text{ for }z\gg 1\ ,\\\ (2+\ln
z)&\text{for }z\ll 1\ .\end{array}\right.\ .$ (17)
The Barr-Zee two-loop contributions, according to Eq.(15), are suppressed by
the muon mass and the loop factor, and thus the large $r_{\mathcal{H}}$ and
the small $m_{\mathcal{H}}$ are necessary. The lower limit of the light Higgs
boson mass is around $44$ GeV for $r_{h}\simeq\tan\beta$ from the LEP Opal01 ,
but the light Higgs boson keeps the same lower limit of the SM Higgs boson,
$113.5$ GeV, for $r_{H}\simeq\tan\beta.$ The case for $r_{h}\simeq\tan\beta$
is taken. We can approach these analyses in the 2HDM since the VEVs of the
Higgs triplets have the small size. Besides, the doubly charged scalar
$\chi^{++}$ in the inner loop gives the main contribution to the deviaton due
to its double electric charge. The lower limit of the doubly charged scalar,
around $120$ GeV from the Tevatron CDF04 and the LEP L303 , is considered.
Figure 5: $\Delta a_{\mu}$ as a function of the light Higgs boson mass $m_{h}$
at the two-loop level for various values of $\tan\beta$.
Fig.5 shows the predictions for the Barr-Zee two-loop contribution, $\Delta
a_{\mu}^{(\text{two})}$, as a function of the light Higgs boson mass $m_{h}$.
To predict the two-loop contribution $\Delta a_{\mu}^{(\text{two})}$, we
assume a coupling $\lambda_{+}$ of the same size as the SM Higgs quartic
coupling for the SM Higgs of 120 GeV. The predictions barely reside in the
allowed region.
## V Electron Electric Dipole Moment
The EDM of fermions predicted by the standard model is extremely small
compared to the present experimental bounds. Another mechanism beyond the SM
has been required to induce the sizable EDM. There are also explicit CP
violation interactions related to the Barr-Zee two-loop mechanism Smba90 ;
Dar99 ; Jheo08 for the EDM in this model. Since the interaction must involve
the CP violation, it is comprised of only the CP-odd Higgs (pseudoscalar) $A$.
The irreducible CP phase appears in the diagonalization101010The detailed
process for diagonalization of mass matrix by the unitary transformation can
be found in Ref.Jheo08 . of the mass matrix for the Higgs triplets in the
Higgs potential of (6). If we introduce the new phenomenological couplings,
the relevant interaction Lagrangian is given by
$\frac{\sqrt{2}m_{\mu}r_{A}}{v}\overline{e}i\gamma^{5}Ae-\frac{\lambda_{-}v}{\sqrt{2}}A\left(\chi_{\ell}\chi_{\ell}-\chi_{h}\chi_{h}\right),$
where $r_{A}=\tan\beta$ is the rotational angle, and
$\lambda_{-}=\lambda\sin\delta$ where $\sin\delta$ is the CP-violation effect
which comes from combinations of the complex quartic couplings in the
potential of (6). The Barr-Zee diagrams were well calculated in many papers to
induce the sizable electric dipole moment, and the result is identical to the
Barr-Zee two-loop contribution of the anomalous magnetic moment, except for
CP-violation effect. The electron electric dipole moment results in
$\displaystyle\left(\frac{d_{e}}{e}\right)^{\gamma}$
$\displaystyle=-\sum_{\chi}\frac{\alpha
m_{e}}{32\pi^{3}}\frac{Q_{\chi}^{2}r_{A}\lambda_{-}}{m_{A}^{2}}\left[F\left(\frac{M_{\chi_{\ell}}^{2}}{m_{A}^{2}}\right)-F\left(\frac{M_{\chi_{h}}^{2}}{m_{A}^{2}}\right)\right]$
$\displaystyle=-9.25\times
10^{-27}\cdot\left(\frac{200\text{GeV}}{m_{A}}\right)^{2}\lambda\sin\delta\tan\beta\left[F\left(\frac{M_{\chi_{\ell}}^{2}}{m_{A}^{2}}\right)-F\left(\frac{M_{\chi_{h}}^{2}}{m_{A}^{2}}\right)\right]\text{
\ }(cm),$ (18)
where the two-loop function is given in Eq.(16) ; also note that $\sum
Q_{\chi}^{2}=5$ due to singly and doubly charged scalars from the Higgs
triplets. The electron EDM results in the difference between two contributions
from the light and heavy scalars, $\chi_{\ell}$ and $\chi_{h}$. The
contribution from the heavy scalar is neglected, since the two-loop function
behaves like $F\left(z\right)\rightarrow 0$ as $z\rightarrow\infty$.
Figure 6: Numerical estimates of the EDMs as a function of the CP-odd (or
pseudoscalar) Higgs boson mass for various values of $\tan\beta$ and
$M_{\chi}$. Also shown the predictions for CP-phase, $0.1\leq\sin\delta\leq 1$
$(6^{\circ}\leq\arg(A)\leq 90^{\circ})$. The horizonal line indicates the
current 90% C.L. experimental bound Bcr02 .
In order to predict the electron electric dipole moment numerically, we also
assume the coupling $\lambda$ of the same size as the SM Higgs quartic
coupling for the SM Higgs of 120 GeV. Fig.6 shows the predictions of the
electron electric dipole moment as a function of the CP-odd Higgs (or
pseudoscalar) mass with the current 90% C.L. experimental bound Bcr02 . The
sizable contributions are expected for the moderate size of the CP phase,
$0.1\leq\sin\delta\leq 1$ $(6^{\circ}\leq\arg(A)\leq 90^{\circ})$.
## VI Conclusions
The model (Lagrangian) with a peculiar extra $U(1)$, that Barr and Dosner
suggested, has clearly been presented. The gauge charges of the extra $U(1)$
give a strong constraint to build the Lagrangians. $Z^{\prime}$ discovery
limits are estimated and predicted at the Tevatron and the LHC. The discovery
limit at the Tevatron (CDF detector) gives the lower limit of the extra $U(1)$
symmetry breaking scale, approximately $200\sim 800$ GeV. The muon anomalous
magnetic moment could be explained at the one-loop level for a Yukawa coupling
around $0.05$. If we allow masses of the new particles to be more than 1 TeV,
the larger Yukawa coupling is possible. However, smaller Yukawa couplings are
prohibited by the discovery limits of new particles at the Tevatron and the
LEP. The muon anomalous magnetic moment could also be explained at the two-
loop level, but the region of parameters is very narrow. There are explicit
CP-violation interactions in this model. A sizable electron electric dipole
moment is expected for a moderately sized CP phase, $0.1\leq\sin\delta\leq 1$,
$(6^{\circ}\leq\arg(A)\leq 90^{\circ})$ via the Barr-Zee mechanism.
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arxiv-papers
| 2008-11-03T13:35:19 |
2024-09-04T02:48:58.585394
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jae Ho Heo",
"submitter": "Jae Ho Heo",
"url": "https://arxiv.org/abs/0811.0298"
}
|
0811.0536
|
11institutetext: Institut de Physique Théorique, CEA - IPhT, CNRS, URA 2306
F-91191 Gif-sur-Yvette, France
Faculty of Engineering, Takushoku University - Hachioji, Tokyo 193-0985, Japan
Research School of Physical Sciences, Australian National University -
Canberra, ACT 0200, Australia
Conformation (statistics and dynamics) Structural transitions in nanoscale
materials Analytical theories
# Ribbon polymers in poor solvents: layering transitions in annular and
tubular condensates
Y. Y. Suzuki 1122 D. R. M. Williams 33112233
###### Abstract
We study the structures of a ribbon or ladder polymer immersed in poor
solvents. The anisotropic bending rigidity coupled with the surface tension
leads ribbon polymers to spontaneous formation of highly anisotropic
condensates in poor solvents. Unlike ordinary flexible polymers these
condensates undergo a number of distinct layering transitions as a function of
chain length or solvent quality, and the size of condensates becomes non-
monotonic function of chain length. We show that the fluctuations of the
condensates are in general small and these condensates are stable.
###### pacs:
36.20.Ey
###### pacs:
64.70.Nd
###### pacs:
87.15.ad
Recently considerable attention has been devoted to the physics of
semiflexible polymers, i.e., chains which have significant bending stiffness.
These kinds of polymers exhibit interesting physics in the form of liquid
crystalline phases and are of great importance in biology and biophysics. For
example, DNA is the most well-known semiflexible polymer. Much of the interest
has focused on semiflexible chains with isotropic bending elasticity[1, 2, 3,
4, 5, 6, 7]. Macroscopically this corresponds to modeling the chain as a
cylinder with a circular cross section. In many cases, however, the cross
section is highly anisotropic[8, 9, 10, 11, 12].
The anisotropic polymer can be modeled as a ribbon, of length $L$, width $w$,
and thickness $t$ assuming $L\gg w>t$ (fig.1) with anisotropic bending
elasticity. Indeed, if the ribbon is composed of an isotropic material, the
ratio of the elastic constants for bend in the easy and hard directions is
$\epsilon_{e}/\epsilon_{h}=(t/w)^{2}$ [13] . This strong dependence on the
ratio of $t/w$ implies that the ribbon polymer shows a very anisotropic
elastic response. On the chemical scale, the chains do not normally consist of
an isotropic material, and details of chemical bonding are important for bend,
thus, even higher ratios of $\epsilon_{e}/\epsilon_{h}$ are possible.
[width=7cm]fig1.eps
Figure 1: Model for anisotropic semiflexible polymer
Some of the solution properties, in particular the liquid-crystalline
behavior, have been studied theoretically [10]. Here we study the structure of
a single ribbon polymer immersed in a poor solvent. Assuming the system is
well below the $\Theta$ temperature, the chain collapses completely and forms
a condensate which is effectively a polymer melt.
For isotropic semiflexible chains such as DNA, the condensate often forms a
toroid [1, 2, 14, 15, 16, 17, 18, 19, 20, 21] and sometimes a globule[22, 23,
24, 25] through intermediate states[28, 29, 30]. For ribbon polymers, we show
in this paper that the condensates forms an annulus. The annulus formation
manages to avoid bend in the hard direction, while reducing contact of the
chain with the solvent. The novel character of these annular condensates is
that multilayer annuli can form and they undergo sudden changes as a function
of chain length or surface tension. This is in marked contrast to the case of
ordinary flexible or semiflexible polymers where no dramatic changes occur.
This quantization is a direct consequence of the anisotropic elasticity.
To model the system, we introduce two surface tensions $\gamma_{h}$ and
$\gamma_{e}$ which corresponds to the surface energies of contact between the
top (narrow surface) of the polymer and the solvent, and the surface energies
of contact between the side (flat and wide surface) of the polymer and the
solvent, respectively. If the chain consists of anisotropic molecules, any
ratio of $\gamma_{e}/\gamma_{h}$ is possible in principle. In this paper,
however, we assume $\gamma_{e}/\gamma_{h}\approx 1$.
A large number of parameters appear in this model. Once final results have
been obtained, it is useful to ignore numerical prefactors and to make some
crude estimates. Then, we write $w\approx t\approx a$ and
$\gamma_{e}\approx\gamma_{h}\approx gkT/a^{2}$, where $g$ is of order unity
and $a$ is a length of order of one Angstrom. It is convenient to define two
bare persistence lengths, $l_{e}\equiv\epsilon_{e}/kT$ and
$l_{h}\equiv\epsilon_{h}/kT$ which are essentially the scales on which a free
chain bends in the two directions due to thermal fluctuations.
We begin by examining the shape of condensates when we slowly lengthen the
chain. For short chains, bend costs elastic energy, then the chain more-or-
less remains as a rod. For slightly longer chains, however, the overlap
induced by bend compensates the elastic energy by reducing its surface energy,
therefore, the chain may form a ring. For a circular ring of radius $R$ it is
easy to show that the bending energy is ${1\over 2}\epsilon_{e}L/R^{2}$. The
change in surface energy in bending into a ring is $4\pi
Rw\gamma_{e}-2Lw\gamma_{e}$. Minimising it over $R$ and equating the free
energies gives the critical length,
$L^{*}=(3\sqrt{3}\pi/2)\sqrt{\epsilon_{e}/(\gamma_{e}w)}$ at which ring
formation occurs. The chain overlaps by $1/2$ turn, at $L=L^{*}$. Our interest
here is in the case $L\gg L^{*}$ so the chain winds around many times.
The simplest way the chain can pack is to form a disk with a hole in the
middle (annulus) as shown in fig.2. This annulus has height $w$ and inner and
outer radii $R_{i}$ and $R_{o}$, respectively. The total energy consists of
three terms: (1) surface energy of the top and bottom surfaces, (2) surface
energy of the inner and outer exposed sections of the annulus, (3) bending
energy of the chain. This last term is due to bending only in the easy
direction.
[width=6cm]fig2.eps
Figure 2: Annular condensate for short ribbon polymer in poor solvent
Because the volume of the chain is conserved, $R_{o}$ and $R_{i}$ are related
by
$tLw=w\pi(R_{o}^{2}-R_{i}^{2}).$ (1)
There is thus only one variable to minimize over (say $R_{i}$). Also, since
the area of the upper and lower surfaces is fixed (i.e., both are always of
area $tL$), there are really only two relevant terms in the total energy. The
lateral surface energy is easily calculated as
$2\pi\gamma_{e}w(R_{o}+R_{i}).$ (2)
The bending energy is calculated by
$\frac{\epsilon_{e}}{2}\int_{0}^{L}ds\left(\frac{d\theta}{ds}\right)^{2},$ (3)
where $s$ is the arc length and $\theta$ is the angle between the tangent of
the ribbon curve and some fixed direction in the plane of the disk.
In one complete turn of the ribbon, $\theta$ increases by $2\pi$ and $s$
increases by $2\pi r$ where $r$ is the distance of the chain from the center
of the annulus. This gives $d\theta/ds=1/r$. During the same change in $s$ the
radius increases by one ribbon thickness, i.e., $dr/ds=t/(2\pi r)$. This
allows us to write the bending energy as
$\pi\epsilon_{e}t^{-1}\ln(R_{o}/R_{i})$. The total energy is then
$U=\frac{\pi\epsilon_{e}}{t}\ln\left({R_{o}\over
R_{i}}\right)+2\pi\gamma_{e}w(R_{o}+R_{i}),$ (4)
subject to $R_{o}^{2}=R_{i}^{2}+tL/\pi$.
It is convenient at this point to introduce two characteristic lengths. One is
$\Lambda\equiv\epsilon_{e}/(2\gamma_{e}wt)$, and the other is
$\Gamma\equiv\sqrt{Lt/\pi}$. For long chains, $\Gamma$ is much larger than
$\Lambda$, and we shall see $R_{i}\approx\Lambda$ and $R_{o}\approx\Gamma$. In
this limit, $\Lambda\ll\Gamma$, the annulus is “fat”, i.e., it has a small
hole in the middle. In the opposite limit, $\Lambda\gg\Gamma$, the hole is
large and the annulus is “thin”. It looks like a ring.
In the long-chain limit, $R_{i}\ll\Gamma$, the total energy is approximated by
$U={\rm const.}+2\pi\gamma_{e}wR_{i}-\frac{\pi\epsilon_{e}}{t}\ln\left({\pi
R_{i}\over\sqrt{Lt}}\right),$ (5)
which has a minimum at $R_{i}^{*}=\Lambda$. Using our crude estimates give
$R_{i}^{*}\approx l_{e}/g$, i.e., the internal radius is somewhat less than
the bare persistence length. In this limit the internal radius of the annulus
is independent of chain length, whereas the external radius is approximately
$R_{o}\approx(tL/\pi)^{1/2}$. A more accurate expression for $R_{i}^{*}$ can
be found by expanding in ascending powers of $\alpha\equiv\Lambda/\Gamma$.
This gives $R_{i}^{*}=\Lambda(1-\alpha+{3\over 2}\alpha^{3}-...)$. From this
it is clear that, even though $R_{i}^{*}$ becomes asymptotically $\Lambda$,
the approach is rather slow as $L^{-1/2}$.
In the limit $\Lambda\gg\Gamma$, we can again expand the total energy and find
the minimum at $R_{i}^{*}=(\Gamma^{2}\Lambda/2)^{1/3}$, or more crudely,
$R_{i}^{*}\approx(Ll_{e}a/g)^{1/3}$. The crossover between the thin and fat
regimes occurs at $\Gamma=\Lambda$. A crude estimate of this gives
$L_{c}=a(L^{*}/a)^{4}$. This implies that the chain must be rather long to be
in the fat regime.
The results just derived correspond to the case of one layer. The novel point
about this system is that it can choose to form more than one layer. This
decreases the surface area exposed by the top and bottom surfaces but comes at
the expense of extra surface energy on the inside and outside surfaces and
extra bend both in the hard and easy directions. For second layer to form the
chain must bend in the hard direction. The best way of doing this is to join
the two layers together at the outer radius (fig.3).
[width=4cm]fig3.eps
Figure 3: tubular condensate for long ribbon polymer in poor solvent
We need to calculate the bending energy of the chain in the hard direction.
The chain must bend between the two layers. It must jump a distance $w$ in the
vertical direction while winding a distance $\Omega=2\pi R_{o}$ in arc length.
If we let $\phi$ be the angle made by the chain with the plane of the lower
annulus then the bending energy can be written as
$U_{h}=\frac{\epsilon_{h}}{2}\int_{0}^{\Omega}ds\left({d\phi\over
ds}\right)^{2}.$ (6)
This is to be minimized subject to a constraint that the chain jumps up a
distance $w$, i.e., $\int_{0}^{\Omega}ds\phi=w$. Here we have made the very
good approximation that the angle $\phi$ is always small. Using the method of
Lagrange multipliers and the Euler-Lagrange equations we find that the minimum
energy trajectory coresponds to the solution of the differential equation:
$d^{2}\phi/ds^{2}={\rm const}$. Using the boundary conditions
$\phi(0)=0=\phi(\Omega)$ the constraint gives
$\phi(s)=\frac{6w}{\Omega}\left\\{\frac{s}{\Omega}-\left(\frac{s}{\Omega}\right)^{2}\right\\},$
(7)
and the bending energy becomes
$U_{h}={6\epsilon_{h}w^{2}\over\Omega^{3}}.$ (8)
It is fairly clear from this formula that the chain wants to spread the bend
over as much length as possible, so that it is best to bend at the outer
radius of the annulus and $\Omega=2\pi R_{o}$.
We now examine the transition from one to two layers. The one-layer energy is
$U_{1}={\pi\epsilon_{e}\over t}\ln\left({R_{o}\over
R_{i}}\right)+2\pi\gamma_{e}w(R_{o}+R_{i})+2\gamma_{h}Lt,$ (9)
subject to $R_{o}^{2}=R_{i}^{2}+tL/\pi$. The two-layer energy is
$\displaystyle U_{2}$ $\displaystyle=$ $\displaystyle{2\pi\epsilon_{e}\over
t}\ln\left({R_{o}\over R_{i}}\right)+4\pi\gamma_{e}w(R_{o}+R_{i})$ (10)
$\displaystyle+\gamma_{h}Lt+{6\epsilon_{h}w^{2}\over(2\pi R_{o})^{3}},$
subject to $R_{o}^{2}=R_{i}^{2}+tL/2\pi$. Here $\gamma_{h}Lt$ is the surface
tension of the “top” of annulus.
The transition from one to two layers occurs when $U_{1}=U_{2}$. To solve this
equation we need to minimize $U_{1}$ over $R_{i}$ and similarly for $U_{2}$.
In principle different $R_{i}$ will be associated with the two geometries.
However, in the limit where $R_{i}\ll(Lt)^{1/2}$ the result is the same for
both, i.e., $R_{i}^{2}\approx\epsilon_{e}/(2\gamma_{e}wt)$. This gives
$\displaystyle U[n]$ $\displaystyle=$
$\displaystyle\frac{n\pi\epsilon_{e}}{t}\left\\{1+\ln\left({2\gamma_{e}tw\over\epsilon_{e}}\sqrt{Lt\over(n\pi)}\right)\right\\}$
(11) $\displaystyle+\frac{2}{n}\gamma_{h}Lt+2\gamma_{e}\pi
w\sqrt{nLt\over\pi}$
$\displaystyle+{3(n-1)\epsilon_{h}w^{2}\over\sqrt{2}\pi^{3}}\left(Lt\over\pi\right)^{-3/2},$
where $n$ is the number of layers ($1$ or $2$). Equating the two energies for
$n=1$ and $n=2$, and ignoring the logarithmic factors, gives
$\displaystyle{\pi\epsilon_{e}\over t}-\gamma_{h}Lt+2(\sqrt{2}-1)\pi
w\gamma_{e}\sqrt{Lt\over\pi}$
$\displaystyle+{3\epsilon_{h}w^{2}\over\sqrt{2}\pi^{3}}\left(Lt\over\pi\right)^{-3/2}=$
$\displaystyle 0.$ (12)
The four terms in this equation represent the four kinds of energy differences
between the two layer and one layer system. In order these are: Easy bend;
top/bottom surfaces; side surfaces; hard bend. The driving force for the
transition to two layers is the top/bottom surface energy. This is opposed by
the other three terms. We are mainly interested here in the case where the
hard bend is the dominant penalty. In this case the transition occurs when
$L^{5}\approx{\epsilon_{h}^{2}w^{4}\over(\gamma_{h}^{2}t^{5})},$ (13)
where we have ignored numerical prefactors. Even more approximately this can
be written $L\approx a(l_{h}/a)^{2/5}$ where $l_{h}$ is the persistence length
in the hard direction. Provided $l_{h}\gg l_{e}$ (it is true when $L\gg
R_{i}$) our approximations are valid.
At the transition the jump in $R_{i}$ is small, since $R_{i}$ is about the
same for both layers. However, there is a sudden jump in $R_{o}$ by a factor
of $1/\sqrt{2}$. In the case of a three layer system, the chain must jump once
on the outside radius and once on the inside radius. Since the jump on the
inside radius must take place over a shorter amount of arc length, its energy
cost is expensive. The energy for $n$ layers is then
$\displaystyle U[n]$ $\displaystyle=$ $\displaystyle{n\pi\epsilon_{e}\over
t}\ln\left(R_{o}\over R_{i}\right)$ (14)
$\displaystyle+n(2\pi\gamma_{e}w)(R_{o}+R_{i})+{2\over n}\gamma_{h}Lt$
$\displaystyle+\frac{6\epsilon_{h}w^{2}}{(2\pi)^{3}}\left(\left[{n\over
2}\right]R_{o}^{-3}+\left[{{n-1}\over 2}\right]R_{i}^{-3}\right),$
with $R_{o}=(R_{i}^{2}+n^{-1}tL/\pi)^{1/2}$, and where $[x]$ indicates the
integer part of $x$.
In order to show the layering transition of equilibrium form in a fat
condition, in fig.4 we plot the outer radius of the condesates $R_{o}$, the
radius of the inner hole of the condensates $R_{i}$, and the quantized height
of the condensate $nw$. This fat condition is always achieved in the long
chain limit. For large $L$, the size of the condensate is roughly proportional
to $\sqrt{L}$.
[width=6cm]fatdisk.eps
Figure 4: Layering transition in a fat condition $\Gamma/\Lambda\approx 10$.
The horizontal axis is the polymer length $L/w$. The upper and the lower
curves represent $R_{o}$ and $R_{i}$ respectively and the middle step line
represents the height of the tubular condensate in unit of ribbon width $w$.
Below the medium chain length, there exists a certain parameter region where
the condensate is thin. We show in fig.5 behavior of the layering transition
in a thin condition.
[width=6cm]thindisk.eps
Figure 5: Layering transition in a thin condition $\Gamma/\Lambda\approx 0.5$.
The horizontal axis is the polymer length $L/w$. The upper two curves
represent $R_{o}$ and $R_{i}$ respectively and the lower step line represents
the height of the tubular condensate in unit of ribbon width $w$.
We have assumed here that the system can form a well-defined annulus. It is
important and interesting to examine the fluctuations around the equilibrium
shape of the system. We will concentrate here on the case of a single layer.
There are two types of fluctuations. The first kind is fluctuations of the
shape in the radial direction, i.e., distortion from a circle. We discuss it
later.
The second kind is fluctuations out of the plane which cause two kinds of
energy penalty. The first is arises from hard bend and the energy penalty is
$\epsilon_{h}/2\int_{0}^{L}ds(d\phi/ds)^{2}$, where $\phi$ is the angle made
by the tangent to the ribbon out of the plane of the annulus. The second
arises from the surface area exposed due to differences in the height $z(s)$
of the layers. This gives rise to an energy penalty $\gamma_{e}\int_{2\pi
R_{1}}^{L}ds|z(s)-z\\{s-2\pi r(s)\\}|.$ Noting that
$z(s)=\int_{0}^{s}ds^{\prime}\phi(s^{\prime})$ and expanding for small
gradients gives $2\pi\ \gamma_{e}\int_{2\pi R_{1}}^{L}ds|\phi(s)|r(s).$ The
total energy penalty is then
$\frac{\epsilon_{h}}{2}\int_{0}^{L}ds\left(d\phi\over
ds\right)^{2}+2\pi\gamma_{e}\int_{2\pi R_{1}}^{L}ds\ r(s)|\phi(s)|.$ (15)
This kind of total energy is peculiar because of the form of the surface
tension term, $|\phi(s)|$. This term is first order and not differentiable
about the equilibrium position $\phi=0$. This means that the usual eigenmode
techniques we might have used to analyse the system are not applicable.
However, a good estimate of the size of the fluctuations and in particular
$\delta z\equiv\sqrt{\langle(z(L)-z(0))^{2}\rangle}\ $ can be found by mapping
the problem onto that of a semiflexible chain in a nematic field. From now on
we ignore all numerical prefactors. For our purposes it is convenient to use
an approach developed by Odijk [31, 32]. He considers a wormlike chain
directed by a parabolic potential $\mu\phi^{2}$, where $\mu$ is a constant.
The chain has a typical angular deviation
$\delta\phi\equiv\sqrt{\langle\phi^{2}\rangle}$ . Such a chain can be viewed
as a series of steps of size $\lambda_{\parallel}=l_{h}(\delta\phi)^{2}$ in
the plane of the annulus and
$\lambda_{z}=\lambda_{\parallel}\delta\phi=l_{h}(\delta\phi)^{3}$
perpendicular to the annulus. A chain of length $L$ undergoes an anisotropic
random walk of $L/\lambda_{\parallel}$ steps in the plane, with each step
travelling a distance $\lambda_{z}$ perpendicular to the plane. This implies
the deviation out of the plane is $\delta z=\sqrt{Ll_{h}}(\delta\phi)^{2}$.
Odijk shows that the strength of the potential and the angular deviations are
related by $(\delta\phi)^{4}=kT/(\mu l_{h})$, so that $\delta
z=\sqrt{kTL/\mu}$. This result cannot be applied directly to our problem,
since we have a confining potential of the form $\gamma_{e}r(s)|\phi(s)|$.
However, the details of the shape of the potential cannot matter for the gross
behaviour of the system. In particular we can replace our potential by a
harmonic one, provided we choose $\mu$ self-consistently, i.e., we choose
$\mu$ so that for a typical value of the angular deviation the two potentials
coincide. Equating the two potentials gives
$\gamma_{e}r(s)|\phi|=\mu\phi^{2}$. This together, with the relation
$(\delta\phi)^{4}=kT/(\mu l_{h})$ gives
$\mu=(\gamma_{e}r(s))^{4/3}(l_{h}/kT)^{1/3}$. Note that since $r$ varies along
the chain, the effective confining potential $\mu$ also varies along the
chain. We can now integrate over the whole annulus to get the variation in
height $\delta z$. Between $r$ and $r+dr$ there is a total length $rdr/t$ and
hence the square of the step in the $z$ direction is
$dz^{2}=(kT/(\gamma_{e}r))^{4/3}l_{h}^{-1/3}t^{-1}rdr$. Integrating over the
whole annulus gives:
$\delta
z=\frac{kT}{(\gamma_{e}l_{h})^{2/3}}\sqrt{\frac{l_{h}}{t}}\sqrt{R_{o}^{2/3}-R_{i}^{2/3}}.$
(16)
Note that this deviation grows only very slowly with the chain length. In,
particular, for very long chains we have $\delta
z=(kT/(\gamma_{e}l_{h}))^{2/3}(l_{h}/t)^{1/2}t^{1/6}L^{1/6}$. Setting $t=a$
and $\gamma_{e}=gkT/a^{2}$ gives $\delta z\approx ag^{-2/3}(L/l_{h})^{1/6}$.
Thus, in general the overall fluctuations will be less than, or of the same
order as the annular thickness, $w$, and the annulus will be well-defined and
relatively flat. This is the conclusion for one layer.
For many layers this conclusion remains true, since in that case the effective
persistence length scales with the number of layers and the radius also scales
with the number of layers. This implies that the absolute deviation in
thickness remains roughly constant and the relative variation decreases as
more layers are added.
We now discuss fluctuations of radial direction. The simplest mode is one
where the inner radius changes from its equilibrium value, $R_{i}^{*}$. In the
long-chain limit, where $R_{i}^{*}\ll\sqrt{Lt}$ the energy penalty for
deviating from equilibrium is
$\delta U={\pi\epsilon_{e}\over
2t}\left(\frac{R_{i}-R_{i}^{*}}{\Lambda}\right)^{2}.$ (17)
Equating this to $kT/2$ gives a typical size of fluctuation
$\sqrt{\langle(R_{i}-R_{i}^{*})^{2}\rangle}=\Lambda\sqrt{t/(\pi l_{e})}$. For
$l_{e}/t=10$ this gives a relative fluctuation of $13\%$. We expect that the
fluctuations within the plane will be reasonably small.
In conclusion, ribbon polymers which have large bending rigidities in lateral
direction spontaneously form anisotropic condensates in poor solvents. The
size of those peculiar condensates quantized as a variation of the length of
polymers and solvent quality.
When the material is homogeneous, the surface tensions and elastic energies
are determined by geometry of the ribbon. In general, for homogeneous
materials, we may assume $\epsilon_{e}<\epsilon_{h}$ and
$\gamma_{e}\approx\gamma_{h}$, which is the case we discussed in this paper.
On the chemical scale, however, other anisotropic conditions are possible in
principle. The ribbon polymers with $\epsilon_{e}<\epsilon_{h}$ and
$w\gamma_{e}<t\gamma_{h}$[33], form a different type of anisotropic
condensates in poor solvents, that will be discussed somewhere else.
###### Acknowledgements.
The authors thank Jean-Louis Sikorav for discussions and comments on the
manuscript. This work was initiated with support by NTT Basic Research
Laboratories. Y.Y.S. wishes to acknowledge CEA, IPhT and Takushoku University,
RISE for financial support.
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|
arxiv-papers
| 2008-11-04T15:25:15 |
2024-09-04T02:48:58.596819
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Y. Y. Suzuki and D. R. M. Williams",
"submitter": "Yasuo Y. Suzuki",
"url": "https://arxiv.org/abs/0811.0536"
}
|
0811.0544
|
# Asymptotic Behavior of Individual Orbits of Discrete Systems
Nguyen Van Minh Department of Mathematics, University of West Georgia,
Carrollton, GA 30118. USA vnguyen@westga.edu
###### Abstract.
We consider the asymptotic behavior of bounded solutions of the difference
equations of the form $x(n+1)=Bx(n)+y(n)$ in a Banach space $\mathbb{X}$,
where $n=1,2,...$, $B$ is a linear continuous operator in $\mathbb{X}$, and
$(y(n))$ is a sequence in $\mathbb{X}$ converging to $0$ as $n\to\infty$. An
obtained result with an elementary proof says that if
$\sigma(B)\cap\\{|z|=1\\}\subset\\{1\\}$, then every bounded solution $x(n)$
has the property that $\lim_{n\to\infty}(x(n+1)-x(n))=0$. This result extends
a theorem due to Katznelson-Tzafriri. Moreover, the techniques of the proof
are furthered to study the individual stability of solutions of the discrete
system. A discussion on further extensions is also given.
###### Key words and phrases:
Katznelson-Tzafriri Theorem, discrete system, individual orbit, stability,
asymptotically almost periodic.
###### 1991 Mathematics Subject Classification:
Primary: 47D06; Secondary: 47A35; 39A11
The author is grateful to the anonymous referee for his carefully reading the
manuscript and pointing out several inaccuracies and suggestions to improve
the presentation of this paper.
## 1\. Introduction, Notations and Preliminaries
Suppose that $T$ is a power-bounded linear continuous operator in a given
complex Banach space $\mathbb{X}$, i.e.,
$\sup_{n\in\mathbb{N}}\|T^{n}\|<\infty$. In [12, Theorem 1] it is proven that
$\lim_{n\to\infty}\|T^{n+1}-T^{n}\|=0$ if
$\sigma(T)\cap\\{|z|=1\\}\subset\\{1\\}$. As noted in [20], this assertion is
actually equivalent to a little weaker one that for each $x_{0}\in\mathbb{X}$,
$\lim_{n\to\infty}\|T^{n+1}x_{0}-T^{n}x_{0}\|=0$ if
$\sigma(T)\cap\\{|z|=1\\}\subset\\{1\\}$. An elegant proof of this assertion,
which we refer to as Katznelson-Tzafriri Theorem, was given in [19]. There are
numerous works on extensions and applications of this result of which to name
a few the reader is referred to e.g. [1], [4], [5], [6] [7], [8], [10], [11],
[13], [14], [16], [17], [19], and their references.
It is the first purpose of this note to extend the Katznelson-Tzafriri Theorem
to difference equations of the form
(1.1) $x(n+1)=Bx(n)+y(n),\quad x(n)\in\mathbb{X},n\in\mathbb{N},$
where $x(n)\in\mathbb{X}$, $B$ is a linear continuous operator acting in
$\mathbb{X}$ that is not necessarily assumed to be power-bounded,
$y(n)\in\mathbb{X}$ is a sequence satisfying $\lim_{n\to\infty}y(n)=0.$ Our
main result is Theorem 2.1 that is proven by an elementary method which can be
furthered to study the stability of individual solutions of (1.1). A Tauberian
theorem (Theorem 2.8) is stated and then used to prove Theorem 2.10 on the
asymptotical stability of individual solutions of (1.1). This result may be
seen as the discrete version of several results in [3, 7, 13, 15], and it
complements a result on strong stability of solutions in [21]. For a more
complete account of results and methods in this direction the reader is
referred to [4, 8, 17].
In this note we will use the following notations:
$\mathbb{N}=\\{1,2,\cdots\\}$, $\mathbb{Z}$ \- the set of all integers,
$\mathbb{R}$ \- the set of reals, $\mathbb{C}$ \- the complex plane with $\Re
z$ denoting the real part of $z\in\mathbb{C}$, $\mathbb{X}$ \- a given complex
Banach space. A sequence in $\mathbb{X}$ will be denoted by
$(x(n))_{n=1}^{\infty}$, or, simply by $(x(n))$, and the spaces of sequences
$\displaystyle l^{\infty}(\mathbb{X})$ $\displaystyle:=$
$\displaystyle\\{(x(n))\subset\mathbb{X}|\
\sup_{n\in\mathbb{N}}\|x(n)\|<\infty\\}$ $\displaystyle c_{0}$
$\displaystyle:=$ $\displaystyle\\{(x(n))\subset\mathbb{X}|\
\lim_{n\to\infty}x(n)=0\\}$
are equipped with sup-norm. The shift operator $S$ acts in
$l^{\infty}(\mathbb{X})$ as follows:
$\displaystyle Sx(n)=x(n+1),\quad n\in\mathbb{N},x\in l^{\infty}(\mathbb{X}).$
In this paper, for a complex Banach space $\mathbb{X}$, the space of all
bounded linear operators acting in $\mathbb{X}$ is denoted by $L(\mathbb{X})$;
$\rho(B),\sigma(B),R\sigma(B),Ran(B)$ denote the resolvent set, spectrum,
residual spectrum, range of $B\in L(\mathbb{X})$, respectively. It is well
known that the operator $S$ defined as above is a contraction. Consider the
quotient space $\mathbb{Y}:=l^{\infty}(\mathbb{X})/c_{0}$ with the induced
norm. The equivalent class containing a sequence $x\in l^{\infty}(\mathbb{X})$
will be denoted by $\bar{x}$. Since $S$ leaves $c_{0}$ invariant it induces a
bounded linear operator $\bar{S}$ acting in $\mathbb{Y}$. Moreover, one notes
that $\bar{S}$ is a surjective isometry. As a consequence,
$\sigma(\bar{S})\subset\Gamma$, where $\Gamma$ denotes the unit circle in the
complex plane. We will use the following estimate for the resolvent of the
isometry $\bar{S}$ whose proof can be easily obtained:
(1.2) $\|R(\lambda,\bar{S})\|\leq\frac{1}{||\lambda|-1|},\quad\mbox{for all}\
|\lambda|\not=1.$
## 2\. Main Results
### 2.1. Katznelson-Tzafriri Theorem for Individual Orbits
Consider the difference equation (1.1) with $(y(n))\in c_{0}$. A main result
of this note is the following
###### Theorem 2.1.
Let $B$ be any linear continuous operator acting in $\mathbb{X}$ such that
$\sigma(B)\cap\Gamma\subset\\{1\\}$, and let $x:=(x(n))_{n=1}^{\infty}$ be a
bounded solution of (1.1). Then,
(2.1) $\lim_{n\to\infty}[x(n+1)-x(n)]=0.$
The theorem is an immediate consequence of several lemmas that may be of
independent interest.
###### Lemma 2.2.
Assume that $\bar{x}$ is any point in $\mathbb{Y}$, and the complex function
$g(\lambda):=R(\lambda,\bar{S})\bar{x}$ has the point
$\lambda=\xi_{0}\in\Gamma$ as an isolated singular point. Then, $\xi_{0}$ is
either a removable singular point or a pole of first order.
###### Proof.
Without loss of generality we may assume that $\xi_{0}=1$. Consider $\lambda$
in a small neighborhood of $1$ in the complex plane. We will express
$\lambda=e^{z}$ with $|z|<\delta_{0}$. Choose a small $\delta_{0}>0$ such that
if $|z|<\delta_{0}$, then
(2.2) $\frac{1}{|1-|\lambda||}\leq\frac{2}{|\Re z|}.$
Notice that if $0<|z|<\delta_{0}$, then
(2.3)
$\|R(\lambda,\bar{S})\bar{x}\|\leq\frac{1}{|1-|\lambda||}\|\bar{x}\|\leq\frac{2}{|\Re
z|}\|\bar{x}\|.$
Set $f(z)=R(e^{z},\bar{S})\bar{x}$ with $|z|<\delta_{0}$. Since $1$ is a
singular point of $\|R(\lambda,\bar{S})\bar{x}\|$, $0$ is a singular point of
$f(z)$ in $\\{|z|<\delta_{0}\\}$ . For each $n\in\mathbb{Z}$ and
$0<r<\delta_{0}$, we have
$\displaystyle\|\frac{1}{2\pi
i}\int_{|z|=r}z^{n}\left(1+\frac{z^{2}}{r^{2}}\right)f(z)dz\|$
$\displaystyle\hskip
56.9055pt\leq\frac{1}{2\pi}\int_{|z|=r}|z^{n}\left(1+\frac{z^{2}}{r^{2}}\right)|\cdot\|f(z)\|\cdot|dz|.$
If $z=re^{i\varphi}$, where $\varphi$ is real, then one has
(2.4) $\displaystyle|z^{n}\left(1+\frac{z^{2}}{r^{2}}\right)|$
$\displaystyle=$ $\displaystyle
r^{n}|1+e^{2i\varphi}|=r^{n}|e^{-i\varphi}+e^{i\varphi}|$ $\displaystyle=$
$\displaystyle r^{n}2|cos\varphi|=2r^{n-1}|\Re z|.$
Therefore,
(2.5) $\displaystyle|\frac{1}{2\pi
i}\int_{|z|=r}z^{n}\left(1+\frac{z^{2}}{r^{2}}\right)f(z)dz|$
$\displaystyle\leq$ $\displaystyle\frac{1}{2\pi}\int_{|z|=r}2r^{n-1}|\Re
z|\frac{2}{|\Re z|}\cdot|dz|$ $\displaystyle=$ $\displaystyle\frac{2\cdot
2r^{n-1}}{2\pi}\int_{|z|=r}|dz|$ $\displaystyle=$ $\displaystyle{4r^{n}}.$
Consider the Laurent series of $f(z)$ at $z=0$,
(2.6) $f(z)=\sum_{n=-\infty}^{\infty}a_{n}z^{n},$
where
(2.7) $a_{n}=\frac{1}{2\pi i}\int_{|z|=r}\frac{f(z)dz}{z^{n+1}},\quad
n\in\mathbb{Z}.$
From (2.5) it follows that
$\displaystyle\|a_{-(n+1)}+r^{-2}a_{-(n+3)}\|$ $\displaystyle=$
$\displaystyle\|\frac{1}{2\pi i}\int_{|z|=r}z^{n}f(z)dz+\frac{1}{2\pi
i}\int_{|z|=r}\frac{z^{n+2}}{r^{2}}f(z)dz\|$ $\displaystyle=$
$\displaystyle\|\frac{1}{2\pi
i}\int_{|z|=r}z^{n}\left(1+\frac{z^{2}}{r^{2}}\right)f(z)dz\|$
$\displaystyle\leq$ $\displaystyle 4r^{n}.$
Therefore,
(2.8) $\|r^{2}a_{-(n+1)}+a_{-(n+3)}\|\leq 4r^{n+2},\quad n\in\mathbb{Z}.$
Letting $r$ tend to $0$ in (2.8), we come up with $a_{-k}=0$ for all $k\geq
2$. This shows that $z=0$ is a removable singular point (when $a_{-1}=0$) or a
pole of first order of $f(z)$. This yields that the complex function
$g(\lambda):=R(\lambda,\bar{S})\bar{x}$ has $\lambda=1$ as a removable
singular point or a pole of first order. The lemma is proven. ∎
Before proceeding we introduce a new notation: let $0\not=z\in\mathbb{C}$ such
that $z=re^{i\varphi}$ with reals $r=|z|,\varphi$, and let $F(z)$ be any
complex function. Then, (with $s$ larger than $r$) we define
(2.9) $\lim_{\lambda\downarrow z}F(\lambda):=\lim_{s\downarrow
r}F(se^{i\varphi}).$
That is, we consider the limit as $\lambda$ approaches $z$ in a special
direction corresponding to the ray $\arg\lambda=\arg z$.
###### Lemma 2.3.
Let $\xi_{0}\in\Gamma$ be an isolated singular point of
$g(\lambda)=R(\lambda,\bar{S})\bar{x}$ with a given $\bar{x}\in\mathbb{Y}$.
Then, this singular point $\xi_{0}$ is removable provided that
(2.10)
$\lim_{\lambda\downarrow\xi_{0}}(\lambda-\xi_{0})R(\lambda,\bar{S})\bar{x}=0.$
###### Proof.
As shown in Lemma 2.2, $\xi$ is either a removable singular point or pole of
first order. Without loss of generality we may assume that $\xi_{0}=1$ for the
reader’s convenience. Then, the Laurent series of $g(\lambda)$ is of the form
(2.11)
$g(\lambda)=\sum_{n=0}^{\infty}(\lambda-1)^{n}b_{n}+\frac{1}{\lambda-1}b_{-1}.$
We need to show that under condition (2.10) the coefficient $b_{-1}=0$. In
fact,
$\displaystyle 0=\lim_{\lambda\downarrow 1}g(\lambda)$ $\displaystyle=$
$\displaystyle\lim_{\lambda\downarrow 1}(\lambda-1)R(\lambda,\bar{S})\bar{x}$
$\displaystyle=$ $\displaystyle\lim_{\lambda\downarrow
1}(\lambda-1)\left(\sum_{n=0}^{\infty}(\lambda-1)^{n}b_{n}+\frac{1}{\lambda-1}b_{-1}\right)$
$\displaystyle=$ $\displaystyle\lim_{\lambda\downarrow
1}\left(\sum_{n=0}^{\infty}(\lambda-1)^{n+1}b_{n}+\frac{\lambda-1}{\lambda-1}b_{-1}\right)$
$\displaystyle=$ $\displaystyle b_{-1}.$
This shows $\xi_{0}$ is removable. The lemma is proven. ∎
###### Definition 2.4.
Let $(x(n))$ be a bounded sequence in $\mathbb{X}$. The notation $\sigma(x)$
stands for the set of all non-removable singular points of the complex
function $g(\lambda):=R(\lambda,\bar{S})\bar{x}$. This set may be referred to
as the spectrum of $x$, an analog of a similar concept in [3]. Obviously,
$\sigma(x)$ is a closed subset of $\Gamma$.
###### Lemma 2.5.
Let $x:=(x(n))$ be a bounded solution of equation (1.1). Then,
(2.12) $\sigma(x)\subset\sigma(B)\cap\Gamma.$
###### Proof.
Consider $R(\lambda,\bar{S})\bar{x}$ for all $|\lambda|\not=1$. Since $x$ is a
bounded solution of (1.1) and $\bar{y}=0$ we have
(2.13) $\displaystyle R(\lambda,\bar{S})\bar{S}\bar{x}$ $\displaystyle=$
$\displaystyle R(\lambda,\bar{S})\bar{B}\bar{x}+R(\lambda,\bar{S})\bar{y}$
$\displaystyle=$ $\displaystyle\bar{B}R(\lambda,\bar{S})\bar{x}$
On the other hand, the identity $\lambda
R(\lambda,\bar{S})\bar{x}-\bar{x}=R(\lambda,\bar{S})\bar{S}\bar{x}$ gives
(2.14) $\lambda
R(\lambda,\bar{S})\bar{x}-\bar{x}=R(\lambda,\bar{S})\bar{S}\bar{x}=\bar{B}R(\lambda,\bar{S})\bar{x},$
so,
$\displaystyle\bar{x}$ $\displaystyle=$ $\displaystyle\lambda
R(\lambda,\bar{S})\bar{x}-\bar{B}R(\lambda,\bar{S})\bar{x}$ $\displaystyle=$
$\displaystyle(\lambda-\bar{B})R(\lambda,\bar{S})\bar{x}.$
Obviously, $R(\lambda,\bar{S})\bar{x}$ is analytic on
$\mathbb{C}\backslash\Gamma$. Moreover, if $|\xi_{1}|=1$ and
$\xi_{1}\not\in\sigma(B)\cap\Gamma$, (as we can easily check that
$\sigma(\bar{B})=\sigma(B)$), in a small neighborhood $U(\xi_{1})$ of
$\xi_{1}$ we have
(2.15) $\displaystyle R(\lambda,\bar{S})\bar{x}$ $\displaystyle=$
$\displaystyle(\lambda-\bar{B})^{-1}\bar{x},\quad\lambda\in
U(\xi_{1})\backslash\Gamma.$
This shows that $g(\lambda)=R(\lambda,\bar{S})\bar{x}$ is analytically
extendable to a neighborhood of $\xi_{1}$, that is, $\xi_{1}\not\in\sigma(x)$.
The lemma is proven. ∎
Proof of Theorem 2.1: The identity $R(\lambda,\bar{S})\bar{S}\bar{x}=\lambda
R(\lambda,\bar{S})\bar{x}-\bar{x}$ gives
$\displaystyle R(\lambda,\bar{S})(\bar{S}\bar{x}-\bar{x})$ $\displaystyle=$
$\displaystyle R(\lambda,\bar{S})\bar{S}\bar{x}-R(\lambda,\bar{S})\bar{x}$
$\displaystyle=$ $\displaystyle\lambda
R(\lambda,\bar{S})\bar{x}-\bar{x}-R(\lambda,\bar{S})\bar{x}$ $\displaystyle=$
$\displaystyle(\lambda-1)R(\lambda,\bar{S})\bar{x}-\bar{x}.$
Therefore,
(2.16) $\displaystyle
h(\lambda):=(\lambda-1)R(\lambda,\bar{S})(\bar{S}\bar{x}-\bar{x})$
$\displaystyle=$
$\displaystyle(\lambda-1)^{2}R(\lambda,\bar{S})\bar{x}-(\lambda-1)\bar{x}.$
By Lemmas 2.5, 2.2, $\sigma(Sx-x)\subset\sigma(B)\cap\Gamma\subset\\{1\\}$,
$h(\lambda)$ is extendable analytically to the whole complex plane with only
possible exception at $1$. Since $g(\lambda):=R(\lambda,\bar{S})\bar{x}$ has
$1$ as a either removable singular point or a pole of first order we have
$\displaystyle\lim_{\lambda\to 1}(\lambda-1)^{2}R(\lambda,\bar{S})\bar{x}$
$\displaystyle=$ $\displaystyle 0.$
Consequently,
$\displaystyle\lim_{\lambda\to
1}(\lambda-1)R(\lambda,\bar{S})(\bar{S}\bar{x}-\bar{x})$ $\displaystyle=$
$\displaystyle\lim_{\lambda\to
1}[(\lambda-1)^{2}R(\lambda,\bar{S})\bar{x}-(\lambda-1)\bar{x}]$
$\displaystyle=$ $\displaystyle\lim_{\lambda\to
1}(\lambda-1)^{2}R(\lambda,\bar{S})\bar{x}-\lim_{\lambda\to
1}(\lambda-1)\bar{x}$ $\displaystyle=$ $\displaystyle 0.$
By Lemma 2.3, $h(\lambda)$ has $\lambda=1$ as a removable singular point, so
$h(\lambda)$ is extendable to an entire function. For $|\lambda|>1$, by (1.2)
we have
$\displaystyle\limsup_{|\lambda|\to\infty}\|h(\lambda)\|$ $\displaystyle=$
$\displaystyle\limsup_{|\lambda|\to\infty}\|(\lambda-1)R(\lambda,\bar{S})(\bar{S}\bar{x}-\bar{x})\|$
$\displaystyle\leq$
$\displaystyle\limsup_{|\lambda|\to\infty}\frac{|\lambda|+1}{|\lambda|-1}\cdot\|\bar{S}\bar{x}-\bar{x}\|$
$\displaystyle=$ $\displaystyle\|\bar{S}\bar{x}-\bar{x}\|.$
This shows that $h(\lambda)$ is bounded on the complex plane, so, as a bounded
entire function it should be a constant by Liouville’s Theorem. In turn, it is
identically equal to zero because $h(1):=\lim_{\lambda\to 1}h(\lambda)=0$.
Since $R(\lambda,\bar{S})$ is injective for each $\lambda\not=1$, we have
$\bar{S}\bar{x}-\bar{x}=0$. Therefore, $(Sx-x)\in c_{0}$, that is, (2.1). The
theorem is proven.
###### Remark 2.6.
In the remark following Theorem 2.8 we will give an alternative proof of
Theorem 2.1 in a more general context. However, the above proof seems to be
more elementary.
### 2.2. Stability of Individual Orbits
We define ${\mathcal{M}}_{\bar{x}}$ as the smallest closed subspace of
$\mathbb{Y}:=l^{\infty}(\mathbb{X})/c_{0}$ spanned by
$\\{\bar{S}^{n}\bar{x},n\in\mathbb{Z}\\}$. Consider the restriction
$\bar{S}|_{{\mathcal{M}}_{\bar{x}}}$ that is also a surjective isometry.
###### Lemma 2.7.
Let $x:=(x(n))\in l^{\infty}(\mathbb{X})$. Then, the following assertions
hold:
1. i)
$\sigma(x)=\emptyset$ if and only if $x\in c_{0}$;
2. ii)
If $\sigma(\bar{x})\not=\emptyset$, then
$\sigma(x)=\sigma(\bar{S}|_{{\mathcal{M}}_{\bar{x}}})$.
###### Proof.
(i): If $\sigma(x)=\emptyset$, the function
$g(\lambda):=R(\lambda,\bar{S})\bar{x}$ can be extended to an entire function.
Using exactly the argument in the proof of Theorem 2.1 we come up with the
boundedness of the complex function
$t(\lambda):=(\lambda-1)R(\lambda,\bar{S})\bar{x}$, so by Liouville’s Theorem
$t(\lambda)$ is a constant. And thus, $t(\lambda)=\lim_{\lambda\to
1}(\lambda-1)g(\lambda)=0$. The injectiveness of $R(\lambda,\bar{S})$ for each
$|\lambda|\not=1$ yields that $\bar{x}=0$. The converse is clear.
(ii): By (i), $\bar{x}\not=0$, so
$\rho(\bar{S}|_{{\mathcal{M}}_{\bar{x}}})\not=\emptyset$. Let
$\xi_{0}\in\rho(\bar{S}|_{{\mathcal{M}}_{\bar{x}}})$. Then, since for
$|\lambda|\not=1$
$R(\lambda,\bar{S})\bar{x}=R(\lambda,\bar{S}|_{{\mathcal{M}}_{\bar{x}}})\bar{x}$
it is clear that $\xi_{0}$ is a regular point of $g(\lambda)$.
Conversely, let $\xi_{0}$ be a regular point of $g(\lambda)$. Without loss of
generality we may assume $|\xi_{0}|=1$, otherwise it is already in
$\rho(\bar{S}|_{{\mathcal{M}}_{\bar{x}}})$. We will show that
$\xi_{0}\in\rho(\bar{S}|_{{\mathcal{M}}_{\bar{x}}})$ by proving that the
equation
(2.17) $\xi_{0}v-\bar{S}v=w$
has a unique solution $v\in{\mathcal{M}}_{\bar{x}}$ for each given
$w\in{\mathcal{M}}_{\bar{x}}$. First, we show that there is at least one
solution. In fact, we note that for each $n\in\mathbb{Z}$ the set of regular
points of $g(\lambda)=R(\lambda,\bar{S})\bar{x}$ is the same as that of
$\bar{S}^{n}g(\lambda)=R(\lambda,\bar{S})\bar{S}^{n}\bar{x}$. And in turn, by
the property of holomorphic functions, the set of all regular points of
$g(\lambda)=R(\lambda,\bar{S})\bar{x}$ must be part of that of the function
$k(\lambda)=R(\lambda,\bar{S})w$, so $k(\lambda)=R(\lambda,\bar{S})w$ is
analytically extendable to a neighborhood of $\xi_{0}$. In particular,
$\lim_{\lambda\to\xi_{0}}k(\lambda)=v\in{\mathcal{M}}_{\bar{x}}$, so
$\displaystyle\lim_{(|\lambda|>1),\ \lambda\to\xi_{0}}[\lambda
R(\lambda,\bar{S})w-R(\lambda,\bar{S})\bar{S}w]$ $\displaystyle=$
$\displaystyle w$ $\displaystyle\xi_{0}v-\bar{S}v$ $\displaystyle=$
$\displaystyle w.$
To show that equation (2.17) has a unique solution in
${\mathcal{M}}_{\bar{x}}$ we can show that the homogeneous equation
$\xi_{0}v-\bar{S}v=0$ has only a trivial solution in
${\mathcal{M}}_{\bar{x}}$. In fact, let $v_{0}\in{\mathcal{M}}_{\bar{x}}$ be a
solution of this equation. Then, for each $|\lambda|>1$, using the identity
$R(1,A)=(I-A)^{-1}=\sum_{n=0}^{\infty}A^{n}$ for each $\|A\|<1$ and
$\bar{S}^{n}v_{0}=\xi_{0}^{n}v_{0}$ we have
(2.18) $\displaystyle R(\lambda,\bar{S})v_{0}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{1}{\lambda^{n+1}}\bar{S}^{n}v_{0}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{1}{\lambda^{n+1}}\xi_{0}^{n}v_{0}$
$\displaystyle=$ $\displaystyle\frac{1}{\lambda-\xi_{0}}v_{0}.$
Since $v_{0}\in{\mathcal{M}}_{\bar{x}}$, this function must, as above, be
extendable analytically to a neighborhood of $\xi_{0}$, and this is possible
only if $v_{0}=0$. Summing up, we have that
$\xi_{0}\in\rho({\mathcal{M}}_{\bar{x}})$, so the lemma is proven. ∎
###### Theorem 2.8.
Let $(x(n))$ be a bounded sequence such that the set $\sigma(x)$ of all non-
removable singular points of $g(\lambda)=R(\lambda,\bar{S})\bar{x}$ is
countable, and let the following condition holds for each
$\xi_{0}\in\sigma(\bar{x})$
(2.19)
$\lim_{\lambda\downarrow\xi_{0}}(\lambda-\xi_{0})R(\lambda,\bar{S})\bar{x}=0.$
Then,
(2.20) $\lim_{n\to\infty}x(n)=0.$
###### Proof.
We have to show (2.20), that is, $\bar{x}=0$, or equivalently,
${\mathcal{M}}_{\bar{x}}$ is trivial. Suppose to the contrary that it is not.
Then, by Lemma 2.7,
$\sigma(x)=\sigma(\bar{S}|_{{\mathcal{M}}_{\bar{x}}})\not=\emptyset$. Since
$\sigma(x)$ is a non-empty closed subset of $\Gamma$ and is countable, it has
an isolated point, say $\xi_{0}$, so $\xi_{0}$ is an isolated singular point
for $g(\lambda)$. By Lemma 2.2 this isolated singular point must be either a
removable singular point or a pole of first order. Since $\xi_{0}$ is a pole
of first order of the resolvent
$R(\lambda,\bar{S}|_{{\mathcal{M}}_{\bar{x}}})$, by a well known result in
Functional Analysis111We actually avoid applying Geldfand Theorem in this
case. (see e.g. [18, Theorem 5.8 A, p. 306], or, [22, Theorem 3, p. 229])
$\xi_{0}$ must be an eigenvalue of $\bar{S}|_{{\mathcal{M}}_{\bar{x}}}$ with a
non-zero eigenvector $w_{0}$. As in the proof of Lemma 2.7, (see 2.18)), for
each $|\lambda|\not=1$ we have
(2.21) $R(\lambda,\bar{S})w_{0}=\frac{1}{\lambda-\xi_{0}}w_{0}.$
On the other hand, by Lemma 2.3 $\xi_{0}$ is a removable singular point for
$g(\lambda)$, so is for $R(\lambda,\bar{S})w_{0}$. This is possible only if
$w_{0}=0$, contradicting that $w_{0}$ is a non-zero vector. This proves the
theorem. ∎
###### Remark 2.9.
An alternative proof of Theorem 2.1 is a direct application of Lemma 2.5 and
Theorem 2.8. As another consequence of Theorem 2.8 we have the following on
the strong asymptotical stability of solutions of (1.1).
###### Theorem 2.10.
For equation (1.1) assume that $(y(n))\in c_{0}$, and the operator $B$ in
equation (1.1) has $\sigma(B)\cap\Gamma$ as a countable set. Then, the
following holds for each bounded solution $(x(n))$ of (1.1)
(2.22) $\lim_{n\to\infty}x(n)=0,$
provided that for each $\xi_{0}\in\sigma(B)\cap\Gamma$ the following condition
holds
(2.23)
$\lim_{\lambda\downarrow\xi_{0}}(\lambda-\xi_{0})R(\lambda,\bar{S})\bar{x}=0.$
###### Proof.
This theorem is an immediate consequence of Lemma 2.5 and Theorem 2.8. ∎
## 3\. Discussion
Theorem 2.1 may be seen as an extension of the following result due to
Katznelson-Tzafriri (see [12, Theorem 1]).
###### Theorem 3.1.
Let $T$ be a power bounded linear operator in a Banach space $\mathbb{X}$ such
that $\sigma(T)\cap\Gamma\subset\\{1\\}$. Then,
(3.1) $\lim_{n\to\infty}(T^{n+1}-T^{n})=0.$
In fact, as noted in [20] this theorem is equivalent to a weaker one
###### Theorem 3.2.
Let $T$ be a power bounded linear operator in a Banach space $\mathbb{X}$ such
that $\sigma(T)\cap\Gamma\subset\\{1\\}$. Then, for each $x_{0}\in\mathbb{X}$
(3.2) $\lim_{n\to\infty}(T^{n+1}x_{0}-T^{n}x_{0})=0.$
Obviously, our Theorem 2.1 extends Theorem 3.2.
As an immediate consequence of Theorem 2.10 we have the following corollary:
###### Corollary 3.3.
Let $B\in L(\mathbb{X})$ be a power bounded operator such that
$\sigma(B)\cap\Gamma$ is a countable set. Moreover, assume that for each
$\xi_{0}\in\sigma(B)\cap\Gamma$ the following holds for each
$x_{0}\in\mathbb{X}$
(3.3) $\lim_{\lambda\downarrow\xi_{0}}(\lambda-\xi_{0})R(\lambda,B)x_{0}=0.$
Then, for every $x_{0}\in\mathbb{X}$
(3.4) $\lim_{n\to\infty}B^{n}x_{0}=0.$
###### Proof.
Let $x(n)=B^{n}x_{0}$. Then, $(x(n))$ is a bounded solution of (1.1) with
$(y(n))=0$. Therefore, if $|\lambda|>1$, $\lambda\in\rho(B)$ and
$\lambda\in\rho(\bar{S})$, so by (2.15) (and the proof of Lemma 2.5),
$\displaystyle\lim_{\lambda\downarrow\xi_{0}}\|(\lambda-\xi_{0})R(\lambda,\bar{S})\bar{x}\|$
$\displaystyle=$
$\displaystyle\lim_{\lambda\downarrow\xi_{0}}\|(\lambda-\xi_{0})R(\lambda,\bar{B})\bar{x}\|$
$\displaystyle\leq$
$\displaystyle\lim_{\lambda\downarrow\xi_{0}}\sup_{n\in\mathbb{N}}\\{\|(\lambda-\xi_{0})R(\lambda,B)B^{n}x_{0}\|\\}$
$\displaystyle\leq$
$\displaystyle\lim_{\lambda\downarrow\xi_{0}}\sup_{n\in\mathbb{N}}\\{\|B^{n}\|\\}\cdot\|(\lambda-\xi_{0})R(\lambda,B)x_{0}\|$
(3.6) $\displaystyle\leq$
$\displaystyle\lim_{\lambda\downarrow\xi_{0}}\|(\lambda-\xi_{0})R(\lambda,B)x_{0}\|=0.$
Therefore, by Theorem 2.10, $x(n)=B^{n}x_{0}\to 0$. ∎
###### Remark 3.4.
Condition (3.3) is satisfied if $R\sigma(B)\cap\Gamma=\emptyset$, and hence
Corollary 3.3 yields the discrete version of the Arendt-Batty-Ljubich-Vu
Theorem [2, Theorem 5.1], [10, Corollary 3.3], [21]). In fact, since $B$ is
power-bounded one can easily show that there exists a positive constant $C$
such that
(3.7) $\|R(\lambda,B)\|\leq\frac{C}{||\lambda|-1|},\quad\mbox{for}\
|\lambda|>1.$
Next, since $R\sigma(B)\cap\Gamma=\emptyset$, for all
$\xi_{0}\in\sigma(B)\cap\Gamma$, the range of $(\xi_{0}-B)$ is dense in
$\mathbb{X}$. Therefore, for each $x_{0}\subset\mathbb{X}$ there is a sequence
$(x_{0}^{n})\in Ran(\xi_{0}-B)$ such that $x_{0}=\lim_{n\to\infty}x_{0}^{n}$.
Then, $x^{n}_{0}=(\xi_{0}-B)y^{n}_{0}$ for some sequence
$(y^{n}_{0})\subset\mathbb{X}$. By our definition of the limit as
$\lambda\downarrow\xi_{0}$ we have
$|\lambda-\xi_{0}|=||\lambda|-|\xi_{0}||=||\lambda|-1|\to 0$, so in view of
(3.7), for each fixed $n$ we have
(3.8)
$\displaystyle\lim_{\lambda\downarrow\xi_{0}}\|(\lambda-\xi_{0})R(\lambda,B)x^{n}_{0}\|$
$\displaystyle=$
$\displaystyle\lim_{\lambda\downarrow\xi_{0}}\|(\lambda-\xi_{0})R(\lambda,B)(\xi_{0}-B)y^{n}_{0}\|$
$\displaystyle=$
$\displaystyle\lim_{\lambda\downarrow\xi_{0}}\|(\lambda-\xi_{0})R(\lambda,B)[(\lambda-B)y^{n}_{0}+(\xi_{0}-\lambda)y^{n}_{0}]\|$
$\displaystyle=$
$\displaystyle\lim_{\lambda\downarrow\xi_{0}}\left(|\lambda-\xi_{0}|\cdot\|y^{n}_{0}\|+\|(\lambda-\xi_{0})R(\lambda,B)(\xi_{0}-\lambda)y^{n}_{0}\|\right)$
$\displaystyle\leq$ $\displaystyle
0+\lim_{\lambda\downarrow\xi_{0}}\|\lambda-\xi_{0}\|^{2}\cdot\|R(\lambda,B)\|\cdot\|y^{n}_{0}\|$
$\displaystyle\leq$
$\displaystyle\lim_{\lambda\downarrow\xi_{0}}||\lambda|-|\xi_{0}||^{2}\cdot\frac{C}{||\lambda|-1|}\cdot\|y^{n}_{0}\|$
$\displaystyle=$
$\displaystyle\lim_{\lambda\downarrow\xi_{0}}\frac{||\lambda|-1|^{2}\cdot
C}{||\lambda|-1|}\cdot\|y^{n}_{0}\|=0.$
By (3.7) for every fixed $n$
(3.9)
$\limsup_{\lambda\downarrow\xi_{0}}\|(\lambda-\xi_{0})R(\lambda,B)(x_{0}-x^{n}_{0})\|\leq
C\|x_{0}-x_{0}^{n}\|.$
Finally, for each $n\in\mathbb{N}$ from (3.8) and (3.9) we have
$\displaystyle\limsup_{\lambda\downarrow\xi_{0}}\|(\lambda-\xi_{0})R(\lambda,B)x_{0}\|$
$\displaystyle\leq$
$\displaystyle\limsup_{\lambda\downarrow\xi_{0}}[\|(\lambda-\xi_{0})R(\lambda,B)x^{n}_{0}$
$\displaystyle+(\lambda-\xi_{0})R(\lambda,B)(x_{0}-x^{n}_{0})\|]$
$\displaystyle\leq$ $\displaystyle C\|x_{0}-x_{0}^{n}\|.$
Since $\|x_{0}^{n}-x_{0}\|\to 0$ as $n\to\infty$, we have that (3.3) holds for
any $x_{0}\in\mathbb{X}$.
Let us define a so-called Condition H for a closed subspace ${\mathcal{M}}$ of
$l^{\infty}(\mathbb{X})$ by the following axioms:
1. i)
${\mathcal{M}}$ is bi-invariant under translation $S$, that is,
${\mathcal{M}}=\\{x\in l^{\infty}(\mathbb{X}):Sx\in{\mathcal{M}}\\}$;
2. ii)
If $x:=(x(n))\in{\mathcal{M}}$ and $A\in L(\mathbb{X})$, then
$y:=(Ax(n))\in{\mathcal{M}}$;
3. iii)
$c_{0}\subset{\mathcal{M}}$,
As an example of such a closed subspace ${\mathcal{M}}$ of
$l^{\infty}(\mathbb{X})$ that satisfies Condition H one can take the space
$AAP(\mathbb{N},\mathbb{X})$ of all asymptotic almost periodic sequences. If
we replace $c_{0}$ by ${\mathcal{M}}$, we will arrive at various analogs of
Theorems 2.1, 2.10 and 2.8. Note that the proofs of these analogs are
identically similar to those of the mentioned theorems. Below are the
statements of analogs of the mentioned theorems in case
${\mathcal{M}}=AAP(\mathbb{N},\mathbb{X})$.
Recall that a sequence $(x(n))$ is said to be asymptotically almost periodic
if $x(n)=y(n)+z(n)$ for all $n\in\mathbb{N}$ where $(y(n))\in c_{0}$ and
$(z(n))$ is an almost periodic sequence. An almost periodic sequence on
$\mathbb{N}$ is the restriction to $\mathbb{N}$ of an almost periodic sequence
on $\mathbb{Z}$. In turn, an almost periodic sequence on $\mathbb{Z}$ is
defined to be an element of the following subspace
$\overline{span\\{(\lambda^{n}y_{0})_{n\in\mathbb{Z}},\lambda\in\Gamma,y_{0}\in\mathbb{X}\\}}$
of $l^{\infty}(\mathbb{X})$. In the following, by abusing notations, $\bar{x}$
denotes the equivalent class of
$l^{\infty}(\mathbb{X})/AAP(\mathbb{N},\mathbb{X})$ containing $x$, $\bar{S}$
denotes the the operator acting in
$l^{\infty}(\mathbb{X})/AAP(\mathbb{N},\mathbb{X})$ induced by $S$.
###### Theorem 3.5.
Let $B$ be any linear continuous operator acting in $\mathbb{X}$ such that
$\sigma(B)\cap\Gamma\subset\\{1\\}$, and let $x:=(x(n))_{n=1}^{\infty}$ be a
bounded solution of (1.1) in which $(y(n))\in AAP(\mathbb{N},\mathbb{X})$.
Then, the sequence $(y(n))$, defined as $y(n):=x(n+1)-x(n)$ for all
$n\in\mathbb{N}$, is asymptotically almost periodic.
###### Theorem 3.6.
Let $(x(n))$ be a bounded sequence such that the set
$\sigma_{AAP(\mathbb{N},\mathbb{X})}(x)$ of all non-removable singular points
of $g(\lambda)=R(\lambda,\bar{S})\bar{x}$ is countable, and let the following
condition hold for each $\xi_{0}\in\sigma_{AAP(\mathbb{N},\mathbb{X})}(x)$
(3.10)
$\lim_{\lambda\downarrow\xi_{0}}(\lambda-\xi_{0})R(\lambda,\bar{S})\bar{x}=0.$
Then, $(x(n))$ is asymptotically almost periodic.
###### Theorem 3.7.
For equation (1.1) assume that $(y(n))\in AAP(\mathbb{N},\mathbb{X})$, and the
operator $B$ in equation (1.1) has $\sigma(B)\cap\Gamma$ as a countable set.
Then, each bounded solution $(x(n))$ of (1.1) is asymptotically almost
periodic, provided that for each $\xi_{0}\in\sigma(B)\cap\Gamma$ the following
holds
(3.11)
$\lim_{\lambda\downarrow\xi_{0}}(\lambda-\xi_{0})R(\lambda,\bar{S})\bar{x}=0.$
## References
* [1] G. R. Allan, T. J. Ransford, Power-dominated elements in a Banach algebra. Studia Math. 94 (1989), 63-79.
* [2] W. Arendt, C.J.K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc. 306(1988), 837-852.
* [3] W. Arendt, C.J.K. Batty, Asymptotically almost periodic solutions of inhomogeneous Cauchy problems on the half-line. Bull. London Math. Soc., 31 (1999), 291-304.
* [4] W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, 96, Birkh user Verlag, Basel, 2001.
* [5] W. Arendt, J. Pruss, Vector-valued Tauberian theorems and asymptotic behavior of linear Volterra equations. SIAM J. Math. Anal. 23 (1992), 412-448.
* [6] B. Basit, A. J. Pryde, Ergodicity and stability of orbits of unbounded semigroup representations. J. Aust. Math. Soc. 77 (2004), 209-232.
* [7] C. J. K. Batty, Jan van Neerven, Frank Rabiger, Local spectra and individual stability of uniformly bounded $C_{0}$-semigroups. Trans. Amer. Math. Soc. 350 (1998), 2071-2085.
* [8] R. Chill, Y. Tomilov, Stability of operators semigroups: ideas and results. In ”Perspectives in Operator Theory”. Banach Center Publications, Vol. 75. (2007), pp. 71-109.
* [9] J. Esterle, E. Strouse, F. Zouakia, Stabilité asymptotique de certains semigroupes d’opérateurs et idéaux primaires de $L^{1}(\mathbb{R}_{+})$. J. Operator Theory 28 (1992), 203-227.
* [10] J. Esterle, E. Strouse, F. Zouakia, Theorems of Katznelson-Tzafriri type for contractions. J. Funct. Anal. 94 (1990), 273-287.
* [11] N. Kalton, S. Montgomery-Smith, K. Oleszkiewicz, Y. Tomilov, Power-bounded operators and related norm estimates. J. London Math. Soc. (2) 70 (2004), 463-478.
* [12] Y. Katznelson, L. Tzafriri, On power bounded operators. J. Funct. Anal. 68 (1986), 313–328.
* [13] R. deLaubenfels, Vu Qu c Phong, Stability and almost periodicity of solutions of ill-posed abstract Cauchy problems. Proceedings of the A.M.S., 125 (1997), 235-241.
* [14] Nguyen Van Minh, Katznelson-Tzafriri type theorems for individual solutions of evolution equations. Proceedings of the A.M.S. 136 (2008), 1749-1755.
* [15] Nguyen Van Minh, A new approach to the spectral theory of functions and the Loomis-Arendt-Batty-Vu Theory. Submitted. Preprint in ArXiv.org at the URL: http://arxiv.org/abs/math.FA/0609652
* [16] H. Mustafayev, The Banach algebra generated by a $C_{0}$-semigroup. C. R. Math. Acad. Sci. Paris 342 (2006), no. 8, 575–578.
* [17] J. M. A. M. van Neerven, ” The asymptotic Behaviour of Semigroups of Linear Operator”, Birkha$\ddot{u}$ser Verlag. Basel. Boston. Berlin, Operator Theory, Advances and Applications Vol.88 1996.
* [18] A. E. Taylor, ”Introduction to Functional Analysis”. John Wiley & Sons. New York -London, 1958.
* [19] Vu Quoc Phong, Theorems of Katznelson-Tzafriri type for semigroups of operators. J. Funct. Anal. 103 (1992), 74–84.
* [20] Vu Quoc Phong, A short proof of the Y. Katznelson’s and L. Tzafriri’s theorem. Proceedings of the A.M.S., 115 (1992), pp. 1023-1024.
* [21] Vu Quoc Phong, Yu. I. Lyubich, A spectral criterion for almost periodicity for one-parameter semigroups. (Russian) Teor. Funktsii Funktsional. Anal. i Prilozhen. No. 47 (1987), 36-41; translation in J. Soviet Math. 48 (1990), no. 6, 644-647.
* [22] K. Yosida, ”Functional Analysis”. Springer. Berlin-Heidelberg - New York. 1977.
|
arxiv-papers
| 2008-11-04T15:56:45 |
2024-09-04T02:48:58.602923
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nguyen Van Minh",
"submitter": "Minh Nguyen",
"url": "https://arxiv.org/abs/0811.0544"
}
|
0811.0738
|
# Gamma ray astrophysics: the EGRET results
D J Thompson Astrophysics Science Division, NASA Goddard Space Flight Center,
Greenbelt, MD 20771 USA David.J.Thompson@nasa.gov
###### Abstract
Cosmic gamma rays provide insight into some of the most dynamic processes in
the Universe. At the dawn of a new generation of gamma-ray telescopes, this
review summarizes results from the Energetic Gamma Ray Experiment Telescope
(EGRET) on the Compton Gamma Ray Observatory, the principal predecessor
mission studying high-energy photons in the 100 MeV energy range. EGRET viewed
a gamma-ray sky dominated by prominent emission from the Milky Way, but
featuring an array of other sources, including quasars, pulsars, gamma-ray
bursts, and many sources that remain unidentified. A central feature of the
EGRET results was the high degree of variability seen in many gamma-ray
sources, indicative of the powerful forces at work in objects visible to
gamma-ray telescopes.
###### pacs:
95.55.Ka, 95.85.Pw, 98.70.Rz
## 1 Introduction: gamma rays from the Universe
For most of history, humans have learned about the cosmos by viewing the light
that our eyes can detect. Only in the Twentieth Century did it become clear
that vast amounts of information arrive in different channels, most
prominently the invisible forms of electromagnetic radiation. This information
revolution started with radio, which can reach the Earth’s surface, and then
exploded with the space age discoveries that essentially every part of the
spectrum carries unique information about conditions and processes in distant
parts of the Universe. Most of these forms of radiation are blocked by the
Earth’s atmosphere and therefore require space observatories or some form of
indirect detection.
Gamma rays represent the high-energy end of the electromagnetic spectrum,
comprising photons with the highest frequencies or shortest wavelength.
Because gamma rays are so energetic, they are usually defined by their
energies, which are typically higher than 100 kilo electron Volts (keV),
although the line between X-rays and gamma rays is not a sharp one. Often
X-rays are considered to be those photons produced by atomic or thermal
processes, while gamma rays are those involving nuclear or nonthermal
processes. Above 106 eV (1 MeV), and with no upper limit to energy, all
photons are gamma rays.
As might be imagined, gamma rays are produced by energetic phenomena. In fact,
only the lowest-energy gamma rays, those associated with radioactivity, have
natural sources on Earth. Cosmic sources of gamma rays extend to vastly higher
energies, reflecting extreme physical conditions and powerful collisions.
Astrophysical settings for gamma-ray production include supernovae, pulsars,
and quasars, as well as the interstellar and intergalactic medium. Although
gamma rays are absorbed in the atmosphere, the Universe is largely transparent
to these high-energy photons out to high redshift. Gamma rays thus provide a
valuable probe of the largest energy transfers throughout much of the
Universe. Such phenomena can be expected to be important in understanding the
forces of change on the largest scale.
The present review is focused on one important segment of gamma-ray
astrophysics: the results from the Energetic Gamma Ray Experiment Telescope
(EGRET) that flew on the Compton Gamma Ray Observatory (CGRO) during its
1991-2000 life. EGRET provided the first detailed all-sky observations of
high-energy gamma rays (with typical energies in the 100 MeV to 1000 MeV
range). With the advent of two successors to EGRET, the Italian Astro-
rivelatore Gamma a Immagini Leggero (AGILE) mission and the international
Fermi Gamma-ray Telescope (formerly the Gamma-ray Large Area Space Telescope,
GLAST) mission, the time seems appropriate to examine the observational
foundation of these two current missions.
The outline of this review is as follows:
1\. Introduction
2\. EGRET in the context of other gamma-ray telescopes
Historical
CGRO
EGRET: Detecting high-energy gamma rays
3\. An overview of the gamma-ray sky
4\. Galactic diffuse
5\. Gamma-ray sources: the third EGRET catalogue
6\. Galactic sources
Pulsars
Binary Sources
Other Sources
7\. Extragalactic sources
Blazars
Other galaxies
Diffuse extragalactic radiation
Gamma-ray bursts
8\. Local sources
The Moon
Solar flares
Primordial black holes
9\. Open questions for AGILE and Fermi
## 2 EGRET in the context of other gamma-ray telescopes
### 2.1 Before the Compton Observatory
The potential for studying the sky with gamma rays was identified fifty years
ago (e.g. Morrison 1958). A decade later the first definitive detections of
gamma rays from space came with the OSO-3 discovery of gamma radiation from
the plane of our Galaxy (Clark et al1968). In following years, a number of
fairly small balloon-borne and satellite missions began to reveal aspects of
the gamma-ray sky. Some examples:
* •
Browning, Ramsden and Wright (1971) found pulsed high-energy gamma radiation
from the Crab Pulsar, using a gamma-ray telescope carried on a balloon.
* •
The U.S. Small Astronomy Satellite 2 (SAS-2, Fichtel et al1975) showed that
high-energy gamma rays help trace the structure of our Galaxy, the Milky Way
(Hartman et al1979). It also discovered a second gamma-ray pulsar, the Vela
Pulsar (Thompson et al1975) and the first unidentified gamma-ray source,
$\gamma$195+5 (Kniffen et al1975), which later was called Geminga (Bignami et
al1983).
* •
The European COS-B satellite (Bignami et al1975) produced the first catalogues
of high-energy gamma-ray sources, most of which were not identified with
objects seen at other wavelengths (Hermsen et al1977, Swanenburg et al1981).
In addition, it found the first extragalactic gamma-ray source, the nearby
quasar 3C273 (Swanenburg et al1978) and made the first gamma-ray observations
of molecular clouds as spatially-extended sources (Caraveo et al1980).
* •
VELA satellites operated by the U.S. military discovered short-duration cosmic
gamma-ray bursts, a completely new phenomenon (Klebesadel et al1973).
* •
The Third High Energy Astrophysical Observatory (HEAO-3) carried a low-energy
gamma-ray telescope with high spectral resolution (Mahoney et al1980) that
detected the 0.5 MeV positron-electron annihilation line coming from the
Galactic Center region (Riegler et al1981), confirming previous reports from
balloon instruments (e.g. Leventhal et al1978).
Figure 1: Some space gamma-ray missions, showing energy coverage and the time
frame of the mission.
During this same period, a parallel branch of gamma-ray astrophysics using
ground-based detectors was developing. At gamma-ray energies above about 1011
eV (100 GeV), cosmic photons are too scarce to be detected by satellite
detectors. The Earth’s atmosphere itself can be used, however, as a detector
for these very high energy (VHE) photons. When such photons collide with the
material at the top of the atmosphere, they produce showers of particles
moving faster than the local speed of light, thereby emitting Cherenkov
radiation in the optical and ultraviolet. The flashes of light from these
interactions can be detected with Atmospheric Cherenkov Telescopes (ACTs) on
the ground, providing an indirect way to conduct gamma-ray astrophysics. A
milestone in VHE gamma-ray astrophysics was reached in 1989 with the high-
confidence detection of the Crab Nebula (but not the pulsar) with the Whipple
Observatory ACT (Weekes et al1989). Although VHE studies are beyond the scope
of the present review, this rapidly-developing field is highly complementary
to space-based gamma-ray studies. Weekes (2003) provides a broad review of
gamma-ray astrophysics, with emphasis on the VHE field, while Chadwick et
al(2008) and Aharonian et al(2008) present recent summaries of results.
Each of these programs provided a glimpse of the gamma-ray Universe,
confirming the potential of this field for helping understand high-energy
cosmic phenomena. None of them gave a comprehensive picture. Some viewed only
portions of the sky. Some studied only a limited portion of the huge energy
range falling under the gamma-ray banner. Most were operated for only limited
durations. Figure 1 shows schematically the time frames and energy ranges of
many of these space-based gamma-ray telescopes. NASA’s Great Observatories
program offered the opportunity to make a major advance across much of the
gamma ray portion of the spectrum.
### 2.2 The Compton Gamma Ray Observatory
The Compton Gamma Ray Observatory (CGRO), shown in Figure 2, was the second of
NASA’s Great Observatories, following the Hubble Space Telescope. Although the
original plan was to have all four operating simultaneously, circumstances
delayed the launches of the Chandra X-ray Observatory (originally called AXAF)
and the Spitzer Space Telescope (Infrared, originally called SIRTF) until the
Compton Observatory mission was essentially complete. CGRO itself was launched
on the Space Shuttle Atlantis on 5 April 1991 and operated successfully until
it was de-orbited on 4 June 2000.
Figure 2: The Compton Gamma Ray Observatory just before its release by the
Shuttle in April. 1991.
CGRO carried four gamma-ray telescopes, each with its own energy range,
detection technique, and scientific goals. Together these four instruments
covered energies from less than 15 keV to more than 30 GeV, over six orders of
magnitude in the electromagnetic spectrum. The three lower-energy telescopes
were:
* •
Burst and Transient Source Explorer (BATSE, Principal Investigator, G.
Fishman, NASA Marshall Space Flight Center). BATSE was the smallest of the
CGRO instruments, consisting of one module located on each corner of the
spacecraft. Each BATSE unit included a large flat NaI(Tl) scintillator pointed
with its face away from the center of the observatory and a smaller thicker
scintillator for spectral measurements, combining to cover an energy range
from 15 keV to over 1 MeV. Important results from BATSE included the mapping
of over 2700 gamma-ray bursts, showing an isotropic distribution on the sky. A
summary of BATSE results is given by Fishman (1995).
* •
Oriented Scintillation Spectrometer Experiment (OSSE, Principal Investigator
J. Kurfess, Naval Research Laboratory). OSSE used four large, collimated
scintillator detectors to study low-energy gamma-rays, 60 keV - 10 MeV. OSSE
mapped the 0.5 MeV line from positron annihilation and provided detailed
measurements of many hard X-ray/soft-gamma-ray sources. Kurfess (1996)
summarized many of the important results from OSSE.
* •
Imaging Compton Telescope (COMPTEL, Principal Investigator V. Schönfelder, Max
Planck Institute for Extraterrestrial Physics). COMPTEL detected medium-energy
gamma rays using a Compton scattering technique, effective between 0.8 MeV and
30 MeV. Among its results, COMPTEL mapped the distribution of radioactive
Aluminum-26 in the Galaxy, showing the locations of newly formed material. The
summary by Schönfelder et al(1996) describes many of the COMPTEL results.
This brief summary covers only a few of the many important results from these
CGRO instruments. Schönfelder (2001) reviewed the entire field of gamma-ray
astrophysics, with particular emphasis on this energy range.
### 2.3 EGRET: Detecting High-Energy Gamma Rays
In the energy range above 10 MeV, the principal interaction process for gamma
rays is pair production, a direct example of the Einstein equation E = mc2.
The photon energy is converted into an electron and its antiparticle, a
positron. This process can take place in the field of an atomic nucleus or in
a strong magnetic field, but not in free space, in order to conserve energy
and momentum. These high-energy gamma rays cannot be reflected or refracted; a
gamma-ray telescope actually detects the electron and positron.
Figure 3: Schematic diagram of a pair production telescope.
The Energetic Gamma Ray Experiment Telescope (EGRET) was the high-energy
instrument on the Compton Observatory, covering the energy range 20 MeV to 30
GeV (Hughes et al1980; Fichtel et al1983). The Co-Principal Investigators
represented the three major contributors to the EGRET hardware: C. Fichtel,
NASA Goddard Space Flight Center (GSFC); R. Hofstadter, Stanford University
(SU); and K. Pinkau, Max Planck Institute for Extraterrestrial Physics (MPE).
The operational concept of EGRET, similar in most respects to the designs of
other high-energy gamma-ray telescopes, is shown in Figure 3. The two key
challenges for any such telescope are: (1) identify the gamma-ray interaction
in the presence of a huge background of charged particles (cosmic rays, solar
particles, and trapped radiation); and (2) measure the gamma-ray arrival time,
arrival direction, and energy.
The process works as follows:
1. 1.
A gamma ray enters EGRET. It first passes through the Anticoincidence System
without producing a signal.
2. 2.
The gamma ray interacts in one of 28 thin tantalum sheets. This interaction
converts the gamma ray into an electron and a positron via pair production.
3. 3.
The spark chamber Tracker records the paths of the electron and positron,
allowing EGRET to see the pair interaction and to determine the arrival
direction of the gamma ray.
4. 4.
The electron and positron pass through two scintillator detectors operated in
a time-of-flight (TOF) configuration. The TOF signal confirms the direction of
the particles and triggers the readout of the spark chambers.
5. 5.
The electron and positron enter the Calorimeter, producing an electromagnetic
shower, which measures the energies of the particles and therefore the energy
of the original gamma ray.
6. 6.
Unwanted cosmic-ray particles produce signals in the Anticoincidence System,
which tell the electronics not to trigger the spark chamber. The
Anticoincidence System rejects nearly all unwanted signals produced by cosmic
rays that enter EGRET.
Figure 4: Composite photo showing a cutaway view of EGRET. The major
subsystems are identified.
Figure 4 shows EGRET as it appeared before integration onto the Compton Gamma
Ray Observatory. Here is some information about the key subsystems:
* •
The Anticoincidence System consists of a single dome of plastic scintillator,
read out by 24 photomultiplier tubes mounted around the bottom. This subsystem
was built by the MPE group.
* •
The Tracker is made of 36 wire grid spark chambers, interleaved with the
converter plates. The active area of the spark chambers is 81 cm x 81 cm. This
subsystem was the work of the group at GSFC.
* •
The TOF trigger system, also from GSFC, has two four by four arrays of plastic
scintillator tiles, each read out by a single photomultiplier tube.
* •
The Calorimeter, called the Total Absorption Shower Counter (TASC), was made
of 36 NaI crystals bonded together and read out by 16 photomultiplier tubes.
The SU group built the TASC.
The primary EGRET calibration was carried out at the Stanford Linear
Accelerator Center (SLAC), using a gamma-ray beam varying in energy from 15
MeV to 10 GeV. The beam scanned the active area of EGRET at a variety of
angles out to 40∘ from the instrument axis. The calibration thus covered
essentially the entire phase space to be observed in orbit (Thompson et
al1993). Analysis of both the calibration and flight data concentrated on
recognizing and measuring the individual pair-production events. An initial
selection of events was made in software, removing many triggers that did not
produce tracks consistent with being electron-positron pairs. Events the
software could not resolve were reviewed by data analysts and scientists,
leading to a final data set almost entirely free of charged particle
contamination.
The basic properties of EGRET are shown in Table 1. The single-photon angular
resolution, or point spread function (PSF) is energy-dependent. The angle
$\theta$ in degrees containing 67% of the gamma rays from a delta function
source with energy E in MeV is given by: $\theta$ = 5∘.85 E-0.534.
Table 1: Performance Characteristics of EGRET.
* Property | Value
---|---
Energy Range | 20 MeV - $>$ 10 GeV
Peak Effective Area | 1500 cm2 at 500 MeV
Energy Resolution | 15% FWHM
Off-axis effective area | 25% of peak at 30∘
Timing accuracy | $<$ 100 $\mu$sec absolute
Observations of the Compton Gamma Ray Observatory ranged in duration from a
few days to a few weeks, reflecting the paucity of cosmic gamma-ray photons. A
listing of the observations, along with other information about CGRO, can be
found at the CGRO Science Support Center Web site,
http://cossc.gsfc.nasa.gov/docs/cgro/index.html.
Because the EGRET spark chambers were gas detectors, their performance
deteriorated with time due to gas aging. The system carried enough replacement
gas for four complete refills of the chamber, all of which were used. The
performance of the instrument was recalibrated in flight, using constant
sources such as the diffuse Galactic emission and bright pulsars as reference
sources. The preliminary in-flight calibration was described by Esposito et
al1999, and the final performance analysis was described by Bertsch et al2001.
## 3 An overview of the high-energy gamma-ray sky
In its nine-year lifetime, EGRET detected over 1,500,000 celestial gamma rays.
One photon at a time, EGRET built up a picture of the entire high-energy
gamma-ray sky. Figure 5 shows the summed map above 100 MeV, in Galactic
coordinates. The Milky Way runs horizontally across the center of the figure,
and the Galactic center lies at the center of the map.
Figure 5: The sky seen with EGRET, shown in Galactic coordinates. In this
false color image, the Galactic Center lies in the middle of the image.
This image provides a striking contrast to the view of the sky at visible
wavelengths. Some of the key features that will be discussed in following
sections are:
* •
The Milky Way is extremely bright, particularly toward the inner part of the
Galaxy.
* •
The brightest persistent sources are pulsars.
* •
Many of the bright sources away from the Galactic plane are highly variable
types of Active Galactic Nuclei in the blazar class.
* •
The Moon is visible and outshines the Sun most of the time.
* •
Many of the sources are not identified with known objects.
## 4 Galactic diffuse gamma-ray emission
The high-energy gamma-ray sky is dominated by the bright Galactic ridge. This
component was the first one predicted and detected in gamma-ray astronomy,
because our galaxy is known to be filled with high-energy particles, magnetic
fields, photon fields, and interstellar gas. The spatial distribution of the
gamma radiation traces galactic structure as determined from radio and other
measurements. These basic features were known in the SAS-2 and COS-B era,
along with the realization that the cosmic ray flux cannot be uniform
throughout the Galaxy and still be consistent with these measurements. The
EGRET data provided detailed information, but also introduced a puzzle that
remains unresolved.
The physical processes that produce EGRET-energy gamma rays are familiar ones
to the particle physics community:
* •
Inelastic collisions of cosmic ray particles with the interstellar gas (mostly
hydrogen) produce secondary particles, particularly charged and neutral $\pi$
mesons. The neutral pions decay almost immediately into two gamma rays. In the
center of mass reference frame, each gamma ray has half the energy of the
$\pi^{\circ}$, or about 67 MeV. Because the cosmic rays typically have a broad
range of high energies, the energy spectrum of the gamma rays resulting from
nucleon-nucleon collisions is spread out into a broad peak rather than a
narrow line.
* •
Cosmic ray electrons colliding with photons can boost the photon energies into
the gamma-ray band by inverse Compton scattering. The principal targets are
the optical and infrared photons found throughout the Galaxy.
* •
Another cosmic-ray electron process involves collisions with the interstellar
gas, producing gamma rays through bremsstrahlung.
* •
Both nucleon and electron cosmic rays can in principle produce gamma rays
through interactions with magnetic fields by synchrotron radiation. In
practice, the fluxes expected from synchrotron radiation with known particles
and magnetic fields are small compared the the other sources.
Figure 6: Diffuse Galactic spectrum (Hunter et al1997). Calculated components
are NN: nucleon-nucleon interactions ( $\pi^{\circ}$-decay); EB: electron
bremsstrahlung; IC: Inverse Compton; ID: Isotropic diffuse (see section 7.3).
The EGRET energy spectrum of the gamma radiation from the Galactic Center
region is shown in Figure 6, along with the calculated source components
(Hunter et al1997). Below 100 MeV, electron bremsstrahlung is the principal
component, while at higher energies the nucleon-nucleon $\pi^{\circ}$-decay
source is the most important. The expected “bump” compared to a power law
spectrum is clearly visible.
Building on the work of Bertsch et al(1993), Hunter et al(1997) carried out
detailed modeling of the Galactic radiation. Measured cosmic-ray intensities
were combined with a three-dimensional model of the photon and gas
distribution, using 21 cm radio measurements to trace neutral hydrogen (HI)
and carbon monoxide (CO) transition radio measurements as a tracer of
molecular hydrogen (H2). Coupling between the matter and cosmic ray densities
was assumed, based on arguments of dynamic balance between matter, magnetic
fields, and cosmic rays. This model reproduced most features of the observed
gamma radiation.
Visible in Figure 6 is one unexpected feature of the EGRET observations: the
flux above 1 GeV exceeds the model prediction by a significant amount (well
beyond any known measurement uncertainties). This discrepancy has become known
as the “GeV excess.”
An alternative modeling approach, called GALPROP, has been developed by
Strong, Moskalenko, and Reimer (2000, 2004b). This model emphasizes cosmic-ray
propagation calculations and a larger inverse Compton contribution to the
gamma radiation. Although the basic version of GALPROP does not reproduce the
GeV excess, these authors have shown that plausible assumptions about cosmic
ray densities in the Galaxy higher than the local values can reproduce the
EGRET data.
De Boer (2005) has invoked a new component of the gamma radiation coming from
annihilation of supersymmetric dark matter as the source of the GeV excess.
Bergström et al(2006) conclude, however, that this dark matter model is
inconsistent with measurements of antiprotons. Stecker, Hunter, and Kniffen
(2008) argue in favor of a miscalibration of the EGRET detector at high
energies as an explanation. No consensus exists. It is perhaps ironic that the
one component of the gamma-ray sky that was thought to be understood has left
a mystery at the end of the EGRET mission.
On a more local scale, Hunter et al(1994), Digel, Hunter and Mukherjee (1995),
and Digel et al(1996) conducted gamma-ray studies of the nearby Ophiuchus,
Orion, and Cepheus regions. By comparing the EGRET maps to maps of CO clouds,
they were able to trace the ratio of molecular hydrogen column density to
integrated CO intensity. The gamma-ray intensities were found consistent with
models based on the local flux measurements of cosmic rays.
Another issue with the diffuse radiation is that the matter content of the
Galaxy is not necessarily completely measured. Infrared data suggest the
presence of unseen gas clouds that could require an additional component in
the diffuse Galactic gamma-ray model (Grenier et al2005).
## 5 Gamma-ray sources: the third EGRET catalogue
Individual gamma-ray sources appear as excesses above the modeled diffuse
emission. The EGRET analysis process used a maximum likelihood method to
compare probabilities of fitting a given region of the sky with and without a
source (Mattox et al1996). The most complete analysis of the sky by the EGRET
team was the third EGRET catalogue (3EG: Hartman et al1999). Starting from the
Hunter et al(1997) diffuse model, the 3EG analysis examined each viewing
period plus combinations of viewing periods, from the start of the mission up
through the end of 1995, using multiple energy ranges. Because EGRET was
operated only intermittently after this time, the total exposure added by
later phases of the mission contributed little to the overall mapping of the
sky.
Figure 7 summarizes the 3EG results on gamma-ray sources, with the source
locations shown in Galactic coordinates. In this figure, the symbol size
indicates the peak source brightness. The gamma-ray sky is highly variable, so
not all sources were seen at all times. The census of the 271 3EG gamma-ray
sources was:
* •
94 sources show a probable or possible association with the class of Active
Galactic Nuclei known as blazars.
* •
Five pulsars appear in the catalogue.
* •
The Large Magellanic Cloud was detected as an extended gamma-ray source.
* •
One solar flare was bright enough to be seen in the source analysis
* •
170 sources, well over half the total, had no identification with known
astrophysical objects.
Following the publication of the 3EG catalogue, extensive efforts were made to
identify individual sources or source populations. A few of the 3EG sources
appear to be artifacts of the analysis (Thompson et al2001). The following
sections describe the detected source classes and efforts at identification,
as well as some sources that did not appear in the catalogue.
Recently, Cassandjian and Grenier (2008) have developed a new catalogue of
EGRET sources, based on a new model of the diffuse emission (Grenier et
al2005). This catalogue, which contains only 188 sources, incorporates many of
the 3EG sources into the diffuse radiation as gas concentrations, particularly
at intermediate Galactic latitudes. It does, however, remove a number of
identified sources, some of which showed evidence of time variability.
Changing the diffuse model would not be expected to eliminate time-variable
sources. This catalogue should probably be considered as an alternative
analysis rather than a replacement for the 3EG catalogue. Because it is so
new, it has not received the same level of study as the 3EG catalogue.
Figure 7: Map of source locations for the third EGRET catalogue (Hartman et
al1999), shown in Galactic coordinates.
## 6 Galactic gamma-ray sources
Particularly along the Galactic Plane, the modeling of the diffuse gamma
radiation strongly affects the calculated properties, or even the existence,
of sources. For this reason, the third EGRET catalogue adopted different
confidence levels for including sources as detections. Within 10 degrees of
the Galactic plane, a statistical significance of 5 $\sigma$ was required; at
higher latitudes the requirement was 4 $\sigma$. Even with this more stringent
requirement, Figure 7 shows a clear population of sources concentrated along
the plane.
### 6.1 Pulsars
The first high-energy gamma-ray source class was rotation-powered pulsars,
starting with the Crab and Vela, seen by SAS-2 and COS-B. EGRET expanded the
number of gamma-ray pulsars to at least 6, with several other good candidates.
A summary of the EGRET results on pulsars is given by Thompson (2004). These
rapidly-rotating neutron stars, originally seen in the radio (Hewish et
al1968), have strong magnetic, electric, and gravitational fields. Particles
accelerated to high energies in the magnetospheres of pulsars can interact
near the pulsar to produce gamma rays through curvature radiation, synchrotron
radiation, or inverse Compton scattering.
The telescopes on the Compton Gamma Ray Observatory identified seven or more
gamma-ray pulsars, some with very high confidence and others with less
certainty. Figure 8 shows the light curves from the seven highest-confidence
gamma-ray pulsars in five energy bands: radio, optical, soft X-ray ($<$1 keV),
hard X-ray/soft gamma ray ($\sim$10 keV - 1 MeV), and hard gamma ray (above
100 MeV). Based on the detection of pulsations, all seven of these are
positive detections in the gamma-ray band.
Figure 8: Light curves of seven gamma-ray pulsars in five energy bands. Each
panel shows one full rotation of the neutron star. Adapted from Thompson
(2004).
Some important features of these pulsar light curves are:
* •
They are not the same at all wavelengths. Some combination of the geometry and
the emission mechanism is energy-dependent. In soft X-rays, for example, the
emission in some cases appears to be thermal, probably from the surface of the
neutron star; thermal emission is not the origin of radio or gamma radiation.
* •
Not all seven are seen at the highest energies. PSR B1509$-$58 is seen up to
10 MeV by COMPTEL (Kuiper et al. 1999), but not at higher energies by EGRET.
* •
The six seen by EGRET all have a common feature - they show a double peak in
their light curves. Because these high-energy gamma rays are associated with
energetic particles, it seems likely that the particle acceleration and
interactions are taking place along a large hollow cone or other surface.
Models in which emission comes from both magnetic poles of the neutron star
appear less probable in light of the prevalence of double pulses.
In addition to the six high-confidence pulsar detections above 100 MeV, three
additional radio pulsars may have been seen by EGRET: PSR B1046$-$58, PSR
B0656+14, and PSR J0218+4232, the only millisecond pulsar with evidence of
gamma-ray emission (Kuiper et al 2000, 2002).
Figure 9: Multiwavelength spectra of seven gamma-ray pulsars. Updated from
Thompson et al. (2004)
Although the pulsations identify sources as rotating neutron stars, the
observed energy spectra reflect the physical mechanisms that accelerate
charged particles and help identify interaction processes that produce the
pulsed radiation. Broadband spectra for the seven highest-confidence gamma-ray
pulsars are shown in Figure 9. The presentation in $\nu$Fν format (or E2 times
the photon number spectrum) indicates the observed power per frequency
interval across the spectrum. In all cases, the maximum power output is in the
gamma-ray band. Other noteworthy features on this figure are:
* •
The distinction between the radio emission (which originates from a coherent
process) and the high-energy emission (probably from individual charged
particles in an incoherent process) is visible for some of these pulsars,
particularly Crab and Vela.
* •
Vela, Geminga, and B1055$-$52 all show evidence of a thermal component in
X-rays, thought to be from the hot neutron star surface.
* •
The gamma-ray spectra of known pulsars are typically flat, with most having
photon power-law indices of about 2 or less between 30 MeV and several GeV.
Energy breaks are seen in the 1-4 GeV band for several of these pulsars. These
changes in spectral index may be related to the calculated surface magnetic
field of the pulsar. The lowest-field pulsars have no visible break in the
EGRET energy range; the existence of a spectral change is deduced from the
absence of TeV detections of pulsed emission. The highest-field pulsar among
these, B1509$-$58, is seen only up to the COMPTEL energy band.
* •
No pulsed emission is seen above 30 GeV, the upper limit of the EGRET
observations, except for a recent detection of the Crab by the MAGIC telescope
(Teshima, 2008). The nature of the high-energy cutoff is an important feature
of pulsar models.
* •
Although Figure 9 shows a single spectrum for each pulsar, the pulsed energy
spectrum varies with pulsar phase. A study of the EGRET data by Fierro et al.
(1998) of the phase-resolved emission of the three brightest gamma-ray pulsars
(Vela, Geminga, Crab) showed no simple pattern of variation of the spectrum
with phase that applied to all three pulsars. A broadband study of the Crab by
Kuiper et al. (2001) indicated the presence of multiple emission components,
including one that peaks in the 0.1 - 1 MeV range for the bridge emission
between the two peaks in the light curve. Improved measurements and modeling
of the phase-resolved spectra of pulsars can be expected to be a powerful tool
for study of the emission processes.
The measured spectra can be integrated to determine the energy flux of each
pulsar. Except for the Crab and PSR B1509$-$58, whose luminosity peaks lie in
the $\sim$100 keV - 1 MeV range, the energy flux for the other gamma-ray
pulsars is dominated by the emission above 10 MeV. The energy flux can be
converted to an estimated luminosity by using the measured distance to the
pulsar and an assumed emission solid angle. For simplicity, I assume an
emission into one steradian. This value is unlikely to be the same for all
pulsars but provides a simple reference point for comparison. A significant
uncertainty in such calculations is introduced by the distance estimate. Table
2 summarizes these results for the ten pulsars. P is the pulsar spin period in
seconds. $\tau$ is the estimated age of the pulsar. $\dot{E}$ is the rate the
pulsar is losing energy as it slows down. FE is the energy flux seen at high
energies (X-rays and gamma rays). The distance d is given in kiloparsecs. LHE
is the calculated high-energy luminosity of the pulsar. $\eta$ is the
efficiency for conversion of spin-down energy loss into high energy radiation.
Table 2: Summary Properties of the Highest-Confidence and Candidate Gamma-Ray
Pulsars
Name | P | $\tau$ | $\dot{E}$ | FE | d | LHE | $\eta$
---|---|---|---|---|---|---|---
| (s) | (Ky) | (erg/s) | (erg/cm2s) | (kpc) | (erg/s) | (E$>$1 eV)
Crab | 0.033 | 1.3 | 4.5 $\times$ 1038 | 1.3 $\times$ 10-8 | 2.0 | 5.0 $\times$ 1035 | 0.001
B1509$-$58 | 0.150 | 1.5 | 1.8 $\times$ 1037 | 8.8 $\times$ 10-10 | 4.4 | 1.6 $\times$ 1035 | 0.009
Vela | 0.089 | 11 | 7.0 $\times$ 1036 | 9.9 $\times$ 10-9 | 0.3 | 8.6 $\times$ 1033 | 0.001
B1706$-$44 | 0.102 | 17 | 3.4 $\times$ 1036 | 1.3 $\times$ 10-9 | 2.3 | 6.6 $\times$ 1034 | 0.019
B1951+32 | 0.040 | 110 | 3.7 $\times$ 1036 | 4.3 $\times$ 10-10 | 2.5 | 2.5 $\times$ 1034 | 0.007
Geminga | 0.237 | 340 | 3.3 $\times$ 1034 | 3.9 $\times$ 10-9 | 0.16 | 9.6 $\times$ 1032 | 0.029
B1055$-$52 | 0.197 | 530 | 3.0 $\times$ 1034 | 2.9 $\times$ 10-10 | 0.72 | 1.4 $\times$ 1033 | 0.048
B1046$-$58 | 0.124 | 20 | 2.0 $\times$ 1036 | 3.7 $\times$ 10-10 | 2.7 | 2.6 $\times$ 1034 | 0.013
B0656+14 | 0.385 | 100 | 4.0 $\times$ 1034 | 1.6 $\times$ 10-10 | 0.3 | 1.3 $\times$ 1032 | 0.003
J0218+4232 | 0.002 | 460,000 | 2.5 $\times$ 1035 | 9.1 $\times$ 10-11 | 2.7 | 6.4 $\times$ 1033 | 0.026
One trend, first noted by Arons (1996), can be derived from this table. Figure
10 shows the efficiency of each pulsar as a function of the open field line
voltage, the potential that can be developed by the rotating neutron star. The
efficiency increases as the open field line voltage decreases. The implication
of this figure is that there must be a limit to gamma-ray production by
pulsars, because the efficiency shown in Figure 10 is approaching 1. Pulsars
with lower open field line voltage must either turn off or reach some
saturation value.
Figure 10: Calculated pulsed high-energy (X-ray and gamma ray) efficiencies of
the known and candidate gamma-ray pulsars, as a function of the open field
line voltage. Circles: high-confidence gamma-ray pulsars. Triangles: lower-
confidence gamma-ray pulsars.
Despite trends such as the one shown in Figure 10, each one of the nine high-
confidence and lower-confidence pulsars seen by EGRET has at least one unique
feature. Some examples:
* •
The Crab is the only gamma-ray pulsar to have its light curve aligned with the
light curves seen at all other wavelengths. This is the youngest gamma-ray
pulsar and the one with the largest spin-down luminosity. It is also the least
efficient of these in converting spin-down luminosity into high-energy
radiation.
* •
PSR B1951+32 is the only one of these to have a spectrum extending to the
limits of the EGRET energy range with no evidence of a spectral cutoff. Of the
high-confidence pulsars, it is the only one not bright enough to appear as a
source in the 3EG catalogue.
* •
PSR B1706$-$44 is unique in having a gamma-ray spectrum with two power laws
and a change of slope at about 1 GeV (compared to spectral cutoffs seen in
others).
* •
Geminga is the only radio-quiet gamma-ray pulsar found by EGRET.
* •
PSR B1055$-$52 shows the highest efficiency of any of these pulsars for
conversion of spin-down energy into gamma radiation.
* •
PSR B1046$-$58 is the only one of these with no pulsed X-ray counterpart.
* •
PSR J0218+4232 is the only millisecond pulsar with apparent gamma-ray
emission. It is also the only one of these pulsars with another candidate
gamma-ray source (a blazar) close enough spatially to cause source confusion.
Despite the wide range of information about these pulsars from their timing
properties and their measured gamma-ray characteristics, the fact that a
special feature can be found for each one limits our ability to draw broad
conclusions.
Stimulated by these observations of gamma-ray pulsars, theorists have carried
out extensive modeling of these high-energy neutron stars, but without
reaching a consensus on where the particles are being accelerated in the
star’s magnetosphere or how these particles interact to produce the gamma
radiation. Some examples of theoretical high-energy pulsar modeling are
Sturrock (1971), Ruderman and Sutherland (1975), Romani (1996), Harding and
Muslimov (2005), Cheng and Zhang (1998), and Hirotani (2008). A useful
overview of such theoretical work and the implications for pulsar population
studies is given by Harding et al(2007).
In addition to the gamma-ray pulsars identified by their periodic emission,
there are several potential associations of known pulsars with EGRET sources.
Some radio pulsars discovered after the end of the CGRO mission are
positionally consistent with 3EG sources and have enough energy to power the
observed gamma radiation (e.g. Kramer et al. 2003, Halpern et al. 2001). A
related case is 3EG J1835+5918, with properties suggesting an association with
the isolated neutron star RX J1836.2+5925 (Mirabal and Halpern 2001; Reimer et
al2001; Halpern et al2002). Because gamma-ray pulsars appear to have
significant timing noise, searching for pulsations in the EGRET data by
extrapolating timing solutions back in time involves too many trials to
produce high-confidence results, as illustrated by the unsuccessful search for
PSR J2229+6114, found in the error box of 3EG J2227+6122 (Thompson et al2002)
### 6.2 Binary sources
The EGRET data showed some indication of gamma radiation from binary sources,
although the case is far from certain. Cen X-3 , an accretion-powered X-ray
binary, may have shown a flare during an EGRET observation, including some
evidence of variability at the 4.8 second spin period of the neutron star
(Vestrand et al1997). Two sources in the 3EG catalogue, 3EG J0241+6103 and 3EG
J1824$-$1514 are positionally consistent with high-mass X-ray binary systems
(HMXB) often placed in the microquasar class, LSI +61∘303 (Kniffen et al1997;
Tavani et al1998) and LS 5039 (Paredes et al2000) respectively. The detection
of TeV radiation from both these HMXBs, showing orbitally modulated emission
for LSI +61∘303 (Albert et al2006) and periodic emission for LS 5039
(Aharonian et al2007), indicates that such sources can accelerate particles to
energies well beyond those needed to produce gamma rays in the EGRET energy
band. The evidence that the EGRET sources are actually these binary systems
remains largely circumstantial, however, and searches for other binaries in
the EGRET data have been unsuccessful.
### 6.3 Other Galactic sources
Most of the 3EG sources along the Galactic plane remain unidentified,
including one that flared up to be the second brightest source in the gamma-
ray sky for less than one month (Tavani et al1997). Two general approaches
have been followed in order to shed light on the possible nature of these
sources:
* •
The characteristics of the Galactic EGRET sources as a class, based on their
spatial and spectral properties, was investigated by several authors. Starting
with the earlier 2EG catalogue (Thompson et al1995), Mukherjee et al(1995),
Kanbach et al(1996) and Merck et al(1996) compared the characteristics of
unidentified sources with Galactic tracers. McLaughlin et al(1996) developed a
method for characterizing gamma-ray source variability and found that some of
the catalogued Galactic sources appear to be variable. Özel and Thompson
(1996) constructed log N-log S distributions for 2EG sources, showing that the
number of unidentified sources N with flux greater than S at high Galactic
latitudes had an isotropic distribution, consistent with being either quite
local or extragalactic. Similar analyses were carried out for the 3EG
catalogue. Reimer and Thompson (2000) and Bhattacharya et al(2003) studied the
log N-log S distributions. Spatial-statistical considerations and variability
studies suggest there is a population of Galactic and variable GeV gamma-ray
emitters among the unidentified EGRET sources (Nolan et al2003). A population
of steady gamma-ray sources with different properties from those close to the
Galactic Plane, possibly associated with the nearby Gould Belt complex of gas
and stars, was suggested by Gehrels et al(2000). As noted by Cassandjian and
Grenier (2008), many of these may be gas clouds that were not modeled in the
EGRET analysis. A valuable summary of 3EG source characteristics, along with
some cautions about limitations, is given by Reimer (2001). The strongest
conclusion from these many studies was that the EGRET Galactic sources
comprise more than one population.
* •
Correlations of known Galactic populations with the positions of the EGRET
sources offered another approach to trying to discern their nature. Using
various statistical techniques, various authors found indications of
associations of EGRET sources with star forming regions or groups of hot and
massive stars (e.g. Kaaret and Cottam 1996; Romero et al1999), supernova
remnants (e.g. Sturner and Dermer 1995; Esposito et al1996), or pulsar wind
nebulae (e.g. Roberts, Romani and Kawai 2001). Torres et al(2003) provide an
excellent summary of the observational and theoretical possibilities for
supernova remnants as EGRET sources. Pulsar populations may also explain a
fraction of the Galactic unidentified sources (Yadigaroglu and Romani 1997).
All these classes of sources are plausible, because they have the potential to
accelerate particles to high energies in an environment in which the particles
can interact to produce gamma rays. There are several issues with these
analyses:
* –
All these sources tend to be located in the same regions. Particularly
considering the possibility that more than one type of gamma-ray source may be
present, separating the classes is difficult.
* –
There is no unique spectral or timing signature for most of these sources, in
the absence of pulsations or orbital periods (which have not been found).
* –
The EGRET error boxes are too large to make unique associations possible. In
fact, none of these approaches found a clear example of one source that would
serve as a prototype for the class.
Ultimately, all these efforts proved valuable in pointing to young, active
Galactic objects as likely gamma-ray sources.
## 7 Extragalactic gamma-ray sources
As seen in figure 7, EGRET sources are seen in all directions in the sky. The
detection of the nearby quasar 3C273 by COS-B (Swanenburg et al1978) had long
suggested that extragalactic sources would be a significant component of the
high-energy sky (e.g. Bignami et al1979).
### 7.1 Blazars
Just three months after the launch of the Compton Observatory, a Target of
Opportunity pointing (requested by the OSSE team in search of emission from a
supernova in the Virgo cluster) produced the first big surprise in the EGRET
data. The observation that was expected to produce an observation of 3C273 was
instead dominated by a very bright source about 10 degrees away, positionally
consistent with a more distant quasar, 3C279, with a redshift z = 0.536
(Hartman et al1992). This detection had two immediate implications:
* •
The gamma-ray sky is variable, at least on long time scales, because COS-B had
not seen this source.
* •
The idea of nearby AGN as candidate gamma-ray sources was at best incomplete.
The fact that 3C279 was part of the blazar subclass of AGN, thought to be
powered by supermassive black holes and having powerful jets of particles and
radiation pointed toward the Solar System (Blandford and Rees 1978; Blandford
and Königl 1979), suggested that gamma rays might be a valuable probe of jet
sources.
The recognition of short-term variability of 3C279 (on a scale of days,
Kniffen et al1993) reinforced the idea that jets were a likely source of the
EGRET-detected gamma rays. The rapid variability required a compact emitting
region, and such a region should be opaque to gamma rays due to gamma-gamma
pair production, the interaction of a high-energy gamma ray with a lower-
energy photon to produce an electron-positron pair, $\gamma$ \+ $\gamma$ = e+
\+ e-, unless most of the photons are moving in the same direction, as in a
jet. Some example calculations of this process are those of Mattox et al(1993)
and Sikora et al(1994). The ultimate confirmation of such gamma-ray sources as
blazars came with multiwavelength campaigns that found correlated variability
of gamma-ray flares with flares seen at other wavelengths. The first example
was a flare seen in PKS 1406$-$076 in January 1993, shown in Figure 11, where
an optical flare was seen nearly simultaneous with the gamma-ray flare (Wagner
et al1995). Such correlated variability is a valuable source of information
about how such jets are formed, how they are collimated, and how they carry
energy.
Figure 11: Optical and gamma-ray flare in PKS 1406$-$076 (Wagner et al1995)
Figure 12: Spectral Energy Distribution for 3C279 during a bright state in
January 1996 (just before a large flare) with models (Hartman et al2001a).
Modeled components, from left to right in the figure: synchrotron radiation,
thermal radiation from the accretion disk, synchrotron-self-Compton radiation,
Compton radiation from scattering of accretion disk photons, and Compton
radiation from scattering of photons from gas clouds.
Continuing EGRET observations revealed a series of bright, well-localized
sources positionally consistent with prominent blazars. Many of these were
known as Optically Violently Variable (OVV) quasars or BL Lacertae (BL Lac)
objects. Although the term “blazar” is not uniquely defined, it typically
encompasses those Active Galactic Nuclei with the following characteristics:
radio-loud, with flat radio spectrum; significant polarization in optical
and/or radio; significant variability. Blazars are seen across the
electromagnetic spectrum, with a characteristic two-peak Spectral Energy
Distribution (SED), as illustrated in Figure 12 (Hartman et al2001a). At lower
frequencies, from radio to optical or sometimes X-rays, the emission is
thought to be dominated by synchrotron radiation of high-energy electrons in
the jet. The upper peak, extending from X-rays upward, is thought to be
primarily inverse Compton scattering of low-energy photons by the same
population of high-energy electrons that produces the lower-energy synchrotron
radiation. The source of the photons to be upscattered can be the synchrotron
radiation itself (Synchrotron Self-Compton) or some outside source of photons
(External Compton). Extensive modeling of blazars has evolved following the
EGRET discoveries. An example of such modeling is shown in Figure 12. In many
cases, the gamma-ray emission is the dominant observable output of these
blazars.
Figure 13: Correlated multiwavelength emission from 3C279 during the large
flare of January-February 1996 (Wehrle et al1998)
Some of the EGRET blazars of particular interest are:
* •
3C279. This first blazar recognized by EGRET was prominent many times during
the CGRO mission. It was the first to be involved in a concentrated
multiwavelength campaign (Maraschi et al1994). A later multiwavelength
campaign in January-February 1996, shown in Figure 13, captured a dramatic
flare, seen at multiple wavelengths (Wehrle et al1998). The gamma-ray flux
increased by over a factor of 10 during this flare. As a point of interest,
the limited optical coverage of this flare came about because most of the
optical astronomers who monitor this source were attending a blazar workshop
at the time and were not at their telescopes.
* •
PKS 1622$-$297\. Located not far from the Galactic Center region, this lesser-
known blazar was not seen in any of the first 17 EGRET observations of this
region, then flared up in the Summer of 1995 to be the brightest gamma-ray
source seen by EGRET to that time, and the one showing the fastest time
variability, doubling in flux in less than 8 hours, with the limitation being
just the photon statistics (Mattox et al1997)
* •
Mkn 421. This well-known BL Lac object appeared in an early EGRET observation
(Lin et al1992). Its detection helped stimulate the TeV gamma-ray astronomy
community, and it became the first blazar detected at TeV energies (Punch et
al1992).
* •
PKS 0528+134. This blazar, actually the first one that appeared in the EGRET
data although not identified until later, is one of at least five seen by
EGRET with redshift z $>$ 2.0 (Hunter et al1993). Although it underwent a
strong flare during one 1993 observation (Figure 14, Mukherjee et al1999), it
was also seen to be quite stable during other observations, even in short-
timescale investigations (Wallace et al2000).
* •
PKS 2255$-$282\. After the completion of the third EGRET catalogue, this
blazar was the only new one to appear with high significance in the EGRET data
(Macomb et al1999). The bright gamma-ray flare appeared in early 1998
following a period of rising flux in the submillimeter band (Tornikoski et
al1999).
Figure 14: Light curve for PKS 0528+134 during the nine-year life of EGRET
(Mukherjee et al1999)
The EGRET discoveries of numerous highly-variable gamma-ray blazars provided a
stimulus to this field of study and to other studies of Active Galactic
Nuclei. Terminology in this field has also evolved, and blazars are now
usually classified as Flat-Spectrum Radio Quasars (FSRQs), Low-frequency
Peaked BLLac Objects (LBLs), and High-frequency Peaked BLLac Objects (HBLs).
With blazars as the most numerous class of identified gamma-ray sources, they
offered opportunities for multiwavelength comparisons and classifications.
Some examples:
Figure 15: The blazar sequence concept of Fossati et al(1998). These Spectral
Energy Distributions for blazars show a multiwavelength pattern in which FSRQs
(toward the top) have higher luminosity and peaks at lower energy, while LBLs
and HBLs have lower luminosity but peaks at higher energy. The EGRET gamma-ray
spectra are to the right in this figure. Figure courtesy of G. Fossati.
* •
A number of prominent blazars were not seen by EGRET despite significant
exposures (von Montigny et al1995a).
* •
An effort was made to find a unified model for AGN based on geometry: the
orientation of the black-hole/accretion-disk/torus/jet system relative to the
viewing direction (Urry and Padovani, 1995). The beamed nature of the EGRET
gamma radiation, associated with apparent superluminal motion seen in the
radio, is an important aspect of this scheme, because it emphasizes that
blazars must have jets aligned close to our line of sight.
* •
A comparison of spectral energy distributions led Fosatti (1998) to suggest
the “blazar sequence” in which higher-luminosity blazars (usually FSRQs) have
their synchrotron and Compton peaks at lower energies while lower-luminosity
blazars (LBLs and HBLs) have peaks at increasingly higher energies as their
luminosity decreases (see Figure 15).
* •
Very Long Baseline Interferometry (VLBI) radio studies have been used to
construct time-resolved images of the jets associated with the EGRET blazars.
In addition to finding that many of these blazars have apparent superluminal
motions (a relativistic illusion produced by the fact that the jet is directed
close to our line of sight), a study of a sample of bright blazars suggests
that those blazars seen in gamma rays tend to have higher jet Lorentz factors
than those blazars not seen by EGRET (Figure 16, Kellerman et al2004).
* •
Another important result from VLBI studies is an indication that gamma-ray
flares seen with EGRET occur at approximately the same time that new
components of radio emission (“knots”) emerge from the core of the AGN
(Jorstad et al2001). This result suggests that the gamma radiation is produced
in the parsec-scale region of the jet rather than closer to the black hole at
the center of the AGN.
Summaries of the observational properties of the EGRET blazars were given by
von Montigny et al(1995b) and Mukherjee et al(1997). Mukherjee (2001) reviewed
the EGRET blazar results after the end of the CGRO mission. Efforts to search
the Third EGRET catalogue systematically for blazar-like counterparts were
carried out by Mattox et al(2001) and by Sowards-Emmerd et al(2004). A number
of plausible associations were added to the ones shown in the catalogue.
Figure 17 shows a map of such associations.
Although many studies of these blazars provided useful insights into jets and
other AGN features, others found puzzling results. The Sowards-Emmerd et
alresults concluded that some of the high-Galactic-latitude EGRET sources did
not have clear identifications with blazar-like sources. Nandikotkur et
al(2007) carried out a detailed study of gamma-ray blazar variability using
the entire EGRET data set and found a variety of relationships between flux
and spectral hardening, without a simple pattern. Hartman et al(2001b)
concluded that at least one episode of variability of 3C279 was not correlated
with optical or X-ray variation. No consensus has emerged about whether the
jets are primarily dominated by electrons or by protons.
Figure 16: Histogram of speeds of the fastest component of VLBI-imaged blazar
jets (Kellerman et al2004). The EGRET blazars have a higher average speed than
the others. Figure 17: New candidate identifications for EGRET sources
(Sowards-Emmerd et al2004). The sources are the same as the 3EG catalogue
(Figure 7). The filled circles show potential blazar associations. The sources
shown by open circles do not have blazar counterparts. This analysis did not
extend to declinations south of -40∘, shown by a solid contour. Crosses show
sources that were not classified.
### 7.2 Other Galaxies
As noted in the previous section, not all EGRET sources at high Galactic
latitudes are identified as blazars. Other potential extragalactic sources are
discussed below.
#### 7.2.1 Local Galaxies
The same type of cosmic ray interaction processes that operate in our Galaxy
are likely to operate in other normal galaxies, although most of these are too
far away to be detectable. EGRET was able to set only upper limits on gamma
radiation from the Andromeda Galaxy, M31, for example (Sreekumar et al1994).
One exception is the Large Magellanic Cloud (LMC), reported as an extended
gamma-ray source by Sreekumar et al(1992). Based on radio and optical
observations, the LMC is thought to have a cosmic ray population similar to
that of the Milky Way. The source seen by EGRET was consistent in flux and
spatial extent with resulting from cosmic ray interactions in the LMC. A map
of the EGRET emission and a comparison to the radio emission that traces the
gas content of the LMC is shown in Figure 18.
Figure 18: EGRET gamma radiation from the Large Magellanic Cloud region
(Sreekumar et al1992). The contours of gamma-ray intensity show an extent
consistent with that seen at radio frequencies.
By contrast, Ginzburg (1972) predicted that the Small Magellanic Cloud (SMC)
would not be detectable at the sensitivity of EGRET, because that galaxy is
thought to be unable to sustain a local cosmic ray population. The absence of
significant gamma radiation from the SMC was a critical test of whether cosmic
rays are confined to galaxies or whether they are more universal. The SMC
would only be a gamma-ray source if cosmic rays extended beyond stable
galaxies. The upper limit from EGRET (Sreekumar et al1993) provided the
evidence to prove Ginzburg’s hypothesis that cosmic rays are confined to
galaxies.
#### 7.2.2 Radio Galaxies and Others
Evidence for gamma-ray emission from other extragalactic sources is less
secure. The nearest large radio galaxy, Cen A, may have been seen by EGRET.
There is a catalogued EGRET source positionally consistent with Cen A, and the
energy spectrum appears to be a continuation of the spectrum seen at lower
energies (Sreekumar et al1999). In the absence of any variability correlated
with other wavelengths, however, this identification is not certain. A similar
situation exists for two other radio galaxies. NGC 6251 is located in an EGRET
source error box and is a plausible, but not definite, identification
(Mukherjee et al2002), and 3C111 has been suggested as a possible counterpart
to another EGRET source (Sguera et al2005).
Searches for other extragalactic populations include:
* •
Upper limits on some starburst galaxies were given by Sreekumar et al(1994).
* •
Searches for EGRET gamma rays from clusters of galaxies were reported by
Reimer et al(2003), who also provide a critical analysis of some suggestions
of cluster emission in the EGRET data.
* •
Stacking analyses gave upper limits in examining possible low-level
contributions from radio galaxies (Cillis et al2004) and luminous infrared
galaxies (Cillis et al2005).
### 7.3 Diffuse Extragalactic Radiation
Finding a signal of diffuse extragalactic gamma radiation, sometimes called
the extragalactic gamma-ray background, is probably the most difficult
analysis challenge for a gamma-ray telescope, because this emission is what
remains after all foreground constituents of the radiation are subtracted. For
the EGRET analysis, the foreground included three principal components, dealt
with in separate steps by Sreekumar et al(1998):
* •
Individual sources, as identified for the EGRET catalogue work, were removed
from the analysis;
* •
A residual contribution from the bright Earth limb was eliminated by removing
all photons whose arrival directions were within 4 times the Point Spread
Function from the limb (for a given energy), enlarged from the standard EGRET
cut of 2.5 times the PSF from the limb;
* •
The Galactic diffuse component was taken into account, based on the standard
model developed for EGRET analysis (see section 4).
The third of these steps was the most critical, because the Galactic diffuse
emission exceeds any extragalactic diffuse radiation everywhere on the sky,
including the Galactic poles. The basic approach was to compare the observed
radiation to that expected from the Galactic model and then to extrapolate to
a condition of no Galactic emission. Figure 19 shows such a fit for the
standard EGRET energy bands.
Figure 19: Observed gamma ray flux in many high-latitude locations, comparing
that observed (Y-axis) to the flux expected from the EGRET model of the
Galactic diffuse emission (Sreekumar et al1998). The correlation is strong,
and the offset from zero represents the diffuse emission.
This analysis produced a non-zero residual for all energies. The resulting
extragalactic diffuse spectrum was consistent with a power law with an index
of $-$2.10+/-0.03.
The unresolved gamma radiation after removal of foreground contributions is
certainly not all truly diffuse emission. As with the X-ray background, some
or all of this radiation results from individual sources that are too faint to
be recognized by EGRET. Primary candidates those from the known extragalactic
gamma-ray source classes:
* •
Normal galaxies like the Milky Way must contribute. Pavlidou and Fields (2002)
estimate that up to 30% of the emission at 1 GeV may result from such
galaxies, with a lesser contribution at other energies. With a sample of only
the Milky Way and the LMC, however, any extrapolation must be considered
highly uncertain.
* •
Blazars are strong candidates to be the origin of much of the unresolved
emission. Many authors have estimated this contribution, and the results range
from about 25% to 100% of the total (e.g. Stecker and Salamon 1996; Mukherjee
and Chiang 1999; Chiang and Mukherjee 1998; Mücke and Pohl 2000). The
principal challenges for this calculation are twofold: (1) Blazars seen by
EGRET were almost always in a flaring state, and the duty cycle for such
flaring remains highly uncertain; and (2) the evolution of blazar gamma-ray
emission must be estimated. The wide range of potential contributions to the
diffuse radiation reflects the sensitivity of the calculation to the input
assumptions.
Beyond the contributions from known but unresolved sources, there is no
shortage of other candidate mechanisms for producing a largely isotropic
gamma-ray background on the scale observed by EGRET. Dermer (2007) presents a
recent summary. Some possibilities that have been discussed in recent years
include: unresolved emission from galaxy clusters (Ensslin et al1997),
starburst galaxies (Thompson, Quataert and Waxman 2007), shock waves
associated with large scale cosmological structure formation (Loeb and Waxman
2000; Miniati 2002), distant gamma-ray burst events (Casanova, Dingus and
Zhang 2007), annihilation of weakly interacting massive particles (WIMPs) (e.
g. Ullio et al2002), and cosmic strings (Berezinsky, Hnatyk and Vilenkin
2001).
Strong et al(2000) and Moskalenko and Strong (2000) have argued for a larger
component of the Galactic diffuse emission from inverse Compton scattering of
the Galactic plane photons and the cosmic microwave background. Because the
calculation of the extragalactic emission depends on correct subtraction of
the Galactic contribution, this alternate approach would change the result on
any determination of the extragalactic diffuse emission. A new model of the
Galactic diffuse emission (Strong et al2004a) has provided a new estimate of
the extragalactic diffuse gamma radiation that is lower in flux and steeper
than found by Sreekumar et al(1998) The revised spectrum is not consistent
with a power-law and shows some positive curvature. Both results are shown in
Figure 20.
Figure 20: Spectrum of the diffuse extragalactic gamma radiation. Upper
(lighter) data points, connected by dashed line: Sreekumar et al(1998)
analysis. Lower data points: Strong et al(2004a) analysis.
### 7.4 Gamma ray bursts
Gamma-ray bursts (GRBs) have been described as the brightest explosions in the
Universe, radiating more energy in a few seconds than our sun will emit during
its entire 10 billion year lifetime. Much of this energy is seen as low-energy
gamma rays, with typical energies in a range around 100 keV. GRBs are thought
to be caused by unusually powerful supernovae (collapsars) or possibly an
effect of the merger of two neutron stars or a neutron star and a black hole
(for a recent review, see Meszaros 2006). During the Compton Observatory
mission, BATSE recorded more then 2700 GRBs, averaging nearly one per day. A
small fraction of these bursts were also seen by EGRET, revealing important
characteristics of GRBs.
Figure 21: The long-duration high-energy emission from the gamma-ray burst of
17 Feb. 1994 (Hurley et al1994). Upper panel: energies and arrival times of
individual EGRET photons from the direction of the burst. The horizontal lines
show periods with no data. Lower panel: Ulysses 25-150 keV counts rates during
the same time period. Inset: expanded plot of the early time of the burst,
showing the EGRET photons compared to the BATSE light curve.
EGRET photons above 30 MeV were seen coincident with BATSE low-energy gamma-
ray emission for at least 4 bursts (for a summary, see Dingus 2003). Because
these were the brightest BATSE bursts in the EGRET field of view and EGRET
upper limits from other bursts do not constrain the results, it is possible
that all GRBs have a high-energy component. The combined energy spectrum shows
no break up to 10 GeV, suggesting that the spectrum may extend to even higher
energies. Having a spectrum extend to such high energies assures that GRBs are
nonthermal phenomena.
The importance of these detections is that high-energy gamma rays should not
escape easily from the environment of a GRB due to the high flux of lower-
energy photons. The process of photon-photon pair production, $\gamma$ \+
$\gamma$ = e+ \+ e-, would effectively remove all the high-energy gamma rays,
unless all the photons are moving in the same direction, i.e. beamed, with
high bulk Lorentz factor (e.g. Baring 1994). Lithwick and Sari (2001)
calculate that Lorentz factors of several hundred are needed.
A second aspect of the EGRET GRB observations offers a different challenge to
models of these explosions. In addition to the prompt high-energy emission,
delayed emission from some GRBs was detected by EGRET. One example is shown in
Figure 21, probably the best-known EGRET GRB detection. In this burst, high-
energy emission, including the highest-energy photon associated with a GRB at
18 GeV, was seen nearly 90 minutes after the initial burst (Hurley et al1994).
An example of shorter-term delayed emission was found by Gonzalez et al(2003)
by comparing BATSE data with data from the EGRET TASC calorimeter (see section
2.3). Unlike the primary EGRET gamma-ray telescope, the TASC was an
omnidirectional detector for gamma rays up to energies of nearly 200 MeV. For
GRB941017, shown in Figure 22, the GRB spectrum was seen to evolve from one
dominated by the low-energy BATSE data to one dominated by the higher-energy
EGRET emission over the course of about 200 seconds. The fact that the
spectrum is still rising at the upper limit of the TASC readout suggests that
the delayed emission probably extended to yet higher energies. This burst was
not in the field of view of the EGRET spark chamber, precluding measurements
at higher energies. The relationship of these delayed gamma-ray components of
bursts may be related to afterglows seen at longer wavelengths, but such burst
afterglows were not known for most of the EGRET era, so no measurements were
made.
Figure 22: BATSE and EGRET spectra of GRB941017 (Gonzales et al2003). Crosses:
BATSE; Solid circles: EGRET TASC. The time intervals in panels a. to e. start
about the time of the main burst and extend to 200 seconds later.
## 8 Local gamma-ray sources
Although most objects energetic enough to produce gamma rays detectable by
EGRET are distant, there are two sources visible within the Solar System (in
addition to the Earth itself, since the Earth’s atmosphere is gamma-ray bright
due to cosmic ray interactions, e.g. Petry 2005). These two local sources are
the Moon and the Sun.
### 8.1 The Moon
A remarkable, yet well-understood, curiosity of gamma-ray astrophysics is that
in this part of the electromagnetic spectrum the Moon is brighter than the
quiet Sun (when no solar flares are present). Morris (1984) predicted the Moon
to be a gamma-ray source, using a calculation extrapolated from modeling of
the Earth’s atmospheric gamma radiation. The same sorts of interactions of
cosmic rays with matter that produce the diffuse Galactic gamma radiation
(meson production, Compton scattering, and bremsstrahlung; see section 4) take
place in the lunar surface. Thompson et al(1997) analyzed relevant portions of
the EGRET data in a moving, Moon-centered coordinate system to verify the
prediction, including evidence of the expected variation of the lunar gamma
radiation with the Solar cycle of cosmic ray modulation. Figure 23 shows the
calculated and measured spectra for the Moon.
The same processes that produce the lunar radiation also operate in the
atmosphere of the Sun. All other factors being equal, the fact that the Sun
and Moon subtend the same solid angle from Earth might suggest that they would
be equally bright in gamma rays. The solar magnetic field, however, excludes
some of the Galactic cosmic rays from hitting the solar atmosphere, making the
expected gamma-ray flux from the quiet Sun lower (Hudson, 1989; Seckel et
al1991). The original EGRET analysis found only an upper limit for the Sun,
consistent with these calculations, although a more recent calculation and
analysis of the EGRET data provides some evidence for a weak detection of the
non-flare solar gamma radiation (Orlando and Strong 2008).
Figure 23: Gamma-ray spectrum of the Moon (Thompson et al1996).
### 8.2 Solar flares
Although the quiet Sun is not a bright gamma-ray source, solar flares can
accelerate particles to high enough energies to produce strong gamma-ray
emission. The Compton Observatory was launched near solar maximum, and in
early June of 1991 a number of solar flares were detected by all the
instruments on CGRO, including EGRET (Schneid et al1996). On 1991 June 11
EGRET observed gamma-ray emission up to energies above 1 GeV for over 8 hours
(Kanbach et al1993) following a large solar flare. This flare represented the
most energetic radiation ever seen from the Sun. The energy spectrum confirmed
evidence seen from previous missions that solar flares could accelerate
protons (which collide with ambient solar material to produce neutral pions
that decay into high-energy gamma rays). That spectrum is shown in Figure 24.
Modeling of the particle acceleration and interactions provided the first
evidence for long-term trapping of particles in solar flare regions
(Mandzhavidze and Ramaty, 1992).
Figure 24: Gamma-ray spectrum of a large solar flare (Schneid et al1996.)
### 8.3 Microsecond bursts - the search for Hawking radiation from near-Earth
black holes
Hawking (1974) and Page and Hawking (1976) calculated that tiny black holes
that had formed in the early Universe might be detectable as flashes of high-
energy gamma rays. The process of black hole evaporation, called Hawking
radiation, is negligible for solar mass black holes or larger, but those that
formed with mass about 1014 g would have lost enough mass that their final
evaporation might be seen if they were close enough to the Earth. A search of
the EGRET data for multiple gamma rays occurring within the 1 microsecond live
time of the spark chamber was able to set a new limit on such black holes,
although not extremely restrictive (Fichtel et al1994 ).
## 9 Summary - Open questions for AGILE and Fermi
The Energetic Gamma Ray Experiment Telescope (EGRET) on the Compton Gamma Ray
Observatory provided a dramatic new view of the high-energy Universe,
including the first all-sky mapping of the sky at energies above 30 MeV. The
EGRET observations revealed a wealth of information about Galactic and
extragalactic gamma radiation from both individual and diffuse sources.
Perhaps the single most striking characteristic of the EGRET gamma-ray sky is
its variability, ranging from the extremely rapid flaring of gamma-ray bursts
to the long-term variations seen in some sources such as blazars. As with many
successful missions, however, the questions that were answered led to new
questions for future missions. With AGILE and Fermi now in operation, the time
has come to seek out the solutions to some of these mysteries left behind by
EGRET:
* •
What is the nature of the diffuse Galactic gamma radiation, and in particular
the GeV excess? Does this unexpected finding indicate some exotic new physics
or a more mundane origin such as calibration?
* •
Does the gamma radiation from the Milky Way or its surroundings contain clues
to unseen forms of matter, such as cold dark gas or dark matter? Mapping the
gamma-ray emission with higher precision and comparing those measurements to
information about gas derived from other types of observations is a highly
promising avenue for this research.
* •
What will a larger sample of gamma-ray pulsars reveal about the location of
the particle acceleration and the particle interactions processes under
extreme conditions?
* •
How many more radio-quiet pulsars will be found, and what will those pulsars
say about the neutron star population of our Galaxy?
* •
Which binary systems produce gamma rays, and how do those systems work?
* •
What other classes of Galactic objects have enough energy to produce gamma
rays detectable by the new generation of telescopes?
* •
Will including new, high-quality gamma-ray measurements of blazar spectra and
time variability in multiwavelength studies provide the clues to jet
properties such as composition and possibly to jet formation and collimation?
* •
What will the new gamma-ray measurements of other galaxies tell us about
cosmic rays and matter densities in these systems?
* •
What other types of extragalactic objects are capable of generating detectable
gamma ray fluxes?
* •
Will the new data resolve the diffuse extragalactic radiation as a collection
of discrete sources, or will there be some residual diffuse emission that
demands a new and possibly exotic explanation?
* •
Do most or all gamma-ray bursts have high-energy emission, and what does that
radiation say about the forces at work in these explosive phenomena?
* •
Can high-energy gamma-ray measurements of solar flares shed new light on solar
activity?
* •
What surprises will be found in the gamma-ray sky?
EGRET was a team effort. My thanks go to all those who contributed, but
particularly to the three Co-Principal Investigators: Carl Fichtel, Robert
Hofstadter, and Klaus Pinkau. I also greatly appreciate comments and
suggestions from Bob Hartman and Olaf Reimer. The pulsar section of this paper
made use of the ATNF pulsar catalogue (Manchester et al2005).
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arxiv-papers
| 2008-11-05T13:10:42 |
2024-09-04T02:48:58.613170
|
{
"license": "Public Domain",
"authors": "D J Thompson",
"submitter": "David J. Thompson",
"url": "https://arxiv.org/abs/0811.0738"
}
|
0811.0863
|
# A QCD motivated model for soft processes
A. Kormilitzin E. Levin
###### Abstract
In this talk we give a brief description of a QCD motivated model for both
hard and soft interactions at high energies. In this model the long distance
behaviour of the scattering amplitude is determined by the dipole scattering
amplitude in the saturation domain. All phenomenological parameters for
dipole-proton interaction were fitted from the deep inelastic scattering data
and the soft processes are described with only one new parameter, related to
the wave function of hadron. It turns out that we do not need to introduce the
so called soft Pomeron that has been used in high energy phenomenology for
four decades.
###### Keywords:
High density QCD, saturation, Pomeron structure, diffraction
###### :
13.8.5.-t, 13.85.Hd,11.55.-m,11.55.Bq
## 1 Main ideas and theoretical input
This talk is the brief description of our attempt to find a self-consistent
theoretical approach to soft (long distance) interaction at high energy (see
our paper of Ref.KOLE for full presentation). Our main idea is that there is
no other dimemsionful scales except the saturation scale for high energy
interaction in the entire kinematic region. This idea looks strange since the
traditional approach to soft high energy interaction is based on the
phenomenological soft Pomeron, whose typical dimensionful scale is given by
the slope of the Pomeron trajectory ($\alpha_{P}\,\approx 0.25\,GeV^{-2}$). In
simple words, we would like to replace the soft Pomeron exchange in the
scattering amplitude by the QCD scattering amplitude in the saturation region.
$\alpha_{P}=0$ for this amplitude and the saturation momentum ( $Q_{s}$) is
the only dimensionful scale for it. Since this amplitude governs both short
distances and long distances processes, all needed parameters in a such
amplitude can be found from the DIS on the contrary to soft phenomenology in
which all parameters are extracted from the soft scattering. The idea is not
new and it is scattered in a number of reviews and talks (see references in
Ref.KOLE ). However the first practical realization was suggested in Ref.BATAV
with an encouraging result: it is possible to describe the data on total cross
section in the framework of such ideas. In this talk as well as in our paper
KOLE we wish to check these ideas against the full set of experimental data
in the region of accessible energies and even give some predictions for the
LHC energy range. Our approach is based on two theoretical inputs.
Factorization of short and long distances: Using the dispersion relation we
can prove (see Ref.KOLE that at least at large impact parameters the
scattering amplitude for the BFKL Pomeron can be written in the form
$A_{BFKL}(Q^{2},x;b)\,=\,\stackrel{{\scriptstyle\mbox{ \small short
distances}}}{{A_{BFKL}(Q^{2},x;t=0)}}\,\times\,{\stackrel{{\scriptstyle\mbox{\small
long distances}}}{{S(b)}}}$ (1)
The scattering amplitude of dipole (x,y) near to the saturation boundary has
the formLMP :
$N(Y-Y_{0};x,y)\,=\,1-e^{-\int\,d^{2}x^{\prime}\,d^{2}y^{\prime}\,\,P(Y-Y_{0};x,y;x^{\prime},y^{\prime})\gamma(Y_{0};x^{\prime},y^{\prime})}$
(2)
where $\gamma(Y_{0};x^{\prime},y^{\prime})$ is the scattering amplitude of
dipole $(x^{\prime},y^{\prime})$ with the target at low energy and
$P(Y-Y_{0};x,y;x^{\prime},y^{\prime})$ is the Green’s function of the BFKL
Pomeron.
## 2 The model
Using these two theory inputs we built our model assuming
* •
$\gamma_{mod}(Y_{0};x^{\prime},y^{\prime})\,=\,\,\,\frac{\pi^{2}\,\alpha_{S}(\mu^{2}_{0})}{3}\,r^{2}\,x_{0}G^{DGLAP}(x_{0},\mu^{2}_{0})\,S(b)$;
* •
$S(b)\,=\,\frac{2}{\pi\,R^{2}}(\frac{\sqrt{8}\,b}{R})\,K_{1}(\frac{\sqrt{8}\,b}{R})$\-
the electromagnetic proton form factor;
* •
Instead of the BFKL Pomeron in Eq.2 we use the solution to the DGLAP equation;
* •
The scale of hardness in the DGLAP evolution ia chosen in the form:
$\mu^{2}\,=\,\mu^{2}_{0}\,\,+\,\,\frac{C}{r^{2}}$.
Finally,
$N(Y-Y_{0};x,y)\,=\,1-\exp\left(-\stackrel{{\scriptstyle\mbox{ \small short
distances}}}{{\frac{\pi^{2}\,\alpha_{S}(\mu^{2})}{3}\,r^{2}\,xG^{DGLAP}(x,\mu)}}\,\times\,{\stackrel{{\scriptstyle\mbox{
\small long distances}}}{{S(b)}}}\right)$ (3)
All parameters in Eq.3 we found from fitting the DIS data with
$\chi^{2}$/d.o.f. =1.02. The formulae for calculation of the physical
observables are simple and can be found in Ref. KOLE and the structure of
them is seen from the expression for the total cross section for meson proton
scattering: $\sigma_{tot}\,=\,\int d^{2}bd^{2}r|\Psi(r)|^{2}\,N(Y-Y_{0},r)$
where $\Psi$ is the wave function of a meson which consists of one colourless
dipole with size $r$, and $N$ is dipole scattering amplitude. Therefore, we
need to specify the wave hadronic wave function in terms of t dipoles. As have
been mentioned, we assume that mesons consist of one dipole, for baryon we
have two possibilities for the dipole content: two dipoles or three dipoles.
We used both in our comparison with the experiment. For one dipole in the
hadron we used the simplest Gaussian parameterization, namely
$\Psi(r)\,=\frac{1}{\pi S}\,\exp(-\frac{r^{2}_{i}}{S})$ where $S$ was chosen
from comparison with the experiment. This is the only parameter that we found
from the data on soft scattering. The second input that we need to calculate
the scattering amplitude is to define the energy variable $x$ (or $Y-Y_{0}$).
We did this in the following way, using our the only dimensionful scale
(saturation momentum $Q_{s}$):
$x_{soft}\,=\,\frac{Q^{2}+Q^{2}_{s}(x_{soft})}{s}\,\,\stackrel{{\scriptstyle
Q^{2}\to
0}}{{\rightarrow}}\,\,\frac{Q^{2}_{s}(x_{soft})}{s};\,\,\,\,x_{soft}\,=\,\frac{Q^{2}+Q^{2}_{s}(x_{soft})}{s}\,\,\stackrel{{\scriptstyle
Q^{2}\gg Q^{2}_{s}}}{{\rightarrow}}\,\,x_{Bjorken};$ (4)
## 3 Comparison with the data
Using the model we obtain the description of the experimental data which is
not too bad. We consider that it is much better than we could hope with such a
simple model.
| |
---|---|---
$\sigma_{tot}(\pi-p)$ | $\sigma_{tot}(K-p)$ | $\sigma_{tot}(p-\bar{p})$
Figure 1: |
---|---
$\sigma_{el}(p-\bar{p})$ | $d\sigma_{el}(p\bar{p})/dt$
Figure 2: Model | $\sigma_{tot}$ | $\sigma_{el}$ | $\sigma_{diff}$ | $B_{el}$ | $<|S^{2}|>$
---|---|---|---|---|---
| mb | mb | mb | $GeV^{-2}$ |
Our A (DIS) | 83.0 | 23.54 | 10 | 16.67 | (0.24-0.89)%
Our B (Soft) | 101.3 | 28.84 | 10.5 | 18.4 | (0.24-0.57)%
GLMGLM1 | 110.5 | 25.3 | 11.6 | 20.5 | (0.7 -2)%
GLMMGLM2 | 91.7 | 20.9 | 11.8 | 17.3 | 0.21%
RMKRMK | 88.0 (86.3) | 20.1 (18.1) | 13.3 (16.1) | 19 | (1.2 - 3.2)%
Table 1: Predictions for the LHC
One can see that our model reproduces quite well the energy and $t$ dependence
of total and elastic cross sections. The most essential test for the model,
however, is the dependence of the elastic slope versus energy. Indeed, in
traditional high energy phenomenology this dependence was related to the value
of $\alpha^{\prime}_{P}\approx 0.25\,GeV^{-2}$ which in our model is equal to
zero. Nevertheless, in spite of the fact that input
$\alpha^{\prime}_{P}\,=\,0$ the shadowing corrections ,taken into account in
our master formula of Eq.3, generate effective $\alpha^{\prime}_{p}\approx
0.2\,GeV^{-2}$. For the single diffraction cross section we take into account
the emission of an additional gluon. The low curve in Fig. 3 corresponds to
diffraction production of one dipole while the resulting curve includes the
extra gluon emission. In all figures we plotted our calculations for two sets
of parameterizations: set A (DIS) which gives very good description of DIS
data with $\chi^{2}$/d.o.f. = 1.02; and set B(soft) which leads to worse
description of DIS data ($\chi^{2}/d.o.f$\- 3.6) but predicts the soft
observables closer to the experimental data. Notations 2D and 3D specify the
model for nucleon with two or three dipoles.
|
---|---
Figure 3:
## 4 Conclusions
In the table we show our prediction for the LHC range of energy with
comparison with other models. The most pronounced difference between our
estimates and RMK modelRMK in the value of the survival probability for the
central Higgs production. It is worth mentioning, that in our approach the
small value of $<|S^{2}|>$ stems from Good-Walker mechanism but not from the
enhanced diagrams as in GLMM model GLM2 .
We will be happy if you, thinking about Pomeron, would remember that (i) there
is a possibility to describe the data without introducing soft Pomeron; (ii)
the idea about the saturation momentum as the only dimensional parameter of
the strong interaction works; and (iii) all parameters for strong interaction
can be found from DIS.
We are grateful to Errol Gotsman, Uri Maor and Alex Palatnik for useful
discussions on the subject. This research was supported in part by the Israel
Science Foundation, founded by the Israeli Academy of Science and Humanities,
by BSF grant $\\#$ 20004019 and by a grant from Israel Ministry of Science,
Culture and Sport and the Foundation for Basic Research of the Russian
Federation.
## References
* (1) A. Kormilitzin and E. Levin, “Soft processes at high energy without soft Pomeron: a QCD motivated model,” arXiv:0809.3886 [hep-ph].
* (2) J. Bartels, E. Gotsman, E. Levin, M. Lublinsky and U. Maor, Phys. Rev. D 68 (2003) 054008 [arXiv:hep-ph/0304166]; Phys. Lett. B 556 (2003) 114 [arXiv:hep-ph/0212284].
* (3) E. Levin, J. Miller and A. Prygarin, Nucl.Phys. A (in press); arXiv:0706.2944 [hep-ph].
* (4) E. Gotsman, E. Levin and U. Maor, ”A Soft Interaction Model at Ultra High Energies: Amplitudes, Cross Sections and Survival Probabilities”, arXiv:0708.1506 [hep-ph]
* (5) E. Gotsman, E. Levin, U. Maor and J. Miller, Eur. Phys. J. C,( in press); arXiv:0805.2799 [hep-ph].
* (6) M. G. Ryskin, A. D. Martin and V. A. Khoze, Eur. Phys. J. C54 (2008) 199 [arXiv:0710.2494 [hep-ph]].
|
arxiv-papers
| 2008-11-06T14:07:20 |
2024-09-04T02:48:58.627479
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Kormilitzin and E. Levin (Tel Aviv Un.)",
"submitter": "Eugene Levin",
"url": "https://arxiv.org/abs/0811.0863"
}
|
0811.0906
|
# Spectroscopic implications from the combined analysis of processes with
pseudoscalar mesons111Supported by the Votruba-Blokhintsev Program for
Cooperation of the Czech Republic with JINR (Dubna), the Grant Agency of the
Czech Republic (Grant No.202/08/0984), the Slovak Scientific Grant Agency
(Grant VEGA No.2/0034/09), and the Bogoliubov-Infeld Program for Cooperation
of Poland with JINR (Dubna). 222Talk given at XIII International Conference
Selected Problems of Modern Theoretical Physics, Bogoliubov Laboratory of
Theoretical Physics, JINR, Dubna, Russia, June 23-27, 2008.
Yu.S. Surovtsev333E-mail address: surovcev@thsun1.jinr.ru Bogoliubov
Laboratory of Theoretical Physics, JINR, Dubna, Russia P. Bydžovský444E-mail
address: bydz@ujf.cas.cz Nuclear Physics Institute, ASCR, Řež near Prague,
Czech Republic M. Nagy555E-mail address: fyzinami@unix.savba.sk Institute of
Physics, SAS, Bratislava, Slovakia
(19. 11. 2008)
###### Abstract
In the analysis a status and parameters of the scalar, vector, and tensor
mesonic resonances are obtained and compared with other results. Possible
classification of the resonance states in terms of the SU(3) multiplets is
discussed.
###### pacs:
11.55.Bq, 13.75.Lb, 14.40.Cs
Outline:
* •
Motivation
* •
Method of analysis
* •
Analysis of the isoscalar-scalar sector
* •
Analysis of the isovector $P$-wave of $\pi\pi$ scattering
* •
Analysis of the isoscalar-tensor sector
* •
Spectroscopic implications from the analysis
## I Motivation
The spectroscopy of light mesons plays an important role in understanding the
strong interactions at low energies. Among possibilities to study the spectrum
of light mesons, analysis of the $\pi\pi$ interaction is particularly useful
and, therefore, it has always been an object of continuous theoretical and
experimental investigation PDG08 . Here, we present results of the coupled-
channel analysis of data on processes
$\pi\pi\to\pi\pi,K\overline{K},\eta\eta,\eta\eta^{\prime}$ in the channels
with $I^{G}J^{PC}=0^{+}0^{++}$ and $0^{+}2^{++}$ and on the $\pi\pi$
scattering in the channel with $1^{+}1^{--}$.
The scalar sector is problematic up to now especially as to an assignment of
the discovered mesonic states to quark-model configurations in spite of a big
amount of work devoted to these problems (see, e.g., Ref. Ani06 and
references therein). An exceptional interest to this sector is supported by
the fact that there, possibly indeed, we deal with a glueball $f_{0}(1500)$
(see, e.g., Ref. PDG08 ; Ams96 ).
Investigation of vector mesons is up-to-date subject due to their role in
forming the electromagnetic structure of particles and because our knowledge
about these mesons is still too incomplete (e.g., in the Particle Data Group
tables PDG08 (PDG) the mass of $\rho(1450)$ is ranging from 1250 to 1582
MeV).
In the tensor sector, among the thirteen discussed resonances, the nine states
($f_{2}(1430)$, $f_{2}(1565)$, $f_{2}(1640)$, $f_{2}(1810)$, $f_{2}(1910)$,
$f_{2}(2000)$, $f_{2}(2020)$, $f_{2}(2150)$, $f_{2}(2220)$) must be confirmed
in various experiments and analyses. For example, in the analysis of
$p\overline{p}\to\pi\pi,\eta\eta,\eta\eta^{\prime}$, five resonances –
$f_{2}(1920)$, $f_{2}(2000)$, $f_{2}(2020)$, $f_{2}(2240)$ and $f_{2}(2300)$ –
have been obtained, one of which, $f_{2}(2000)$, is a candidate for the
glueball Ani05 .
In our analysis, we have used both a model-independent method KMS96 , based on
the first principles (analyticity and unitarity) directly applied to analysis
of experimental data, and the multichannel Breit–Wigner forms. The former
approach permits us to introduce no theoretical prejudice to extracted
parameters of resonances, however, it is limited with the possibility to use
only three coupled channels. Therefore, in more general cases, one has to use,
e.g., the Breit–Wigner approach. Considering the obtained disposition of
resonance poles on the Riemann surface, obtained coupling constants with
channels, and resonance masses we draw particular conclusions about nature of
the investigated states.
## II Method of analysis
In both methods of analysis, we parametrized the $S$-matrix elements
$S_{\alpha\beta}$ where $\alpha,\beta=1,2,\cdots,n$ denote channels, using the
Le Couteur-Newton relations LeCou . This relations express the $S$-matrix
elements of all coupled processes in terms of the Jost matrix determinant
$d(k_{1},\cdots,k_{n})$ that is a real analytic function with the only square-
root branch-points at the channel momenta $k_{\alpha}=0$.
In the model-independent approach, the $S$-matrix is determined on the 4- and
8-sheeted Riemann surfaces for the 2- and 3-channel cases, respectively. The
matrix elements $S_{\alpha\beta}$ have the right-hand cuts along the real axis
of the $s$ complex plane ($s$ is the invariant total energy squared), starting
at the coupled-channels thresholds $s_{i}$ ($i=1,2,3$), and the left-hand cuts
related to the crossed channels. The Riemann-surface sheets are numbered
according to the signs of analytic continuations of the channel momenta
$k_{i}=\sqrt{s-s_{i}}/{2}~{}~{}~{}~{}(i=1,2,3)$, as shown in Table 1.
Table 1: Signs of channel momenta on the eight sheets of the Rieman surface in the 3-channel case. sheet: | I | II | III | IV | V | VI | VII | VIII
---|---|---|---|---|---|---|---|---
Im $k_{1}$ | $+$ | $-$ | $-$ | $+$ | $+$ | $-$ | $-$ | $+$
Im $k_{2}$ | $+$ | $+$ | $-$ | $-$ | $-$ | $-$ | $+$ | $+$
Im $k_{3}$ | $+$ | $+$ | $+$ | $+$ | $-$ | $-$ | $-$ | $-$
The model-independent method which essentially utilizes an uniformizing
variable can be used only for the 2-channel case and under some conditions for
the 3-channel one. Only in these cases we obtain a simple symmetric (easily
interpreted) picture of the resonance poles and zeros of the $S$-matrix on an
uniformization plane. The important branch points, corresponding to the
thresholds of the coupled channels and to the crossing ones, are taken into
account in the uniformizing variable.
The resonance representations on the Riemann surfaces are obtained with the
help of formulas from Ref. KMS96 , expressing analytic continuations of the
$S$-matrix elements to unphysical sheets in terms of those on sheet I that
have only the zeros of resonances (beyond the real axis), at least, around the
physical region. Then, starting from the resonance zeros on sheet I, one can
obtain an arrangement of poles and zeros of resonance on the whole Riemann
surface.
In the 2-channel case, we obtain three types of resonances described by a pair
of conjugate zeros on sheet I: (a) in $S_{11}$, (b) in $S_{22}$, (c) in each
of $S_{11}$ and $S_{22}$.
In the 3-channel case, we obtain seven types of resonances corresponding to
seven possible situations when there are resonance zeros on sheet I only in
$S_{11}$ – (a); $S_{22}$ – (b); $S_{33}$ – (c); $S_{11}$ and $S_{22}$ – (d);
$S_{22}$ and $S_{33}$ – (e); $S_{11}$ and $S_{33}$ – (f); and $S_{11}$,
$S_{22}$, and $S_{33}$ – (g).
A resonance of every type is represented by a pair of complex-conjugate
clusters (of poles and zeros on the Riemann surface). Note that whereas the
cases (a), (b) and (c) can be simply related to the representation of
resonances by the Breit-Wigner forms, the cases (d), (e), (f) and (g) are
practically lost at that description. The cluster type is related to the
nature of state. For example, if we consider the $\pi\pi$, $K\overline{K}$,
and $\eta\eta$ channels, then a resonance which is coupled relatively more
strongly to the $\pi\pi$ channel than to the $K\overline{K}$ and $\eta\eta$
ones is described by the cluster of type (a). If the resonance is coupled more
strongly to the $K\overline{K}$ and $\eta\eta$ channels than to the $\pi\pi$
one, then it is represented by the cluster of type (e) (say, the state with
the dominant $s{\bar{s}}$ component). The flavour singlet (e.g., glueball)
must be represented by the cluster of type (g) (of type (c) in the 2-channel
consideration) as a necessary condition for the ideal case, if this state lies
above the thresholds of considered channels.
We can distinguish, in a model-independent way, a bound state of colourless
particles (e.g., $K\overline{K}$ molecule) and a $q{\bar{q}}$ bound state.
Just as in the 1-channel case, the existence of the particle bound-state means
the presence of the pole on the real axis under the threshold on the physical
sheet, so in the 2-channel case, the existence of the particle bound-state in
channel 2 ($K\overline{K}$ molecule) that, however, can decay into channel 1
($\pi\pi$ decay), would imply the presence of a pair of complex conjugate
poles on sheet II under the second-channel threshold without the corresponding
shifted pair of poles on sheet III.
In the 3-channel case, the bound-state in channel 3 ($\eta\eta$) that,
however, can decay into channels 1 ($\pi\pi$ decay) and 2 ($K\overline{K}$
decay), is represented by the pair of complex conjugate poles on sheet II and
by shifted poles on sheet III under the $\eta\eta$ threshold without the
corresponding poles on sheets VI and VII. This test KMS96 ; MPe93 is a
multichannel analogue of the known Castillejo–Dalitz–Dyson poles in the one-
channel case. According to this test, earlier in Ref. KMS96 , the
interpretation of the $f_{0}(980)$ state as the $K\overline{K}$ molecule has
been rejected because this state is represented by the cluster of type (a) in
the 2-channel analysis of processes $\pi\pi\to\pi\pi,K\overline{K}$ and,
therefore, it does not satisfy the necessary condition to be the
$K\overline{K}$ molecule.
## III Analysis of the isoscalar-scalar sector
Considering the $S$-waves of processes
$\pi\pi\to\pi\pi,K\overline{K},\eta\eta,\eta\eta^{\prime}$ in the model-
independent method, we performed two variants of the 3-channel analysis:
variant I: the combined analysis of
$\pi\pi\to\pi\pi,K\overline{K},\eta\eta\,$;
variant II: analysis of $\pi\pi\to\pi\pi,K\overline{K},\eta\eta^{\prime}$.
Influence of the $\eta\eta^{\prime}$-channel in variant I and the
$\eta\eta$-channel in variant II are taken into account via the background.
Here, the left-hand cuts are neglected in the Riemann-surface structure
assuming that contributions on these cuts are also included in the background.
Under neglecting the $\pi\pi$-threshold branch point (however, unitarity on
the $\pi\pi$-cut is taken into account), the uniformizing variable is
$w=\frac{k_{2}+k_{3}}{\sqrt{m_{\eta}^{2}-m_{K}^{2}}}~{}~{}~{}~{}{\rm
for~{}variant~{}I},$ (1)
and
$w^{\prime}=\frac{k_{2}^{\prime}+k_{3}^{\prime}}{\sqrt{\frac{1}{4}(m_{\eta}+m_{\eta^{\prime}})^{2}-m_{K}^{2}}}~{}~{}~{}~{}{\rm
for~{}variant~{}II}.$ (2)
The quantities related to variant II are primed.
On the $w$-plane, the Le Couteur-Newton relations are 666 Other authors have
also used the parameterizations with the Jost functions in analyzing the
$S$-wave $\pi\pi$ scattering in the one-channel approach Boh80 and in the
two-channel one MPe93 .
$\displaystyle
S_{11}=\frac{d^{*}(-w^{*})}{d(w)},~{}~{}~{}~{}~{}~{}~{}~{}S_{22}=\frac{d(-w^{-1})}{d(w)},~{}~{}~{}~{}~{}~{}~{}~{}S_{33}=\frac{d(w^{-1})}{d(w)},$
(3) $\displaystyle
S_{11}S_{22}-S_{12}^{2}=\frac{d^{*}({w^{*}}^{-1})}{d(w)},~{}~{}~{}~{}~{}~{}~{}~{}S_{11}S_{33}-S_{13}^{2}=\frac{d^{*}(-{w^{*}}^{-1})}{d(w)}\,,$
(4)
where the $d$-function is assumed in the form
$d=d_{B}d_{res},$ (5)
and the resonance part is
$d_{res}(w)=w^{-\frac{M}{2}}\prod_{r=1}^{M}(w+w_{r}^{*})$ (6)
with $M$ the number of resonance zeros. The background part is taken as
$d_{B}=\mbox{exp}[-i\sum_{n=1}^{3}\frac{k_{n}}{m_{n}}(\alpha_{n}+i\beta_{n})],$
(7)
where
$\displaystyle\alpha_{n}=a_{n1}+a_{n\sigma}\frac{s-s_{\sigma}}{s_{\sigma}}\theta(s-s_{\sigma})+a_{nv}\frac{s-s_{v}}{s_{v}}\theta(s-s_{v}),$
(8)
$\displaystyle\beta_{n}=b_{n1}+b_{n\sigma}\frac{s-s_{\sigma}}{s_{\sigma}}\theta(s-s_{\sigma})+b_{nv}\frac{s-s_{v}}{s_{v}}\theta(s-s_{v})$
(9)
with $s_{\sigma}$ the $\sigma\sigma$ threshold and $s_{v}$ a combined
threshold of many opened channels in the vicinity of 1.5 GeV (e.g.,
$\eta\eta^{\prime},~{}\rho\rho,~{}\omega\omega$).
In variant II, the terms
$a_{n\eta}^{\prime}\frac{s-4m_{\eta}^{2}}{4m_{\eta}^{2}}\theta(s-4m_{\eta}^{2})~{}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}~{}b_{n\eta}^{\prime}\frac{s-4m_{\eta}^{2}}{4m_{\eta}^{2}}\theta(s-4m_{\eta}^{2})$
(10)
should be added to $\alpha^{\prime}_{n}$ and $\beta^{\prime}_{n}$ to account
for an influence of the $\eta\eta$-channel.
As the data, we use the results of phase analyses given for phase shifts of
the amplitudes $\delta_{ab}$ and for moduli of the $S$-matrix elements
$\eta_{ab}=|S_{ab}|$ ($a,b=$1-$\pi\pi$, 2-$K\overline{K}$, 3-$\eta\eta$ or
$\eta\eta^{\prime}$):
$S_{aa}=\eta_{aa}e^{2i\delta_{aa}},~{}~{}~{}~{}~{}S_{ab}=\eta_{ab}e^{i\phi_{ab}}.$
(11)
If below the $\eta\eta$-threshold there is the 2-channel unitarity, then the
relations
$\eta_{11}=\eta_{22},~{}~{}\eta_{12}=(1-{\eta_{11}}^{2})^{1/2},~{}~{}\phi_{12}=\delta_{11}+\delta_{22}$
(12)
are fulfilled in this energy region.
The $\pi\pi$ scattering data, which range from the threshold up to 1.89 GeV,
are taken from Ref. Hya73 ; expd1 777Note that there are alternative data,
e.g., one of the solutions of the phase analysis in Ref. Grayer and the
recent phase analysis in Ref. Kaminski_data which are in accordance with each
other, but which differ from those used here, especially in the $f_{0}(980)$
region of energy. Analysis with these data should be performed separately.
This work is in progress.. For $\pi\pi\to K\overline{K}$, practically all the
accessible data are used expd2 . For $\pi\pi\to\eta\eta$, we used data for
$|S_{13}|^{2}$ from the threshold to 1.72 GeV expd3 . For
$\pi\pi\to\eta\eta^{\prime}$, the data for $|S_{13}|^{2}$ from the threshold
to 1.813 GeV are taken from Ref. expd4 . We included all the five resonances
discussed below 1.9 GeV.
In variant I, we got satisfactory description: for the $\pi\pi$ scattering,
$\chi^{2}/\mbox{NDF}\approx 1.35$; for $\pi\pi\to K\overline{K}$,
$\chi^{2}/\mbox{NDF}\approx 1.77$; for $\pi\pi\to\eta\eta$,
$\chi^{2}/\mbox{N.exp.points}\approx 0.86$. The total $\chi^{2}/\mbox{NDF}$ is
$345.603/(301-40)\approx 1.32$. From possible resonance representations by
pole-clusters, the analysis selects the following one: the $f_{0}(600)$ is
described by the cluster of type (a); $f_{0}(1370)$, type (c); $f_{0}(1500)$,
type (g); $f_{0}(1710)$, type (b); and the $f_{0}(980)$ is represented only by
the pole on sheet II and shifted pole on sheet III in both variants. The
background parameters are: $a_{11}=0.2006$, $a_{1\sigma}=0.0146$, $a_{1v}=0$,
$b_{11}=0$, $b_{1\sigma}=-0.01025$, $b_{1v}=0.0542$, $a_{21}=-0.6986$,
$a_{2\sigma}=-1.4207$, $a_{2v}=-5.958$, $b_{21}=0.047$, $b_{2\sigma}=0$,
$b_{2v}=6.888$, $b_{31}=0.6511$, $b_{3\sigma}=0.3404$, $b_{3v}=0$;
$s_{\sigma}=1.638~{}{\rm GeV}^{2}$, $s_{v}=2.084~{}{\rm GeV}^{2}$.
In variant II, we got the following description: for the $\pi\pi$ scattering
$\chi^{2}/\mbox{NDF}\approx 1.0$! for $\pi\pi\to K\overline{K}$
$\chi^{2}/\mbox{NDF}\approx 1.62$; for $\pi\pi\to\eta\eta^{\prime}$
$\chi^{2}/\mbox{N.exp.points}\approx 0.36$. The total $\chi^{2}/\mbox{NDF}$ is
$282.682/(293-38)\approx 1.11$! In this case, the $f_{0}(600)$ is described by
the cluster of type (a′); $f_{0}(1370)$, type (b′); $f_{0}(1500)$, type (d′);
and $f_{0}(1710)$, type (c′). The background parameters are:
$a_{11}^{\prime}=0.0111$, $a_{1\eta}^{\prime}=-0.058$,
$a_{1\sigma}^{\prime}=0$, $a_{1v}^{\prime}=0.0954$,
$b_{11}^{\prime}=b_{1\eta}^{\prime}=b_{1\sigma}^{\prime}=0$,
$b_{1v}^{\prime}=0.047$, $a_{21}^{\prime}=-3.439$,
$a_{2\eta}^{\prime}=-0.4851$, $a_{2\sigma}^{\prime}=1.7622$,
$a_{2v}^{\prime}=-5.158$, $b_{21}^{\prime}=0$, $b_{2\eta}^{\prime}=-0.7524$,
$b_{2\sigma}^{\prime}=2.6658$, $b_{2v}^{\prime}=1.836$,
$b_{31}^{\prime}=0.5545$, $s_{\sigma}=1.638~{}{\rm GeV}^{2}$,
$s_{v}=2.126~{}{\rm GeV}^{2}$.
Figure 1: The phase shift and module of the $S$-matrix element in the $S$-wave
$\pi\pi$-scattering. The solid curve corresponds to variant I and the dashed
curve to variant II.
Figure 2: The phase shift and module of the $S$-matrix element in $S$-wave of
$\pi\pi\to K\overline{K}$. The solid curve corresponds to variant I and the
dashed curve to variant II.
Figure 3: The squared modules of the $\pi\pi\to\eta\eta$ (upper figure) and
$\pi\pi\to\eta\eta^{\prime}$ (lower figure) $S$-wave matrix elements.
In Figures 1-3, we show results of fitting to the experimental data and in
Table 2 we indicate the obtained pole clusters for resonances on the eight
sheets of the complex energy plane $\sqrt{s}$, on which the 3-channel
$S$-matrix is determined ($\\!\sqrt{s_{r}}={\rm E}_{r}-i\Gamma_{r}\\!$).
Table 2: Pole clusters for the $f_{0}$-resonances in variants I and II. Sheet | II | III | IV | V | VI | VII | VIII
---|---|---|---|---|---|---|---
variant I
$\\!\\!f_{0}(600)\\!\\!$ | $\\!\\!{\rm E}_{r}\\!\\!$ | 598.2$\pm$13 | 585.8$\\!\pm\\!$14 | | | 505.8$\\!\pm\\!$16 | 518.2$\\!\pm\\!$15 |
| $\\!\Gamma_{r}\\!$ | 583$\\!\pm\\!$18 | 583$\\!\pm\\!$18 | | | 583$\\!\pm\\!$18 | 583$\\!\pm\\!$18 |
$\\!f_{0}(980)\\!$ | $\\!{\rm E}_{r}\\!$ | 1013.1$\\!\pm\\!$4 | 983.6$\\!\pm\\!$9 | | | | |
| $\\!\Gamma_{r}\\!$ | 34.1$\\!\pm\\!$6 | 57.4$\\!\pm\\!$10 | | | | |
$\\!f_{0}(1370)\\!$ | $\\!{\rm E}_{r}\\!$ | | | | 1398.2$\\!\pm\\!$16 | 1398.2$\\!\pm\\!$18 | 1398.2$\\!\pm\\!$18 | 1398.2$\\!\pm\\!$13
| $\\!\Gamma_{r}\\!$ | | | | 287.4$\\!\pm\\!$17 | 270.6$\\!\pm\\!$15 | 155$\\!\pm\\!$9 | 171.8$\\!\pm\\!$7
$\\!f_{0}(1500)\\!$ | $\\!{\rm E}_{r}\\!$ | 1502.6$\\!\pm\\!$11 | 1479.5$\\!\pm\\!$13 | 1502.6$\\!\pm\\!$12 | 1496.7$\\!\pm\\!$12 | 1498$\\!\pm\\!$16 | 1496.8$\\!\pm\\!$12 | 1502.6$\\!\pm\\!$10
| $\\!\Gamma_{r}\\!$ | 357.1$\\!\pm\\!$15 | 139.4$\\!\pm\\!$12 | 238.7$\\!\pm\\!$13 | 139.9$\\!\pm\\!$14 | 191.2$\\!\pm\\!$17 | 87.36$\\!\pm\\!$11 | 356.5$\\!\pm\\!$14
$\\!f_{0}(1710)\\!$ | $\\!{\rm E}_{r}\\!$ | | 1708.2$\\!\pm\\!$12 | 1708.2$\\!\pm\\!$10 | 1708.2$\\!\pm\\!$13 | 1708.2$\\!\pm\\!$15 | |
| $\\!\Gamma_{r}\\!$ | | 142.3$\\!\pm\\!$9 | 160.3$\\!\pm\\!$8 | 323.3$\\!\pm\\!$14 | 305.3$\\!\pm\\!$13 | |
variant II
$\\!f_{0}(600)\\!$ | $\\!{\rm E}_{r}\\!$ | 616.5$\\!\pm\\!$8 | 621.8$\\!\pm\\!$10 | | | 598.3$\\!\pm\\!$11 | 593$\\!\pm\\!$12 |
| $\\!\Gamma_{r}\\!$ | 563$\\!\pm\\!$11 | 563$\\!\pm\\!$12 | | | 563$\\!\pm\\!$14 | 563$\\!\pm\\!$13 |
$\\!f_{0}(980)\\!$ | $\\!{\rm E}_{r}\\!$ | 1009.3$\\!\pm\\!$3 | 986$\\!\pm\\!$6 | | | | |
| $\\!\Gamma_{r}\\!$ | 32$\\!\pm\\!$4 | 58$\\!\pm\\!$5.5 | | | | |
$\\!f_{0}(1370)\\!$ | $\\!{\rm E}_{r}\\!$ | | 1394.3$\\!\pm\\!$9 | 1394.3$\\!\pm\\!$11 | 1412.7$\\!\pm\\!$13 | 1412.7$\\!\pm\\!$14 | |
| $\\!\Gamma_{r}\\!$ | | 236.3$\\!\pm\\!$10 | 255.7$\\!\pm\\!$12 | 255.7$\\!\pm\\!$12 | 236.3$\\!\pm\\!$19 | |
$\\!f_{0}(1500)\\!$ | $\\!{\rm E}_{r}\\!$ | 1498.3$\\!\pm\\!$11 | 1502.4$\\!\pm\\!$9 | 1498.3$\\!\pm\\!$12 | 1498.3$\\!\pm\\!$13 | 1494.6$\\!\pm\\!$11 | 1498.3$\\!\pm\\!$14 |
| $\\!\Gamma_{r}\\!$ | 198.8$\\!\pm\\!$14 | 236.8$\\!\pm\\!$11 | 193$\\!\pm\\!$9 | 198.8$\\!\pm\\!$11 | 194$\\!\pm\\!$8 | 193$\\!\pm\\!$10 |
$\\!f_{0}(1710)\\!$ | $\\!{\rm E}_{r}\\!$ | | | | 1726.1$\\!\pm\\!$12 | 1726.1$\\!\pm\\!$13 | 1726.1$\\!\pm\\!$12 | 1726.1$\\!\pm\\!$10
| $\\!\Gamma_{r}\\!$ | | | | 140.2$\\!\pm\\!$9 | 111.6$\\!\pm\\!$8 | 84.2$\\!\pm\\!$8 | 112.8$\\!\pm\\!$7
The $f_{0}(1370)$ and $f_{0}(1710)$ are represented by the pole clusters
corresponding to states with the dominant $s\bar{s}$ component; $f_{0}(1500)$,
with the dominant glueball component.
Note a surprising result obtained for the $f_{0}(980)$. This state lies
slightly above the $K\overline{K}$ threshold and is described by the pole on
sheet II and by the shifted pole on sheet III under the $\eta\eta$ threshold
without the corresponding poles on sheets VI and VII, as it was expected for
standard clusters. This corresponds to the description of the $\eta\eta$ bound
state.
Masses and total widths of states should be calculated from the pole
positions. If, when calculating these quantities, the resonance part of
amplitude is taken in the form
$T^{res}=\frac{\sqrt{s_{r}}\Gamma_{el}}{m_{res}^{2}-s_{r}-i\sqrt{s_{r}}\Gamma_{tot}},$
(13)
we obtain values of masses and total widths of the $f_{0}$-resonances,
presented in Table 3.
Table 3: Masses and total widths of the $f_{0}$-resonances (all in MeV). | Variant I | Variant II
---|---|---
State | $m_{res}$ | $\Gamma_{tot}$ | $m_{res}$ | $\Gamma_{tot}$
$f_{0}(600)$ | 835.3 | 1166 | 834.9 | 1126
$f_{0}(980)$ | 1013.7 | 68.2 | 1009.8 | 64
$f_{0}(1370)$ | 1408.7 | 343.6 | 1417.5 | 511
$f_{0}(1500)$ | 1544 | 714 | 1511.4 | 398
$f_{0}(1710)$ | 1715.7 | 321 | 1729.8 | 225.6
## IV Analysis of the isovector $P$-wave of $\pi\pi$ scattering
In this sector we applied both the model-independent method and multichannel
Breit–Wigner forms. We analyzed data in Ref. expd5 ; Hya73 , for the
inelasticity parameter ($\eta$) and phase shift of the $\pi\pi$-scattering
amplitude ($\delta$) ($S(\pi\pi\to\pi\pi)=\eta\exp(2i\delta)$), introducing
three ($\rho(770)$, $\rho(1250)$ and $\rho(1550-1780)$), four (the indicated
ones plus $\rho(1860-1910)$) and five (the indicated four plus $\rho(1450)$)
resonances SB_NPA08 .
### IV.1 The Model-Independent Analysis
Since in the data for the $P$-wave $\pi\pi$ scattering a deviation from
elasticity is observed in the near-threshold region of the $\omega\pi$
channel, we considered explicitly the thresholds of the $\pi\pi$ and
$\omega\pi$ channels and the left-hand one at $s=0$ in the uniformizing
variable:
$v=\frac{(m_{\omega}+m_{\pi^{0}})/2~{}\sqrt{s-4m_{\pi^{+}}^{2}}+m_{\pi^{+}}~{}\sqrt{s-(m_{\omega}+m_{\pi^{0}})^{2}}}{\sqrt{s\left[\left((m_{\omega}+m_{\pi^{0}})/2\right)^{2}-m_{\pi^{+}}^{2}\right]}}.$
(14)
Influence of other channels which couple to the $\pi\pi$ one is supposed to be
taken into account via the background.
On the $v$-plane, the resonance part of the 2-channel $S$-matrix element of
$\pi\pi$-scattering $S_{res}$ has no cuts and has the form
$S_{res}=\frac{d(-v^{-1})}{d(v)},$ (15)
where $d(v)$ represents the contribution of resonances SB_NPA08 .
The background part is
$\displaystyle
S_{bg}=\exp\left[2i\left(\sqrt{\frac{s-4m_{\pi^{+}}^{2}}{s}}\right)^{3}\left(\alpha_{0}+\alpha_{1}~{}\frac{s-s_{1}}{s}~{}\theta(s-s_{1})+\right.\right.$
$\displaystyle\left.\left.\alpha_{2}~{}\frac{s-s_{2}}{s}~{}\theta(s-s_{2})\right)\right]\,,$
(16)
where $\alpha_{i}=a_{i}+ib_{i}$, $s_{1}$ is the threshold of 4$\pi$ channel
noticeable in the $\rho$-like meson decays and $s_{2}$ is the threshold of
$\rho 2\pi$ channel. Due to allowing for the left-hand branch-point at $s=0$
in the $v$-variable, $a_{0}=b_{0}=0$. Furthermore, $b_{1}=0$ which is related
to the experimental fact that the $P$-wave $\pi\pi$ scattering is elastic also
above the 4$\pi$-channel threshold up to about the $\omega\pi^{0}$ threshold.
In Figure 4 we present results of fitting to the data with three, four and
five resonances.
Figure 4: The phase shift of amplitude and module of the $S$-matrix element
for the $P$-wave $\pi\pi$-scattering in the model-independent approach.
We obtained satisfactory description with the total $\chi^{2}/\mbox{NDF}$
equal to $~{}291.76/(183-15)=1.74$, $~{}278.50/(183-19)=1.70$, and
$~{}266.14/(183-23)=1.66$ for the case of three, four and five resonances,
respectively.
The $\rho(770)$ is described by the cluster of type (a) and the others by type
(b). The background parameters are: $a_{1}=0.0093\pm 0.0199$, $a_{2}=0.0618\pm
0.0305$, and $b_{2}=-0.0135\pm 0.0371$ for the three-resonance,
$a_{1}=0.0017\pm 0.2118$, $a_{2}=0.0433\pm 0.3552$, and $b_{2}=-0.0044\pm
0.4782$ for the four-resonance, and $a_{1}=0.0256\pm 0.0186$, $a_{2}=0.0922\pm
0.0335$, and $b_{2}=0.0011\pm 0.0478$ for the five-resonance descriptions. The
positive sign of $b_{2}$ in the last case is more natural from the physical
point of view.
Though the description can be considered, practically, as the same in all
three cases, careful comparison of the obtained parameters and energy
dependence of the fitted quantities suggests that the resonance $\rho(1900)$
is desired and that the $\rho(1450)$ might be also included improving slightly
the description (at all events, its existence does not contradict to the
data).
In Table 4, we show the pole clusters of the $\rho$-like states on the lower
$\sqrt{s}$-half-plane (in MeV) (the conjugate poles on the upper half-plane
are not shown).
Table 4: Pole clusters distributed on the sheets II, III, and IV for the case with five $\rho$-like resonances. $\sqrt{s_{r}}$ in MeV is given. | II | III | IV
---|---|---|---
$\rho(770)$ | $\\!765.8\pm 0.6-i(73.3\pm 0.4)\\!$ | $\\!778.2\pm 9.1-i(68.9\pm 3.9)\\!$ |
$\rho(1250)$ | | $\\!1251.4\pm 11.3-i(130.9\pm 9.1)\\!$ | $\\!1251\pm 11.1-i(130.5\pm 9.2)\\!$
$\rho(1470)$ | | $\\!1469.4\pm 10.6-i(91\pm 12.9)\\!$ | $\\!1465.4\pm 12.1-i(99.8\pm 15.6)\\!$
$\rho(1600)$ | | $\\!1634\pm 20.1-i(144.7\pm 23.8)\\!$ | $\\!1592.9\pm 7.9-i(73.7\pm 11.7)\\!$
$\rho(1900)$ | | $\\!1882.8\pm 24.8-i(112.4\pm 25.2)\\!$ | $\\!1893\pm 21.9-i(93.4\pm 19.9)\\!$
Masses and total widths of the obtained $\rho$-states can be calculated from
the pole positions on sheets II and IV for resonances of type (a) and (b),
respectively. The obtained values are shown in Table 5.
Table 5: Calculated masses and total widths of the $\rho$-states (all in MeV). | $m_{res}$ | $\Gamma_{tot}$
---|---|---
$\rho(770)$ | 769.3$\pm$0.6 | 146.6$\pm$0.9
$\rho(1250)$ | 1257.8$\pm$11.1 | 261$\pm$18.3
$\rho(1470)$ | 1468.8$\pm 12.1$ | 199.6$\pm$31.2
$\rho(1600)$ | 1594.6$\pm 8$ | 147.4$\pm$23.4
$\rho(1900)$ | 1895.3$\pm$21.9 | 186.8$\pm$39.8
### IV.2 The Breit–Wigner Analysis
We used the 5-channel Breit–Wigner forms in constructing the Jost matrix
determinant $d(k_{1},\cdots,k_{5})$. The resonance poles and zeros in the
$S$-matrix are generated utilizing the Le Couteur–Newton relation
$S_{11}=\frac{d(-k_{1},\cdots,k_{5})}{d(k_{1},\cdots,k_{5})}\;,$ (17)
where $k_{1}$, $k_{2}$, $k_{3}$, $k_{4}$, and $k_{5}$ are the momenta of
$\pi\pi$, $\pi^{+}\pi^{-}2\pi^{0}$, $2\pi^{+}2\pi^{-}$, $\eta 2\pi$, and
$\omega\pi^{0}$ channels, respectively. The Jost function is taken as
$d=d_{res}d_{bg}\,,$ (18)
where the resonance part is
$d_{res}(s)=\prod_{r}\left[M_{r}^{2}-s-i\sum_{j=1}^{5}\rho_{rj}^{3}~{}R_{rj}~{}f_{rj}^{2}\right]$
(19)
with $\rho_{rj}=k_{j}(s)/k_{j}(M_{r}^{2})$ and $f_{rj}^{2}/M_{r}$ the partial
width of a resonance of mass $M_{r}$. $R_{rj}$ is a Blatt–Weisskopf barrier
factor:
$R_{rj}=\frac{1+\frac{1}{4}(\sqrt{M_{r}^{2}-4m_{j}^{2}}~{}r_{rj})^{2}}{1+\frac{1}{4}(\sqrt{s-4m_{j}^{2}}~{}r_{rj})^{2}}$
(20)
with radius $r_{rj}=0.7035$ fm for all resonances in all channels as a result
of our analysis. Furthermore, we have assumed that the widths of resonance
decays to $\pi^{+}\pi^{-}2\pi^{0}$ and $2(\pi^{+}\pi^{-})$ channels are
related each other by relation: $f_{r2}=f_{r3}/\sqrt{2}$. This relation is
well justified with a 5-10% accuracy, for example, by calculations of the
$\rho^{0}$-meson decays in some variant of the chiral model Ach05 .
The background part of the Jost function is
$d_{bg}=\exp\left[-i\left(\sqrt{\frac{s-4m_{\pi^{+}}^{2}}{s}}\right)^{3}\left(\alpha_{0}+\alpha_{1}~{}\frac{s-s_{1}}{s}~{}\theta(s-s_{1})\right)\right]\;,$
(21)
where $\alpha_{i}=a_{i}+ib_{i}$ and $s_{1}$ is the threshold of the $\rho
2\pi$ channel.
In Figure 5, results of fitting to the data are shown and in Table 6, the
$\rho$-like resonance parameters are presented. We obtained equally reasonable
description in all three cases: the total
$\chi^{2}/\mbox{NDF}=316.21/(183-17)=1.87$, $314.69/(183-22)=1.92$, and
$303.10/(183-27)=1.91$ for the case of three, four, and five resonances,
respectively.
Figure 5: The phase shift of amplitude and module of the $S$-matrix element for the $P$-wave $\pi\pi$-scattering for the case of five resonances in the Breit-Wigner approach. Table 6: The $\rho$-like resonance parameters in the Breit-Wigner analysis (all in MeV). State | $\rho(770)$ | $\rho(1250)$ | $\rho(1450)$ | $\rho(1600)$ | $\rho(1900)$
---|---|---|---|---|---
$M$ | 777.69$\\!\pm\\!$0.32 | 1249.8$\\!\pm\\!$15.6 | 1449.9$\\!\pm\\!$12.2 | 1587.3$\\!\pm\\!$4.5 | 1897.8$\\!\pm\\!$38
$f_{r1}$ | 343.8$\\!\pm\\!$0.73 | 87.7$\\!\pm\\!$7.4 | 56.9$\\!\pm\\!$5.4 | 248.2$\\!\pm\\!$5.2 | 47.3$\\!\pm\\!$12
$f_{r2}$ | 24.6$\\!\pm\\!$5.8 | 186.3$\\!\pm\\!$39.9 | 100.1$\\!\pm\\!$18.7 | 240.2$\\!\pm\\!$8.6 | 73.7
$f_{r3}$ | 34.8$\\!\pm\\!$8.2 | 263.5$\\!\pm\\!$56.5 | 141.6$\\!\pm\\!$26.5 | 339.7$\\!\pm\\!$12.5 | 104.3
$f_{r4}$ | | 231.8$\\!\pm\\!$111 | 141.2$\\!\pm\\!$98 | 141.8$\\!\pm\\!$33 | 9
$f_{r5}$ | | 231$\\!\pm\\!$115 | 150$\\!\pm\\!$95 | 108.6$\\!\pm\\!$40.4 | 10
$\Gamma_{tot}$ | $\approx$154.3 | $>$175 | $>$52 | $>$168 | $>$10
The background parameters for the five-resonance description are:
$a_{0}=-0.00121\pm 0.0018$, $a_{1}=-0.1005\pm 0.011$, and $b_{1}=0.0012\pm
0.006$. The background parameters for the other two cases can be found in Ref.
SB_NPA08 .
In order to look at consistency of the description, we checked if the obtained
formula for the $\pi\pi$-scattering amplitude gives a value of the scattering
length consistent with the results of other approaches (Table 7). It seems
that the satisfactory agreement we obtained is not accidental, because in the
energy region from the $\pi\pi$ threshold to about 500 MeV (where the
experimental data appear) there are no opened channels. Therefore, at the
adequate representation of the amplitude, its continuation to the threshold is
unique.
Table 7: Comparison of the $\pi\pi$ scattering length from various approaches. $a_{1}^{1}[10^{-3}m_{\pi^{+}}^{-3}]$ | References | Remarks
---|---|---
$33.9\pm 2.02$ | This paper | Breit–Wigner analysis
$34$ | NJL92 | Local NJL model
$37$ | Osi06 | Non-local NJL model
$37.9\pm 0.5$ | Cap08 | Roy equations using ChPT
$39.6\pm 2.4$ | Kam03 | Roy equations
$38.4\pm 0.8$ | Pel05 | Forward dispersion relations
## V Analysis of isoscalar-tensor sector
In analysis of the processes $\pi\pi\to\pi\pi,K\overline{K},\eta\eta$, we
considered explicitly also the channel $(2\pi)(2\pi)$. Here it is impossible
to use the uniformizing-variable method. Therefore, using the Le Couteur-
Newton relations, we generate the resonance poles by some 4-channel Breit-
Wigner forms. The $d(k_{1},k_{2},k_{3},k_{4})$-function is taken as
$d=d_{B}d_{res}$, where the resonance part is
$d_{res}(s)=\prod_{r}\left[M_{r}^{2}-s-i\sum_{j=1}^{4}\rho_{rj}^{5}R_{rj}f_{rj}^{2}\right]$
(22)
with $\rho_{rj}=2k_{j}/\sqrt{M_{r}^{2}-4m_{j}^{2}}$ and $f_{rj}^{2}/M_{r}$ the
partial width. The Blatt–Weisskopf barrier factor for a tensor particle is
$R_{rj}=\frac{9+\frac{3}{4}(\sqrt{M_{r}^{2}-4m_{j}^{2}}~{}r_{rj})^{2}+\frac{1}{16}(\sqrt{M_{r}^{2}-4m_{j}^{2}}~{}r_{rj})^{4}}{9+\frac{3}{4}(\sqrt{s-4m_{j}^{2}}~{}r_{rj})^{2}+\frac{1}{16}(\sqrt{s-4m_{j}^{2}}~{}r_{rj})^{4}},$
(23)
with radii of 0.943 fm for all resonances in all channels, except for
$f_{2}(1270)$ and $f_{2}(1960)$ for which they are: for $f_{2}(1270)$, 1.498,
0.708, and 0.606 fm in the channels $\pi\pi$, $K\overline{K}$, and $\eta\eta$,
respectively; for $f_{2}(1960)$, 0.296 fm in the channel $K\overline{K}$.
The background part has the form
$d_{B}=\mbox{exp}\left[-i\sum_{n=1}^{3}\left(\frac{2k_{n}}{\sqrt{s}}\right)^{5}(a_{n}+ib_{n})\right]$
(24)
with
$\displaystyle
a_{1}=\alpha_{11}+\frac{s-4m_{K}^{2}}{s}~{}\alpha_{12}~{}\theta(s-4m_{K}^{2})+\frac{s-s_{v}}{s}~{}\alpha_{10}~{}\theta(s-s_{v})),$
(25) $\displaystyle
b_{n}=\beta_{n}+\frac{s-s_{v}}{s}~{}\gamma_{n}~{}\theta(s-s_{v}).$ (26)
$s_{v}\approx 2.274$ GeV2 is a combined threshold of the channels
$\eta\eta^{\prime}$, $\rho\rho$, and $\omega\omega$.
The data for the $\pi\pi$ scattering are taken from an energy-independent
analysis by Hyams et al. Hya73 . The data for $\pi\pi\to
K\overline{K},\eta\eta$ are taken from works Lin92 .
We obtained a satisfactory description with ten resonances $f_{2}(1270)$,
$f_{2}(1430)$, $f_{2}^{\prime}(1525)$, $f_{2}(1580)$, $f_{2}(1730)$,
$f_{2}(1810)$, $f_{2}(1960)$, $f_{2}(2000)$, $f_{2}(2240)$, and $f_{2}(2410)$
(the total $\chi^{2}/\mbox{NDF}=161.147/(168-65)\approx 1.56$) and with eleven
states adding one more resonance $f_{2}(2020)$ which is needed in the combined
analysis of processes $p\overline{p}\to\pi\pi,\eta\eta,\eta\eta^{\prime}$
Ani05 . In our analysis, the description with eleven resonances is practically
the same as that with ten resonances: the total
$\chi^{2}/\mbox{NDF}=156.617/(168-69)\approx 1.58$.
The obtained resonance parameters are shown in Table 8 for the cases of ten
and eleven states.
Table 8: The resonance parameters in the tensor sector for ten and eleven states (in MeV). State | $M$ | $f_{r1}$ | $f_{r2}$ | $f_{r3}$ | $f_{r4}$ | $\Gamma_{tot}$
---|---|---|---|---|---|---
ten states
$f_{2}(1270)$ | 1275.3$\\!\pm\\!$1.8 | 470.8$\\!\pm\\!$5.4 | 201.5$\\!\pm\\!$11.4 | 90.4$\\!\pm\\!$4.76 | 22.4$\\!\pm\\!$4.6 | $\approx$212
$f_{2}(1430)$ | 1450.8$\\!\pm\\!$18.7 | 128.3$\\!\pm\\!$45.9 | 562.3$\\!\pm\\!$142 | 32.7$\\!\pm\\!$18.4 | 8.2$\\!\pm\\!$65 | $>$230
$f_{2}^{\prime}(1525)$ | 1535$\\!\pm\\!$8.6 | 28.6$\\!\pm\\!$8.3 | 253.8$\\!\pm\\!$78 | 92.6$\\!\pm\\!$11.5 | 41.6$\\!\pm\\!$160 | $>$49
$f_{2}(1565)$ | 1601.4$\\!\pm\\!$27.5 | 75.5$\\!\pm\\!$19.4 | 315$\\!\pm\\!$48.6 | 388.9$\\!\pm\\!$27.7 | 127$\\!\pm\\!$199 | $>$170
$f_{2}(1730)$ | 1723.4$\\!\pm\\!$5.7 | 78.8$\\!\pm\\!$43 | 289.5$\\!\pm\\!$62.4 | 460.3$\\!\pm\\!$54.6 | 107.6$\\!\pm\\!$76.7 | $>$182
$f_{2}(1810)$ | 1761.8$\\!\pm\\!$15.3 | 129.5$\\!\pm\\!$14.4 | 259$\\!\pm\\!$30.7 | 469.7$\\!\pm\\!$22.5 | 90.3$\\!\pm\\!$90 | $>$177
$f_{2}(1960)$ | 1962.8$\\!\pm\\!$29.3 | 132.6$\\!\pm\\!$22.4 | 333$\\!\pm\\!$61.3 | 319$\\!\pm\\!$42.6 | 65.4$\\!\pm\\!$94 | $>$119
$f_{2}(2000)$ | 2017$\\!\pm\\!$21.6 | 143.5$\\!\pm\\!$23.3 | 614$\\!\pm\\!$92.6 | 58.8$\\!\pm\\!$24 | 450.4$\\!\pm\\!$221 | $>$299
$f_{2}(2240)$ | 2207$\\!\pm\\!$44.8 | 136.4$\\!\pm\\!$32.2 | 551$\\!\pm\\!$149 | 375$\\!\pm\\!$114 | 166.8$\\!\pm\\!$104 | $>$222
$f_{2}(2410)$ | 2429$\\!\pm\\!$31.6 | 177$\\!\pm\\!$47.2 | 411$\\!\pm\\!$196.9 | 4.5$\\!\pm\\!$70.8 | 460.8$\\!\pm\\!$209 | $>$170
eleven states
$f_{2}(1270)$ | 1276.3$\\!\pm\\!$1.8 | 468.9$\\!\pm\\!$5.5 | 201.6$\\!\pm\\!$11.6 | 89.9$\\!\pm\\!$4.79 | 7.2$\\!\pm\\!$4.6 | $\approx$210.5
$f_{2}(1430)$ | 1450.5$\\!\pm\\!$18.8 | 128.3$\\!\pm\\!$45.9 | 562.3$\\!\pm\\!$144 | 32.7$\\!\pm\\!$18.6 | 8.2$\\!\pm\\!$63 | $>$230
$f_{2}^{\prime}(1525)$ | 1534.7$\\!\pm\\!$8.6 | 28.5$\\!\pm\\!$8.5 | 253.9$\\!\pm\\!$79 | 89.5$\\!\pm\\!$12.5 | 51.6$\\!\pm\\!$155 | $>$49.5
$f_{2}(1565)$ | 1601.5$\\!\pm\\!$27.9 | 75.5$\\!\pm\\!$19.6 | 315$\\!\pm\\!$50.6 | 388.9$\\!\pm\\!$28.6 | 127$\\!\pm\\!$190 | $>$170
$f_{2}(1730)$ | 1719.8$\\!\pm\\!$6.2 | 78.8$\\!\pm\\!$43 | 289.5$\\!\pm\\!$62.6 | 460.3$\\!\pm\\!$545. | 108.6$\\!\pm\\!$76. | $>$182.4
$f_{2}(1810)$ | 1760$\\!\pm\\!$17.6 | 129.5$\\!\pm\\!$14.8 | 259$\\!\pm\\!$32. | 469.7$\\!\pm\\!$25.2 | 90.3$\\!\pm\\!$89.5 | $>$177.6
$f_{2}(1960)$ | 1962.2$\\!\pm\\!$29.8 | 132.6$\\!\pm\\!$23.3 | 331$\\!\pm\\!$61.5 | 319$\\!\pm\\!$42.8 | 62.4$\\!\pm\\!$91.3 | $>$118.6
$f_{2}(2000)$ | 2006$\\!\pm\\!$22.7 | 155.7$\\!\pm\\!$24.4 | 169.5$\\!\pm\\!$95.3 | 60.4$\\!\pm\\!$26.7 | 574.8$\\!\pm\\!$211 | $>$193
$f_{2}(2020)$ | 2027$\\!\pm\\!$25.6 | 50.4$\\!\pm\\!$24.8 | 441$\\!\pm\\!$196.7 | 58$\\!\pm\\!$50.8 | 128$\\!\pm\\!$190 | $>$107
$f_{2}(2240)$ | 2202$\\!\pm\\!$45.4 | 133.4$\\!\pm\\!$32.6 | 545$\\!\pm\\!$150.4 | 381$\\!\pm\\!$116 | 168.8$\\!\pm\\!$103 | $>$222
$f_{2}(2410)$ | 2387$\\!\pm\\!$33.3 | 175$\\!\pm\\!$48.3 | 395$\\!\pm\\!$197.7 | 24.5$\\!\pm\\!$68.5 | 462.8$\\!\pm\\!$211 | $>$168
The background parameters for ten resonances are: $\alpha_{11}=-0.07805$,
$\alpha_{12}=0.03445$, $\alpha_{10}=-0.2295$, $\beta_{1}=-0.0715$,
$\gamma_{1}=-0.04165$, $\beta_{2}=-0.981$, $\gamma_{2}=0.736$,
$\beta_{3}=-0.5309$, $\gamma_{3}=0.8223$; and for eleven resonances are:
$\alpha_{11}=-0.0755$, $\alpha_{12}=0.0225$, $\alpha_{10}=-0.2344$,
$\beta_{1}=-0.0782$, $\gamma_{1}=-0.05215$, $\beta_{2}=-0.985$,
$\gamma_{2}=0.7494$, $\beta_{3}=-0.5162$, $\gamma_{3}=0.786$.
In Figures 6 and 7 we show results of fitting to the data.
Figure 6: The phase shift and module of the $\pi\pi$-scattering $D$-wave
$S$-matrix element.
Figure 7: The squared modules of the $\pi\pi\to K\overline{K}$ (upper figure)
and $\pi\pi\to\eta\eta$ (lower figure) $D$-wave $S$-matrix elements.
## VI Spectroscopic implications from the analysis
In the combined model-independent analysis of data on the
$\pi\pi\to\pi\pi,K\overline{K},\eta\eta,\eta\eta^{\prime}$ processes in the
channel with $I^{G}J^{PC}=0^{+}0^{++}$, an additional confirmation of the
$\sigma$-meson with mass 835 MeV is obtained (the pole position on sheet II is
$598-i583$ MeV). This value of mass corresponds most near to the one ($\sim
860$ MeV) of Ref. Tornqvist and rather accords with prediction
($m_{\sigma}\approx m_{\rho}$) on the basis of mended symmetry by S. Weinberg
Wei90 . Note that our values of $E_{r}$ and $\Gamma_{r}$ for the
$f_{0}(600)$-pole position are larger than those obtained in the dispersive
analysis of data on only the $\pi\pi$ scattering, see Ref. G-MKP08 and
reference therein.
Indication for $f_{0}(980)$ to be the $\eta\eta$ bound state is obtained. From
the point of view of the quark structure, this is the 4-quark state. Maybe,
this is consistent somehow with arguments in favour of the 4-quark nature of
$f_{0}(980)$ 20 .
The ${f_{0}}(1370)$ and $f_{0}(1710)$ have the dominant $s{\bar{s}}$
component. Conclusion about the ${f_{0}}(1370)$ agrees quite well with the one
drawn by the Crystal Barrel Collaboration 21 where the ${f_{0}}(1370)$ is
identified as $\eta\eta$ resonance in the $\pi^{0}\eta\eta$ final state of the
${\bar{p}}p$ annihilation at rest. Conclusion about the $f_{0}(1710)$ is quite
consistent with the experimental facts that this state is observed in
$\gamma\gamma\to K_{S}{\bar{K}}_{S}$ 22 and not observed in
$\gamma\gamma\to\pi^{+}\pi^{-}$ 23 .
As to the $f_{0}(1500)$, we suppose that it is practically the eighth
component of octet mixed with the glueball being dominant in this state. Its
biggest width among the enclosing states tells also in behalf of its glueball
nature 24 .
We propose the following assignment of scalar mesons below 1.9 GeV to lower
nonets, excluding the $f_{0}(980)$ as the $\eta\eta$ bound state. The lowest
nonet: the isovector $a_{0}(980)$, the isodoublet $K_{0}^{*}(900)$, and
$f_{0}(600)$ and $f_{0}(1370)$ as mixtures of the eighth component of octet
and the SU(3) singlet. Then the Gell-Mann–Okubo (GM-O) formula
$3m_{f_{8}}^{2}=4m_{K_{0}^{*}}^{2}-m_{a_{0}}^{2}\,,$ (27)
gives $m_{f_{8}}=872$ MeV ($m_{\sigma}=835\pm 14$ MeV). In the relation for
masses of nonet
$m_{\sigma}+m_{f_{0}(1370)}=2m_{K_{0}^{*}}\,,$ (28)
the left-hand side is about 25 % bigger than the right-hand one.
The next nonet: $a_{0}(1450)$, $K_{0}^{*}(1450)$, and $f_{0}(1500)$ and
$f_{0}(1710)$. From the GM-O formula, we get $m_{f_{8}}\approx 1450$ MeV. In
the relation
$m_{f_{0}(1500)}+m_{f_{0}(1710)}=2m_{K_{0}^{*}(1450)}\,,$ (29)
the left-hand side is about 12 % bigger than the right-hand one.
Now an adequate mixing scheme should be found.
In the vector sector, the obtained value of mass for the $\rho(770)$ is
smaller in the model-independent approach, $769.3$ MeV, and a little bit
bigger in the Breit–Wigner one, $777.69\pm 0.32$ MeV, than the averaged value
cited in the PDG tables PDG08 , $775.49\pm 0.34$ MeV. However, it also occurs
in analysis of some reactions (see PDG tables). The obtained value of the
total width in the first case ($146.6$ MeV) is in a good agreement with the
averaged PDG one ($149.4\pm 1.0$ MeV) and it is a little bit bigger in the
second case ($\approx 154.3$ MeV) than the averaged PDG value, however, this
is encountered also in other analyses (see PDG tables). Note that predicted
widths of the $\rho(770)$ decays to the $4\pi$-modes are significantly larger
than, e.g., the ones evaluated in the chiral model of some mesons based on the
hidden local symmetry added with the anomalous terms Ach05 .
The first $\rho$-like meson has the mass 1257.8$\pm$11 MeV in the model-
independent analysis and 1249.8$\pm$15.6 MeV in the Breit–Wigner one. These
values differ significantly from the mass (1459$\pm$11 MeV) of the first
$\rho$-like meson cited in the PDG tables. The $\rho(1250)$ was discussed
actively some time ago 25 and later the evidence for its existence was
obtained in SB_NPA08 ; 26 .
If the $\rho(1250)$ is interpreted as the first radial excitation of the
$1^{+}1^{--}$ $q{\bar{q}}$ state, then it lies down well on the corresponding
linear trajectory with an universal slope on the $(n,M^{2})$ plane (n is the
radial quantum number of the $q{\bar{q}}$ state)27 , whereas the $\rho(1450)$
turns out to be considerably higher than this trajectory. The $\rho(1250)$ and
the isodoublet $K^{*}(1410)$ are well located to the octet of the first radial
excitations. The mass of the latter should be by about 150 MeV larger than the
mass of the former. Then the GM-O formula
$3m_{\omega_{8}^{\prime}}^{2}=4m_{{K^{*}}^{\prime}}^{2}-m_{\rho^{\prime}}^{2}$
(30)
gives $m_{\omega_{8}^{\prime}}=1460$ MeV, that is fairly good compatible with
the mass of the first $\omega$-like meson $\omega(1420)$, for which one
obtains the values in range 1350-1460 MeV (see PDG tables).
Existence of the $\rho(1450)$ (along with $\rho(1250)$) does not contradict to
the data. In the $q{\bar{q}}$ picture, it might be the first ${}^{3}D_{1}$
state with, possibly, the isodoublet $K^{*}(1680)$ in the corresponding octet.
From the GM-O formula, we should obtain the value 1750 MeV for the mass of the
eighth component of this octet. This corresponds to one of the observations of
the second $\omega$-like meson with masses from 1606 to 1840 MeV that is cited
in the PDG tables under the $\omega(1650)$.
The third $\rho$-like meson has the mass about 1600 MeV rather than 1720 MeV
cited in the PDG tables PDG08 .
As to the $\rho(1900)$, in this energy region there are practically no data on
the $P$-wave of $\pi\pi$ scattering. The model-independent analysis testifies
in favour of existence of this state, whereas the Breit–Wigner analysis gives
the same description with and without the $\rho(1900)$.
The suggested picture for the first two $\rho$-like mesons is consistent with
predictions of the quark model 28 . In Ref. 29 the discussed mass spectrum
for radially excited $\rho$ and $K^{*}$ mesons was obtained using rather
simple mass operator. If the existence of the $\rho(1250)$ is confirmed, some
quark potential models, e.g., in Ref. 30 , will require substantial revisions,
because the first $\rho$-like meson is usually predicted about 200 MeV higher
than this state. To the point, the first $K^{*}$-like meson is obtained in the
indicated quark model at 1580 MeV, whereas the corresponding very well
established resonance has the mass of only 1410 MeV.
In the tensor sector, we carried out two analysis – without and with the
$f_{2}(2020)$. We do not obtain $f_{2}(1640)$, $f_{2}(1910)$ and
$f_{2}(2150)$, however, we see $f_{2}(1450)$ and $f_{2}(1730)$ which are
related to the statistically-valued experimental points.
Usually one assigns the states $f_{2}(1270)$ and $f_{2}^{\prime}(1525)$ to the
ground tensor nonet. To the second nonet, one could assign $f_{2}(1600)$ and
$f_{2}(1760)$ though for now the isodoublet member is not discovered. If
$a_{2}(1730)$ is the isovector of this octet and if $f_{2}(1600)$ is almost
its eighth component, then, from the GM-O formula, we expect this isodoublet
mass at about 1633 MeV. Then the relation for masses of nonet would be
fulfilled with a 3% accuracy. Karnaukhov et al. 31 observed the strange
isodoublet with yet indefinite remaining quantum numbers and with mass
$1629\pm 7$ MeV in the mode $K_{s}^{0}\pi^{+}\pi^{-}$. This state might be the
tensor isodoublet of the second nonet.
The states $f_{2}(1963)$ and $f_{2}(2207)$ together with the isodoublet
$K_{2}^{*}(1980)$ could be put into the third nonet. Then in the relation for
masses of nonet
$M_{f_{2}(1963)}+M_{f_{2}(2207)}=2M_{K_{2}^{*}(1980)},$ (31)
the left-hand side is only 5.3 % bigger than the right-hand one. If one
consider $f_{2}(1963)$ as the eighth component of octet, the GM-O formula
$M_{a_{2}}^{2}=4M_{K_{2}^{*}(1980)}^{2}-3M_{f_{2}(1963)}^{2}$ (32)
gives $M_{a_{2}}=2030$ MeV. This value coincides with the one for
$a_{2}$-meson obtained in works 32 . This state is interpreted as a second
radial excitation of the $1^{-}2^{++}$-state on the basis of consideration of
the $a_{2}$ trajectory on the $(n,M^{2})$ plane Ani05 .
As to $f_{2}(2000)$, the presence of the $f_{2}(2020)$ in the analysis with
eleven resonances helps to interpret $f_{2}(2000)$ as the glueball. In the
case of ten resonances, the ratio of the $\pi\pi$ and $\eta\eta$ widths is in
the limits obtained in Ref. Ani05 for the tensor glueball on the basis of the
1/N-expansion rules. However, the $K\overline{K}$ width is too large for the
glueball. At practically the same description of processes with the
consideration of eleven resonances as in the case of ten, their parameters
have varied a little, except for the ones for $f_{2}(2000)$ and $f_{2}(2410)$.
Mass of the latter has decreased by about 40 MeV. As to $f_{2}(2000)$, its
$K\overline{K}$ width has changed significantly. Now all the obtained ratios
of the partial widths are in the limits corresponding to the glueball.
The question of interpretation of the $f_{2}(1450)$, $f_{2}(1730)$,
$f_{2}(2020)$ and $f_{2}(2410)$ is open.
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* (35) A.V. Anisovich, V.V. Anisovich, and A.V. Sarantsev, Phys. Rev. D 62, 051502 (2000).
* (36) E. van Beveren, G. Rupp, T.A. Rijken, and C. Dullemond, Phys. Rev. D 27, 1527 (1983).
* (37) S.B. Gerasimov and A.B. Govorkov, Z. Phys. C 13, 43 (1982); ibid. 29, 61 (1985).
* (38) S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985).
* (39) V.M. Karnaukhov et al., Yad. Fiz. 63, 652 (2000).
* (40) A.V. Anisovich et al., Phys. Lett. B 452, 173 (1999); ibid. 452, 187 (1999); ibid. 517, 261 (2001).
|
arxiv-papers
| 2008-11-06T09:42:27 |
2024-09-04T02:48:58.633490
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yu.S. Surovtsev, P. Bydzovsky, M. Nagy",
"submitter": "Petr Bydzovsky",
"url": "https://arxiv.org/abs/0811.0906"
}
|
0811.0938
|
# An improved formulation of the relativistic hydrodynamics equations in 2D
Cartesian coordinates
Thorsten Kellerman1, Luca Baiotti1,2, Bruno Giacomazzo1 and Luciano
Rezzolla1,3,4 1 Max-Planck-Institut für Gravitationsphysik, Albert Einstein
Institut, Golm, Germany 2 Graduate School of Arts and Sciences, University of
Tokyo, Komaba, Meguro-ku, Tokyo, 153-8902, Japan 3 Department of Physics,
Louisiana State University, Baton Rouge, LA 70803 USA 4 INFN, Sezione di
Trieste, Trieste, Italy
###### Abstract
A number of astrophysical scenarios possess and preserve an overall
cylindrical symmetry also when undergoing a catastrophic and nonlinear
evolution. Exploiting such a symmetry, these processes can be studied through
numerical-relativity simulations at smaller computational costs and at
considerably larger spatial resolutions. We here present a new flux-
conservative formulation of the relativistic hydrodynamics equations in
cylindrical coordinates. By rearranging those terms in the equations which are
the sources of the largest numerical errors, the new formulation yields a
global truncation error which is one or more orders of magnitude smaller than
those of alternative and commonly used formulations. We illustrate this
through a series of numerical tests involving the evolution of oscillating
spherical and rotating stars, as well as shock-tube tests.
###### pacs:
04.25.D,04.40.Dg
## 1 Introduction
Numerical simulations assuming and enforcing axisymmetry are particularly
useful to study at higher resolution and smaller computational costs those
astrophysical scenarios whose evolution is expected to possess and preserve
such a symmetry. On the other side, the numerical solution of systems of
equations expressed in coordinates adapted to the symmetry has often posed
serious difficulties, because of the coordinate singularity present on the
symmetry axis. The “cartoon” method, proposed by Alcubierre et al. [1], allows
to exploit the advantages of reduced computational resource requirements,
while adopting Cartesian coordinates, which are non-singular.
The “cartoon” method proves particularly useful in the numerical evolution of
smooth functions, like the metric quantities of the Einstein equations.
However, because of the interpolations necessary to impose the axisymmetric
conditions on a Cartesian grid, the “cartoon” approach is not considered to be
accurate enough to describe the shocks which generically develop when matter
is present. As a consequence, general-relativistic codes employing the
“cartoon” method have adopted cylindrical coordinates for the evolution of the
matter (and magnetic field) variables [2, 3, 4, 5, 6]. All the cited works
adopt the same formulation for the hydrodynamical equations in cylindrical
coordinates. In the present article, we propose a slightly different
formulation, which has proven to reduce the numerical errors, especially in
the vicinity of the symmetry axis.
More specifically, we have written the Whisky2D code, which solves the
general-relativistic hydrodynamics equations in a flux-conservative form and
in cylindrical coordinates. This of course brings in $1/r$ singular terms,
which must be dealt with appropriately. In the above-referenced works, the
flux operator is expanded and the $1/r$ terms, not containing derivatives, are
moved to the right hand side of the equation (the source term), so that the
left hand side assumes a form identical to the one of the three-dimensional
(3D) Cartesian formulation. We call this the standard formulation. An other
possibility is not to split the flux operator and to redefine the conserved
variables, via a multiplication by $r$. We call this the new formulation. The
new equations are solved with the same methods as in the Cartesian case. From
a mathematical point of view, one would not expect differences between the two
ways of writing the differential operator, but, of course, a difference is
present at the numerical level. Our tests show that the new formulation yields
results with a global truncation error which is one or more orders of
magnitude smaller than those of alternative and commonly used formulations.
Here we perform a series of tests to ascertain the convergence behaviour of
the two formulations. We then show that the new formulation produces results
which are generally more accurate, with a truncation error which can be
several orders of magnitude smaller.
The paper in organized as follows. In Section 2 we remind the essentials of
the “cartoon” approach for the evolution of the geometrical variables, while
in Section 3 we review the flux-conservative formulation of relativistic
hydrodynamics. We write down the relativistic flux-conservative hydrodynamics
equations for axisymmetric formulations and we illustrate the two possible
ways to write the singular term. In Section 4, we present several tests that
compare the two formulations. We begin with the conservation of rest mass and
angular momentum in the Cowling approximation and in full-spacetime evolution.
Then the eigenfrequencies of uniformly rotating neutron-star models are
compared with the results of a perturbative code. The last test examines the
differences between the two formulations with respect to an analytic solution
of an extreme shock case, which mimics the reflection of a cold and very fast
gas at the symmetry axis.
We have used a spacelike signature $(-,+,+,+)$, with Greek indices running
from 0 to 3, Latin indices from 1 to 3 and the standard convention for the
summation over repeated indices. Unless explicitly stated, all the quantities
are expressed in the system of dimensionless units in which $c=G=M_{\odot}=1$.
## 2 Evolution of the Einstein equations
The logical and algorithmic structures of the Whisky2D code presented here
follow closely the ones of the CCATIE [7] and Whisky codes [8], which solve
the same set of equations in 3D and using Cartesian coordinates. In what
follows we provide only a brief overview of the set of equations for the
evolution of the fields (within the “cartoon” prescription [1]) and for the
evolution of the fluid variables, referring the interested reader to refs. [7,
9, 10] for a more detailed discussion. As for the other codes mentioned above,
also Whisky2D is based on the Cactus Computational Toolkit [11].
More specifically, we evolve a conformal-traceless “$3+1$” formulation of the
Einstein equations [12, 13, 14], in which the spacetime is decomposed into 3D
spacelike slices, described by a metric $\gamma_{ij}$, its embedding in the
full spacetime, specified by the extrinsic curvature $K_{ij}$, and the gauge
functions $\alpha$ (lapse) and $\beta^{i}$ (shift), which specify a coordinate
frame (see [15] for a general description of the $3+1$ split). The particular
system which we evolve transforms the standard ADM variables as follows. The
3-metric $\gamma_{ij}$ is conformally transformed via
$\Phi=\frac{1}{12}\ln\det\gamma_{ij},\qquad\tilde{\gamma}_{ij}=e^{-4\Phi}\gamma_{ij},$
(1)
and the conformal factor $\Phi$ evolved as an independent variable, whereas
$\tilde{\gamma}_{ij}$ is subject to the constraint
$\det\tilde{\gamma}_{ij}=1$. The extrinsic curvature is subjected to the same
conformal transformation and its trace $\tr K_{ij}$ is evolved as an
independent variable. That is, in place of $K_{ij}$ we evolve:
$K\equiv\tr
K_{ij}=g^{ij}K_{ij},\qquad\tilde{A}_{ij}=e^{-4\Phi}(K_{ij}-\frac{1}{3}\gamma_{ij}K),$
(2)
with $\tr\tilde{A}_{ij}=0$. Finally, new evolution variables
$\tilde{\Gamma}^{i}=\tilde{\gamma}^{jk}\tilde{\Gamma}^{i}_{jk}$ (3)
are introduced, defined in terms of the Christoffel symbols of the conformal
3-metric.
The Einstein equations specify a well known set of evolution equations for the
listed variables and are given by
$\displaystyle\hskip 28.45274pt(\partial_{t}-{\cal
L}_{\beta})\;\tilde{\gamma}_{ij}=-2\alpha\tilde{A}_{ij},$ (4)
$\displaystyle\hskip 28.45274pt(\partial_{t}-{\cal
L}_{\beta})\;\Phi=-\frac{1}{6}\alpha K,$ (5) $\displaystyle\hskip
28.45274pt(\partial_{t}-{\cal
L}_{\beta})\;\tilde{A}_{ij}=e^{-4\Phi}[-D_{i}D_{j}\alpha+\alpha(R_{ij}-8\pi
S_{ij})]^{TF}+\alpha(K\tilde{A}_{ij}-2\tilde{A}_{ik}\tilde{A}^{k}{}_{j}),$ (6)
$\displaystyle\hskip 28.45274pt(\partial_{t}-{\cal
L}_{\beta})\;K=-D^{i}D_{i}\alpha+\alpha\left[\tilde{A}_{ij}\tilde{A}^{ij}+\frac{1}{3}K^{2}+4\pi(\rho_{{}_{\rm
ADM}}+S)\right],$ (7) $\displaystyle\hskip
28.45274pt\partial_{t}\tilde{\Gamma}^{i}=\tilde{\gamma}^{jk}\partial_{j}\partial_{k}\beta^{i}+\frac{1}{3}\tilde{\gamma}^{ij}\partial_{j}\partial_{k}\beta^{k}+\beta^{j}\partial_{j}\tilde{\Gamma}^{i}-\tilde{\Gamma}^{j}\partial_{j}\beta^{i}+\frac{2}{3}\tilde{\Gamma}^{i}\partial_{j}\beta^{j}$
$\displaystyle\hskip
56.9055pt-2\tilde{A}^{ij}\partial_{j}\alpha+2\alpha\left(\tilde{\Gamma}^{i}{}_{jk}\tilde{A}^{jk}+6\tilde{A}^{ij}\partial_{j}\Phi-\frac{2}{3}\tilde{\gamma}^{ij}\partial_{j}K-8\pi\tilde{\gamma}^{ij}S_{j}\right),$
(8)
where $R_{ij}$ is the three-dimensional Ricci tensor, $D_{i}$ is the covariant
derivative associated with the three metric $\gamma_{ij}$, “TF” indicates the
trace-free part of tensor objects and ${\rho}_{{}_{\rm ADM}}$, $S_{j}$ and
$S_{ij}$ are the matter source terms defined as
$\displaystyle\rho_{{}_{\rm ADM}}$ $\displaystyle\equiv
n_{\alpha}n_{\beta}T^{\alpha\beta},$ $\displaystyle S_{i}$
$\displaystyle\equiv-\gamma_{i\alpha}n_{\beta}T^{\alpha\beta},$ (9)
$\displaystyle S_{ij}$
$\displaystyle\equiv\gamma_{i\alpha}\gamma_{j\beta}T^{\alpha\beta},$
where $n_{\alpha}\equiv(-\alpha,0,0,0)$ and $T^{\alpha\beta}$ is the stress-
energy tensor for a perfect fluid (see Section 3).
Four elliptic constraint equations, which are usually referred to as
Hamiltonian and momentum constraints,
$\displaystyle\hskip 28.45274pt\mathcal{H}\equiv
R^{(3)}+K^{2}-K_{ij}K^{ij}-16\pi\rho_{{}_{\rm ADM}}=0\,,$ (10)
$\displaystyle\hskip 28.45274pt\mathcal{M}^{i}\equiv
D_{j}(K^{ij}-\gamma^{ij}K)-8\pi S^{i}=0\,,$ (11)
should be satisfied within each spacelike slice. Here
$R^{(3)}=R_{ij}\gamma^{ij}$ is the Ricci scalar on a 3D timeslice. Additional
constraints are given by
$\displaystyle\hskip 28.45274pt\det\tilde{\gamma}_{ij}=1,\hskip
56.9055pt\tr\tilde{A}_{ij}=0,\hskip
56.9055pt\tilde{\Gamma}^{i}=\tilde{\gamma}^{jk}\tilde{\Gamma}^{i}_{jk}\,,$
(12)
with the last two equations of (12) being enforced algebraically. The
remaining constraint in (12) and the constraints $\mathcal{H}$ and
$\mathcal{M}^{i}$ are not actively enforced and can be used as monitors of the
accuracy of our numerical solution.
We specify the gauges in terms of the standard ADM lapse function, $\alpha$,
and shift vector, $\beta^{a}$ [16]. We evolve the lapse according to the
“$1+\log$” slicing condition [17]:
$\partial_{t}\alpha-\beta^{i}\partial_{i}\alpha=-2\alpha(K-K_{0}),$ (13)
where $K_{0}$ is the initial value of the trace of the extrinsic curvature and
equals zero for the maximally sliced initial data we consider here. The code
uses a hyperbolic $\tilde{\Gamma}$-driver condition [18]
$\displaystyle\partial_{t}\beta^{i}-\beta^{j}\partial_{j}\beta^{i}$
$\displaystyle=$ $\displaystyle\frac{3}{4}\alpha B^{i}\,,$ (14)
$\displaystyle\partial_{t}B^{i}-\beta^{j}\partial_{j}B^{i}$ $\displaystyle=$
$\displaystyle\partial_{t}\tilde{\Gamma}^{i}-\beta^{j}\partial_{j}\tilde{\Gamma}^{i}-\eta
B^{i}\,,$ (15)
where $\eta$ is a parameter which acts as a damping coefficient (see
discussion in ref. [19]).
Two routes are possible when solving numerically the Einstein equations in
axisymmetric spacetimes. One route consists in using coordinates that exploit
the symmetry and enforce its preservation already at a mathematical level,
such as cylindrical coordinates. This advantage is counterbalanced by the fact
that such coordinates are usually singular somewhere (e.g., on the axis for
cylindrical coordinates) and that regularization conditions are therefore
necessary (see [20, 21] and references therein for a recent discussion).
The second route consists, instead, in using Cartesian coordinates and in
exploiting the fact that these coincide with the cylindrical ones in one
plane, namely the $(x,z)$ plane (for concreteness we will assume hereafter
that the Cartesian and the cylindrical $z$-axes coincide). The chief
advantages of this approach, which is usually referred to as the “cartoon”
method [1], are the absence of the need of regularization conditions and the
easiness of implementation, through a simple dimensional reduction from fully
3D codes in Cartesian coordinates. However, these advantages are
counterbalanced by at least two disadvantages. The first one is that the
method still essentially requires the use of a 3D domain covered with
Cartesian coordinates, although one of the three dimensions, namely the
$y$-direction, has a very small extent. The second one is that, in order to
compute the spatial derivatives in the $y$-direction appearing in the Einstein
equations, a number of high-order interpolations onto the $x$-axis are
necessary (see discussion below) and these can amount to a significant portion
of the time spent for each evolution to the new timelevel. In practice, the
spatial derivatives in the $y$-direction are computed exploiting the fact that
all quantities are constant on cylinders and thus the value of a variable
$\Psi$ at a generic position $({x},{y},{z})$ off the $(x,z)$ plane can be
computed from the corresponding value $\Psi(\tilde{x},0,\tilde{z})$ on the
$(x,z)$ plane, where
$\tilde{x}=({x}+{y})^{1/2}\,,\qquad\tilde{z}={z}\,.$ (16)
Clearly, since the solution of the evolution equations is computed only on the
$(x,z)$ plane, interpolations (with truncation errors smaller than that of the
finite-difference operators) are needed at all the positions
$(\tilde{x},y=0,\tilde{z})$.
Overall the “cartoon” method represents the choice for many codes and it has
been implemented with success in many applications, e.g., [1, 2, 3, 4, 22, 23,
5, 6] to cite a few.
## 3 Evolution of the relativistic hydrodynamics equations
An important feature of multidimensional non-vacuum numerical-relativity codes
that solve the coupled Einstein–hydrodynamics equations in Cartesian
coordinates is the adoption of a conservative formulation of the hydrodynamics
equations [24, 25]. In such a formulation, the set of conservation equations
for the stress-energy tensor $T^{\mu\nu}$ and for the matter current density
$J^{\mu}$, that is
$\displaystyle\nabla_{\mu}J^{\mu}=0\,,$ (17)
$\displaystyle\nabla_{\mu}T^{\mu\nu}=0\,,$ (18)
is written in a hyperbolic, first-order “flux-conservative” form of the type
[26]
$\partial_{t}{\mathbf{q}}+\partial_{i}{\mathbf{f}}^{(i)}({\mathbf{q}})={\mathbf{s}}({\mathbf{q}})\,,$
(19)
where ${\mathbf{f}}^{(i)}({\mathbf{q}})$ and ${\mathbf{s}}({\mathbf{q}})$ are
the flux vectors and source terms, respectively [27]. Note that the right-hand
side (the source terms) depends only on the metric, on its first derivatives
and on the stress-energy tensor. Furthermore, while the system (19) is not
strictly hyperbolic, strong hyperbolicity is recovered in a flat spacetime,
where ${\mathbf{s}}({\mathbf{q}})=0$.
As shown by [25], in order to write the system (17)–(18) in the form of system
(19), the primitive hydrodynamical variables (i.e. the rest-mass density
$\rho$, the pressure $p$ measured in the rest-frame of the fluid, the fluid
3-velocity $v^{i}$ measured by a local zero-angular momentum observer, the
specific internal energy $\epsilon$ and the Lorentz factor $W$) are mapped to
the so called conserved variables ${\mathbf{q}}\equiv(D,S^{i},\tau)$ via the
relations
$\displaystyle D$ $\displaystyle\equiv$ $\displaystyle\sqrt{\gamma}\rho W\,,$
$\displaystyle S^{i}$ $\displaystyle\equiv$ $\displaystyle\sqrt{\gamma}\rho
hW^{2}v^{i}\,,$ (20) $\displaystyle\tau$ $\displaystyle\equiv$
$\displaystyle\sqrt{\gamma}\left(\rho hW^{2}-p\right)-D\,,$
where $h\equiv 1+\epsilon+p/\rho$ is the specific enthalpy and
$W\equiv(1-\gamma_{ij}v^{i}v^{j})^{-1/2}$.
The advantage of a flux-conservative formulation is that it allows to use
high-resolution shock-capturing (HRSC) schemes, which are based on Riemann
solvers and which are essential for a correct representation of shocks. This
is particularly important in astrophysical simulations, where large shocks are
expected. In this approach, all variables ${\bf q}$ are represented on the
numerical grid by cell-integral averages. The function is then reconstructed
within each cell, usually through piecewise polynomials, in a way that
preserves the conservation of the variables ${\bf q}$. This gives two values
at each cell boundary, which are then used as initial data for the
(approximate) Riemann problem, whose solution gives the flux through the cell
boundary.
As in the Whisky code, the evolution equations are here integrated in time
using the method of line [28], which reduces the partial differential
equations (19) to a set of ordinary differential equations that can be evolved
using standard numerical methods, such as Runge-Kutta or the iterative Cranck-
Nicolson schemes [29, 30]. Furthermore, the Whisky2D code implements several
reconstruction methods, such as Total-Variation-Diminishing (TVD) methods,
Essentially-Non-Oscillatory (ENO) methods [31] and the Piecewise-Parabolic-
Method (PPM) [32]. Also, a variety of approximate Riemann solvers can be used,
starting from the Harten-Lax-van Leer-Einfeldt (HLLE) solver [33], over to the
Roe solver [34] and the Marquina flux formula [35] (see [8, 9] for a more
detailed discussion).
The ability of properly evolving large gradients moving at relativistic speeds
represents one of the main motivations that make this formulation the choice
for all of the present 3D numerical-relativity codes solving the relativistic
hydrodynamics equations on Eulerian grids (see refs. [36, 37, 38] for some of
the most recent examples and ref. [39] for an alternative Lagrangian method).
However, when considered within the axisymmetric approach used here, the use
of a flux-conservative formulation in Cartesian coordinates may suffer from a
potentially very serious disadvantage. In fact, the interpolations required by
the “cartoon” method may be highly inaccurate when discontinuities in the
fluid variables appear. To confront this problem, Shibata [2] has made the
useful suggestion of writing the relativistic hydrodynamics equations in
cylindrical coordinates, while keeping the solution of the Einstein equations
in Cartesian coordinates. This approach has the obvious advantage that it does
not require interpolation and that it exploits, at the mathematical level, the
symmetries of the system, thus guaranteeing a better conservation of mass and
angular momentum. However, the use of cylindrical coordinates for the
evolution of the fluid variables also comes with an undesirable property: the
coordinates are degenerate at the symmetry axis and the equations are no
longer free of singularities. As we will comment in the following Section,
this drawback can be compensated through a suitable formulation of the
equations and a proper setup of the numerical grid.
### 3.1 A new formulation of the hydrodynamics equations
As mentioned in the previous Section, following ref. [3], we write the
relativistic hydrodynamics equations (17)–(18) in a first-order form in space
and time using cylindrical coordinates $(r,\phi,z)$. However, as an important
difference from the approach suggested in ref. [3], we do not introduce source
terms that contain coordinate singularities. Rather, we re-define the
conserved quantities in such a way to remove the singular terms, which are the
largest source of truncation error, also when evaluated far from the axis.
We illustrate our approach by using as a representative example the continuity
equation. This is the simplest of the five hydrodynamical equations but
already contains all the basic elements necessary to illustrate the new
formulation. We start by using the definitions for the conserved variables (3)
to write eq. (17) generically as
$\partial_{t}(\sqrt{\gamma}\rho W)+\partial_{i}\left[\sqrt{\gamma}\rho
W\left(\alpha v^{i}-\beta^{i}\right)\right]=0\,,$ (21)
which in cylindrical coordinates takes the form
$\hskip 28.45274pt\partial_{t}(\sqrt{\tilde{\gamma}}\rho
W)+\partial_{r}\left[\sqrt{\tilde{\gamma}}\rho W\left(\alpha
v^{r}-\beta^{r}\right)\right]+\partial_{z}\left[\sqrt{\tilde{\gamma}}\rho
W\left(\alpha v^{z}-\beta^{z}\right)\right]=0\,,$ (22)
where $\sqrt{\tilde{\gamma}}$ is the determinant of the 3-metric in
cylindrical coordinates and where we have enforced the condition of
axisymmetry $\partial_{\phi}=0$. Because any $\phi$-constant plane in
cylindrical coordinates can be mapped into the $(x,z)$ plane in Cartesian
coordinates, we consider equation (22) as expressed in Cartesian coordinates
and restricted to the $y=0$ plane, i.e.,
$\hskip 56.9055pt\partial_{t}(xD)+\partial_{x}\left[xD\left(\alpha
v^{x}-\beta^{x}\right)\right]+\partial_{z}\left[xD\left(\alpha
v^{z}-\beta^{z}\right)\right]=0\,,$ (23)
where we have exploited the fact that for any vector of components $A^{i}$ on
this plane $A^{r}=A^{x},\ A^{\phi}=A^{y}$ and ${\tilde{\gamma}}=x^{2}\gamma$,
with $\gamma$ being the determinant of the 3-metric in Cartesian coordinates.
Equation (23) represents the prototype of the formulation proposed here, which
we will refer to hereafter as the “new” formulation to contrast it with the
formulation adopted so far, e.g., in ref. [3], for the solution of the
relativistic hydrodynamics equations in axisymmetry and in Cartesian
coordinates. The only, but important, difference with respect to the
“standard” formulation is that in the latter the derivative in the
$x$-direction is written out explicitly and becomes part of the source term
${\mathbf{s}}({\mathbf{q}})$, i.e.,
$\hskip 56.9055pt\partial_{t}(D)+\partial_{x}\left[D\left(\alpha
v^{x}-\beta^{x}\right)\right]+\partial_{z}\left[D\left(\alpha
v^{z}-\beta^{z}\right)\right]=-\frac{D\left(\alpha
v^{x}-\beta^{x}\right)}{x}\,.$ (24)
Even though the right-hand-side of eq. (24) is never evaluated at $x=0$
(because no grid points are located at $x=0$), both the numerator and the
denominator of the right-hand-side of eq. (24) are very small for $x\simeq 0$,
so that small round-off errors in the evaluation of the right-hand-side can
increase the overall truncation error. Stated differently, the right-hand-side
of eq. (24) becomes stiff for $x\simeq 0$ and this opens the door to the
problems encountered in the numerical solution of hyperbolic equations with
stiff source terms [40].
What was done for the continuity equation (23) can be extended to the other
hydrodynamics equations which, for the conservation of momentum in the $x$\-
and $z$-directions, take the form
$\displaystyle\hskip-56.9055pt\frac{1}{\alpha
x\sqrt{\gamma}}\biggl{\\{}\partial_{t}\left(xS_{A}\right)+\partial_{x}\left[x\left(S_{A}\left(\alpha
v^{x}-\beta^{x}\right)+\alpha\sqrt{\gamma}p\delta^{x}_{A}\right)\right]+$
$\displaystyle\hskip 113.81102pt\partial_{z}\left[x\left(S_{A}\left(\alpha
v^{z}-\beta^{z}\right)+\alpha\sqrt{\gamma}p\delta^{z}_{A}\right)\right]\biggr{\\}}=$
$\displaystyle\hskip-35.56593pt\left[T^{00}\left(\frac{1}{2}\beta^{l}\beta^{m}\partial_{A}\gamma_{lm}-\alpha\partial_{A}\alpha\right)+T^{0i}\beta^{l}\partial_{A}\gamma_{il}+T^{0}_{\
i}\partial_{A}\beta^{i}+\frac{1}{2}T^{lm}\partial_{A}\gamma_{lm}\right]\,,$
(25) with $A=x,z$. Similarly, the evolution of the conserved angular momentum
$S_{\phi}=xS_{y}$ is expressed as
$\displaystyle\hskip-56.9055pt\frac{1}{\alpha
x\sqrt{\gamma}}\biggl{\\{}\partial_{t}\left(x^{2}S_{y}\right)+\partial_{x}\left[x^{2}S_{y}\left(\alpha
v^{x}-\beta^{x}\right)\right]+\partial_{z}\left[x^{2}S_{y}\left(\alpha
v^{z}-\beta^{z}\right)\right]\biggr{\\}}=0\,,$ (26)
while the equation of the energy conservation is given by
$\displaystyle\hskip-56.9055pt\frac{1}{\alpha
x\sqrt{\gamma}}\biggl{\\{}\partial_{t}\left(x\tau\right)+\partial_{x}\left[x\left(\tau\left(\alpha
v^{x}-\beta^{x}\right)+pv^{x}\right)\right]+\partial_{z}\left[x\left(\tau\left(\alpha
v^{z}-\beta^{z}\right)+pv^{z}\right)\right]\biggr{\\}}=$ $\displaystyle\hskip
0.0ptT^{00}\left(\beta^{i}\beta^{j}K_{ij}-\beta^{i}\partial_{i}\alpha\right)+T^{0i}\left(-\partial_{i}\alpha+2\beta^{j}K_{ij}\right)+T^{ij}K_{ij}\,.$
(27)
The changes made to the formulation are rather simple but, as we will show in
Section 4, these can produce significant improvements on the overall accuracy
of the simulations with a truncation error at least one order of magnitude
smaller for all of the tests considered. Because of its simplicity, the
changes in the new formulation of the equations can be implemented
straightforwardly in codes written using the standard formulation.
Finally, we note that both eq. (24) and eq. (23) are written in a flux-
conservative form in the sense that the source term does not contain first-
order spatial derivatives of the conserved variables. More precisely, eq. (23)
is written in a flux-conservative form, while eq. (24) is written in a “flux-
balanced” form, as it is typical for flux-conservative equations written in
curvilinear coordinates [26]. The same is true also for eqs. (25)–(27) and for
the corresponding equations presented in ref. [3], which are incorrectly
classified as non flux-conservative.
### 3.2 Equation of state
In whatever coordinate system they are written, the system of hydrodynamics
equations can be closed only after specifying an additional equation, the
equation of state (EOS), which relates the pressure to the rest-mass density
and to the energy density. The code has been written to use any EOS, but all
the tests so far have been performed using either an (isentropic) polytropic
EOS
$\displaystyle p$ $\displaystyle=$ $\displaystyle K\rho^{\Gamma}\,,$ (28)
$\displaystyle e$ $\displaystyle=$ $\displaystyle\rho+\frac{p}{\Gamma-1}\,,$
(29)
or an “ideal-fluid” EOS
$p=(\Gamma-1)\rho\,\epsilon\,.$ (30)
Here, $e$ is the energy density in the rest frame of the fluid, $K$ the
polytropic constant (not to be confused with the trace of the extrinsic
curvature defined earlier) and $\Gamma$ the adiabatic exponent. In the case of
the polytropic EOS (28), $\Gamma=1+1/N$, where $N$ is the polytropic index and
the evolution equation for $\tau$ does not need to be solved. In the case of
the ideal-fluid EOS (30), on the other hand, non-isentropic changes can take
place in the fluid and the evolution equation for $\tau$ needs to be solved.
Note that the polytropic EOS (28) is isentropic and thus does not allow for
the formation of physical shocks, in which entropy (and internal energy) can
be increased locally (shock heating).
Table 1: Equilibrium properties of the initial stellar models. The different
columns refer respectively to: the ratio of the polar to equatorial coordinate
radii $r_{p}/r_{e}$, the central rest-mass density $\rho_{c}$, the
gravitational mass $M$, the rest mass $M_{0}$, the circumferential equatorial
radius $R_{e}$, the angular velocity $\Omega$, the maximum angular velocity
for a star of the same rest mass $\Omega_{{}_{\rm K}}$, the ratio $J/M^{2}$
where $J$ is the angular momentum, the ratio of rotational kinetic energy to
gravitational binding energy $T/|W|$. All models have been computed with a
polytropic EOS with $K=100$ and $\Gamma=2$.
| $r_{p}/r_{e}$ | $\rho_{c}$ | $M$ | $M_{0}$ | $R_{e}$ | $\Omega$ | $\Omega_{K}$ | $J/M^{2}$ | $T/|W|$
---|---|---|---|---|---|---|---|---|---
| | ($\times 10^{-3}$) | ($M_{\odot}$) | ($M_{\odot}$) | | | ($\times 10^{-2})$ | |
A | 1.00 | 1.28 | 1.400 | 1.506 | 9.586 | 0.000 | 3.987 | 0.000 | 0.000
B | 0.67 | 1.28 | 1.651 | 1.786 | 12.042 | 0.253 | 3.108 | 0.594 | 0.081
## 4 Numerical tests
In order to test the stability properties of the new formulation and compare
its accuracy with that of the formulation first presented in [3] and then
used, among others, in [4, 6, 41, 42], we have implemented both of them in
Whisky2D. After this paper was published we were made aware via a private
communication [43] that the numerical approach followed in [23] is in practice
very similar to our new formulation, although the version in cylindrical
coordinates of the equations discussed in [23] is not in a flux-conservative
form; information about such numerical implementation was not given in [23]
and therefore not available to us at the time this work was written.
The initial data, in particular, has been produced as solution of the Einstein
equations for axisymmetric and stationary stellar configurations [44], using
the EOS (28) with $\Gamma=2$ and polytropic constant $K=100$, in order to
produce stellar models that are, at least qualitatively, representative of
what is expected from observations of neutron stars. Our attention has been
restricted to two illustrative models representing a nonrotating star and a
rapidly rotating star having equatorial and polar (coordinate) radii in a
ratio $r_{p}/r_{e}=0.67$. The relevant properties of these stellar models are
reported in Table 1.
All the numerical results presented hereafter have been obtained with the
following fiducial numerical set-up: the reconstruction of the values at the
boundaries of the computational cells is made using the PPM method [32], while
the HLLE algorithm is used as an approximate Riemann solver [33]. The lapse
function is evolved with the “$1+\log$” slicing condition given by eq. (13),
while the shift is evolved using a version of the hyperbolic
$\tilde{\Gamma}$-driver condition (14) in which the advection terms for the
variables $\beta^{i},\,B^{i}$ and $\tilde{\Gamma}^{i}$ are set to zero. The
time evolution is made with a method-of-line approach [28] and a third-order
Runge-Kutta integration scheme (our CFL factor is usually chosen between $0.3$
and $0.5$). A third-order Lagrangian interpolation is adopted to implement the
“cartoon” method. For the matter variables we use “Dirichlet”boundary
conditions (i.e., the solution at the outer boundary is always kept to be the
initial one), while for the field variables we adopt outgoing Sommerfeld
boundary conditions.
We typically present results at four different resolutions:
$\bar{h}=0.4M,\,\bar{h}/2,\,\bar{h}/4,\,3\bar{h}/16$ and $\bar{h}/8$, which
correspond to about $25,\,50,\,100,\,133$ and $200$ points across the stellar
radius, respectively. The computational domain extends to $20\,M$ both in the
$x$ and $z$ directions, and a reflection symmetry is applied across the
equatorial (i.e., $z=0$) plane. Finally, we remark that in contrast with the
interesting analysis of [45], we could not find signs of numerical
instabilities when using the above numerical prescriptions for either of the
two formulations considered.
### 4.1 Oscillating Neutron stars: fixed spacetime
The first set of tests we discuss has been carried out by simulating
relativistic polytropic stars in equilibrium and in a fixed spacetime (i.e. in
the Cowling approximation). In this case the Einstein equations are not
evolved and the truncation error is in general smaller because it is produced
uniquely from the evolution of the hydrodynamics equations.
Figure 1: Evolution of the central rest-mass density for rapidly rotating
stars (model B in Table 1) evolved within the Cowling approximation. The left
panel refers to the use of the standard formulation, while the right one to
the new formulation. Note the different scales in the two panels and note that
in both cases the amplitude of the oscillations decreases with increasing
resolution, while keeping the same phase.
Although the stars are in equilibrium, oscillations are triggered by the
first-order truncation error at the center and the surface of the star (our
hydrodynamical evolution schemes are only first order at local extrema). Both
the amplitude of the oscillations and the rate of the secular change in their
amplitude converge to zero at nearly second order with increasing grid
resolution [46, 47]. The genuine dynamics produced by the truncation error can
then be studied either when the spacetime is held fixed (i.e., in the Cowling
approximation) or when the spacetime is evolved through the solution of the
Einstein equations. This is shown in Fig. 1, which reports the evolution of
the central rest-mass density for rapidly rotating stars (model B in Table 1)
evolved within the Cowling approximation. The left panel refers to the
standard formulation, while the right one to the new formulation. Note that in
both cases the amplitude of the oscillations decreases at roughly second order
with increasing resolution, while keeping the same phase. This is a clear
signature that the oscillations corresponds to proper eigenmodes of the
simulated star. However, the difference of the secular evolution between the
standard formulation and the new one is rather remarkable. The latter, in
fact, is much more accurate and the well-known secular increase in the central
density is essentially absent in the new formulation.
Quantities that are particularly useful to assess the accuracy of the two
formulations are the rest mass and the angular momentum which we compute as
[48]
$\displaystyle\hskip-56.9055ptM_{0}=2\pi\,\int_{V_{*}}\sqrt{\gamma}\rho
Wx\,dx\,dz\,,$ (31)
$\displaystyle\hskip-56.9055ptJ_{z}\,=\,2\pi\,\epsilon_{zj}^{k}\int_{V}\left(\frac{1}{8\pi}\tilde{A}^{j}_{k}+x^{j}S_{k}+\frac{1}{12\pi}x^{j}K_{,k}-\frac{1}{16\pi}x^{j}\tilde{\gamma}^{lm}_{,k}\tilde{A}_{lm}\right)e^{6\Phi}x\,dx\,dz\,,$
(32)
where $V_{*}$ is the coordinate volume occupied by the star and $V$ is
coordinate volume of the computational domain.
Figure 2: Time derivative of the average of the rest mass $M_{0}$, normalized
to the initial value $M_{0}(t=0)$, for evolutions in a fixed spacetime
(Cowling approximation). The average ${\rm d}\langle
M_{0}/M_{0}(t=0)\rangle/{\rm d}t$ is computed between the initial value and a
time $t=25\,{\rm ms}$, corresponding to about $30$ oscillations. The left
panel refers to a nonrotating star (model A in Table 1), while the right panel
to a rapidly rotating star (model B in Table 1). Indicated with squares are
the numerical values obtained with the standard formulation of the
hydrodynamics equations, while triangles are used for the new one. Also
indicated with a dot-dashed line is the slope for a second-order convergence
rate. Figure 3: Time derivative of average of the angular momentum normalized
to the initial value ${\rm d}\langle J/J(t=0)\rangle/{\rm d}t$ (cf., Fig. 2)
for a rapidly rotating star (model B of Table 1). Indicated with squares are
the numerical values obtained with the standard formulation of the
hydrodynamics equations, while triangles are used for the new one; a dot-
dashed line is the slope for a second-order convergence rate.
Figure 2 shows the dependence on the inverse of the resolution of the error in
the conservation of the rest mass for a nonrotating model as computed in the
Cowling approximation (left panel) or in a fully dynamical simulation (right
panel). Since the evolution of the rest mass shows, in addition to a secular
evolution, small oscillations (i.e., of $\sim 3\times 10^{-9}$ for the highest
resolution and of $\sim 3\times 10^{-6}$ for the lowest resolution) the
calculation of the rest mass at a given time can be somewhat ambiguous. To
tackle this problem and to avoid the measurement to be spoiled by the
oscillations, we perform a linear fit of the evolution of $M_{0}$, normalized
to the initial value $M_{0}(t=0)$, between the initial value and a time
$t=25\,{\rm ms}$ (corresponding to about $30$ oscillations) and we take as the
time derivative of the mass the coefficient of the linear fit: ${\rm d}\langle
M_{0}/M_{0}(t=0)\rangle/{\rm d}t$. Fig. 2, in particular, reports in a
logarithmic scale ${\rm d}\langle M_{0}/M_{0}(t=0)\rangle/{\rm d}t$ as a
function of the inverse of the resolution $h$. Indicated with squares are the
numerical values obtained with the standard formulation of the hydrodynamics
equations, while triangles are used for the new one. Also indicated with a
long-short-dashed line is the slope for a second-order convergence rate.
Note that although we use a third-order method for the reconstruction (namely,
PPM), we do not expect third-order convergence. This is also due to the fact
that the reconstruction schemes are only first-order accurate at local extrema
(i.e. at the centre and at the surface of the star), thus increasing the
overall truncation error. Similar estimates were obtained also using the
Whisky code in 3D Cartesian coordinates [8, 9].
Clearly both the new and the standard methods provide a convergence rate which
is close to two. However, and this is the most important result of this work,
the new method yields a truncation error which is several orders of magnitude
smaller than the old one. More specifically, in the case of the rest mass, the
conservation is more accurate of about four orders of magnitude. We believe
that this is essentially due to the rewriting of the source terms in the flux-
conservative formulation which in the new formulation does not have any
coordinate-singular term (i.e. $\propto 1/x$).
Note also that, because the new formulation is intrinsically more accurate, it
also suffers more easily from the contamination of errors which are not
directly related to the finite-difference operators. [The one made in the
calculation of the integral (32) is a relevant example but it is not the only
one]. This may be the reason why, in general, at lower resolutions the new
formulation has convergence rate which is not exactly two and appears over-
convergent (see right panel of Fig. 2). However, as the resolution is
increased and the finite-difference errors become the dominant ones, a clearer
trend in the convergence rate is recovered.
Another way of measuring the accuracy of the two formulations is via the
comparison of the evolution of the angular momentum. While this quantity is
conserved to machine precision in the case of a nonrotating star, this does
not happen for rotating stars and the error can be of a few percent in the
case of very low resolution and of a very rapidly rotating star. This is shown
in Fig. 3 for the stellar model B of Table 1 and it reports in a logarithmic
scale the time derivative of the average of the angular momentum $J$
normalized to the initial value $J(t=0)$. In analogy with Fig. 2, in order to
remove the small-scale oscillations we first perform a linear fit of the
evolution of $J$ between the initial value and a time $t=25\,{\rm ms}$ and
take the coefficient of the fit as the time derivative of the angular
momentum: ${\rm d}\langle J/J(t=0)\rangle/{\rm d}t$. Indicated with squares
are the numerical values obtained with the standard formulation of the
hydrodynamics equations, while triangles are used for the new one; a dot-
dashed line shows the slope for a second-order convergence rate.
It is simple to recognize from Fig. 3 that also for the angular momentum
conservation the new formulation yields a truncation error which is two or
more orders of magnitude smaller, with a clear second-order convergence being
recovered at sufficiently high resolution.
Figure 4: The same as in Fig. 1 but for a full-spacetime evolution. The left
panel refers to the standard formulation, while the right one to the new
formulation. Note the different scale between the two panels.
### 4.2 Oscillating Neutron stars: dynamical spacetime
Also the second set of tests we discuss is based on the evolution of
relativistic polytropic stars in equilibrium, but now the evolution is
performed in a dynamical spacetime, thus with the coupling of Einstein and
hydrodynamics equations. The truncation error in this case is given by the
truncation error coming from the solution of both the field equations and the
hydrodynamics equations. The results of our calculations are summarized in
Figs. 4–6, which represent the equivalents of Figs. 1–3 for full-spacetime
evolutions. Because the results are self-explanatory and qualitatively similar
to the ones discussed for the evolutions with fixed spacetimes, we will
comment on them only briefly.
Figure 5: The same as in Fig. 2, but for full-spacetime evolutions. The left
panel refers to a nonrotating star (model A in Table 1), while the right panel
to a rapidly rotating star (model B in Table 1). Figure 6: The same as Fig. 3
but for a rapidly rotating star evolved in a dynamical spacetime.
In particular, Figs. 5–6 highlight that while the overall truncation error in
dynamical spacetimes is essentially unchanged for the standard formulation, it
has increased in the case of the new formulation. This is particularly evident
at very low resolutions, where the new formulation seems to be hyper-
convergent. However, despite a truncation error which is larger than the one
for fixed spacetimes, the figures also indicate that the new formulation does
represent a considerable improvement over the standard one and that its
truncation error is at least two orders of magnitude smaller. Most
importantly, the conservation properties of the numerical scheme have greatly
improved and the secular increase in the rest mass, is also considerably
suppressed. This is clearly shown in Fig. 4, where the secular increase is
suppressed almost quadratically with resolution. More precisely, for both
approaches the growth rate of the central rest-mass density for the coarse
resolution is $\sim 12$ times larger than the corresponding one for the high
resolution. However, at the highest resolution, the growth rate for the
standard formulation is $\sim 10$ times larger than the one of the new
formulation.
### 4.3 Calculation of the eigenfrequencies
As mentioned in the previous Section, although in equilibrium, the simulated
stars undergo oscillations which are triggered by the nonzero truncation
error. It is possible to consider these oscillations not as a numerical
nuisance, on the contrary it is possible to exploit them to perform a check on
the consistency of a full nonlinear evolution with a small perturbation (the
truncation error) with the predictions of perturbation theory [46, 47].
Furthermore, when used in conjunction with highly accurate codes, these
oscillations can provide important information on the stellar oscillations
within regimes, such as those of very rapid or differential rotation, which
are not yet accessible via perturbative calculations [49].
Figure 7: Power spectral density (in arbitrary units) of the maximum rest-
mass density evolution in the new and standard formulation (solid and dashed
lines, respectively). The simulations are relative to a nonrotating star
(model A in Table 1) with the left panel referring to an evolution with a
fixed spacetime and the right one to an evolution with a dynamical spacetime.
The spectra are calculated from the simulations at the highest resolution and
cover $25\,{\rm ms}$ of evolution. For both panels the insets show a
magnification of the spectra near the $F$-mode and the comparison with the
perturbative estimate as calculated with the numerical code described in ref.
[50].
In this Section we use such oscillations, and in particular the fundamental
$\ell=0$ quasi-radial $F$-mode, to compare the accuracy of the two
formulations against the perturbative predictions. This is summarized in Fig.
7 which reports the power spectral density (in arbitrary units) of the maximum
rest-mass density evolution (cf., Figs. 1 and 3) in the new and standard
formulation (solid and dashed lines, respectively). The simulations are
relative to a nonrotating star (model A in Table 1) with the left panel
referring to an evolution with a fixed spacetime, while the right one to an
evolution with a dynamical spacetime. The specific spectra shown are
calculated from the simulations at the highest resolution and cover an
interval of $25\,{\rm ms}$. It is quite apparent that the two formulations
yield spectra which are extremely similar, with a prominent $F$-mode at about
$2.7\,{\rm kHz}$ and $1.4\,{\rm kHz}$ for the fixed and dynamical spacetime
evolutions, respectively. The spectra also show the expected quasi-radial
overtones at roughly multiple integers of the $F$-mode, the first of which has
a comparable power in the case of Cowling evolution, while it is reduced of
about $50\%$ in the full spacetime evolution. Indeed, the spectra in the two
formulations are so similar that it is necessary to concentrate on the
features of the $F$-mode to appreciate the small differences. These are shown
in the insets of the two panels which report, besides a magnification of the
spectra near the $F$-mode, also the perturbative estimate $F_{\rm pert}$, as
calculated with the perturbative code described in ref. [50].
Figure 8: Relative difference between the numerical and perturbative
eigenfrequencies of the $F$-mode for the two formulations (solid lines for the
new one and dashed lines for the standard one). The differences are computed
for different resolutions and refer to the nonrotating model A of Table 1 when
evolved in a fixed spacetime (left panel) and in a dynamical one (right
panel). Indicated with a dot-dashed line is the slope for a second-order
convergence rate.
To provide a more quantitative assessment of the accuracy with which the two
formulations reproduce the perturbative result we have computed the
eigenfrequency of the $F$-mode, which we indicate as $F_{\rm num}$, by
performing a Lorentzian fit to the power spectrum with a window of $0.2$ kHz.
We remark that it is essential to make use of a Lorentzian function for the
fit as this reflects the expected functional behaviour and increases the
accuracy of the fit significantly. Shown in Fig. 8 is the absolute value of
the relative difference between the numerical and perturbative
eigenfrequencies of the $F$-mode, $|1-F_{\rm num}/F_{\rm pert}|$ for the two
formulations (solid lines for the new one and dashed lines for the standard
one). The differences are computed for different resolutions with
$\bar{h}=0.4M,\,\bar{h}/2$ and $\bar{h}/4$ and refer to the nonrotating mode A
of Table 1 when evolved in a fixed spacetime (left panel) and in a dynamical
one (right panel). Indicated with a dot-dashed line is the slope for a second-
order convergence rate. This helps to see that both formulations yield an
almost second-order convergent measure of the eigenfrequencies of the
$F$-mode, with the new formulation having a truncation error which is always
smaller than the one coming from the standard formulation. Given the
importance of an accurate measurement of the eigenfrequencies to study the
mode properties of compact stars, we believe that Figs. 7 and 8 provide an
additional evidence of the advantages of the new formulation.
Finally, we note that a behaviour similar to the one shown in Fig. 7– 8 has
been found also for rotating stars although in this case the comparison is
possible only for evolutions within the Cowling approximation since we lack a
precise perturbative estimate of the eigenfrequency for model B of Table 1 for
a dynamical spacetime.
### 4.4 Cylindrical Shock Reflection
One of the most important properties of HRSC schemes is their capability of
handling the formation of discontinuities, such as shocks, which are often
present and play an important role in many astrophysical scenarios. Tests
involving shocks formation are usually quite demanding and codes that are not
flux-conservative can also show numerical instabilities or difficulties in
converging to the exact solution of the problem. Since both the new and the
standard formulation solve the relativistic hydrodynamics equations as written
in a flux-conservative form, they are both expected to be able to correctly
resolve the formation of shocks, although each with its own truncation error.
In the following test we consider one of such discontinuous flows and show
that the new formulation provides a higher accuracy with respect to the
standard one, stressing once again the importance of the definition of the
conserved variables.
More specifically, we consider a one-dimensional test, first proposed by [51],
describing the reflection of a shock wave in cylindrical coordinates. The
initial data consist of a pressureless gas with uniform density
$\rho_{0}=1.0$, radial velocity $v^{x}_{0}=0.999898$, corresponding to an
initial Lorentz factor $W_{0}=70.0$ and an internal energy which is taken to
be small and proportional to the initial Lorentz factor, i.e.,
$\epsilon=10^{-5}(W_{0})$. During the evolution an ideal-fluid EOS (30) is
used with a fixed adiabatic index $\Gamma=4/3$. The symmetry condition at
$x=0$ produces a compression and generates an outgoing shock in the radial
direction. The analytic solution for the values of pressure, density, gas and
shock velocities are given in [51]. From them one can determine the position
$x_{{}_{\rm S}}$ of the shock front at any time $\bar{t}$
$x_{{}_{\rm S}}\,=\,\frac{(\Gamma-1)W_{0}|v^{x}_{0}|}{W_{0}+1}\bar{t}\,.$ (33)
This can then be used to compare the accuracy of the two formulations.
In the left panel of Fig. 9 we show the value of the radial component of the
velocity $v_{x}$ as a function of $x$ at a time $\bar{t}=0.002262\,{\rm ms}$
and for a resolution of $h/M_{\odot}=6.25\times 10^{-5}$. The solid line
represent the analytic solution, the short-dashed line the numerical solution
computed with the new formulation and the long-dashed line the one obtained
with the standard formulation. As it is evident from the inset, the position
of the shock is very well captured by both formulations, but the new one is
closer to the exact one at this time.
Figure 9: Left panel: Comparison of the velocity profiles for the two
formulations in the solution of the axisymmetric shock-tube test with a
resolution of $h=6.25\times 10^{-5}\,M_{\odot}$ The solid line shows the exact
position after a time $\bar{t}=0.002262\,{\rm ms}$, while the short-dashed and
the long-dashed lines represent the solutions with the new and the standard
formulations, respectively. Right panel: Comparison of the error in the
determination of the position of the shock in the two formulations. Note the
first-order convergence rate as expected for discontinuous flows.
To compare with the exact prediction given by expression (33), we compute the
numerical position of the shock as the middle of the region where the value of
the velocity moves from the pre-shock value $v^{+}_{x}$ to the post-shock one
$v^{-}_{x}$ (in practice, we fit a straight line between the last point of the
constant post-shock state and the first point of the constant pre-shock state
and evaluate the position at which this function has value
$(v^{+}_{x}+v^{-}_{x})/2$.). The right panel of Fig. 9, shows the relative
error $1-(x_{{}_{\rm S}})_{\rm num}/(x_{{}_{\rm S}})_{\rm anal}$ in the
position of the shock at time $\bar{t}=0.002622\,{\rm ms}$ and for five
different resolutions: $\bar{h}=0.01\,M_{\odot}$, $\bar{h}/8$, $\bar{h}/40$,
$\bar{h}/80$ and $\bar{h}/160$. Indicated with a dashed line is the error
computed when using the standard formulation, while indicated with a solid
line is the error coming from the new formulation. Note that both formulations
show a first-order convergence, as expected for HRSC schemes in the presence
of a discontinuous flow, but, as for the other tests, also in this case the
new formulation has a smaller truncation error. A similar behaviour is shown
also by other quantities in this test but these are not reported here.
It is useful to note that the difference between the two formulations in this
test is smaller than in the previous ones, being of a factor of a few only and
not of orders of magnitude. We believe this is due in great part to the fact
that, in contrast with what happens for stars, the solution in the most
troublesome part of the numerical domain (i.e. near $x\sim 0$, $z\sim 0$) is
not characterized by particularly large values of the fields or of the fluid
variables. In support of this conjecture is the evidence that at earlier
times, when the shock is closer to the axis, both the absolute errors and the
difference between the two formulations are larger.
## 5 Conclusion
A number of astrophysical scenarios can be very conveniently studied
numerically by assuming they possess and preserve a rotation symmetry around
an axis. Such an assumption reduces the number of spatial dimensions to be
considered and thus the computational costs. This, in turn, allows for a more
sophisticated treatment of the physical and astrophysical processes taking
place and, as a result, for more realistic simulations.
We have presented a new numerical code developed to solve in Cartesian
coordinates the full set of general relativistic hydrodynamics equations in a
dynamical spacetime and in axisymmetry. More specifically, the new code solves
the Einstein equations by using the “cartoon” method, while HRSC schemes are
used to solve the hydrodynamic equations written in a conservative form. An
important feature of the code is the use of a novel formulation of the
equations of relativistic hydrodynamics in cylindrical coordinates. More
specifically, by exploiting a suitable definition of the conserved variables,
we removed from the source of the flux-conservative equations those terms that
presented coordinate singularities at the axis and that are usually retained
in the standard formulation of the equations. Despite their simplicity, the
changes made to the standard formulation can produce significant improvements
on the overall accuracy of the simulations with a truncation error which is
often several orders of magnitude smaller.
In order to assess the validity of the new formulation and compare its
accuracy with that of the formulation which is most commonly used in Cartesian
coordinates, we have performed several tests involving the evolution of
oscillating spherical and rotating stars, as well as shock-tube tests. In all
cases considered we have shown that the codes implementing the two
formulations yield the expected convergence rate but also that the new
formulation is always more accurate, often considerably more accurate, than
the standard one.
In view of its simplicity, the new formulation of the equations can be
implemented straightforwardly in codes written using the standard formulation
and we recommend its use for all situations in which an axisymmetric problem
needs to be investigated in full general relativity and in Cartesian
coordinates.
It is a pleasure to thank Shin’ichirou Yoshida for providing us with the
perturbative eigenfrequencies and Pedro Montero, Olindo Zanotti and Toni Font
for useful discussions. The computations were performed on the clusters
Peyote, Belladonna and Damiana of the AEI. This work was supported in part by
the DFG grant SFB/Transregio 7 and by the JSPS Postdoctoral Fellowship For
Foreign Researchers, Grant-in-Aid for Scientific Research (19-07803).
## 6 Appendix A
In what follows we recall the essential features of the “cartoon” method for
the solution of the field equations in Cartesian coordinates. Consider
therefore the computational domain to have extents $0\leq x,z\leq d_{max}$ and
$-\Delta y\leq y\leq\Delta y$, where $d_{max}$ refers to the location of the
outer boundary. Reflection symmetry with respect to the $z=0$-plane can
additionally be assumed. The Einstein equations are then solved only on the
$y=0$-plane with the derivatives in the $y$-direction being computed with
second-order centred stencils using the points at $-\Delta y,\,0\,,\Delta y$.
Taking into account axisymmetry, the rotation in the $(x,y)$ plane is defined
as
$\displaystyle
R(\phi)_{j}^{i}\,=\,\left(\begin{array}[]{ccc}\cos({\phi})&-\sin({\phi})&0\\\
\sin({\phi})&\cos({\phi})&0\\\ 0&0&1\\\ \end{array}\right)\,,$ (37)
where $R(\phi)^{-1}\,=\,R(-\phi)$ and the rotation angle is defined as
$\phi=\tan^{-1}(\pm\Delta y/\sqrt{x^{2}+(\Delta y)^{2}})$.
As commented in the main text, the values of all the quantities on the
$\pm\Delta y$ planes are computed via rotations of the corresponding values on
the $y=0$-planes. More specifically, the components of an arbitrary vector
field $T_{i}$ on the $\pm\Delta y$ planes are computed via a $\phi$-rotation
as
$\displaystyle T_{x}=T^{(0)}_{x}\cos(\phi)-T^{(0)}_{y}\sin(\phi)\,,$ (38)
$\displaystyle T_{y}=T^{(0)}_{x}\sin(\phi)+T^{(0)}_{y}\cos(\phi)\,,$ (39)
$\displaystyle T_{z}=T^{(0)}_{z}\,,$ (40)
where $T^{(0)}_{i}$ denote the corresponding components at
$(\sqrt{x^{2}+(\Delta{y})^{2}},0,z)$, which are computed via a Lagrangian
interpolation from the neighboring points on the $x$-axis. Similarly, the
components of an arbitrary tensor field $T_{ij}$ tensor will be computed as
$\displaystyle
T_{xx}=T^{(0)}_{xx}\cos^{2}(\phi)-T^{(0)}_{xy}\sin(2\phi)+T^{(0)}_{yy}\sin^{2}(\phi)\,,$
(41) $\displaystyle
T_{xy}=\frac{1}{2}T^{(0)}_{xx}\sin(2\phi)-T^{(0)}_{xy}\cos(2\phi)+\frac{1}{2}T^{(0)}_{yy}\sin(2\phi)\,,$
(42) $\displaystyle
T_{yy}=T^{(0)}_{xx}\sin^{2}(\phi)-T^{(0)}_{xy}\sin(2\phi)+T^{(0)}_{yy}\cos^{2}(\phi)\,,$
(43) $\displaystyle T_{xz}=T^{(0)}_{xz}\cos(\phi)-T^{(0)}_{yz}\sin(\phi)\,,$
(44) $\displaystyle T_{yz}=T^{(0)}_{xz}\sin(\phi)+T^{(0)}_{yz}\cos(\phi)\,,$
(45) $\displaystyle T_{zz}=T^{(0)}_{zz}\,.$ (46)
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|
arxiv-papers
| 2008-11-06T13:19:25 |
2024-09-04T02:48:58.643089
|
{
"license": "Public Domain",
"authors": "Thorsten Kellermann, Luca Baiotti, Bruno Giacomazzo and Luciano\n Rezzolla",
"submitter": "Thorsten Kellermann",
"url": "https://arxiv.org/abs/0811.0938"
}
|
0811.1086
|
# Translational Invariance and the Anisotropy of the Cosmic Microwave
Background
Sean M. Carroll, Chien-Yao Tseng and Mark B. Wise California Institute of
Technology, Pasadena, CA 91125
( - 16:56)
###### Abstract
Primordial quantum fluctuations produced by inflation are conventionally
assumed to be statistically homogeneous, a consequence of translational
invariance. In this paper we quantify the potentially observable effects of a
small violation of translational invariance during inflation, as characterized
by the presence of a preferred point, line, or plane. We explore the imprint
such a violation would leave on the cosmic microwave background anisotropy,
and provide explicit formulas for the expected amplitudes $\langle
a_{lm}a_{l^{\prime}m^{\prime}}^{*}\rangle$ of the spherical-harmonic
coefficients.
††preprint: CALT-68-2705
## I Introduction
Inflationary cosmology, originally proposed as a solution to the horizon,
flatness, and monopole problems Guth:1980zm ; Linde:Albrecht , provides a very
successful mechanism for generating primordial density perturbations. During
inflation, quantum vacuum fluctuations in a light scalar field are redshifted
far outside the Hubble radius, imprinting an approximately scale-invariant
spectrum of classical density perturbations quantumfluct ; inflationreview .
Models that realize this scenario have been widely discussed Linde:1983gd ;
Linde:1993cn ; Lyth:1998xn . The resulting perturbations give rise to large-
scale structure and temperature anisotropies in the cosmic microwave
background, in excellent agreement with observation COBE ; BOOMERANG ; ACBAR ;
CBI ; VSA ; ARCHEOPS ; DASI ; MAXIMA ; WMAP3 .
If density perturbations do arise from inflation, they provide a unique window
on physics at otherwise inaccessible energy scales. In a typical inflationary
model (although certainly not in all of them), the amplitude of density
fluctuations is of order $\delta\sim(E/M_{\rm P})^{2}$, where $E^{4}$ is the
energy density during inflation and $M_{\rm P}$ is the (reduced) Planck mass.
Since we observe $\delta\sim 10^{-5}$, it is very plausible that inflation
occurs near the scale of grand unification, and not too far from scales where
quantum gravity is relevant. Since direct experimental probes provide very few
constraints on physics at such energies, it makes sense to be open-minded
about what might happen during the inflationary era.
In a previous paper Ackerman , henceforth “ACW,” the possibility that
rotational invariance was violated by a small amount during the inflationary
era was explored (see also Gumrukcuoglu:2006xj ; ArmendarizPicon:2007nr ;
Pereira:2007yy ; Gumrukcuoglu:2007bx ; Akofor:2007fv ; Yokoyama:2008xw ;
Kanno:2008gn ). ACW suggested a simple, model-independent form for the power
spectrum of fluctuations in the presence of a small violation of statistical
isotropy, characterized by a preferred direction in space, and computed the
imprint such a violation would leave on the anisotropy of the cosmic microwave
background radiation. A toy model of a dynamical fixed-norm vector field
Kostelecky:1989jw ; Jacobson:2000xp ; Jacobson:2004ts ; Carroll:2004ai ;
Eling:2003rd ; Kostelecky:2003fs with a spacelike expectation value was
presented, which illustrated the validity of the model-independent arguments.
The spacelike vector model is not fully realistic due to the presence of
instabilities Himmetoglu , and furthermore it does not provide a mechanism for
turning off the violation of rotational invariance at the end of the
inflationary era. Nevertheless, it still provides a useful check of the
general argument that the terms which violate rotational invariance should be
scale invariant. An inflationary era that violates rotational invariance
results in a definite prediction, in terms of a few free parameters, for the
deviation of the microwave background anisotropy that can be compared with the
data Pullen:2007tu ; Groeneboom:2008fz ; ArmendarizPicon:2008yr .
The results of ACW can be thought of as one step in a systematic exploration
of the ways in which inflationary perturbations could deviate by small amounts
from the standard picture, analogously to how the $STU$ parameters of particle
physics PDG parameterize deviations from the Standard Model, or how the
Parameterized Post-Newtonian (PPN) formalism of gravity theory parameterizes
deviations from general relativity Will . In cosmology, the fiducial model is
characterized by primordial Gaussian perturbations that are statistically
homogeneous and isotropic, with an approximately scale-free spectrum. Even in
the absence of an underlying dynamical model, it is useful to quantify how
well existing and future experiments constrain departures from this paradigm.
Deviations from a scale-free spectrum are quantified by the spectral index
$n_{s}$ and its derivatives; deviations from Gaussianity are quantified by the
parameter $f_{NL}$ of the three-point function (and its higher-order
generalizations) Linde:1996gt ; Hu:2001fa ; Acquaviva:2002ud ;
Maldacena:2002vr ; Lyth:2005fi ; Chen:2006nt . The remaining features of the
fiducial model, statistical homogeneity and isotropy, are derived from the
spatial symmetries of the underlying dynamics.
There is another important motivation for studying deviations from pure
statistical isotropy of cosmological perturbations: a number of analyses have
suggested evidence that such deviations might exist in the real world
Rakic:2007ve . These include the “axis of evil” alignment of low multipoles
deOliveiraCosta:2003pu ; Schwarz:2004gk ; Copi:2003kt ; Land:2005ad ;
Gordon:2005ai ; Land:2006bn ; Copi:2006tu ; Hajian:2007pi ; Jain:1999 , the
existence of an anomalous cold spot in the CMB Cruz:2004ce ; Rudnick:2007kw ;
Smith:2008tc , an anomalous dipole power asymmetry Eriksen:2003db ;
Hansen:2004vq ; Eriksen:2007pc ; Gordon:2006ag ; Erickcek:2008sm , a claimed
“dark flow” of galaxy clusters measured by the Sunyaev-Zeldovich effect
Kashlinsky:2008ut , as well as a possible detection of a quadrupole power
asymmetry of the type predicted by ACW in the WMAP five-year data
Groeneboom:2008fz . In none of these cases is it beyond a reasonable doubt
that the effect is more than a statistical fluctuation, or an unknown
systematic effect; nevertheless, the combination of all of them is suggestive
Ralston:2004 . It is possible that statistical isotropy/homogeneity is
violated at very high significance in some specific fashion that does not
correspond precisely to any of the particular observational effects that have
been searched for, but that would stand out dramatically in a better-targeted
analysis.
The isometries of a flat Robertson-Walker cosmology are defined by $E(3)$, the
Euclidean group in three dimensions, which is generated by the three
translations $R^{3}$ and the spatial rotations $O(3)$. Our goal is to break as
little of this symmetry as is possible in a consistent framework. A preferred
vector, considered by ACW Ackerman , leaves all three translations unbroken,
as well as an $O(2)$ representing rotations around the axis defined by the
vector. If we break some subgroup of the translations, there are three minimal
possibilities, characterized by preferred Euclidean submanifolds in space. A
preferred point breaks all of the translations, and preserves the entire
rotational $O(3)$. A preferred line leaves one translational generator
unbroken, as well as one rotational generator around the axis defined by the
line. Finally, a preferred plane leaves the two translations within the plane
unbroken, as well as a single rotation around an axis perpendicular to that
plane. We will consider each of these possibilities in this paper.
A random variable $\phi({\bf x})$ is statistically homogeneous (or
translationally invariant) if all of its correlation functions
$\langle\phi({\bf x}_{1})\phi({\bf x}_{2})\cdots\rangle$ depend only on the
differences ${\bf x}_{i}-{\bf x}_{j}$, and is statistically isotropic (or
rotationally invariant) about some point ${\bf z}_{*}$ if the correlations
depend only on dot products of any of the vectors $({\bf x}_{i}-{\bf z}_{*})$
and $({\bf x}_{i}-{\bf x}_{j})$. The Fourier transform of the two-point
function $\langle\phi({\bf x}_{1})\phi({\bf x}_{2})\rangle$ depends on two
wavevectors ${\bf k}$ and ${\bf q}$, and will be translationally invariant if
it only has support when ${\bf k}=-{\bf q}$. We will show how to perform a
systematic expansion in powers of ${\bf p}={\bf k}+{\bf q}$. ACW showed how a
small violation of rotational invariance during inflation would be manifested
in a violation of statistical isotropy of the CMB; here we perform a
corresponding analysis for a small violation of translational invariance.
At energies accessible to laboratory experiments, translational invariance
plays a pivotal role, since it is responsible for the conservation of
momentum. Here we are specifically concerned with the possibility that
translational invariance may have been broken during inflation by an effect
that disappeared after the inflationary era ended. Such a phenomenon could
conceivably arise from the presence of some sort of source that remained in
our Hubble patch through inflation, although we do not consider any specific
models along those lines.
## II Setup For a Special Point
In the standard inflationary cosmology the primordial density perturbations
$\delta({\bf x})$ have a Fourier transform ${\tilde{\delta}}({\bf k})$,
defined by
$\delta({\bf x})=\int d^{3}ke^{i{\bf k}\cdot{\bf x}}{\tilde{\delta}}({\bf
k}),$ (1)
and the power spectrum $P(k)$ is defined by
$\langle{\tilde{\delta}}({\bf k}){\tilde{\delta}}({\bf
q})\rangle=P(k)\delta^{3}({\bf k}+{\bf q}).$ (2)
so that
$\langle\delta({\bf x})\delta({\bf y})\rangle=\int d^{3}ke^{i{\bf k}\cdot({\bf
x}-{\bf y)}}P(k).$ (3)
The Dirac delta function in Eq. (2) implies that modes with different
wavenumbers are uncoupled. This is a consequence of translational invariance
during the inflationary era, while the fact that the power spectrum $P(k)$
only depends on the magnitude of the vector ${\bf k}$ is a consequence of
rotational invariance.
Suppose that during the inflationary era translational invariance is broken by
the presence of a special point with comoving coordinates ${\bf z_{*}}$. This
is reflected in the statistical properties of the density perturbation
$\delta({\bf x})$. It is possible that the violation of translational
invariance impacts the classical background for the inflation field during
inflation and this induces a one point function,
$\langle\delta({\bf x})\rangle=G\left[|{\bf x}-{{\bf z}_{*}}|\right].$ (4)
Throughout this paper we will assume that this classical piece is small
(consistent with current data) and concentrate on the two-point function,
which now takes the form
$\langle\delta({\bf x})\delta({\bf y})\rangle=F\left[|{\bf x}-{\bf y}|,|{\bf
x}-{{\bf z}_{*}}|,|{\bf y}-{{\bf z}_{*}}|,({\bf x}-{{\bf z}_{*}})\cdot({\bf
y}-{{\bf z}_{*}})\right]\,,$ (5)
where $F$ is symmetric under interchange of ${\bf x}$ and ${\bf y}$. This is
the most general form of the two point correlation that is invariant under the
transformations ${\bf x}\rightarrow{\bf x}+{\bf a}$, ${\bf y}\rightarrow{\bf
y}+{\bf a}$, ${\bf{{\bf z}_{*}}}\rightarrow{\bf{{\bf z}_{*}}}+{\bf a}$, and
rotational invariance about ${\bf z_{*}}$
It is convenient to work with a form for $\langle\delta({\bf x})\delta({\bf
y})\rangle$ that is analogous to Eq. (3). We write,
$\langle\delta({\bf x})\delta({\bf y})\rangle=\int d^{3}k\int
d^{3}q~{}e^{i{\bf k}\cdot({\bf x}-{\bf z_{*}})}e^{i{\bf q}\cdot({\bf y}-{\bf
z_{*}})}P_{t}(k,q,{\bf k}\cdot{\bf q}),$ (6)
where $P_{t}$ is symmetric under interchange of $\bf k$ and $\bf q$. This is
equivalent to Eq. (5) and is the most general form for the density
perturbation’s two-point correlation that breaks statistical translational
invariance by the presence of a special point ${\bf z}_{*}$, preserving
rotational invariance about that point. In the limit where the violations of
translational invariance are small and can be neglected, the replacement
$P_{t}(|{\bf k}|,|{\bf q}|,{\bf k}\cdot{\bf q})\rightarrow P(k)\delta^{3}({\bf
k}+{\bf q})$ is valid.
We assume (as is consistent with the data) that violations of translational
invariance are small and hence that $P_{t}$ is strongly peaked about ${\bf
k}=-{\bf q}$. Hence we introduce the variables ${\bf p}={\bf k}+{\bf q}$,
${\bf l}=({\bf k}-{\bf q})/2$ and to expand in ${\bf p}$ using, for example,
$k=|{\bf l}+{\bf p}/2|=l+{{\bf p}\cdot{\bf l}\over 2l}-{({\bf p}\cdot{\bf
l})^{2}\over 8l^{3}}+{p^{2}\over 8l}+...$ (7)
It is convenient to introduce $U_{t}={\rm ln}P_{t}$ and expand $U_{t}$ to
quadratic order in ${\bf p}$, neglecting the higher order terms since $P_{t}$
and hence $U_{t}$ is dominated by wavevectors ${\bf p}$ near ${\bf p}=0$,
$P_{t}(|{\bf k}|,|{\bf q}|,{\bf k}\cdot{\bf
q})=e^{U_{t}(l,l,-l^{2})-A(l){p^{2}/2}-B(l)({\bf p}\cdot{\bf
l})^{2}/(2l^{2})+\ldots}\simeq P_{t}(l,l,-l^{2})e^{-A(l){p^{2}}/2-B(l)({\bf
p}\cdot{\bf l})^{2}/(2l^{2})}.$ (8)
Note that there are no terms linear in ${\bf p}$ because the symmetry under
interchange of ${\bf k}$ and ${\bf q}$ implies symmetry under ${\bf
l}\rightarrow-{\bf l}$ and ${\bf p}\rightarrow{\bf p}$.
Plugging the expansion of $P_{t}$ in Eq. (8) into Eq. (6) yields
$\langle\delta({\bf x})\delta({\bf y})\rangle=\int d^{3}l~{}e^{i{\bf
l}\cdot({\bf x}-{\bf y})}P_{t}(l,l,-l^{2})\int
d^{3}p~{}e^{-A(l){p^{2}}/2-B(l)({\bf p}\cdot{\bf l})^{2}/(2l^{2})}e^{i{\bf
p}\cdot{\bf z}},$ (9)
where ${\bf z}=({\bf x}+{\bf y}-2{\bf z_{*}})/2$. The integral over $d^{3}p$
can be performed by completing the square in the argument of the exponential.
Introducing the $3\times 3$ matrix,
$C_{ij}=A(l)\delta_{ij}+B(l){l_{i}l_{j}\over l^{2}}$ (10)
we find that
$\int d^{3}p~{}e^{-A(l){p^{2}}/2-B(l)({\bf p}\cdot{\bf
l})^{2}/(2l^{2})}e^{i{\bf p}\cdot{\bf z}}=\sqrt{{{(2\pi)^{3}}\over{\rm
det}C}}e^{-z^{T}C^{-1}z/2}\simeq\sqrt{{{(2\pi)^{3}}\over{\rm
det}C}}(1-z^{T}C^{-1}z/2).$ (11)
Using this expression the two-point function can be written as
$\langle\delta({\bf x})\delta({\bf y})\rangle=\int d^{3}l~{}e^{i{\bf
l}\cdot({\bf x}-{\bf y})}P_{t}(l,l,-l^{2})\sqrt{{{(2\pi)^{3}}\over{\rm
det}C}}\left(1-{z^{T}C^{-1}z\over 2}+\ldots\right),$ (12)
where the ellipses represent terms higher than quadratic order in the
components of $\bf z$. It is straightforward to solve for $C^{-1}$ and ${\rm
det}C$ in terms of the functions $A$ and $B$ . We find that ${\rm
det}C=A^{3}+A^{2}B$ and
$C^{-1}_{ij}={1\over A}\delta_{ij}-{B\over A(A+B)}{l_{i}l_{j}\over l^{2}}.$
(13)
The part of the two point correlation that is rotationally invariant is the
usual power spectrum $P(l)$, so
$P(l)=\sqrt{{{(2\pi)^{3}}\over{\rm det}C}}P_{t}(l,l,-l^{2}).$ (14)
Next we construct some mathematical examples that illustrate how the term
proportional to $z^{2}$ is suppressed when $P_{t}$ is very strongly peaked at
$p=0$. Without any violation of translational invariance, $P_{t}(|{\bf
k}|,|{\bf q}|,{\bf k}\cdot{\bf q})=P(k)\delta^{3}({\bf k}+{\bf
q})=c/k^{3}\delta^{3}({\bf k}+{\bf q})$ for a scale-invariant Harrison
Zeldovich power spectrum, where $c$ is some constant. We want to construct a
form for $P_{t}$ that reduces to the standard Harrisson-Zeldovich spectrum
with translational and rotational invariance as a parameter
$d\rightarrow\infty$. The three dimensional delta function can be written as
$\delta^{3}({\bf k}+{\bf
q})=\lim_{d\rightarrow\infty}\left(\frac{d}{\sqrt{\pi}}\right)^{3}e^{-d^{2}({\bf
k}+{\bf q})^{2}}$ (15)
Therefore, we might try writing $P_{t}$ as
$c/k^{3}\left(\frac{d}{\sqrt{\pi}}\right)^{3}e^{-d^{2}({\bf k}+{\bf q})^{2}}$
with $d$ a large number. However, this $P_{t}$ is not symmetric under the
interchange of $\bf k$ and $\bf q$ because $k^{3}$ is not.
There are many possible ways to resolve this problem. We might imagine
replacing $k^{3}$ by $k^{3/2}q^{3/2}$, $(k+q)^{3}/8$, $|{\bf k}-{\bf
q}|^{3}/8$, $kq(k+q)/2$, $(kq)^{1/2}(k+q)^{2}/4$, $({\bf k}\cdot{\bf
q})(k+q)/2$ $\cdots$, or any linear combinations of these. With ${\bf p}={\bf
k}+{\bf q}$, ${\bf l}=({\bf k}-{\bf q})/2$, we have
$k^{3/2}q^{3/2}=l^{3}\left(1-\frac{3({\bf p}\cdot{\bf
l})^{2}}{4l^{4}}+\frac{3p^{2}}{8l^{2}}\right),\frac{1}{8}(k+q)^{3}=l^{3}\left(1-\frac{3({\bf
p}\cdot{\bf l})^{2}}{8l^{4}}+\frac{3p^{2}}{8l^{2}}\right)~{}~{}\cdots,$ (16)
to second order in ${\bf p}$. Therefore, at quadratic order in ${\bf p}$, the
most general form of a function which is symmetric under the interchange of
$\bf k$ and $\bf q$ and reduces to $k^{3}$ when $\bf k=-\bf q$ is
$l^{3}\left(1-a\frac{({\bf p}\cdot{\bf
l})^{2}}{l^{4}}-b\frac{p^{2}}{l^{2}}\right),$ (17)
with two parameters $a$ and $b$ that are independent of $l$. Hence we arrive
at the following form for $P_{t}(|{\bf k}|,|{\bf q}|,{\bf k}\cdot{\bf q})$,
$P_{t}(|{\bf k}|,|{\bf q}|,{\bf k}\cdot{\bf
q})=\frac{1}{l^{3}}c\left(1+a\frac{({\bf p}\cdot{\bf
l})^{2}}{l^{4}}+b\frac{p^{2}}{l^{2}}\right)\left(\frac{d}{\sqrt{\pi}}\right)^{3}e^{-d^{2}p^{2}},$
(18)
which gives the familiar translationally (and rotationally) invariant density
perturbations with a Harrison-Zeldovich spectrum as $d\rightarrow\infty$.
Plugging into Eq. (6), the two-point function becomes
$\langle\delta({\bf x})\delta({\bf y})\rangle=c(1-\frac{z^{2}}{4d^{2}})\int
d^{3}l~{}e^{i{\bf l}\cdot({\bf x}-{\bf
y})}\frac{1}{l^{3}}\left(1+\frac{1}{2d^{2}}\frac{a+3b}{l^{2}}\right).$ (19)
We can construct another example which also gives dependence on $({\bf
l}\cdot{\bf z})^{2}$. First notice that the three dimensional delta function
can be written as another form,
$\delta^{3}({\bf
p})=\lim_{d\rightarrow\infty}\left(\frac{d}{\sqrt{2\pi}}\right)^{3}\sqrt{{\rm
det}U}e^{-\frac{d^{2}}{2}p^{i}U_{ij}p^{j}},$ (20)
where $U_{ij}=2(\delta_{ij}+fl_{i}l_{j}/l^{2})$ and $f$ is an arbitrary
parameter independent of $\bf l$. So another possible choice for $P_{t}$ that
has the correct limiting behavior as $d\rightarrow\infty$ is
$P_{t}(|{\bf k}|,|{\bf q}|,{\bf k}\cdot{\bf
q})=\frac{1}{l^{3}}c(1+a\frac{({\bf p}\cdot{\bf
l})^{2}}{l^{4}}+b\frac{p^{2}}{l^{2}})\left(\frac{d}{\sqrt{2\pi}}\right)^{3}\sqrt{{\rm
det}U}e^{-\frac{d^{2}}{2}p^{i}U_{ij}p^{j}}.$ (21)
This gives,
$\langle\delta({\bf x})\delta({\bf y})\rangle=\int d^{3}l~{}e^{i{\bf
l}\cdot({\bf x}-{\bf
y})}\frac{1}{l^{3}}c\left(1+\frac{a+(3+2f)b}{2(1+f)d^{2}l^{2}}\right)\left[1-\frac{z^{2}}{4d^{2}}+\frac{f}{4(1+f)d^{2}}\frac{({\bf
l}\cdot{\bf z})^{2}}{l^{2}}\right].$ (22)
Since observable $|{\bf z}|$’s can be as large as our horizon, we need the
parameter $d$ to be of that order (or larger) for the leading two terms of the
expansion in $z$ to be a good approximation in Eq. (19) and (22).
The form we have derived in this section is plausible but is not the most
general. For example, it could be that the Fourier transform of the two point
function has the usual form plus a small piece that is proportional to a small
parameter $\epsilon$. That is,
$P_{t}(|{\bf k}|,|{\bf q}|,{\bf k}\cdot{\bf q})={c\over k^{3}}\delta^{3}({\bf
k}+{\bf q})+\epsilon P^{\prime}_{t}(|{\bf k}|,|{\bf q}|,{\bf k}\cdot{\bf q})$
(23)
If $\epsilon$ is small then the effects of the violation of translational
invariance in Eq. (23) is small even when $P_{t}^{\prime}$ is not strongly
peaked about ${\bf k}=-{\bf q}$.
In the next section we discuss how the violation of translational invariance
during the inflationary era by the presence of a special point at fixed
comoving coordinate impacts the anisotropy of the microwave background. Then
in section IV we generalize the results of this section to the possibility
that the violation of translation invariance during the inflationary era
occurs because of a special line or plane during the inflationary era.
## III Microwave Background Anisotropy with a Special Point
We are interested in a quantitative understanding of how the second term in
Eq. (12) changes the prediction for the microwave background asymmetry from
the conventional translationally invariant one. The multipole moments of the
microwave background radiation are defined by
$a_{lm}=\int{\rm d}\Omega_{\bf e}Y_{l}^{m}({\bf e}){\Delta T\over T}({\bf
e}).$ (24)
(Note that our definition differs from the conventional one111To shift our
results to what the the usual definition gives, $a_{lm}\rightarrow
a_{lm}^{*}$. in which the complex conjugate of $Y_{l}^{m}$ appears in the
integral.) Since the violation of translational invariance vanishes after the
inflationary era ends, the anisotropy of the microwave background temperature
$T$ along the direction of the unit vector $\bf e$ is related to the
primordial fluctuations by
${\Delta T\over T}({\bf e})=\int d^{3}k\sum_{l}\left({2l+1\over
4\pi}\right)(-i)^{l}P_{l}({\hat{\bf k}}\cdot{\bf e}){\tilde{\delta}}({\bf
k})\Theta_{l}(k),$ (25)
where $P_{l}$ is the Legendre polynomial of order $l$ and $\Theta_{l}(k)$ is a
known real function of the magnitude of the wave vector ${\bf k}$ that
includes, for example, the effects of the transfer function.
We are interested in computing $\langle
a_{lm}a_{l^{\prime}m^{\prime}}^{*}\rangle$ to first order in the small
correction that violates translational invariance. This is related to the two-
point function in momentum space via
$\left\langle
a_{lm}a^{*}_{l^{\prime}m^{\prime}}\right\rangle=(-i)^{l-l^{\prime}}\int
d^{3}kd^{3}q~{}Y_{l}^{m}(\hat{\bf
k}){Y_{l^{\prime}}^{m^{\prime}}}^{*}(\hat{\bf
q})\Theta_{l}(k)\Theta_{l^{\prime}}(q)\langle\tilde{\delta}({\bf
k})\tilde{\delta}^{*}({\bf q})\rangle.$ (26)
From Eq. (12) to Eq.(14), we have
$\langle\delta({\bf x})\delta({\bf y})\rangle=\int d^{3}l~{}e^{i{\bf
l}\cdot({\bf x}-{\bf y})}P_{0}(l)+\frac{({\bf x}+{\bf y}-2{\bf
z_{*}})^{2}}{4}\int d^{3}l~{}e^{i{\bf l}\cdot({\bf x}-{\bf y})}P_{1}(l)+\int
d^{3}l~{}e^{i{\bf l}\cdot({\bf x}-{\bf y})}P_{2}(l)\frac{\left[\bf l\cdot({\bf
x}+{\bf y}-2{\bf z_{*}})\right]^{2}}{4l^{2}}$ (27)
where
$\displaystyle P_{1}(l)$ $\displaystyle=$
$\displaystyle-\frac{P_{0}(l)}{2A(l)}$ (28) $\displaystyle P_{2}(l)$
$\displaystyle=$
$\displaystyle\frac{B(l)}{2A(l)\left[A(l)+B(l)\right]}P_{0}(l)$ (29)
The models in Section II had $P_{1,2}(l)$ proportional to $P_{0}(l)$. The
special point ${\bf z_{*}}$ is characterized by three parameters; the
magnitude of its distance from our location and two parameters for its
direction (with respect to our location). Hence the corrections to the
correlations $\langle a_{lm}a_{l^{\prime}m^{\prime}}^{*}\rangle$ are
characterized by just five parameters. The Fourier transform of Eq. (27)
yields
$\displaystyle\langle\tilde{\delta}({\bf k})\tilde{\delta}^{*}({\bf
q})\rangle$ $\displaystyle=$
$\displaystyle\int{d^{3}x\over(2\pi)^{3}}\int{d^{3}y\over(2\pi)^{3}}~{}e^{-i{\bf
k}\cdot{\bf x}}e^{i{\bf q}\cdot{\bf y}}\left\langle\delta({\bf x})\delta({\bf
y})\right\rangle$ (30) $\displaystyle=$ $\displaystyle P_{0}(k)\delta^{3}({\bf
k}-{\bf q})+\frac{(i\nabla_{\bf k}-i\nabla_{\bf q}-2{{\bf
z}_{*}})^{2}}{4}P_{1}(k)\delta^{3}({\bf k}-{\bf q})$ $\displaystyle+$
$\displaystyle\sum_{i,j=1}^{3}\frac{1}{4}\left(i\frac{\partial}{\partial
k_{i}}-i\frac{\partial}{\partial
q_{i}}-2z_{*}^{i}\right)\left(i\frac{\partial}{\partial
k_{j}}-i\frac{\partial}{\partial
q_{j}}-2z_{*}^{j}\right)\left[P_{2}(k)\frac{k_{i}k_{j}}{k^{2}}\delta^{3}({\bf
k}-{\bf q})\right].$
We therefore define
$\langle a_{lm}a_{l^{\prime}m^{\prime}}^{*}\rangle=\langle
a_{lm}a_{l^{\prime}m^{\prime}}^{*}\rangle_{0}+(-i)^{l-l^{\prime}}\Delta_{1}(l,m;l^{\prime},m^{\prime})+(-i)^{l-l^{\prime}}\Delta_{2}(l,m;l^{\prime},m^{\prime}),$
(31)
where the subscript $0$ denotes the usual translationally invariant piece,
$\langle
a_{lm}a_{l^{\prime}m^{\prime}}^{*}\rangle_{0}=\delta_{l,l^{\prime}}\delta_{m,m^{\prime}}\int_{0}^{\infty}{\rm
d}kk^{2}P_{0}(k)\Theta_{l}(k)^{2}.$ (32)
and the correction coming from $P_{1}(k)$ is given by
$\displaystyle\Delta_{1}(l,m;l^{\prime},m^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{4}\int d^{3}kP_{1}(k)\left[-Y_{l}^{m}(\hat{\bf
k})\Theta_{l}(k)\textrm{\boldmath$\nabla_{k}$}^{2}\left(Y_{l^{\prime}}^{m^{\prime}*}({\hat{\bf
k}})\Theta_{l^{\prime}}(k)\right)-Y_{l^{\prime}}^{m^{\prime}*}(\hat{\bf
k})\Theta_{l^{\prime}}(k)\textrm{\boldmath$\nabla_{k}$}^{2}\left(Y_{l}^{m}({\hat{\bf
k}})\Theta_{l}(k)\right)\right.$ (33)
$\displaystyle\left.+2\textrm{\boldmath$\nabla_{k}$}\left(Y_{l}^{m}({\hat{\bf
k}})\Theta_{l}(k)\right)\cdot\textrm{\boldmath$\nabla_{k}$}\left(Y_{l^{\prime}}^{m^{\prime}*}({\hat{\bf
k}})\Theta_{l^{\prime}}(k)\right)+4z_{*}^{2}~{}Y_{l}^{m}(\hat{\bf
k})Y_{l^{\prime}}^{m^{\prime}*}(\hat{\bf
k})\Theta_{l}(k)\Theta_{l^{\prime}}(k)\right.$
$\displaystyle\left.+4iY_{l^{\prime}}^{m^{\prime}*}({\hat{\bf
k}})\Theta_{l^{\prime}}(k)~{}{{\bf
z}_{*}}\cdot\textrm{\boldmath$\nabla_{k}$}\left(Y_{l}^{m}({\hat{\bf
k}})\Theta_{l}(k)\right)-4iY_{l}^{m}({\hat{\bf k}})\Theta_{l}(k)~{}{{\bf
z}_{*}}\cdot\textrm{\boldmath$\nabla_{k}$}\left(Y_{l^{\prime}}^{m^{\prime}*}({\hat{\bf
k}})\Theta_{l^{\prime}}(k)\right)\right].$
It is convenient to break up $\Delta_{1}(l,m;l^{\prime},m^{\prime})$ into the
parts quadratic in ${\bf z}_{*}$, linear in ${\bf z}_{*}$, and independent of
${\bf z}_{*}$, by writing
$\Delta_{1}(l,m;l^{\prime},m^{\prime})=\Delta_{1}^{(2)}(l,m;l^{\prime},m^{\prime})+\Delta_{1}^{(1)}(l,m;l^{\prime},m^{\prime})+\Delta_{1}^{(0)}(l,m;l^{\prime},m^{\prime}).$
(34)
The quadratic piece is relatively simple,
$\Delta_{1}^{(2)}(l,m;l^{\prime},m^{\prime})=\delta_{l,l^{\prime}}\delta_{m,m^{\prime}}~{}z_{*}^{2}\int_{0}^{\infty}{\rm
d}kk^{2}\Theta_{l}(k)^{2}P_{1}(k).$ (35)
The term linear in ${\bf z}_{*}$ is the most complicated. It can be evaluated
using the identity
$i{\textrm{\boldmath$\nabla_{k}$}}(\Theta_{l}(k)Y_{l}^{m}({\hat{\bf
k}}))=i\hat{\bf k}\left({\partial\Theta_{l}(k)\over\partial
k}\right)Y_{l}^{m}({\hat{\bf k}})+{1\over k}{\hat{\bf k}}{\bf\times}({\bf
L}_{\bf k}Y_{l}^{m}({\hat{\bf k}}))\Theta_{l}(k),$ (36)
where ${\bf L}_{\bf k}$ acts as the angular momentum operator in Fourier
space,
${\bf L}_{\bf k}=-i{\bf
k}{\textrm{\boldmath$\times$}}{\textrm{\boldmath$\nabla_{k}$}}.$ (37)
It is convenient to divide $\Delta_{1}^{(1)}(l,m;l^{\prime},m^{\prime})$ into
a piece coming from the first term in Eq. (36) and a term coming from the
second term in Eq. (36),
$\Delta_{1}^{(1)}(l,m;l^{\prime},m^{\prime})=\Delta_{1}^{(1)}(l,m;l^{\prime},m^{\prime})_{a}+\Delta_{1}^{(1)}(l,m;l^{\prime},m^{\prime})_{b}.$
(38)
To evaluate $\Delta_{1}^{(1)}(l,m;l^{\prime},m^{\prime})_{a,b}$, we express
the components of ${\bf z}_{*}$ in terms of its “spherical components,”
$z_{+}=-{z_{*1}-iz_{*2}\over\sqrt{2}},~{}~{}~{}~{}z_{-}={z_{*1}+iz_{*2}\over\sqrt{2}},~{}~{}~{}~{}z_{0}=z_{*3},$
(39)
and express the components ${\bf\hat{k}}$ in terms of the spherical harmonics
$Y_{1}^{m}({\bf{\hat{k}}})$. This gives
$\Delta_{1}^{(1)}(l,m;l^{\prime},m^{\prime})_{a}=i\int_{0}^{\infty}{\rm
d}k~{}k^{2}P_{1}(k)\left(\Theta_{l^{\prime}}(k){\partial\Theta_{l}(k)\over\partial
k}-\Theta_{l}(k){\partial\Theta_{l^{\prime}}(k)\over\partial
k}\right)\left(z_{+}\chi^{(a)+}_{lm;l^{\prime}m^{\prime}}+z_{-}\chi^{(a)-}_{lm;l^{\prime}m^{\prime}}+z_{0}\chi^{(a)0}_{lm;l^{\prime}m^{\prime}}\right),$
(40)
where
$\chi^{(a)0}_{l,m;l^{\prime},m^{\prime}}=\left[(l-m+1)(l+m+1)\over(2l+1)(2l+3)\right]^{1/2}\delta_{l+1,l^{\prime}}\delta_{m,m^{\prime}}+\left[(l-m)(l+m)\over(2l-1)(2l+1)\right]^{1/2}\delta_{l-1,l^{\prime}}\delta_{m,m^{\prime}},$
(41)
$\chi^{(a)+}_{l,m;l^{\prime},m^{\prime}}={1\over\sqrt{2}}\left[(l+m+1)(l+m+2)\over(2l+1)(2l+3)\right]^{1/2}\delta_{l+1,l^{\prime}}\delta_{m+1,m^{\prime}}-{1\over\sqrt{2}}\left[(l-m)(l-m-1)\over(2l-1)(2l+1)\right]^{1/2}\delta_{l-1,l^{\prime}}\delta_{m+1,m^{\prime}},$
(42)
and
$\chi^{(a)-}_{l,m;l^{\prime},m^{\prime}}=\chi^{(a)+}_{l,-m;l^{\prime},-m^{\prime}}.$
(43)
For $\Delta_{1}^{(1)}(l,m;l^{\prime},m^{\prime})_{b}$ we write
$\Delta_{1}^{(1)}(l,m;l^{\prime},m^{\prime})_{b}=\Delta_{1}^{(1)^{\prime}}(l,m;l^{\prime},m^{\prime})_{b}+{\Delta_{1}^{(1)^{\prime}}(l^{\prime},m^{\prime};l,m)_{b}}^{*},$
(44)
and find that
$\Delta_{1}^{(1)^{\prime}}(l,m;l^{\prime},m^{\prime})_{b}=-i\int_{0}^{\infty}{\rm
d}k~{}kP_{1}(k)\Theta_{l}(k)\Theta_{l^{\prime}}(k)\left(z_{+}\chi^{(b)+}_{lm;l^{\prime}m^{\prime}}+z_{-}\chi^{(b)-}_{lm;l^{\prime}m^{\prime}}+z_{0}\chi^{(b)0}_{lm;l^{\prime}m^{\prime}}\right),$
(45)
where
$\chi^{(b)0}_{l,m;l^{\prime},m^{\prime}}=l\left[(l-m+1)(l+m+1)\over(2l+1)(2l+3)\right]^{1/2}\delta_{l+1,l^{\prime}}\delta_{m,m^{\prime}}-(l+1)\left[(l-m)(l+m)\over(2l-1)(2l+1)\right]^{1/2}\delta_{l-1,l^{\prime}}\delta_{m,m^{\prime}},$
(46)
$\chi^{(b)+}_{l,m;l^{\prime},m^{\prime}}={l\over\sqrt{2}}\left[(l+m+1)(l+m+2)\over(2l+1)(2l+3)\right]^{1/2}\delta_{l+1,l^{\prime}}\delta_{m+1,m^{\prime}}+{l+1\over\sqrt{2}}\left[(l-m)(l-m-1)\over(2l-1)(2l+1)\right]^{1/2}\delta_{l-1,l^{\prime}}\delta_{m+1,m^{\prime}},$
(47)
and
$\chi^{(b)-}_{l,m;l^{\prime},m^{\prime}}=\chi^{(b)+}_{l,-m;l^{\prime},-m^{\prime}}.$
(48)
Then we evaluate the term independent of ${\bf z}_{*}$ in
$\Delta_{1}(l,m;l^{\prime},m^{\prime})$. Using integration by parts, we know
$\displaystyle\int
d^{3}kP_{1}(k)\textrm{\boldmath$\nabla_{k}$}\left(Y_{l}^{m}({\hat{\bf
k}})\Theta_{l}(k)\right)\cdot\textrm{\boldmath$\nabla_{k}$}\left(Y_{l^{\prime}}^{m^{\prime}*}({\hat{\bf
k}})\Theta_{l^{\prime}}(k)\right)$ $\displaystyle=\int
d^{3}k\left[-Y_{l^{\prime}}^{m^{\prime}*}({\hat{\bf
k}})\Theta_{l^{\prime}}(k)\frac{\partial P_{1}(k)}{\partial k}\hat{\bf
k}\cdot\textrm{\boldmath$\nabla_{k}$}\left(Y_{l}^{m}({\hat{\bf
k}})\Theta_{l}(k)\right)-P_{1}(k)Y_{l^{\prime}}^{m^{\prime}*}({\hat{\bf
k}})\Theta_{l^{\prime}}(k)\textrm{\boldmath$\nabla_{k}$}^{2}\left(Y_{l}^{m}({\hat{\bf
k}})\Theta_{l}(k)\right)\right]$ (49)
Another familiar result of spherical harmonics is
$-\nabla^{2}_{\bf k}Y_{l}^{m}({\hat{\bf k}})\Theta_{l}(k)=\left[-{1\over
k^{2}}{\partial\over\partial k}\left(k^{2}{\partial\Theta_{l}(k)\over\partial
k}\right)+{l(l+1)\over k^{2}}\Theta_{l}(k)\right]Y_{l}^{m}({\hat{\bf k}}).$
(50)
Combining Eq. (36), (III), and (50) implies that,
$\Delta_{1}^{(0)}(l,m;l^{\prime},m^{\prime})=\delta_{l,l^{\prime}}\delta_{m,m^{\prime}}\int_{0}^{\infty}{\rm
d}k\left[-P_{1}(k)\Theta_{l}(k){\partial\over\partial
k}\left(k^{2}{\partial\Theta_{l}(k)\over\partial
k}\right)+l(l+1)P_{1}(k)\Theta_{l}(k)^{2}-\frac{1}{2}k^{2}\frac{\partial
P_{1}(k)}{\partial k}\frac{\partial\Theta_{l}(k)}{\partial
k}\Theta_{l}(k)\right]$ (51)
The next step is to calculate the correction coming from $P_{2}(k)$.
$\displaystyle\Delta_{2}(l,m;l^{\prime},m^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{4}\int
d^{3}kP_{2}(k)\left[\vphantom{\sum_{i,j=1}^{3}\frac{k_{i}k_{j}}{k^{2}}}4(\hat{\bf
k}\cdot{{\bf z}_{*}})^{2}~{}Y_{l}^{m}(\hat{\bf
k})Y_{l^{\prime}}^{m^{\prime}*}(\hat{\bf
k})\Theta_{l}(k)\Theta_{l^{\prime}}(k)\right.$ (52)
$\displaystyle\left.+4i\left(\hat{\bf k}\cdot{{\bf
z}_{*}}\right)\left(Y_{l^{\prime}}^{m^{\prime}*}({\hat{\bf
k}})\Theta_{l^{\prime}}(k)~{}\hat{\bf
k}\cdot\textrm{\boldmath$\nabla_{k}$}\left(Y_{l}^{m}({\hat{\bf
k}})\Theta_{l}(k)\right)-Y_{l}^{m}({\hat{\bf k}})\Theta_{l}(k)~{}\hat{\bf
k}\cdot\textrm{\boldmath$\nabla_{k}$}\left(Y_{l^{\prime}}^{m^{\prime}*}({\hat{\bf
k}})\Theta_{l^{\prime}}(k)\right)\right)\right.$
$\displaystyle\left.-\sum_{i,j=1}^{3}\frac{k_{i}k_{j}}{k^{2}}\left(Y_{l}^{m}(\hat{\bf
k})\Theta_{l}(k)\frac{\partial}{\partial k_{i}}\frac{\partial}{\partial
k_{j}}\left(Y_{l^{\prime}}^{m^{\prime}*}({\hat{\bf
k}})\Theta_{l^{\prime}}(k)\right)+Y_{l^{\prime}}^{m^{\prime}*}(\hat{\bf
k})\Theta_{l^{\prime}}(k)\frac{\partial}{\partial
k_{i}}\frac{\partial}{\partial k_{j}}\left(Y_{l}^{m}({\hat{\bf
k}})\Theta_{l}(k)\right)\right)\right.$
$\displaystyle\left.\vphantom{\sum_{i,j=1}^{3}\frac{k_{i}k_{j}}{k^{2}}}+2~{}\left(\hat{\bf
k}\cdot\textrm{\boldmath$\nabla_{k}$}\left(Y_{l}^{m}({\hat{\bf
k}})\Theta_{l}(k)\right)\right)\left(\hat{\bf
k}\cdot\textrm{\boldmath$\nabla_{k}$}\left(Y_{l^{\prime}}^{m^{\prime}*}({\hat{\bf
k}})\Theta_{l^{\prime}}(k)\right)\right)\right].$
We also break ${\Delta_{2}}(l,m;l^{\prime},m^{\prime})$ into terms quadratic
in ${\bf z_{*}}$, linear and containing no factors of ${\bf z_{*}}$.
${\Delta_{2}}(l,m;l^{\prime},m^{\prime})={\Delta}_{2}^{(2)}(l,m;l^{\prime},m^{\prime})+{\Delta}_{2}^{(1)}(l,m;l^{\prime},m^{\prime})+{\Delta}_{2}^{(0)}(l,m;l^{\prime},m^{\prime})$
(53)
The term quadratic in ${\bf z_{*}}$ can be written as
${\Delta}_{2}^{(2)}(l,m;l^{\prime},m^{\prime})=\xi_{lm;l^{\prime}m^{\prime}}\int_{0}^{\infty}{\rm
d}kk^{2}P_{2}(k)\Theta_{l}(k)\Theta_{l^{\prime}}(k)$ (54)
where
$\xi_{lm;l^{\prime}m^{\prime}}=\int{\rm d}\Omega_{\bf k}({\hat{\bf
k}}\cdot{{\bf z}_{*}})^{2}Y_{l}^{m}(\hat{\bf
k})Y_{l^{\prime}}^{m^{\prime}*}({\hat{\bf k}}),$ (55)
For the computation of $\xi_{l,m;l^{\prime}m^{\prime}}$, we use the
“spherical” components of ${{\bf z}_{*}}$ in Eq. (39).
$\xi_{lm;l^{\prime}m^{\prime}}$ was calculated in Ackerman where violation of
rotational invariance was considered. It is convenient to decompose
$\xi_{lm;l^{\prime}m^{\prime}}$ into coefficients of the quadratic quantities
$z_{i}z_{j}$, via
$\xi_{lm;l^{\prime}m^{\prime}}=z_{+}^{2}\xi_{lm;l^{\prime}m^{\prime}}^{++}+z_{-}^{2}\xi_{lm;l^{\prime}m^{\prime}}^{--}+2z_{+}z_{-}\xi_{lm;l^{\prime}m^{\prime}}^{+-}+2z_{+}z_{0}\xi_{lm;l^{\prime}m^{\prime}}^{+0}+2z_{-}z_{0}\xi_{lm;l^{\prime}m^{\prime}}^{-0}+z_{0}^{2}\xi_{lm;l^{\prime}m^{\prime}}^{00}.$
(56)
ACW Ackerman found that
$\displaystyle\xi_{lm;l^{\prime}m^{\prime}}^{++}$ $\displaystyle=$
$\displaystyle-\delta_{m^{\prime},m+2}\left[\delta_{l^{\prime},l}{\sqrt{(l^{2}-(m+1)^{2})(l+m+2)(l-m)}\over(2l+3)(2l-1)}-{1\over
2}\delta_{l^{\prime},l+2}{\sqrt{{(l+m+1)(l+m+2)(l+m+3)(l+m+4)\over(2l+1)(2l+3)^{2}(2l+5)}}}\right.$
$\displaystyle\left.-{1\over
2}\delta_{l^{\prime},l-2}{\sqrt{{(l-m)(l-m-1)(l-m-2)(l-m-3)\over(2l+1)(2l-1)^{2}(2l-3)}}}\right],$
$\displaystyle\xi_{lm;l^{\prime}m^{\prime}}^{--}$ $\displaystyle=$
$\displaystyle\xi_{l^{\prime}m^{\prime};lm}^{++},$
$\displaystyle\xi_{lm;l^{\prime}m^{\prime}}^{+-}$ $\displaystyle=$
$\displaystyle{1\over
2}\delta_{m^{\prime},m}\left[-2\,\delta_{l^{\prime},l}\frac{(-1+l+l^{2}+m^{2})}{(2l-1)(2l+3)}\right.+\delta_{l^{\prime},l+2}\sqrt{\frac{((l+1)^{2}-m^{2})((l+2)^{2}-m^{2})}{(2l+1)(2l+3)^{2}(2l+5)}}$
$\displaystyle+\left.\delta_{l^{\prime},l-2}\sqrt{\frac{(l^{2}-m^{2})((l-1)^{2}-m^{2})}{(2l-3)(2l-1)^{2}(2l+1)}}\right],$
$\displaystyle\xi_{lm;l^{\prime}m^{\prime}}^{+0}$ $\displaystyle=$
$\displaystyle\frac{\delta_{m^{\prime},m+1}}{\sqrt{2}}\left[\delta_{l^{\prime},l}{(2m+1)\sqrt{(l+m+1)(l-m)}\over(2l-1)(2l+3)}\right.$
$\displaystyle+\left.\delta_{l^{\prime},l+2}\sqrt{\frac{((l+1)^{2}-m^{2})(l+m+2)(l+m+3)}{(2l+1)(2l+3)^{2}(2l+5)}}-\delta_{l^{\prime},l-2}\sqrt{\frac{(l^{2}-m^{2})(l-m-1)(l-m-2)}{(2l-3)(2l-1)^{2}(2l+1)}}\right],$
$\displaystyle\xi_{lm;l^{\prime}m^{\prime}}^{-0}$ $\displaystyle=$
$\displaystyle-\xi_{l^{\prime}m^{\prime};lm}^{+0},$
$\displaystyle\xi_{lm;l^{\prime}m^{\prime}}^{00}$ $\displaystyle=$
$\displaystyle\delta_{m,m^{\prime}}\left[\delta_{l,l^{\prime}}\frac{(2l^{2}+2l-2m^{2}-1)}{(2l-1)(2l+3)}+\delta_{l^{\prime},l+2}\sqrt{\frac{((l+1)^{2}-m^{2})((l+2)^{2}-m^{2})}{(2l+1)(2l+3)^{2}(2l+5)}}\right.$
(57)
$\displaystyle\left.+\delta_{l^{\prime},l-2}\sqrt{\frac{(l^{2}-m^{2})((l-1)^{2}-m^{2})}{(2l-3)(2l-1)^{2}(2l+1))}}\right].$
The term linear in ${{\bf z}_{*}}$ has already been evaluated before.
$\Delta_{2}^{(1)}(l,m;l^{\prime},m^{\prime})=i\int_{0}^{\infty}{\rm
d}k~{}k^{2}P_{2}(k)\left(\Theta_{l^{\prime}}(k){\partial\Theta_{l}(k)\over\partial
k}-\Theta_{l}(k){\partial\Theta_{l^{\prime}}(k)\over\partial
k}\right)\left(z_{+}\chi^{(a)+}_{lm;l^{\prime}m^{\prime}}+z_{-}\chi^{(a)-}_{lm;l^{\prime}m^{\prime}}+z_{0}\chi^{(a)0}_{lm;l^{\prime}m^{\prime}}\right)$
(58)
where all $\chi^{(a)}$’s are given from Eq. (41) to (43).
The term independent of ${{\bf z}_{*}}$ can be evaluated using the identity
$\displaystyle\sum_{i,j=1}^{3}\frac{k_{i}k_{j}}{k^{2}}Y_{l}^{m}(\hat{\bf
k})\Theta_{l}(k)\frac{\partial}{\partial k_{i}}\frac{\partial}{\partial
k_{j}}\left(Y_{l^{\prime}}^{m^{\prime}*}({\hat{\bf
k}})\Theta_{l^{\prime}}(k)\right)$ $\displaystyle=$
$\displaystyle\sum_{i,j=1}^{3}\frac{k_{i}}{k}Y_{l}^{m}(\hat{\bf
k})\Theta_{l}(k)\frac{\partial}{\partial
k_{i}}\left(\frac{k_{j}}{k}\frac{\partial}{\partial
k_{j}}\left(Y_{l^{\prime}}^{m^{\prime}*}({\hat{\bf
k}})\Theta_{l^{\prime}}(k)\right)\right)$ (59) $\displaystyle=$ $\displaystyle
Y_{l}^{m}({\hat{\bf k}})\Theta_{l}(k)\hat{\bf
k}\cdot\textrm{\boldmath$\nabla_{k}$}\left[\hat{\bf
k}\cdot\textrm{\boldmath$\nabla_{k}$}\left(Y_{l^{\prime}}^{m^{\prime}*}({\hat{\bf
k}})\Theta_{l^{\prime}}(k)\right)\right]$
From Eq. (36), we know that
$\hat{\bf
k}\cdot\textrm{\boldmath$\nabla_{k}$}\left(Y_{l^{\prime}}^{m^{\prime}*}({\hat{\bf
k}})\Theta_{l^{\prime}}(k)\right)=\frac{\partial\Theta_{l^{\prime}}(k)}{\partial
k}Y_{l^{\prime}}^{m^{\prime}*}({\hat{\bf k}})$ (60)
and
$\hat{\bf k}\cdot\textrm{\boldmath$\nabla_{k}$}\left[\hat{\bf
k}\cdot\textrm{\boldmath$\nabla_{k}$}\left(Y_{l^{\prime}}^{m^{\prime}*}({\hat{\bf
k}})\Theta_{l^{\prime}}(k)\right)\right]=\frac{\partial^{2}\Theta_{l^{\prime}}(k)}{\partial
k^{2}}Y_{l^{\prime}}^{m^{\prime}*}({\hat{\bf k}})$ (61)
These give
$\Delta_{2}^{(0)}(l,m;l^{\prime},m^{\prime})=\frac{1}{2}\delta_{l,l^{\prime}}\delta_{m,m^{\prime}}\int_{0}^{\infty}{\rm
d}k~{}k^{2}P_{2}(k)\left[\left(\frac{\partial\Theta_{l}(k)}{\partial
k}\right)^{2}-\Theta_{l}(k)\frac{\partial^{2}\Theta_{l}(k)}{\partial
k^{2}}\right]$ (62)
To recap: the modification of the correlations $\langle
a_{lm}a_{l^{\prime}m^{\prime}}^{*}\rangle$ caused by the violation of
translational invariance is defined by Eq. (31). It can be decomposed into two
pieces, $\Delta_{1}(l,m,l^{\prime},m^{\prime})$ and
$\Delta_{2}(l,m,l^{\prime},m^{\prime})$, and each can be expressed as three
components depending on their dependence on ${\bf z}_{*}$ in Eq. (34) and
(53). The quadratic piece in $\Delta_{1}(l,m,l^{\prime},m^{\prime})$ is given
by (35), the ${\bf z}_{*}$-independent piece by (51), and the linear piece by
(38), whose terms are given by (40-48). Meanwhile, the quadratic piece in
$\Delta_{2}(l,m,l^{\prime},m^{\prime})$ is given by (54), the linear piece by
(58), and the ${{\bf z}_{*}}$-independent piece by (62).
While these expressions appear formidable, the good news is that coefficients
at multipole $l$ are only correlated with those at $l-2\leq l^{\prime}\leq
l+2$. The correlation matrix is sparse, making the analysis of CMB data
computationally tractable Groeneboom:2008fz .
## IV Set up for A Special Line or Plane
In this section we extend the results obtained for the case of a preferred
point in space to the cases where translational invariance is broken by a
special line or point. Since many of the steps are similar to the special
point case we will be brief.
To specify the location of a preferred line in space requires a point ${\bf
z}_{*}$ and a unit tangent vector ${\bf n}$. (Note that we place Earth at the
center of our coordinate system, so that the specification of any point
defines a vector pointing from us to the point.) Since any point on the line
will do, without loss of generality we can take ${\bf z}_{*}$ to be the point
closest to us, implying the constraint ${\bf n}\cdot{\bf z}_{*}=0$. This is
illustrated by the diagram on the left in Figure -959.
Figure -959: A preferred line in space can be specified by its closest point,
${\bf z}_{*}$, and a unit tangent vector $\hat{\bf n}$; a preferred plane can
be specified by its closest point and a unit normal vector. The distance
$l({\bf x})$ to any point ${\bf x}$ in space is measured perpendicularly to
the line or plane.
In order to simplify the calculation, we first align the preferred direction
with the $z$ axis. In that case, the rotational invariance about the $z$ axis
and the translational invariance along this preferred direction are left
unbroken. These symmetries imply that the most general form of the two point
correlation of energy density correlations is
$\langle\tilde{\delta}({\bf k})\tilde{\delta}({\bf
q})\rangle=\delta(k_{z}+q_{z})e^{-i({\bf k}_{\perp}+{\bf q}_{\perp})\cdot{\bf
z}_{*}}P_{t}(k_{\perp},q_{\perp},k_{z},{\bf k}_{\perp}\cdot{\bf q}_{\perp}),$
(63)
so that
$\displaystyle\langle\delta({\bf x})\delta({\bf y})\rangle$ $\displaystyle=$
$\displaystyle\int d^{3}k\int d^{3}q~{}e^{i{\bf k}\cdot{\bf x}}e^{i{\bf
q}\cdot{\bf y}}\langle\tilde{\delta}({\bf k})\tilde{\delta}({\bf q})\rangle$
(64) $\displaystyle=$ $\displaystyle\int dk_{z}\int d^{2}k_{\perp}\int
d^{2}q_{\perp}~{}e^{ik_{z}(x_{z}-y_{z})}e^{i{\bf k}_{\perp}\cdot({\bf
x}_{\perp}-{\bf z_{*\perp}})}e^{i{\bf q}_{\perp}\cdot({\bf y}_{\perp}-{\bf
z_{*\perp}})}P_{t}(k_{\perp},q_{\perp},k_{z},{\bf k}_{\perp}\cdot{\bf
q}_{\perp})$
with $P_{t}$ symmetric under the interchange of ${\bf k}_{\perp}$ and ${\bf
q}_{\perp}$. Here we have decomposed the position and wave vectors along the
$z$ axis and the two dimensional subspace perpendicular to that which is
denoted by a subscript $\perp$. In the limit that there is no violations of
translational (and rotational) invariance,
$P_{t}(k_{\perp},q_{\perp},k_{z},{\bf k}_{\perp}\cdot{\bf q})$ reduces to
$P(k)\delta^{2}({\bf k}_{\perp}+{\bf q}_{\perp})$, where $k=\sqrt{{\bf
k_{\perp}}^{2}+k_{z}^{2}}$. We now assume the violations of translational (and
rotational) invariance are small and hence that $P_{t}$ is strongly peaked
about ${\bf k_{\perp}}=-{\bf q_{\perp}}$. We introduce the variables ${\bf
p}_{\perp}={\bf k}_{\perp}+{\bf q}_{\perp}$, ${\bf l}_{\perp}=({\bf
k}_{\perp}-{\bf q}_{\perp})/2$ and follow the same steps in the point case.
Then,
$\displaystyle\langle\delta({\bf x})\delta({\bf y})\rangle=\int dk_{z}\int
d^{2}l_{\perp}~{}e^{ik_{z}(x_{z}-y_{z})}e^{i{\bf l}_{\perp}\cdot({\bf
x}_{\perp}-{\bf
y}_{\perp})}P_{t}(l_{\perp},l_{\perp},k_{z},-l_{\perp}^{2})\cdot$
$\displaystyle\int
d^{2}p_{\perp}~{}e^{-A(l_{\perp},k_{z}){p_{\perp}^{2}}/2-B(l_{\perp},k_{z})({\bf
p}_{\perp}\cdot{\bf l}_{\perp})^{2}/(2l_{\perp}^{2})}e^{i{\bf
p}_{\perp}\cdot{\bf z}_{\perp}}$
where ${\bf z}_{\perp}=({\bf x}_{\perp}+{\bf y}_{\perp}-2{\bf z_{*\perp}})/2$.
Performing the integral over $d^{2}p_{\perp}$, we find that,
$\langle\delta({\bf x})\delta({\bf y})\rangle=\int dk_{z}\int
d^{2}l_{\perp}~{}e^{ik_{z}(x_{z}-y_{z})}e^{i{\bf l}_{\perp}\cdot({\bf
x}_{\perp}-{\bf
y}_{\perp})}P_{t}(l_{\perp},l_{\perp},k_{z},-l_{\perp}^{2})\sqrt{{{(2\pi)^{2}}\over{\rm
det}C}}\left(1-{z_{\perp}^{T}C^{-1}z_{\perp}\over 2}+\ldots\right)$ (66)
where $C_{ij}=A(l_{\perp},k_{z})\delta_{ij}+B(l_{\perp},k_{z})\dfrac{l_{\perp
i}l_{\perp j}}{l_{\perp}^{2}}$ is a $2\times 2$ matrix, ${\rm det}C=A^{2}+AB$,
and
$C^{-1}_{ij}={1\over A}\delta_{ij}-{B\over A(A+B)}{l_{\perp i}l_{\perp j}\over
l_{\perp}^{2}}$ (67)
We can define
$P(l_{\perp},k_{z})=\sqrt{{{(2\pi)^{2}}\over{\rm
det}C}}P_{t}(l_{\perp},l_{\perp},k_{z}-l_{\perp}^{2})$ (68)
and plug in the expression of $C_{ij}^{-1}$ in terms of $A(l_{\perp},k_{z})$
and $B(l_{\perp},k_{z})$. This gives after relabeling, ${\bf
l}_{\perp}\rightarrow{\bf k}_{\perp}$
$\langle\delta({\bf x})\delta({\bf y})\rangle=\int d^{3}k~{}e^{i{\bf
k}\cdot({\bf x}-{\bf
y})}P(k_{\perp},k_{z})\left[1-\frac{z_{\perp}^{2}}{2A}+\frac{B}{2A(A+B)}\frac{({\bf
k}_{\perp}\cdot{\bf z}_{\perp})^{2}}{k_{\perp}^{2}}\right]$ (69)
Note that we want the leading term in the expansion in $z$ to correspond to
the standard cosmology and hence $P(k_{\perp},k_{z})=P(k)$, where
$k=\sqrt{k_{\perp}^{2}+k_{z}^{2}}$. Finally, to make the preferred direction
arbitrary, we replace all position vectors $a_{z}$ with ${\bf n}\cdot{\bf a}$
and also replacing ${\bf a}_{\perp}$ with ${\bf a}-{\bf n}({\bf n}\cdot{\bf
a})$ in Eq. (69).
As in the special point case we note that another way to get a small violation
of translational is if there is a small parameter $\epsilon$ and $P_{t}$ takes
the form,
$P_{t}(k_{\perp},q_{\perp},k_{z},{\bf k}_{\perp}\cdot{\bf q}_{\perp})={c\over
k^{3}}\delta({\bf k}+{\bf q})+\epsilon
P^{\prime}_{t}(k_{\perp},q_{\perp},k_{z},{\bf k}_{\perp}\cdot{\bf q}_{\perp})$
(70)
where $P^{\prime}_{t}$ cannot be expanded in any simple way. This is what
happened in Ref. (Tseng:2009xw ).
A preferred plane can be specified by a point ${\bf z}_{*}$ and a unit normal
vector ${\bf n}$. We can again choose ${\bf z}_{*}$ to be the point on the
plane closest to us, implying a constraint ${\bf n\times}{\bf z}_{*}=0$, as
shown on the right-hand side of Figure -959. Notice that the rotational
invariance about the ${\bf n}$ axis and the translational invariance along the
${\bf n}$ direction are unbroken. These symmetries imply
$\langle\tilde{\delta}({\bf k})\tilde{\delta}({\bf q})\rangle=\delta^{2}({\bf
k}_{\|}+{\bf q}_{\|})e^{-i(k_{n}+q_{n})z_{*n}}P_{t}(k_{\|},k_{n},q_{n})$ (71)
so that
$\displaystyle\langle\delta({\bf x})\delta({\bf y})\rangle$ $\displaystyle=$
$\displaystyle\int d^{3}k\int d^{3}q~{}e^{i{\bf k}\cdot{\bf x}}e^{i{\bf
q}\cdot{\bf y}}\langle\tilde{\delta}({\bf k})\tilde{\delta}({\bf q})\rangle$
(72) $\displaystyle=$ $\displaystyle\int d^{2}k_{\|}\int dk_{n}\int
dq_{n}~{}e^{i{\bf k}_{\|}\cdot({\bf x}_{\|}-{\bf
y}_{\|})}e^{ik_{n}(x_{n}-z_{*n})}e^{iq_{n}(y_{n}-z_{*n})}P_{t}(k_{\|},k_{n},q_{n})$
Here we have decomposed the position and wave vectors along the normal vector
${\bf n}$ and the two dimensional subspace parallel to the plane which is
denoted by a subscript $\|$. Then we change variables $p_{n}=k_{n}+q_{n}$,
$l_{n}=(k_{n}-q_{n})/2$ and perform the integral over $dp_{n}$ to get
$\langle\delta({\bf x})\delta({\bf y})\rangle=\int d^{2}k_{\|}\int
dl_{n}~{}e^{i{\bf k}_{\|}\cdot({\bf x}_{\|}-{\bf
y}_{\|})}e^{il_{n}(x_{n}-y_{n})}P_{t}(k_{\|},l_{n},l_{n})\sqrt{{{2\pi}\over
A}}\left(1-\frac{z_{n}^{2}}{2A}+\ldots\right)$ (73)
After relabeling $l_{n}\rightarrow k_{n}$ and defining
$P(k_{\|},l_{n})=\sqrt{{{2\pi}\over A}}P_{t}(k_{\|},l_{n},l_{n})$ (74)
we have
$\langle\delta({\bf x})\delta({\bf y})\rangle=\int d^{3}k~{}e^{i{\bf
k}\cdot({\bf x}-{\bf y})}P(k_{\|},k_{n})\left[1-\frac{z_{n}^{2}}{2A}\right]$
(75)
Finally, for the reason that we want the leading order term to correspond to
the standard cosmology, we replace $P(k_{\|},k_{n})$ with $P(k)$, where
$k=\sqrt{k_{\|}^{2}+k_{n}^{2}}$.
## V Conclusions
We have investigated the observational consequences of a small violation of
translational invariance on the temperature anisotropies in the cosmic
microwave background. Three cases were investigated, based on the assumption
of a preferred point, line, or plane in space, and a quadratic dependence on
the distance to the preferred locus of points. Explicit formulae were
presented for the correlations $\langle
a_{lm}a^{*}_{l^{\prime}m^{\prime}}\rangle$ between spherical harmonic
coefficients of the CMB temperature field in the case of a special point. The
expressions we have derived may be used to directly compare CMB observations
against the hypothesis of perfect translational invariance during the
inflationary era, as part of a systematic framework for constraining
deviations from the standard paradigm of primordial perturbations. Explicit
expressions for the correlations $\langle
a_{lm}a^{*}_{l^{\prime}m^{\prime}}\rangle$ can also be derived for the special
line and plane cases.
One can also test the hypothesis of perfect translational invariance during
the inflationary era using data on the large scale distribution of galaxies
and clusters of galaxies, using, in the special point case,
$\langle\delta({\bf x})\delta({\bf
y})\rangle=\int{d^{3}l\over(2\pi)^{3}}e^{i{\bf l}\cdot({\bf x}-{\bf
y})}P_{0}(l)+\frac{({\bf x}+{\bf y}-2{\bf
z_{*}})^{2}}{4}\int{d^{3}l\over(2\pi)^{3}}e^{i{\bf l}\cdot({\bf x}-{\bf
y})}P_{1}(l)+\int{d^{3}l\over(2\pi)^{3}}e^{i{\bf l}\cdot({\bf x}-{\bf
y})}P_{2}(l)\frac{\left[\bf l\cdot({\bf x}+{\bf y}-2{\bf
z_{*}})\right]^{2}}{4l^{2}}.$ (76)
The work in Section II suggests that $P_{1}(k)$ and $P_{2}(k)$ are
proportional to $P_{0}(k)$ and so the corrections to the microwave background
anisotropy and the large scale distribution of galaxies are characterized by
five parameters, two are these constants of proportionality and three are the
parameters to specify the sepcial point including the direction and the
magnitude of ${{\bf z}_{*}}$.
## Acknowledgments
This research was supported in part by the U.S. Department of Energy and by
the Gordon and Betty Moore Foundation.
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|
arxiv-papers
| 2008-11-07T07:44:33 |
2024-09-04T02:48:58.657457
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sean M. Carroll, Chien-Yao Tseng and Mark B. Wise",
"submitter": "Sean Carroll",
"url": "https://arxiv.org/abs/0811.1086"
}
|
0811.1132
|
# Hawking radiation from a Vaidya black hole: a semi-classical approach and
beyond
Haryanto M. Siahaana and Triyantab
Theoretical High Energy Physics and Instrumentation Division,
Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung,
Jalan Ganesha 10, Bandung 40132, Indonesia.
a
b anto_102@students.itb.ac.id triyanta@fi.itb.ac.id
###### Abstract:
We derive the Hawking radiation for Vaidya black hole in the tunneling picture
from the corresponding single particle action by the use of the radial null
geodesic and the Hamilton-Jacobi method (beyond semi-classical approximation).
Both results are then analyzed and compared.
Hawking radiation, Vaidya spacetime, semi-classical method and beyond.
††preprint: arXiv:0811.1132 [gr-qc]
## 1 Introduction
In 1974, Hawking startled the physics community by proving that black hole
evaporates particles [1]. It contradicts with classical general relativistic
definition of a black hole, an object that nothing can escape from it [2],
[3]. Hawking’s derivation was a quantum field theoretically. The semi-
classical approach in modeling the black hole radiation as a tunneling effect
was developed in the past decade [4], [5], [6], [7] and attracted many
attentions [e.g. [16]]. This is mostly because of its bold physical picture
and it is easier, mathematically.
There are two ways to perform semi-classical analysis for a black hole
radiation. The first is by the use of radial null geodesic method developed by
Parikh and Wilczek [3], [4]. In this method, one has to get the expression
dr/dt from the radial null geodesic condition, $ds^{2}=d\Omega=0$, for a
metric that has the form
$ds^{2}=-a\left(r\right)dt^{2}+b\left(r\right)dr^{2}+r^{2}d\Omega^{2}$. Then,
the obtained expression is used to calculate the imaginary part of the action.
We then end up with the obtaining of the Hawking temperature
$T_{H}=\left({k\beta}\right)^{-1}$, after equating the transition rate for a
particle in a tunneling process $\Gamma\sim\exp\left[{2{\mathop{\rm
Im}\nolimits}S}\right]$ with the Boltzmann factor $\exp\left[{-\beta
E}\right]$.
The second method was developed by Padmanabhan et al. [5] and it attracted
some applications (e.g. in 2-d stringy black holes [17]). In the method, the
scalar wave function is determined by the ansatz
$\phi\left({r,t}\right)=\exp\left[{{{-iS\left({r,t}\right)}\mathord{\left/{\vphantom{{-iS\left({r,t}\right)}\hbar}}\right.\kern-1.2pt}\hbar}}\right]$
where $S\left({r,t}\right)$ is the action for a single scalar particle.
Inserting this ansatz into the Klein-Gordon equation in a gravitational
background, one yields an equation for the action $S\left({r,t}\right)$ which
can be solved by the Hamilton-Jacobi method. After obtaining the action, one
can get the probability for outgoing and ingoing particles,
$P_{out}=\left|{\phi_{{\rm{out}}}}\right|^{2}$ and
$P_{in}=\left|{\phi_{{\rm{in}}}}\right|^{2}$, respectively. The Hawking
temperature can be obtained by using the ’principle of detailed balance’ [5],
$P_{out}=\exp\left[{-\beta E}\right]P_{in}=\exp\left[{-\beta E}\right]$, since
all particles must be absorbed by the black hole.
An interesting improvement has recently been made by Banerjee and Majhi [8].
They expand the action for single particle in the power series of Planck
constant,
$S\left({r,t}\right)=\sum\nolimits_{n=0}{\alpha_{n}\hbar^{n}S_{n}\left({r,t}\right)}$.
By considering that, for all $n$, $S_{n}\left({r,t}\right)$ and
$S_{n+1}\left({r,t}\right)$ are proportional to each other with the same
proportionality, one can write
$S\left({r,t}\right)=\left({1+\sum\nolimits_{n=1}{\alpha_{n}\hbar^{n}}}\right)S_{0}\left({r,t}\right)$.
By using the method of Padmanabhan et al., the action gives the correction
value of the Hawking temperature. Even though the correction is in principle
negligible, due to the very small of the Planck constant, one could regard
this as an effect of quantum analysis in the semi-classical quantum gravity.
Several developments have been made, including fermionic consideration and the
relation of the method to trace anomaly [15].
In this paper, we consider a more general metric with the mass of black hole
is time ($t$) and radius ($r$) dependent. A semiclassical approach for a
dynamical black hole has also recently worked in [18]. As generally known,
this subject is well described by the Vaidya metric. To get an exact ($t-r$)
dependence of the Hawking temperature, we insist to work in using
Schwarzschild like metric for the Vaidya space time [9] and impose the
condition for a very slowly varying mass of the black hole, $\left|{\partial
m/\partial r}\right|\equiv\left|{m^{\prime}}\right|<<1$ and $\left|{\partial
m/\partial t}\right|\equiv\left|{\dot{m}}\right|<<1$. In this approximation,
the squared $m^{\prime}$ and $\dot{m}$ are well approximated to zero. This
consideration will be useful later in showing the proportionality between each
term of action’s expansion. Our aim is to see the ($t-r$) dependence of
Hawking temperature in the case of varying mass of black hole. We derive it
both in the radial null geodesic and Hamilton-Jacobi method. For Hamilton-
Jacobi method, we directly consider all of the expansion terms of action
(beyond the method of Padmanabhan et al.) as Banerjee and Majhi had done for
some metrics with constant black hole masses.
The organization of our paper is as follows. In the second section, we will
derive the Hawking temperature for a metric with a varying mass by the use of
radial null geodesic method. In the third section, the Hawking temperature is
obtained by the Hamilton Jacobi method, by considering all terms in action
expansion. In the last section, we give a conclusion for our work. For the
rest of this paper, we use the unit dimension: Newton constant, light velocity
in vacuum, and Boltzman constant, $G=c=k_{B}=1$.
## 2 Hawking Temperature from Radial Null Geodesic Method in Vaidya Space-
Time
We start with the metric derived by Farley and D’Eath [9] for Vaidya space-
time
$\displaystyle
ds^{2}=-\left({\frac{{\dot{m}}}{{x\left(m\right)}}}\right)^{2}\left({1-\frac{{2m}}{r}}\right)dt^{2}+\left({1-\frac{{2m}}{r}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}.$
(1)
In the above, the black hole mass $m$ varies with time $t$ and radius $r$,
$m\equiv m\left({r,t}\right)$. $x\left(m\right)$ is an arbitrary function of
mass and $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$ is the metric
of 2-sphere. Later, rather than adopting the law of mass evolution by Hawking
[10], [11], $dm/dt=-C/m^{2}$, we choose a weaker condition, $\partial
m/\partial t\equiv\dot{m}=-C(m)/m^{2}$, which is a partial derivative, in
contrast to the Hawking law that is an exact differentiation. It will be shown
that this choice leads to a better form of Hawking temperature. The variable
$C\left(m\right)$ in this expression counts the number of particles emitted by
the black hole with mass $m$. This variable will increase for decreasing mass
[9] and for early stage of evaporation, we could consider that this variable
is a small valued-function because the evaporated mass is small compared to
the total mass. In this line of thought, putting $x\left(m\right)=-C(m)/m$ is
still valid, at least at the early stage of evaporation. Our purpose in
setting $x\left(m\right)$ in this form is to reduce the variable contained in
the metric and later on, for the case of fixed mass with respect to radius, to
obtain time dependent Hawking temperature [11],
$T_{H}\left(t\right)=\hbar/8\pi m(t)$.
The above considerations lead the metric (1) into the form
$\displaystyle
ds^{2}=-\left({\frac{1}{{m}}}\right)^{2}\left({1-\frac{{2m}}{r}}\right)dt^{2}+\left({1-\frac{{2m}}{r}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}.$
(2)
It has the general form
$\displaystyle
ds^{2}=-F\left({r,t}\right)dt^{2}+G\left({r,t}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}.$
(3)
where in our case $F\left({r,t}\right)\equiv m^{-2}\left({1-2mr^{-1}}\right)$
and $G\left({r,t}\right)\equiv 1-2mr^{-1}$. The general form (3) will simplify
our next calculation. Unless for specific purposes, we will write
$F\left({r,t}\right)$ and $G\left({r,t}\right)$ as $F$ and $G$, respectively,
for the sake brevity.
It turns out that the metric (2) has a coordinate singularity at $r_{h}=2m$
which of course is time dependent. Painleve transformation which is used to
remove coordinate singularity for a metric with time-like Killing vector, is
also applicable in this analysis. Transforming
$\displaystyle dt\to dt-\sqrt{\frac{{1-G}}{{FG}}}dr$ (4)
the metric (3) changes into
$\displaystyle
ds^{2}=-F\left({r,t}\right)dt^{2}+2F\sqrt{\frac{{1-G}}{{FG}}}dtdr+dr^{2}+r^{2}d\Omega^{2},$
(5)
and therefore no coordinate singularity is found. Thus, by such a Painleve
transformation, it understandable that, in principle, a coordinate singularity
in general relativity can be removed only by changing coordinate, without
defining new physical condition or theorem. In the consideration that a
tunneled particle moves on the path with no singularity, (5) will be useful in
describing particle’s dynamics. For radial null geodesic,
$ds^{2}=d\Omega^{2}=0$, the differentiation of radius with respect to time can
be obtained from (5) as
$\displaystyle\frac{{dr}}{{dt}}=\sqrt{\frac{F}{G}}\left({\pm
1-\sqrt{1-G}}\right),$ (6)
where $+(-)$ signs denote outgoing(ingoing) radial null geodesics.
Near the horizon, we can expand the coefficient $F$ and $G$ by the use of
Taylor expansion. Since $F$ and $G$ are $\left({t-r}\right)$ dependent, and we
only need their approximation values for short distances from a point
(horizon), we could apply the Taylor expansion at a fixed time. So, we can
write
$\displaystyle\left.{F\left({r,t}\right)}\right|_{t}\simeq\left.{F^{\prime}\left({r,t}\right)}\right|_{t}\left({r-r_{h}}\right)+\left.{O\left({\left({r-r_{h}}\right)^{2}}\right)}\right|_{t},$
(7)
and
$\displaystyle\left.{G\left({r,t}\right)}\right|_{t}\simeq\left.{G^{\prime}\left({r_{h},t}\right)}\right|_{t}\left({r-r_{h}}\right)+\left.{O\left({\left({r-r_{h}}\right)^{2}}\right)}\right|_{t}.$
(8)
By the approximations (7) and (8) above, the dependence of radius to time in
(6) can be approached by
$\displaystyle\frac{{dr}}{{dt}}\simeq\frac{1}{2}\sqrt{F^{\prime}\left({r_{h},t}\right)G^{\prime}\left({r_{h},t}\right)}\left({r-r_{h}}\right).$
(9)
Now, we discuss the action of outgoing particle through the horizon. In the
original work by Parikh and Wilczek [3], the imaginary action is written as
$\displaystyle{\mathop{\rm Im}\nolimits}S={\mathop{\rm
Im}\nolimits}\int\limits_{r_{{\rm{in}}}}^{r_{{\rm{out}}}}{p_{r}dr}={\mathop{\rm
Im}\nolimits}\int\limits_{r_{{\rm{in}}}}^{r_{{\rm{out}}}}{\int\limits_{0}^{p_{r}}{dp_{r}^{\prime}dr}}={\mathop{\rm
Im}\nolimits}\int\limits_{r_{{\rm{in}}}}^{r_{{\rm{out}}}}{\int\limits_{0}^{H}{\frac{{dH^{\prime}}}{{{\textstyle{{dr}\over{dt}}}}}}}dr.$
(10)
The above expression is due to the Hamilton equation $dr/dt=dH/dp_{r}|_{r}$
where $r$ and $p_{r}$ are canonical variables (in this case, the radial
component of the radius and the momentum). As a reminder, the action of a
tunneled particle in a potential barrier higher than the energy of the
particle itself will be imaginary, $p_{r}=\sqrt{2m\left({E-V}\right)}$.
Different from discussions of several authors for a static black hole mass
[e.g. [3], [5], [6], [7], [8], [12], [13]], the outgoing particle’s energy
must be time dependent for black holes with varying mass. So, the
$dH^{\prime}$ integration at (10) is for all values of outgoing particle’s
energy, say from zero to $+E\left(t\right)$.
By using approximation (9), we can perform the integration (10). For $dr$
integration, we can perform a contour integration for upper half complex plane
to avoid the coordinate singularity $r_{h}$. The result is
$\displaystyle{\mathop{\rm Im}\nolimits}S=\frac{{2\pi
E\left(t\right)}}{{\sqrt{F^{\prime}\left({r_{h},t}\right)G^{\prime}\left({r_{h},t}\right)}}}.$
(11)
Since the tunneling probability is given by
$\Gamma\sim\exp\left[{-{\textstyle{2\over\hbar}}{\mathop{\rm
Im}\nolimits}S}\right]$, equalizing it with the Boltzmann factor
$\exp\left[{-\beta E\left(t\right)}\right]$ for a system with time dependent
energy we obtain
$\displaystyle
T_{H}=\frac{{\hbar\sqrt{F^{\prime}\left({r_{h},t}\right)G^{\prime}\left({r_{h},t}\right)}}}{{4\pi}}.$
(12)
As seen in expression (12), the Hawking temperature $T_{H}$ is time dependent.
We will see it later that it is also radius dependent. Inserting the values of
$F^{\prime}\left({r_{h},t}\right)$ and $G^{\prime}\left({r_{h},t}\right)$ with
the condition of $\left({m^{\prime}}\right)^{2}\simeq 0$, one has
$\displaystyle
T_{H}=\frac{{\hbar\sqrt{\left({m\left({r_{h},t}\right)}\right)^{2}-2m^{\prime}\left({r_{h},t}\right)\left({m\left({r_{h},t}\right)+2m\left({r_{h},t}\right)}\right)}}}{{8\pi\left({m\left({r_{h},t}\right)}\right)^{2}}}.$
(13)
In the above, the mass $m$ is written as $m\left({r_{h},t}\right)$ explicitly
to remind us that the mass and its derivative
$m^{\prime}\left({r_{h},t}\right)$ are evaluated at $r_{h}$ (the radius of
event horizon). Interestingly, for a black hole whose mass is only time
dependent, not radius dependent, $m^{\prime}=0$, we get
$\displaystyle T_{H}=\frac{\hbar}{{8\pi m\left(t\right)}}.$ (14)
and thus, we recover to the time-dependent Hawking temperature. Further
interesting investigation would be to understand the black hole model with
$x(m)=-C(m)/m$, however it will not be discussed in this paper.
## 3 Hawking Temperature from Hamilton-Jacobi Method in Vaidya Space-Time
In this section, we would work in scalar field theory with gravitational
background. We still work in the metric (1) and impose the condition
$\dot{m}={{-C\left(m\right)}\mathord{\left/{\vphantom{{-C\left(m\right)}{m^{2}}}}\right.\kern-1.2pt}{m^{2}}}$
later. In this section, the condition $\left({\dot{m}}\right)^{2}\simeq 0$ and
$\left({x\left(m\right)}\right)^{2}\simeq 0$ are very important in showing the
proportionality between each expansion terms of action. To avoid confusion
with $F\left({r,t}\right)$ and $G\left({r,t}\right)$ that have been used in
the previous section, we rewrite metric (1) as
$\displaystyle
ds^{2}=-f\left({r,t}\right)dt^{2}+g\left({r,t}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},$
(15)
where $f\left({r,t}\right)\equiv\dot{m}(1-2mr^{-1})/x(m)$ and
$g\left({r,t}\right)\equiv\left({1-2mr^{-1}}\right)$.
Massless scalar particles under the gravitational background $g_{\mu\nu}$ obey
the Klein-Gordon equation
$\displaystyle\frac{{-\hbar^{2}}}{{\sqrt{-g}}}\partial_{\mu}\left[{g^{\mu\nu}\sqrt{-g}\partial_{\nu}}\right]\phi=0.$
(16)
For spherical symmetric black hole, we may reduce our attention only to
($r-t$) sector in the space-time, or in other words, we reduce to two
dimensional black hole problems. In this consideration, equation (16) under
the background metric (15) simplifies to
$\displaystyle\partial_{t}^{2}\phi-\frac{1}{{2fg}}\left({\dot{f}g+\dot{g}f}\right)\partial_{t}\phi-\frac{1}{2}\left({f^{\prime}g+fg^{\prime}}\right)\partial_{r}\phi-
fg\partial_{r}^{2}\phi=0.$ (17)
In the above, $f=f\left({r,t}\right)$ and $g=g\left({r,t}\right)$. By the
standard ansatz for scalar wave function
$\phi\left({r,t}\right)=\exp\left[{-{\textstyle{i\over\hbar}}S\left({r,t}\right)}\right]$,
equation (17) leads to the equation for the action $S\left({r,t}\right)$
$\displaystyle\begin{array}[]{l}\left({\frac{{-i}}{\hbar}\left({\frac{{\partial^{2}S}}{{\partial
t^{2}}}}\right)}\right)-\frac{1}{{\hbar^{2}}}\left({\frac{{\partial
S}}{{\partial
t}}}\right)^{2}-\frac{1}{{2fg}}\left({\dot{f}g+\dot{g}f}\right)\left({\frac{{-i}}{\hbar}}\right)\left({\frac{{\partial
S}}{{\partial t}}}\right)\\\
-\frac{1}{2}\left({f^{\prime}g+fg^{\prime}}\right)\left({\frac{{-i}}{\hbar}}\right)\left({\frac{{\partial
S}}{{\partial
r}}}\right)-fg\left({\frac{{-i}}{\hbar}\left({\frac{{\partial^{2}S}}{{\partial
r^{2}}}}\right)-\frac{1}{{\hbar^{2}}}\left({\frac{{\partial S}}{{\partial
r}}}\right)^{2}}\right)=0\\\ \end{array}.$ (20)
Now, our next step is to solve this equation. An approximation method can be
applied by expanding the action in the order of Planck constant power,
$\displaystyle
S\left({r,t}\right)=S_{0}\left({r,t}\right)+\sum\nolimits_{n}{\alpha_{n}\hbar^{n}S_{n}\left({r,t}\right)},$
(21)
for $n=1,2,3,...$. The constant $\alpha_{n}$ is set to keep all the expansion
terms have the action’s dimension. Taking unit dimensions $G=c=k_{B}=1$,
$\alpha_{n}$ would have the dimension of $[m]^{-2n}$ which $m$ refers to the
mass. It is clear that this expansion would lead to a very long equation. Due
to the very small value of the Planck constant, many authors [5], [6], [12],
[13] neglect the terms for $n\geq 1$. This consideration is acceptable, and
including higher terms is just adding correction for semi-classical derivation
of Hawking temperature. By grouping all the terms into the same powers of
$\hbar$, we could write for some lowest rank as
$\displaystyle\hbar^{0}{\rm{}}:{\rm{}}fg\left({\partial_{r}S_{0}}\right)^{2}-\left({\partial_{t}S_{0}}\right)^{2}=0,$
(22)
$\begin{array}[]{l}\hbar^{1}{\rm{}}:{\rm{}}ifg\partial_{r}^{2}S_{0}-i\left({\partial_{t}^{2}S_{0}}\right)+2fg\left({\partial_{r}S_{0}}\right)\left({\partial_{r}S_{1}}\right)+\frac{i}{2}\left({f^{\prime}g+fg^{\prime}}\right)\partial_{t}S_{0}\\\
{\rm{}}+\frac{i}{2}\frac{{\left({\dot{f}g+\dot{g}f}\right)}}{{fg}}\partial_{t}S_{0}-2\left({\partial_{t}S_{0}}\right)\left({\partial_{t}S_{1}}\right)=0,\\\
\end{array}$
$\begin{array}[]{l}\hbar^{2}{\rm{}}:{\rm{}}ifg\partial_{r}^{2}S_{1}-i\partial_{t}^{2}S_{1}+2fg\left({\partial_{r}S_{0}}\right)\left({\partial_{r}S_{2}}\right)+fg\left({\partial_{r}S_{1}}\right)^{2}+\frac{i}{2}\left({f^{\prime}g+fg^{\prime}}\right)\partial_{t}S_{1}\\\
{\rm{}}+\frac{i}{2}\frac{{\left({\dot{f}g+\dot{g}f}\right)}}{{fg}}\partial_{t}S_{1}-2\left({\partial_{t}S_{0}}\right)\left({\partial_{t}S_{2}}\right)-\left({\partial_{t}S_{1}}\right)^{2}=0,\\\
\end{array}$
$\begin{array}[]{l}\hbar^{3}{\rm{}}:{\rm{}}ifg\partial_{r}^{2}S_{2}-i\partial_{t}^{2}S_{2}+2fg\left({\partial_{r}S_{0}}\right)\left({\partial_{r}S_{3}}\right)++2fg\left({\partial_{r}S_{1}}\right)\left({\partial_{r}S_{2}}\right)-2\left({\partial_{t}S_{0}}\right)\left({\partial_{t}S_{3}}\right)\\\
{\rm{}}-2\left({\partial_{t}S_{1}}\right)\left({\partial_{t}S_{2}}\right)+\frac{i}{2}\left({f^{\prime}g+fg^{\prime}}\right)\partial_{t}S_{2}+\frac{i}{2}\frac{{\left({\dot{f}g+\dot{g}f}\right)}}{{fg}}\partial_{t}S_{2}=0,\\\
\end{array}$ $...$ $...$
As obtained by Banerjee and Majhi [8] for the metric that has a time-like
Killing vector, the metric (15) also leads to such a relation:
$\hbar^{0}{\rm{}}:{\rm{}}\partial_{t}S_{0}=\pm\sqrt{fg}\partial_{r}S_{0},$
$\displaystyle\hbar^{1}{\rm{}}:{\rm{}}\partial_{t}S_{1}=\pm\sqrt{fg}\partial_{r}S_{1},$
(23)
$\hbar^{2}{\rm{}}:{\rm{}}\partial_{t}S_{2}=\pm\sqrt{fg}\partial_{r}S_{2},$
$...$ $...$
To get the benefits of the conditions for black holes with slowly varying
mass, $(m^{\prime})^{2}\simeq 0$ and $(\dot{m})^{2}\simeq 0$, the arbitrary
function $x\left(m\right)$ must be taken to be $-m^{\prime}$ [9]. This
mechanism means that we have used the dynamic equation for black hole mass as
$m^{\prime}=C\left(m\right)/m$. Unless these conditions, the set of equations
(23) can not be obtained. Then one can obtain a general pattern for arbitrary
$n$, thus for the terms of $\hbar^{n}$, that is
$\partial_{t}S_{n}=\pm\sqrt{fg}\partial_{r}S_{n}$. From the set of equations
(22), one can see that each $S_{n}\left({r,t}\right)$ is proportional to
$S_{0}\left({r,t}\right)$. By this evidence, we can write the expansion (21)
into
$\displaystyle
S\left({r,t}\right)=S_{0}\left({r,t}\right)+\sum\nolimits_{n}{\alpha_{n}\hbar^{n}S_{0}\left({r,t}\right)}=\left({1+\sum\nolimits_{n}{\alpha_{n}\hbar^{n}}}\right)S_{0}\left({r,t}\right).$
(24)
The extra value
$\sum\nolimits_{n}{\alpha_{n}\hbar^{n}S_{0}\left({r,t}\right)}$ in (24) can be
regarded as the correction term of the semi-classical analysis. So, our next
step is to find the solution of $S_{0}\left({r,t}\right)$ satisfying
$\partial_{t}S_{0}=\pm\sqrt{fg}\partial_{r}S_{0}$.
In the standard Hamilton-Jacobi method, $S_{0}\left({r,t}\right)$ can be
written into two parts, the time part which has the form of $Et$ and the
radius part $\tilde{S}_{0}\left(r\right)$ which is in general a radius
dependent only. Since our metric coefficients are both radius and time
dependent, the standard method would not be applicable. We could generalized
the method by making an ansatz
$\displaystyle
S_{0}\left({r,t}\right)=\int\limits_{0}^{t}{E\left({t^{\prime}}\right)dt^{\prime}}+\tilde{S}_{0}\left({r,t}\right).$
(25)
At the first sight, it seems that the ansatz is rather strange, that is
$S\left({r,t}\right)$ and $\tilde{S}_{0}\left({r,t}\right)$ are both $t$ and
$r$ dependent. The term $\int{E\left({t^{\prime}}\right)dt^{\prime}}$ is more
understandable, since the emitted particle’s energy is continuum and time
dependent. Let see how it works.
From (25), one can write that
$\displaystyle\partial_{t}S_{0}\left({r,t}\right)=E\left(t\right)+\partial_{t}\tilde{S}_{0}\left({r,t}\right)$
(26)
and
$\displaystyle\partial_{r}S_{0}\left({r,t}\right)=\partial_{r}\tilde{S}_{0}\left({r,t}\right).$
(27)
Since $\tilde{S}_{0}\left({r,t}\right)$ is $t$ and $r$ dependent, one can
write
$\displaystyle\frac{{d\tilde{S}_{0}\left({r,t}\right)}}{{dr}}=\partial_{r}\tilde{S}_{0}\left({r,t}\right)+\partial_{t}\tilde{S}_{0}\left({r,t}\right)\frac{{dt}}{{dr}}.$
(28)
Eliminating $dt/dr$ by the use of
${{dr}\mathord{\left/{\vphantom{{dr}{dt}}}\right.\kern-1.2pt}{dt}}=\pm\sqrt{fg}$,
equation (28) can be written as
$\displaystyle\frac{{d\tilde{S}_{0}\left({r,t}\right)}}{{dr}}\mp\frac{1}{{\sqrt{fg}}}\partial_{t}\tilde{S}_{0}\left({r,t}\right)=\partial_{r}\tilde{S}_{0}\left({r,t}\right).$
(29)
Combining the first equation of (23), with equations (26) and (29) where we
should note that the action equation
$-\left({fg}\right)^{-1/2}\partial_{t}S_{0}\left({r,t}\right)=\partial_{r}S_{0}\left({r,t}\right)$
is belong to outgoing particle and with the radial evolution is
$dr/dt=\sqrt{fg}$, then we could write
$\displaystyle\mp\left({fg}\right)^{-1/2}\left({E\left(t\right)+\partial_{t}\tilde{S}_{0}\left({r,t}\right)}\right)=\frac{{d\tilde{S}_{0}\left({r,t}\right)}}{{dr}}\mp\left({fg}\right)^{-1/2}\partial_{t}\tilde{S}_{0}\left({r,t}\right).$
(30)
From (30), we can get the exact differentiation of
$\tilde{S}_{0}\left({r,t}\right)$
$\displaystyle\frac{{d\tilde{S}_{0}\left({r,t}\right)}}{{dr}}=\mp\left({fg}\right)^{-1/2}E\left(t\right),$
(31)
and the solution of $\tilde{S}_{0}\left({r,t}\right)$ can be obtained by
integration
$\displaystyle\tilde{S}_{0}\left({r,t}\right)=\mp
E\left(t\right)\int{\frac{{dr}}{{\sqrt{fg}}}}.$ (32)
After inserting the mass evolution equation,
$\dot{m}=-{{C\left(m\right)}\mathord{\left/{\vphantom{{C\left(m\right)}{m^{2}}}}\right.\kern-1.2pt}{m^{2}}}$,
and the arbitrary function
$x\left(m\right)=-{{C\left(m\right)}\mathord{\left/{\vphantom{{C\left(m\right)}m}}\right.\kern-1.2pt}m}$,
one may identify that $f$ and $g$ are exactly equal to $F$ and $G$ stated in
the previous section. By this equality, the integration (32) can be evaluated
by adopting the value of $\int{\left({FG}\right)^{-1/2}dr}$ as in obtaining
expression (11) from (10) along with it’s approximation method (near horizon
Taylor expansion). The result for the integration (32) is
$\displaystyle\tilde{S}_{0}\left({r,t}\right)=\mp
E\left(t\right)\frac{{i\pi}}{{\sqrt{F^{\prime}G^{\prime}}}}.$ (33)
The equation gives the complete action
$\displaystyle
S\left({r,t}\right)=\left({1+\sum\nolimits_{n}{\alpha_{n}\hbar^{n}}}\right)\left({\int\limits_{0}^{t}{E\left({t^{\prime}}\right)dt^{\prime}}\mp
E\left(t\right)\frac{{i\pi}}{{\sqrt{F^{\prime}G^{\prime}}}}}\right).$ (34)
The signs $+\left(-\right)$ in expression (34) refer to the action for ingoing
(outgoing) particle.
Back to our first ansatz for scalar wave function,
$\phi=\exp\left[{{\textstyle{{-i}\over\hbar}}S\left({r,t}\right)}\right]$, the
wave function for ingoing and outgoing massless scalar particle can be read of
as
$\displaystyle\phi_{in}\left({r,t}\right)=\exp\left[{-\frac{i}{\hbar}\left({1+\sum\nolimits_{n}{\alpha_{n}\hbar^{n}}}\right)\left({\int\limits_{0}^{t}{E\left({t^{\prime}}\right)dt^{\prime}}+E\left(t\right)\frac{{i\pi}}{{\sqrt{F^{\prime}G^{\prime}}}}}\right)}\right]$
(35)
and
$\displaystyle\phi_{out}\left({r,t}\right)=\exp\left[{-\frac{i}{\hbar}\left({1+\sum\nolimits_{n}{\alpha_{n}\hbar^{n}}}\right)\left({\int\limits_{0}^{t}{E\left({t^{\prime}}\right)dt^{\prime}}-E\left(t\right)\frac{{i\pi}}{{\sqrt{F^{\prime}G^{\prime}}}}}\right)}\right]$
(36)
respectively. Consequently, from (35) one can get the ingoing probability of
particle as below
$\displaystyle
P_{in}=\exp\left[{\frac{2}{\hbar}\left({1+\sum\nolimits_{n}{\alpha_{n}\hbar^{n}}}\right)\left({{\mathop{\rm
Im}\nolimits}\int\limits_{0}^{t}{E\left({t^{\prime}}\right)dt^{\prime}}+\frac{{\pi
E\left(t\right)}}{{\sqrt{F^{\prime}G^{\prime}}}}}\right)}\right].$ (37)
This ingoing probability must be equal to unity since all particles including
the massless one are absorbed by the black hole. This consideration gives us
the relation
${\mathop{\rm
Im}\nolimits}\int\limits_{0}^{t}{E\left({t^{\prime}}\right)dt^{\prime}}=-\frac{{\pi
E\left(t\right)}}{{\sqrt{F^{\prime}G^{\prime}}}},$
which leads the outgoing probability
$\displaystyle P_{out}=\exp\left[{-\frac{{4\pi
E\left(t\right)}}{{\hbar\sqrt{F^{\prime}G^{\prime}}}}\left({1+\sum\nolimits_{n}{\alpha_{n}\hbar^{n}}}\right)}\right].$
(38)
Finally, to get the Hawking temperature from the outgoing probability (38), we
equate this probability expression with $\exp\left[{-\beta
E\left(t\right)}\right]$ which in [5] is called ’detailed balance’ principle.
It yields
$\displaystyle
T_{H^{\prime}}=\frac{{\hbar\sqrt{F^{\prime}\left({r_{h},t}\right)G^{\prime}\left({r_{h},t}\right)}}}{{4\pi\left({1+\sum\nolimits_{n}{\alpha_{n}\hbar^{n}}}\right)}}.$
(39)
We use the index $H^{\prime}$ to distinguish the result (39) with that of
(12). Neglecting the correction term $\sum\nolimits_{n}{\alpha_{n}\hbar^{n}}$
in the denominator of (39), one has recovered result (12) which has been
obtained by the radial null geodesics at the previous section.
## 4 Conclusion
We have worked out the Hawking temperature for the metric with no time-like
Killing vector and with the time and radius dependent coefficients. It has
been shown that by the use of mass evolution $\partial m/\partial
t=-C(m)/m^{2}$ and mass dependent function $x(m)=-C(m)/m$, the resulting
Hawking temperature coincides with that for the widely known time dependent
one in the case of radius independent mass. By neglecting the
$\sum\nolimits_{n}{\alpha_{n}\hbar^{n}}$ term in the denominator of the
resulting Hawking temperature for the case of beyond semi-classical approach
as given in Section 3, one recovers the result in the section 2. However, the
equal obtained results in (12) and (38) with neglected correction terms are
actually based on some subtleties. In section 2, we use the $\dot{r}$ that is
obtained from the coordinate after Painleve transformation which differs with
a factor half with untransformed one (directly obtained from Schwarzschild
like metric (3)). This is clear since the particle moves in the path with no
singularity111We thank B. Majhi and R. Banerjee for pointing this out.. Thus,
in section 3 we use the $\dot{r}$ in deriving the action’s solution (29) that
is obtained from Schwarzschild like metric (without Painleve transformation),
since the equation of action comes from it (17). It needs further
investigations to verify whether the results are still the same in the
suitable Painleve coordinate in deriving the wave equation (16) (parallel to
section 3.2 [8]).
The slowly varying mass with respect both to time and radius, considered in
Section 3, affects our result on the Hawking temperature expression and the
proportionality between each terms of action’s expansion. We have restored the
$x(m)$ as $-m^{\prime}$ in section 3 to get the proportionality by using our
slowly varying mass condition. This consideration of course yields an equation
for mass dynamics $m^{\prime}=C\left(m\right)/m$ which needs further
investigations. Of course, our analysis is not valid for the case of highly
radiated black holes. There are several models that have been proposed for
describing black hole radiation. But, the lack of experimental data does not
enable one to compare between the models.
Further investigation on the relationship between ingoing and outgoing
probabilities for the case of black hole with varying mass/energy by the use
of path integral method is intriguing. This work was first performed by Hartle
and Hawking [14] for a black hole with constant mass and constant energy of
outgoing particle. The relation proposed in the paper,
$P_{out}=\exp\left[{-\beta E\left(t\right)}\right]P_{in}$, is an intuitive
manner as it might not be that simple.
## 5 Acknowledgements
HMS thanks R. Kurniadi and A. Nugroho from the Faculty of Mathematics and
Natural Sciences, ITB for useful discussions. HMS also thanks B. Majhi and R.
Banerjee from S. N. Bose National Centre for Basic Sciences for useful
comments.
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|
arxiv-papers
| 2008-11-07T12:33:51 |
2024-09-04T02:48:58.666414
|
{
"license": "Public Domain",
"authors": "Haryanto M. Siahaan and Triyanta",
"submitter": "Haryanto Siahaan Mangaratua",
"url": "https://arxiv.org/abs/0811.1132"
}
|
0811.1271
|
Efficient model chemistries for peptides. II.
Basis set convergence in the B3LYP method. Pablo ECHENIQUE
Instituto de Biocomputación y Física de Sistemas Complejos (BIFI),
and Departamento de Física Teórica, Universidad de Zaragoza,
Pedro Cerbuna 12, E-50009 Zaragoza, Spain
E-mail: echenique.p@gmail.com Gregory A. CHASS
Global Institute Of COmputational Molecular and Materials Science (GIOCOMMS),
and School of Chemistry, University of Wales, Bangor, Gwynedd, LL57 2UW United
Kingdom,
and College of Chemistry, Beijing Normal University, Beijing, 100875, China
PACS: 07.05.Tp; 31.15.Ar; 31.50.Bc; 87.14.Ee; 87.15.Aa; 89.75.-k
Keywords: peptides, quantum chemistry, PES, B3LYP, basis set convergence
Abstract
Small peptides are model molecules for the amino acid residues that are the
constituents of proteins. In any bottom-up approach to understand the
properties of these macromolecules essential in the functioning of every
living being, to correctly describe the conformational behaviour of small
peptides constitutes an unavoidable first step. In this work, we present an
study of several potential energy surfaces (PESs) of the model dipeptide HCO-
L-Ala-NH2. The PESs are calculated using the B3LYP density-functional theory
(DFT) method, with Dunning’s basis sets cc-pVDZ, aug-cc-pVDZ, cc-pVTZ, aug-cc-
pVTZ, and cc-pVQZ. These calculations, whose cost amounts to approximately 10
years of computer time, allow us to study the basis set convergence of the
B3LYP method for this model peptide. Also, we compare the B3LYP PESs to a
previous computation at the MP2/6-311++G(2df,2pd) level, in order to assess
their accuracy with respect to a higher level reference. All data sets have
been analyzed according to a general framework which can be extended to other
complex problems and which captures the nearness concept in the space of model
chemistries (MCs).
## 1 Introduction
In any bottom-up attempt to understand the behaviour of protein molecules (in
particular, the still elusive protein folding process [3, 1, 5, 2, 4]), the
characterization of the conformational preferences of short peptides [13, 12,
7, 11, 6, 9, 10, 8] constitutes an unavoidable first step. Due to the lower
numerical effort required and also to the manageability of their
conformational space, the most frequently studied peptides are the shortest
ones: the _dipeptides_ [14, 17, 16, 15], in which a single amino acid residue
is capped at both the N- and C-termini with neutral peptide groups. Among
them, the most popular choice has been the _alanine_ dipeptide [34, 30, 26,
23, 27, 24, 21, 22, 6, 20, 29, 19, 33, 25, 31, 28, 32, 18], which, being the
simplest chiral residue, shares many similarities with most of the rest of
dipeptides for the minimum computational price.
Although classical force fields [35, 36, 37, 38, 39, 40, 41, 42, 43] are the
only feasible choice for simulating large molecules at present, they have been
reported to yield inaccurate _potential energy surfaces_ (PESs) for dipeptides
[44, 45, 46, 47, 29] and short peptides [48, 6]. Therefore, it is not
surprising that they are widely recognized as being unable to correctly
describe the intricacies of the whole protein folding process [49, 50, 51, 44,
52, 53, 54, 55]. On the other hand, albeit prohibitively demanding in terms of
computational resources, ab initio quantum mechanical calculations [56, 57,
58] are not only regarded as the correct physical description that in the long
run will be the preferred choice to directly tackle proteins (given the
exponential growth of computer power and the advances in the search for
pleasantly scaling algorithms [60, 59]), but they are also used in small
peptides as the reference against which less accurate methods must be compared
[61, 62, 44, 45, 47, 29, 6] in order to, for example, parameterize improved
generations of additive, classical force fields for polypeptides.
However, despite the sound theoretical basis, in practical quantum chemistry
calculations a plethora of approximations must be typically made if one wants
to obtain the final results in a reasonable human time. The exact ‘recipe’
that includes all the assumptions and steps needed to calculate the relevant
observables for any molecular system has been termed _model chemistry_ (MC) by
John Pople. In his own words, a MC is an “approximate but well-defined general
and continuous mathematical procedure of simulation” [63].
After assuming that the particles involved move at non-relativistic velocities
and that the greater weight of the nuclei allows to perform the Born-
Oppenheimer approximation, we are left with the problem of solving the non-
relativistic electronic Schrödinger equation [60]. The two starting
approximations to its exact solution that a MC must contain are, first, the
truncation of the $N$-electron space (in wavefunction-based methods) or the
choice of the functional (in DFT) and, second, the truncation of the one-
electron space, via the LCAO scheme (in both cases). The extent up to which
the first truncation is carried (or the functional chosen in the case of DFT)
is commonly called the _method_ and it is denoted by acronyms such as RHF,
MP2, B3LYP, CCSD(T), FCI, etc., whereas the second truncation is embodied in
the definition of a finite set of atom-centered Gaussian functions termed
_basis set_ [60, 64, 57, 58, 65], which is also designated by conventional
short names, such as 6-31+G(d), TZP or cc-pVTZ(–f). If we denote the method by
a capital $M$ and the basis set by a $B$, the specification of both is
conventionally denoted by $L:=M/B$ and called a _level of the theory_. Typical
examples of this are RHF/3-21G or MP2/cc-pVDZ [56, 57, 58].
Note that, apart from these approximations, which are the most commonly used
and the only ones that are considered in this work, the MC concept may include
a lot of additional features: the heterolevel approximation (explored in a
previous work in this series [34]), protocols for extrapolating to the
infinite-basis set limit [66, 67, 68, 69, 70], additivity assumptions [71, 72,
73, 74], extrapolations of the Møller-Plesset series to infinite order [75],
removal of the so-called _basis set superposition error_ (BSSE) [76, 77, 78,
79, 80, 81, 82], etc. The reason behind most of these techniques being the
urging need to reduce the computational cost of the calculations.
Now, although general applicability is a requirement that all MCs must
satisfy, general accuracy is not mandatory. Actually, the fact is that the
different procedures that conform a given MC are typically parameterized and
tested in very particular systems, which are often small molecules. Therefore,
the validity of the approximations outside that native range of problems must
be always questioned and checked. However, while the approximate computational
cost of a given MC for a particular system is rather easy to predict on the
basis of simple scaling relations, its expected accuracy on a particular
problem could be difficult to predict a priori, specially if we are dealing
with large molecules in which interactions in very different energy scales are
playing a role. The description of the conformational behaviour of peptides
(or, more generally, flexible organic species), via their PESs in terms of the
soft internal coordinates, is one of such problems and the one that is treated
in this work.
To this end, we first describe, in sec. 2, the computational and theoretical
methods used throughout the rest of the document. Then, in sec. 3, we
introduce a basic framework that rationalizes the actual process of evaluating
the efficiency of any MC for a complex problem. These general ideas are used,
in sec. 4, to perform an study of the _density-functional theory_ (DFT) B3LYP
[83, 84, 85, 86] method with the cc-pVDZ, aug-cc-pVDZ, cc-pVTZ, aug-cc-pVTZ,
and cc-pVQZ Dunning’s basis sets [87, 88]. To this end, we apply these levels
of the theory to the calculation the PES of the model dipeptide HCO-L-Ala-NH2
(see fig. 1), and assess their efficiency by comparison with a reference PES.
Finally, in sec. 5, the most important conclusions are briefly summarized.
## 2 Methods
All ab initio quantum mechanical calculations have been performed using the
GAMESS-US program [89, 90] under Linux and on 2.2 GHz PowerPC 970FX machines
with 2 GB RAM memory.
The internal coordinates used for the Z-matrix of the HCO-L-Ala-NH2 dipeptide
in the GAMESS-US input files are the _Systematic Approximately Separable
Modular Internal Coordinates_ (SASMIC) ones introduced in ref. 91. They are
presented in table 1 (see also fig. 1 for reference).
Atom name | Bond length | Bond angle | Dihedral angle
---|---|---|---
H1 | | |
C2 | (2,1) | |
N3 | (3,2) | (3,2,1) |
O4 | (4,2) | (4,2,1) | (4,2,1,3)
C5 | (5,3) | (5,3,2) | (5,3,2,1)
H6 | (6,3) | (6,3,2) | (6,3,2,5)
C7 | (7,5) | (7,5,3) | $\phi:=$(7,5,3,2)
C8 | (8,5) | (8,5,3) | (8,5,3,7)
H9 | (9,5) | (9,5,3) | (9,5,3,7)
H10 | (10,8) | (10,8,5) | (10,8,5,3)
H11 | (11,8) | (11,8,5) | (11,8,5,10)
H12 | (12,8) | (12,8,5) | (12,8,5,10)
N13 | (13,7) | (13,7,5) | $\psi:=$(13,7,5,3)
O14 | (14,7) | (14,7,5) | (14,7,5,13)
H15 | (15,13) | (15,13,7) | (15,13,7,5)
H16 | (16,13) | (16,13,7) | (16,13,7,15)
Table 1: Internal coordinates in Z-matrix form of the protected dipeptide HCO-
L-Ala-NH2 according to the SASMIC scheme introduced in ref. 91. The numbering
of the atoms is that in fig. 1, and the soft Ramachandran angles $\phi$ and
$\psi$ are indicated.
All PESs in this study have been discretized into a regular 12$\times$12 grid
in the bidimensional space spanned by the Ramachandran angles $\phi$ and
$\psi$, with both of them ranging from $-165^{\mathrm{o}}$ to
$165^{\mathrm{o}}$ in steps of $30^{\mathrm{o}}$. To calculate the PES at a
particular level of the theory, we have run constrained energy optimizations
at each point of the grid, freezing the two Ramachandran angles $\phi$ and
$\psi$ at the corresponding values. In order to save computational resources,
the starting structures were taken, when possible, from PESs previously
optimized at a lower level of the theory. All the basis sets used in the study
have been taken from the GAMESS-US internally stored library, and spherical
Gaussian-type orbitals (GTOs) have been preferred, thus having 5 d-type and 7
f-type functions per shell.
Figure 1: Atom numeration of the protected dipeptide HCO-L-Ala-NH2 according
to the SASMIC scheme introduced in ref. 91. The soft Ramachandran angles
$\phi$ and $\psi$ are also indicated.
We have computed 5 PESs, using the DFT B3LYP [83, 84, 85, 86] method with the
cc-pVDZ, aug-cc-pVDZ, cc-pVTZ, aug-cc-pVTZ, and cc-pVQZ Dunning’s basis sets
[87, 88]. The total cost of these calculations in the machines used is around
10 years of computer time.
Also, let us note that the correcting terms to the PES coming from mass-metric
tensors determinants and from the determinant of the Hessian matrix have been
recently shown to be relevant for the conformational behaviour of peptides
[18]. (The latter may be regarded as a residual entropy arising from the
elimination of the hard coordinates from the description.) Although, in this
study, we have included none of these terms, the PES calculated here is the
greatest part of the effective free energy [18], so that it may be considered
as the first ingredient for a further refinement of the study in which the
correcting terms are taken into account. The same may be said about another
important source of error in the calculation of relatives energies in peptide
systems: the already mentioned BSSE [31].
In order to compare the PESs produced by the different MCs, a statistical
criterium (distance) introduced in ref. 92 has been used. Let us recall here
that this _distance_ , denoted by $d_{12}$, profits from the complex nature of
the problem studied to compare any two different potential energy functions,
$V_{1}$ and $V_{2}$. From a working set of conformations (in this case, the
144 points of each PES), it statistically measures the typical error that one
makes in the _energy differences_ if $V_{2}$ is used instead of the more
accurate $V_{1}$, admitting a linear rescaling and a shift in the energy
reference.
Despite having energy units, the quantity $d_{12}$ approximately presents all
properties characteristic of a typical mathematical metric in the space of MCs
(hence the word ‘distance’), such as the possibility of defining a symmetric
version of it and a fulfillment of the triangle inequality (see ref. 92 for
the technical details and sec. 3 for more about the importance of these
facts). It also presents better properties than other quantities customarily
used to perform these comparisons, such as the energy RMSD, the average energy
error, etc., and it may be related to the Pearson’s correlation coefficient
$r_{12}$ by
$d_{12}=\sqrt{2}\,{\sigma}_{2}(1-r_{12}^{2})^{1/2}\ ,$ (1)
where $\sigma_{2}$ is the standard deviation of $V_{2}$ in the working set.
Moreover, due to its physical meaning, it has been argued in ref. 92 that, if
the distance between two different approximations of the energy of the same
system is less than $RT$, one may safely substitute one by the other without
altering the relevant dynamical or thermodynamical behaviour. Consequently, we
shall present the results in units of $RT$ (at $300^{\mathrm{o}}$ K, so that
$RT\simeq 0.6$ kcal/mol).
Finally, if one assumes that the effective energies compared will be used to
construct a polypeptide potential and that it will be designed as simply the
sum of mono-residue ones (more complex situations may be found in real
problems [93]), then, the number $N_{\mathrm{res}}$ of residues up to which
one may go keeping the distance $d_{12}$ between the two approximations of the
the $N$-residue potential below $RT$ is [92]
$N_{\mathrm{res}}=\left(\frac{RT}{d_{12}}\right)^{2}\ .$ (2)
According to the value taken by $N_{\mathrm{res}}$ for a comparison between a
fixed reference PES, denoted by $V_{1}$, and a candidate approximation,
denoted by $V_{2}$, we shall divide the whole accuracy range in sec. 4 in
three regions depending on the accuracy: the _protein region_ , corresponding
to $0<d_{12}\leq 0.1RT$, or, equivalently, to $100\leq
N_{\mathrm{res}}<\infty$; the _peptide region_ , corresponding to
$0.1RT<d_{12}\leq RT$, or $1\leq N_{\mathrm{res}}<100$; and, finally, the
_inaccurate region_ , where $d_{12}>RT$, and even for a dipeptide it is not
advisable to use $V_{2}$ as an approximation to $V_{1}$. Of course, these are
only approximate regions based on the general idea that we are not interested
on the dipeptides as a final system, but only as a mean to approach protein
behaviour from the botton-up. Therefore, not only the error in the dipeptides
must be measured, but it must also be estimated how this discrepancy
propagates to polypeptide systems.
## 3 General framework
The general abstract framework behind the investigation presented in this
study (and also implicitly behind most of the works found in the literature),
may be described as follows:
The objects of study are the _model chemistries_ defined by Pople [63] and
discussed in the introduction. The MCs under scrutiny are applied to a
particular _problem_ of interest, which may be thought to be formed by three
ingredients: the _physical system_ , the _relevant observables_ and the
_target accuracy_. The MCs are then selected according to their ability to
yield numerical values of the relevant observables for the physical system
studied within the target accuracy. The concrete numerical values that one
wants to approach are those given by the _exact model chemistry_ MCε, which
could be thought to be either the experimental data or the exact solution of
the non-relativistic electronic Schrödinger equation [60]. However, the
computational effort needed to perform the calculations required by MCε is
literally infinite, so that, in practice, one is forced to work with a
_reference model chemistry_ MCref, which, albeit different from MCε, is
thought to be close to it. Finally, the set of MCs that one wants to
investigate are compared to MCref and the nearness to it is seen as
approximating the nearness to MCε.
Figure 2: Space $\mathcal{M}$ of all model chemistries. The exact model
chemistry MCε is shown as a black circle, the MP2 reference MC is shown as a
grey-filled circle, and B3LYP MCs as white-filled ones. Both reference PESs
are indicated with an additional circle around the points. The situation
depicted is (schematically) the one found in this study, assuming that MP2 is
a more accurate method than B3LYP to account for the conformational
preferences of peptide systems. The positions of the different MCs have no
relevance, and only the relative measured distances among them are
qualitatively depicted.
These comparisons are commonly performed using a numerical quantity
$\mathcal{D}$ that is a function of the relevant observables. In order for the
intuitive ideas about relative proximity in the $\mathcal{M}$ space to be
captured and the above reasoning to be meaningful, this numerical quantity
$\mathcal{D}$ must have some of the properties of a mathematical distance. In
particular, it is advisable that the _triangle inequality_ is obeyed, so that,
for any model chemistry MC, one has that
$\displaystyle\mathcal{D}(\mathrm{MC}_{\varepsilon},\mathrm{MC})\leq\mathcal{D}(\mathrm{MC}_{\varepsilon},\mathrm{MC}^{\mathrm{ref}})+\mathcal{D}(\mathrm{MC}^{\mathrm{ref}},\mathrm{MC})\
,$ (3a)
$\displaystyle\mathcal{D}(\mathrm{MC}_{\varepsilon},\mathrm{MC})\geq\big{|}\mathcal{D}(\mathrm{MC}_{\varepsilon},\mathrm{MC}^{\mathrm{ref}})-\mathcal{D}(\mathrm{MC}^{\mathrm{ref}},\mathrm{MC})\big{|}\
,$ (3b)
and, assuming that
$\mathcal{D}(\mathrm{MC}_{\varepsilon},\mathrm{MC}^{\mathrm{ref}})$ is small
(and $\mathcal{D}$ is a positive function), we obtain
$\mathcal{D}(\mathrm{MC}_{\varepsilon},\mathrm{MC})\simeq\mathcal{D}(\mathrm{MC}^{\mathrm{ref}},\mathrm{MC})\
,$ (4)
which is the sought result in agreement with the ideas stated at the beginning
of this section.
The distance $d_{12}$ introduced in ref. 92 and summarized in the previous
section, measured in this case on the conformational energy surfaces (the
relevant observable) of the model dipeptide HCO-L-Ala-NH2 (the physical
system), approximately fulfills the triangle inequality and thus captures the
_nearness_ concept in the space $\mathcal{M}$ of model chemistries.
This space, $\mathcal{M}$, containing all possible MCs, is a rather complex
and multidimensional one. For example, two commonly used ‘dimensions’ which
may be thought to parameterize $\mathcal{M}$ are the size of the basis set and
the amount of electron correlation in the model (or the quality of the DFT
functional used). However, since there are many ways in which the size of a
basis set or the electron correlation may be increased and there are
additional approximations that can be included in the MC definition (see sec.
1), the ‘dimensions’ of $\mathcal{M}$ can be considered to be many more than
two.
The definition of a distance, such as the one described in the previous lines,
for a given problem of interest helps to provide a certain degree of structure
into this complex space. In fig. 2 a two-dimensional scheme of the overall
situation found in this study is presented.
## 4 Results
MCs | $d_{12}/RT$ a | $a_{12}$ b | $N_{\mathrm{res}}$ c | $t$ d
---|---|---|---|---
B3LYP/aug-cc-pVTZ | 0.079 | 15.2 | 159.8 | 79.09%
B3LYP/cc-pVTZ | 0.191 | 21.1 | 27.4 | 9.78%
B3LYP/aug-cc-pVDZ | 0.172 | 82.8 | 33.7 | 5.27%
B3LYP/cc-pVDZ | 1.045 | 109.4 | 0.9 | 1.29%
Table 2: Basis set convergence results for the B3LYP MCs investigated in this
work. aDistance with the B3LYP/cc-pVQZ reference in units of $RT$ at
$300^{\mathrm{o}}$ K. bEnergy offset with the reference MC in kcal/mol.
cMaximum number of residues in a polypeptide potential up to which the
corresponding MC may correctly approximate the reference (under the
assumptions in sec. 2). dRequired computer time, expressed as a fraction of
$t_{\mathrm{ref}}$.
Before starting with the results of the calculations, let us introduce the
concept of _efficiency_ of a particular MC that shall be used: It is laxly
defined as a balance between accuracy (in terms of the distance introduced in
sec. 2) and computational cost (in terms of computer time). It can be
graphically extracted from the _efficiency plots_ , where the distance
$d_{12}$ between any given MC and a reference one is shown in units of $RT$ in
the $x$-axis, while, in the $y$-axis, one can find the computer time taken for
each MC (see the following pages for two examples). As a general thumb-rule,
_we shall consider a MC to be more efficient for approximating the reference
when it is placed closer to the origin of coordinates in the efficiency plot_.
This approach is intentionally non-rigorous due to the fact that many factors
exist that influence the computer time but may vary from one practical
calculation to another; such as the algorithms used, the actual details of the
computers (frequency of the processor, size of the RAM and cache memories,
system bus and disk access velocity, operating system, mathematical libraries,
etc.), the starting guesses for the SCF orbitals or the starting structures in
geometry optimizations.
Taking all this into account, the only conclusions that shall be drawn in this
work about the relative efficiency of the MCs studied are those deduced from
strong signals in the plots and, therefore, those that can be extrapolated to
future calculations; in other words, _the small details shall be typically
neglected_.
Figure 3: Efficiency plot of all the B3LYP MCs studied. In the $x$-axis, we
show the distance $d_{12}$, in units of $RT$ at $300^{\mathrm{o}}$ K, between
any given MC and the B3LYP/cc-pVQZ reference (indicated by an encircled
point), while, in the $y$-axis, we present the computer time needed to compute
the whole 12$\times$12 grid in the Ramachandran space of the model dipeptide
HCO-L-Ala-NH2. The different accuracy regions are labeled, and the 10% of the
time $t_{\mathrm{best}}$ taken by the reference MC is also indicated.
In the first part of the study, we compare all B3LYP MCs to the one with the
largest basis set, B3LYP/cc-pVQZ (the highest level of the theory calculated
for this work, depicted in fig. 4) using the distance introduced in sec. 2.
All mentions to the accuracy of any given MC in this part must be understood
as relative to this reference. However, it has been reported that MP2 is a
superior method to B3LYP to account for the conformational behaviour of
peptide systems [94]. Therefore, the absolute accuracy of the B3LYP MCs
calculated here is probably closer to the relative accuracy with respect to
the MP2/6-311++G(2df,2pd) reference in what follows. In this spirit, this part
of the study should be regarded as an investigation of the convergence to _the
infinite basis set B3LYP limit_ , for which the best B3LYP MC here is probably
a good approximation.
Figure 4: Potential energy surface of the model dipeptide HCO-L-Ala-NH2
computed at the B3LYP/cc-pVQZ level of the theory. The PES has been originally
calculated in a 12$\times$12 discrete grid in the space spanned by the
Ramachandran angles $\phi$ and $\psi$ and later smoothed with bicubic splines
for visual convenience. The energy reference has been set to zero. (At this
level of the theory, the absolute energy of the minimum point in the
12$\times$12 grid, located at $(-75^{o},75^{o})$, is $-417.199231353$
hartree).
The results are depicted in fig. 3, and in table 2. We can extract several
conclusions from them:
* •
Regarding the convergence to the infinite basis set limit, we observe that
only the most expensive MC, B3LYP/aug-cc-pVTZ, correctly approximates the
reference for peptides of more than 100 residues. On the other hand, for only
5.27% of the computer time $t_{\mathrm{ref}}$ taken by the reference MC, we
can use B3LYP/aug-cc-pVDZ, which correctly approximates it up to 30-residue
peptides. Finally, the MC with the smallest basis set, B3LYP/cc-pVDZ cannot
properly replace the reference even in dipeptides.
* •
In ref. [34], using Pople’s basis sets [95, 96, 102, 97, 98, 99, 100, 101], we
saw that “the general rule that is sometimes assumed when performing quantum
chemical calculations, which states that ‘the more expensive, the more
accurate’, is rather coarse-grained and relevant deviations from it may be
found.” We recognized that “One may argue that this observation is due to the
unsystematic way in which Pople basis sets can be enlarged and that the
correlation between accuracy and cost will be much higher if, for example,
only Dunning basis sets are used.”, which is definitely observed in fig. 3,
but we argued that this was something to be expected, since “there are two few
Dunning basis sets below a reasonable upper bound on the number of elements to
see anything but a line in the efficiency plot”. In the results presented in
this work, we can see that, even if the correlation between accuracy and cost
is higher in the case of Dunning’s basis sets than in the case of Pople’s, due
to the smaller number of the former, we can still observe that the thumb-rule
‘the more expensive, the more accurate’ breaks also in this case, since the
B3LYP/aug-cc-pVDZ MC is, at the same time, more accurate and less costly than
B3LYP/cc-pVTZ. In general, this idea applies to all the approximations that a
MC may contain (see the introduction for a partial list), and justifies the
systematic search for the most efficient combination of them for a given
problem. This work is our second step (ref. [34] is the first one) in that
path for the particular case of the conformational behaviour of peptide
systems.
* •
The observation in the previous point also suggests that it may be efficient
to include diffuse functions (the ‘aug-’ in aug-cc-pVDZ) in the basis set for
this type of problems.
* •
The error of the studied MCs regarding the differences of energy (as measured
by $d_{12}$) is much smaller than the error in the absolute energies (as
measured by $a_{12}$), suggesting that the largest part of the discrepancy
must be a systematic one.
In the second part of the study, we assess the absolute accuracy of the B3LYP
MCs by comparing them to the (as far as we are aware) highest homolevel in the
literature, the MP2/6-311++ G(2df,2pd) PES in ref. [34]. If one assumes that
this level of the theory may be close enough to the exact result for the given
problem at hand, then this comparison measures the ‘absolute’ accuracy of the
B3LYP MCs, and not only their relative accuracy with respect to the B3LYP
infinite basis set limit, as we did in the previous part. This is the
fundamental difference between figs. 3 and 5.
MCs | $d_{12}/RT$ a | $a_{12}$ b | $N_{\mathrm{res}}$ c | $t$ d
---|---|---|---|---
B3LYP/cc-pVQZ | 1.008 | -457.2 | 0.98 | 1861
B3LYP/aug-cc-pVTZ | 1.029 | -442.0 | 0.94 | 1472
B3LYP/cc-pVTZ | 1.058 | -436.1 | 0.89 | 182
B3LYP/aug-cc-pVDZ | 1.006 | -374.4 | 0.99 | 98
B3LYP/cc-pVDZ | 1.533 | -347.8 | 0.43 | 24
Table 3: Comparison of all the B3LYP MCs investigated in this work with the
MP2/6-311++G(2df,2pd) in ref. 34. aDistance with the MP2/6-311++G(2df,2pd)
reference in units of $RT$ at $300^{\mathrm{o}}$ K. bEnergy offset with the
reference MC in kcal/mol. cMaximum number of residues in a polypeptide
potential up to which the corresponding MC may correctly approximate the
reference (under the assumptions in sec. 2). dComputer time needed for the
calculation of the whole PES, in days.
The results of this part of the study are depicted in fig. 5, and in table 3.
We can extract several conclusions from them:
* •
All B3LYP MCs, including the largest one, B3LYP/cc-pVQZ, lie in the inaccurate
region of the efficiency plot in fig. 5, meaning that they cannot be reliably
used to approximate the MP2/6-311++G(2df,2pd) reference even in the smallest
dipeptides.
* •
Related with the observations in the previous part of the study, we see that
there is no point, if one is worried about absolute accuracy, in going beyond
the aug-cc-pVDZ basis set in B3LYP.
* •
The B3LYP/cc-pVDZ MC again performs significantly worse than the rest,
agreeing with the results in the previous part of the study, and suggesting
that cc-pVDZ may be a too small basis set for the problem tackled here.
* •
Again, the error of the MCs in the differences of energy (as measured by
$d_{12}$) is much smaller than the error in the absolute energies (as measured
by $a_{12}$).
Figure 5: Efficiency plot of all the B3LYP MCs studied. In the $x$-axis, we
show the distance $d_{12}$, in units of $RT$ at $300^{\mathrm{o}}$ K, between
any given MC and the MP2/6-311++G(2df,2pd) reference calculated in ref. 34,
while, in the $y$-axis, we present the computer time needed to compute the
whole 12$\times$12 grid in the Ramachandran space of the model dipeptide HCO-
L-Ala-NH2. The different accuracy regions are labeled
## 5 Conclusions
In this study, we have investigated 5 PESs of the model dipeptide HCO-L-Ala-
NH2, calculated with the B3LYP method, and the cc-pVDZ, aug-cc-pVDZ, cc-pVTZ,
aug-cc-pVTZ, and cc-pVQZ Dunning’s basis sets. We have first assessed the
convergence of the B3LYP MCs to the infinite basis set limit, and then we have
evaluated their absolute accuracy by comparing them to the (as far as we are
aware) highest homolevel in the literature, the MP2/6-311++G(2df,2pd) PES in
ref. [34]. All the comparisons have been performed according to a general
framework which is extensible to further studies, and using a distance between
the different PESs that correctly captures the nearness concept in the space
of MCs. The calculations performed here have taken around 10 years of computer
time.
The main conclusions of the study are the following:
* •
The complexity of the problem (the conformational behaviour of peptides)
renders the correlation between accuracy and computational cost of the
different quantum mechanical algorithms imperfect. This ultimately justifies
the need for systematic studies, such as the one presented here, in which the
most efficient MCs are sought for the particular problem of interest.
* •
Assuming that the MP2/6-311++G(2df,2pd) level of the theory is closer to the
exact solution of the non-relativistic electronic Schrödinger equation than
B3LYP/cc-pVQZ, B3LYP is not a reliable method to study the conformational
behaviour of peptides. Even if, as we emphasize at the end of this section, it
may be dangerous to state that a method that performs well in the particular
model of an alanine residue studied here will also be recommendable for longer
and more complex peptides, we can clearly _reject_ any method that already
fails in HCO-L-Ala-NH2.
* •
If B3LYP is still needed to be used, due to, for example, computational
constraints, aug-cc-pVDZ represents a good compromise between accuracy and
cost.
* •
The error of the studied MCs regarding the differences of energy (as measured
by $d_{12}$) is much smaller than the error in the absolute energies (as
measured by $a_{12}$), suggesting that the largest part of the discrepancy
must be a systematic one.
Finally, let us stress again that the investigation performed here have used
one of the simplest dipeptides. The fact that we have treated it as an
isolated system, the small size of its side chain and also its aliphatic
character, all play a role in the results obtained. Hence, for bulkier
residues included in polypeptides, and, specially for those that contain
aromatic groups, those that are charged or may participate in hydrogen-bonds,
the methods that have proved to be efficient here must be re-tested and the
conclusions drawn about the B3LYP convergence to the infinite basis set limit,
as well as those regarding the comparison between B3LYP and MP2, should be re-
evaluated.
## Acknowledgments
The numerical calculations in this work have been performed thanks to a
computer time grant at the Zaragoza node (Caesaraugusta) of the Spanish
Supercomputing Network (RES). We thank all the support staff there, for the
efficiency at solving the problems encountered. We also thank J. L. Alonso for
illuminating discussions.
This work has been supported by the research projects DGA (Aragón Government,
Spain) E24/3 and MEC (Spain) FIS2006-12781-C02-01. P. Echenique is supported
by a MEC (Spain) postdoctoral contract.
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|
arxiv-papers
| 2008-11-08T16:07:19 |
2024-09-04T02:48:58.676135
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pablo Echenique, Gregory A. Chass",
"submitter": "Pablo Echenique",
"url": "https://arxiv.org/abs/0811.1271"
}
|
0811.1480
|
# Exact Categories
Theo Bühler theo@math.ethz.ch FIM, HG G39.5, Rämistrasse 101, 8092 ETH
Zürich, Switzerland
###### Abstract
We survey the basics of homological algebra in exact categories in the sense
of Quillen. All diagram lemmas are proved directly from the axioms, notably
the five lemma, the $3\times 3$-lemma and the snake lemma. We briefly discuss
exact functors, idempotent completion and weak idempotent completeness. We
then show that it is possible to construct the derived category of an exact
category without any embedding into abelian categories and we sketch Deligne’s
approach to derived functors. The construction of classical derived functors
with values in an abelian category painlessly translates to exact categories,
i.e., we give proofs of the comparison theorem for projective resolutions and
the horseshoe lemma. After discussing some examples we elaborate on Thomason’s
proof of the Gabriel-Quillen embedding theorem in an appendix.
###### keywords:
Exact Categories , Diagram Lemmas , Homological Algebra , Derived Functors ,
Derived Categories , Embedding Theorems
###### MSC:
Primary: 18-02 , Secondary: 18E10, 18E30
###### Contents
1. 1 Introduction
2. 2 Definition and Basic Properties
3. 3 Some Diagram Lemmas
4. 4 Quasi-Abelian Categories
5. 5 Exact Functors
6. 6 Idempotent Completion
7. 7 Weak Idempotent Completeness
8. 8 Admissible Morphisms and the Snake Lemma
9. 9 Chain Complexes and Chain Homotopy
10. 10 Acyclic Complexes and Quasi-Isomorphisms
1. 10.1 The Homotopy Category of Acyclic Complexes
2. 10.2 Boundedness Conditions
3. 10.3 Quasi-Isomorphisms
4. 10.4 The Definition of the Derived Category
5. 10.5 Derived Categories of Fully Exact Subcategories
6. 10.6 Total Derived Functors
11. 11 Projective and Injective Objects
12. 12 Resolutions and Classical Derived Functors
13. 13 Examples and Applications
1. 13.1 Additive Categories
2. 13.2 Quasi-Abelian Categories
3. 13.3 Fully Exact Subcategories
4. 13.4 Frobenius Categories
5. 13.5 Further Examples
6. 13.6 Higher Algebraic $K$-Theory
14. A The Embedding Theorem
1. A.1 Separated Presheaves and Sheaves
2. A.2 Outline of the Proof
3. A.3 Sheafification
4. A.4 Proof of the Embedding Theorem
15. B Heller’s Axioms
## 1 Introduction
There are several notions of exact categories. On the one hand, there is the
notion in the context of additive categories commonly attributed to Quillen
[50] with which the present article is concerned; on the other hand, there is
the non-additive notion due to Barr [3], to mention but the two most prominent
ones. While Barr’s definition is intrinsic and an additive category is exact
in his sense if and only if it is abelian, Quillen’s definition is extrinsic
in that one has to specify a distinguished class of short exact sequences (an
exact structure) in order to obtain an exact category.
From now on we shall only deal with additive categories, so functors are
tacitly assumed to be additive. On every additive category
$\operatorname{\mathscr{A}}$ the class of all split exact sequences provides
the smallest exact structure, i.e., every other exact structure must contain
it. In general, an exact structure consists of kernel-cokernel pairs subject
to some closure requirements, so the class of all kernel-cokernel pairs is a
candidate for the largest exact structure. It is quite often the case that the
class of all kernel-cokernel pairs is an exact structure, but this fails in
general: Rump [52] constructs an example of an additive category with kernels
and cokernels whose kernel-cokernel pairs fail to be an exact structure.
It is commonplace that basic homological algebra in categories of modules over
a ring extends to abelian categories. By using the Freyd-Mitchell full
embedding theorem ([17] and [46]), diagram lemmas can be transferred from
module categories to general abelian categories, i.e., one may argue by
chasing elements around in diagrams. There is a point in proving the
fundamental diagram lemmas directly, and be it only to familiarize oneself
with the axioms. A careful study of what is actually needed reveals that in
most situations the axioms of exact categories are sufficient. An _a
posteriori_ reason is provided by the Gabriel-Quillen embedding theorem which
reduces homological algebra in exact categories to the case of abelian
categories, the slogan is “relative homological algebra made absolute” (Freyd
[16]). More specifically, the embedding theorem asserts that the Yoneda
functor embeds a small exact category $\operatorname{\mathscr{A}}$ fully
faithfully into the _abelian_ category $\operatorname{\mathscr{B}}$ of left
exact functors
$\operatorname{\mathscr{A}}^{\operatorname{op}}\to\operatorname{\mathbf{Ab}}$
in such a way that the essential image is closed under extensions and that a
short sequence in $\operatorname{\mathscr{A}}$ is short exact if and only if
it is short exact in $\operatorname{\mathscr{B}}$. Conversely, it is not hard
to see that an extension-closed subcategory of an abelian category is
exact—this is _the_ basic recognition principle for exact categories. In
appendix A we present Thomason’s proof of the Gabriel-Quillen embedding
theorem for the sake of completeness, but we will not apply it in these notes.
The author is convinced that the embedding theorem should be used to transfer
the intuition from abelian categories to exact categories rather than to prove
(simple) theorems with it. A direct proof from the axioms provides much more
insight than a reduction to abelian categories.
The interest of exact categories is manifold. First of all they are a natural
generalization of abelian categories and there is no need to argue that
abelian categories are both useful and important. There are several reasons
for going beyond abelian categories. The fact that one may _choose_ an exact
structure gives more flexibility which turns out to be essential in many
contexts. Even if one is working with abelian categories one soon finds the
need to consider other exact structures than the canonical one, for instance
in relative homological algebra [29]. Beyond this, there are quite a few
“cohomology theories” which involve functional analytic categories like
locally convex modules over a topological group [30, 9], locally compact
abelian groups [31] or Banach modules over a Banach algebra [32, 25] where
there is no obvious abelian category around to which one could resort. In more
advanced topics of algebra and representation theory, (e.g. filtered objects,
tilting theory), exact categories arise naturally, while the theory of abelian
categories simply does not fit. It is an observation due to Happel [24] that
in guise of _Frobenius categories,_ exact categories give rise to triangulated
categories by passing to the associated stable categories, see section 13.4.
Further fields of application are algebraic geometry (certain categories of
vector bundles), algebraic analysis ($\operatorname{\mathscr{D}}$-modules)
and, of course, algebraic $K$-theory (Quillen’s $Q$-construction [50],
Balmer’s Witt groups [2] and Schlichting’s Grothendieck-Witt groups [53]). The
reader will find a slightly more extensive discussion of some of the topics
mentioned above in section 13.
The author hopes to convince the reader that the axioms of exact categories
are quite convenient for giving relatively painless proofs of the basic
results in homological algebra and that the gain in generality comes with
almost no effort. Indeed, it even seems that the axioms of exact categories
are more adequate for proving the fundamental diagram lemmas than
Grothendieck’s axioms for abelian categories. For instance, it is quite a
challenge to find a complete proof (directly from the axioms) of the snake
lemma for abelian categories in the literature.
That being said, we turn to a short description of the contents of this paper.
In section 2 we state and discuss the axioms and draw the basic consequences,
in particular we give the characterization of pull-back squares and Keller’s
proof of the obscure axiom.
In section 3 we prove the (short) five lemma, the Noether isomorphism theorem
and the $3\times 3$-lemma.
Section 4 briefly discusses quasi-abelian categories, a source of many
examples of exact categories. Contrary to the notion of an exact category, the
property of being quasi-abelian is intrinsic.
Exact functors are briefly touched upon in section 5 and after that we treat
the idempotent completion and the property of weak idempotent completeness in
sections 6 and 7.
We come closer to the heart of homological algebra when discussing admissible
morphisms, long exact sequences, the five lemma and the snake lemma in section
8. In order for the snake lemma to hold, the assumption of weak idempotent
completeness is necessary.
After that we briefly remind the reader of the notions of chain complexes and
chain homotopy in section 9, before we turn to acyclic complexes and quasi-
isomorphisms in section 10. Notably, we give an elementary proof of Neeman’s
crucial result that the category of acyclic complexes is triangulated. We do
not indulge in the details of the construction of the derived category of an
exact category because this is well treated in the literature. We give a brief
summary of the derived category of fully exact subcategories and then sketch
the main points of Deligne’s approach to total derived functors on the level
of the derived category as expounded by Keller [38].
On a more leisurely level, projective and injective objects are discussed in
section 11 preparing the grounds for a treatment of classical derived functors
(with values in an abelian category) in section 12, where we state and prove
the resolution lemma, the comparison theorem and the horseshoe lemma, i.e.,
the three basic ingredients for the classical construction.
We end with a short list of examples and applications in section 13.
In appendix A we give Thomason’s proof of the Gabriel-Quillen embedding
theorem of an exact category into an abelian one. Finally, in appendix B we
give a proof of the folklore fact that under the assumption of weak idempotent
completeness Heller’s axioms for an “abelian” category are equivalent to
Quillen’s axioms for an exact category.
###### Historical Note.
Quillen’s notion of an exact category has its predecessors e.g. in Heller
[26], Buchsbaum [10], Yoneda [60], Butler-Horrocks [13] and Mac Lane [43,
XII.4]. It should be noted that Buchsbaum, Butler-Horrocks and Mac Lane assume
the existence of an ambient abelian category and miss the crucial push-out and
pull-back axioms, while Heller and Yoneda anticipate Quillen’s definition.
According to Quillen [50, p. “92/16/100”], assuming idempotent completeness,
Heller’s notion of an “abelian category” [26, § 3], i.e., an additive category
equipped with an “abelian class of short exact sequences”coincides with the
present definition of an exact category. We give a proof of this assertion in
appendix B. Yoneda’s quasi-abelian $\operatorname{\mathcal{S}}$-categories are
nothing but Quillen’s exact categories and it is a remarkable fact that Yoneda
proves that Quillen’s “obscure axiom” follows from his definition, see [60, p.
525, Corollary], a fact rediscovered thirty years later by Keller in [36,
A.1].
###### Prerequisites.
The prerequisites are kept at a minimum. The reader should know what an
additive category is and be familiar with fundamental categorical concepts
such as kernels, pull-backs, products and duality. Acquaintance with basic
category theory as presented in Hilton-Stammbach [28, Chapter II] or Weibel
[59, Appendix A] should amply suffice for a complete understanding of the
text, up to section 10 where we assume some familiarity with the theory of
triangulated categories.
###### Disclaimer.
This article is written for the reader who _wants_ to learn about exact
categories and knows _why_. Very few motivating examples are given in this
text.
The author makes no claim to originality. All the results are well-known in
some form and they are scattered around in the literature. The _raison d’être_
of this article is the lack of a systematic _elementary_ exposition of the
theory. The works of Heller [26], Keller [36, 38] and Thomason [57] heavily
influenced the present paper and many proofs given here can be found in their
papers.
## 2 Definition and Basic Properties
In this section we introduce the notion of an exact category and draw the
basic consequences of the axioms. We do not use the minimal axiomatics as
provided by Keller [36, Appendix A] but prefer to use a convenient self-dual
presentation of the axioms due to Yoneda [60, § 2] (modulo some of Yoneda’s
numerous $3\times 2$-lemmas and our Proposition 2.12). The author hopes that
the Bourbakists among the readers will pardon this _faux pas_. We will discuss
that the present axioms are equivalent to Quillen’s [50, § 2] in the course of
events. The main points of this section are a characterization of push-out
squares (Proposition 2.12) and the obscure axiom (Proposition 2.16).
###### 2.1 Definition.
Let $\operatorname{\mathscr{A}}$ be an additive category. A _kernel-cokernel
pair_ $(i,p)$ in $\operatorname{\mathscr{A}}$ is a pair of composable
morphisms
$A^{\prime}\xrightarrow{i}A\xrightarrow{p}A^{\prime\prime}$
such that $i$ is a kernel of $p$ and $p$ is a cokernel of $i$. If a class
$\operatorname{\mathscr{E}}$ of kernel-cokernel pairs on
$\operatorname{\mathscr{A}}$ is fixed, an _admissible monic_ is a morphism $i$
for which there exists a morphism $p$ such that
$(i,p)\in\operatorname{\mathscr{E}}$. _Admissible epics_ are defined dually.
We depict admissible monics by $\operatorname{\rightarrowtail}$ and admissible
epics by $\operatorname{\twoheadrightarrow}$ in diagrams.
An _exact structure_ on $\operatorname{\mathscr{A}}$ is a class
$\operatorname{\mathscr{E}}$ of kernel-cokernel pairs which is closed under
isomorphisms and satisfies the following axioms:
* [E0]
For all objects $A\in\operatorname{\mathscr{A}}$, the identity morphism
$1_{A}$ is an admissible monic.
* [E0${}^{\operatorname{op}}$]
For all objects $A\in\operatorname{\mathscr{A}}$, the identity morphism
$1_{A}$ is an admissible epic.
* [E1]
The class of admissible monics is closed under composition.
* [E1${}^{\operatorname{op}}$]
The class of admissible epics is closed under composition.
* [E2]
The push-out of an admissible monic along an arbitrary morphism exists and
yields an admissible monic.
* [E2${}^{\operatorname{op}}$]
The pull-back of an admissible epic along an arbitrary morphism exists and
yields an admissible epic.
Axioms [E2] and [E2${}^{\operatorname{op}}$] are subsumed in the diagrams
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$PO$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime}}$
and
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$PB$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B}$
respectively.
An _exact category_ is a pair
$(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ consisting of an
additive category $\operatorname{\mathscr{A}}$ and an exact structure
$\operatorname{\mathscr{E}}$ on $\operatorname{\mathscr{A}}$. Elements of
$\operatorname{\mathscr{E}}$ are called _short exact sequences_.
###### 2.2 Remark.
Note that $\operatorname{\mathscr{E}}$ is an exact structure on
$\operatorname{\mathscr{A}}$ if and only if
$\operatorname{\mathscr{E}}^{\operatorname{op}}$ is an exact structure on
$\operatorname{\mathscr{A}}^{\operatorname{op}}$. This allows for reasoning by
dualization.
###### 2.3 Remark.
Isomorphisms are admissible monics and admissible epics. Indeed, this follows
from the commutative diagram
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{1_{A}}$$\scriptstyle{f}$$\scriptstyle{\cong}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{f^{-1}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{A}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$
the fact that exact structures are assumed to be closed under isomorphisms and
that the axioms are self-dual.
###### 2.4 Remark (Keller [36, App. A]).
The axioms are somewhat redundant and can be weakened. For instance, let us
assume instead of [E0] and [E0${}^{\operatorname{op}}$] that $1_{0}$, the
identity of the zero object, is an admissible epic. For any object $A$ there
is the pull-back diagram
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{A}}$PB$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{0}}$$\textstyle{0}$
so [E2${}^{\operatorname{op}}$] together with our assumption on $1_{0}$ shows
that [E0${}^{\operatorname{op}}$] holds. Since $1_{0}$ is a kernel of itself,
it is also an admissible monic, so we conclude by [E2] that [E0] holds as
well. More importantly, Keller proves in _loc. cit._ (A.1, proof of the
proposition, step 3), that one can also dispose of one of [E1] or
[E1${}^{\operatorname{op}}$]. Moreover, he mentions (A.2, Remark), that one
may also weaken one of [E2] or [E2${}^{\operatorname{op}}$]—this is a
straightforward consequence of (the proof of) Proposition 3.1.
###### 2.5 Remark.
Keller [36, 38] uses _conflation_ , _inflation_ and _deflation_ for what we
call short exact sequence, admissible monic and admissible epic. This
terminology stems from Gabriel-Roĭter [21, Ch. 9] who give a list of axioms
for exact categories whose underlying additive category is weakly idempotent
complete in the sense of section 7, see Keller’s appendix to [15] for a
thorough comparison of the axioms. A variant of the Gabriel-Roĭter-axioms
appear in Freyd’s book on abelian categories [17, Ch. 7, Exercise G, p. 153]
(the Gabriel-Roĭter-axioms are obtained from Freyd’s axioms by adding the dual
of Freyd’s condition (2)).
###### 2.6 Exercise.
An admissible epic which is additionally monic is an isomorphism.
###### 2.7 Lemma.
The sequence
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]}$$\textstyle{A\oplus
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}0&1\end{smallmatrix}\right]}$$\textstyle{B}$
is short exact.
* Proof.
The following diagram is a push-out square
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$PO$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}0\\\
1\end{smallmatrix}\right]}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]}$$\textstyle{A\oplus B.}$
The top arrow and the left hand arrow are admissible monics by
[E0${}^{\operatorname{op}}$] while the bottom arrow and the right hand arrow
are admissible monics by [E2]. The lemma now follows from the facts that the
sequence in question is a kernel-cokernel pair and that
$\operatorname{\mathscr{E}}$ is closed under isomorphisms. ∎
###### 2.8 Remark.
Lemma 2.7 shows that Quillen’s axiom a) [50, § 2] stating that split exact
sequences belong to $\operatorname{\mathscr{E}}$ follows from our axioms.
Conversely, Quillen’s axiom a) obviously implies [E0] and
[E0${}^{\operatorname{op}}$]. Quillen’s axiom b) coincides with our axioms
[E1], [E1${}^{\operatorname{op}}$], [E2] and [E2${}^{\operatorname{op}}$]. We
will prove that Quillen’s axiom c) follows from our axioms in Proposition
2.16.
###### 2.9 Proposition.
The direct sum of two short exact sequences is short exact.
* Proof.
Let
$A^{\prime}\operatorname{\rightarrowtail}A\operatorname{\twoheadrightarrow}A^{\prime\prime}$
and
$B^{\prime}\operatorname{\rightarrowtail}B\operatorname{\twoheadrightarrow}B^{\prime\prime}$
be two short exact sequences. First observe that for every object $C$ the
sequence
$A^{\prime}\oplus C\operatorname{\rightarrowtail}A\oplus
C\operatorname{\twoheadrightarrow}A^{\prime\prime}$
is exact—the second morphism is an admissible epic because it is the
composition of the admissible epics
$\left[\begin{smallmatrix}1&0\end{smallmatrix}\right]:A\oplus
C\operatorname{\twoheadrightarrow}A$ and
$A\operatorname{\twoheadrightarrow}A^{\prime\prime}$; the first morphism in
the sequence is a kernel of the second one, hence it is an admissible monic.
Now it follows from [E1] that
$A^{\prime}\oplus B^{\prime}\operatorname{\rightarrowtail}A\oplus B$
is an admissible monic because it is the composition of the two admissible
monics $A^{\prime}\oplus B^{\prime}\operatorname{\rightarrowtail}A\oplus
B^{\prime}$ and $A\oplus B^{\prime}\operatorname{\rightarrowtail}A\oplus B$.
It is obvious that
$A^{\prime}\oplus B^{\prime}\operatorname{\rightarrowtail}A\oplus
B\operatorname{\twoheadrightarrow}A^{\prime\prime}\oplus B^{\prime\prime}$
is a kernel-cokernel pair, hence the proposition is proved. ∎
###### 2.10 Corollary.
The exact structure $\operatorname{\mathscr{E}}$ is an additive subcategory of
the additive category $\operatorname{\mathscr{A}}^{\to\to}$ of composable
morphisms of $\operatorname{\mathscr{A}}$. ∎
###### 2.11 Remark.
In Exercise 3.9 the reader is asked to show that $\operatorname{\mathscr{E}}$
is exact with respect to a natural exact structure.
###### 2.12 Proposition.
Consider a commutative square
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{f}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$$\textstyle{B^{\prime}}$
in which the horizontal arrows are admissible monics. The following assertions
are equivalent:
1. (i)
The square is a push-out.
2. (ii)
The sequence
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}i\\\
-f\end{smallmatrix}\right]}$$\textstyle{B\oplus
A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}f^{\prime}&i^{\prime}\end{smallmatrix}\right]}$$\textstyle{B^{\prime}}$
is short exact.
3. (iii)
The square is bicartesian, i.e., both a push-out and a pull-back.
4. (iv)
The square is part of a commutative diagram
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{f}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{p}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$$\textstyle{C}$
with exact rows.
* Proof.
(i) $\Rightarrow$ (ii): The push-out property is equivalent to the assertion
that $\left[\begin{smallmatrix}f^{\prime}&i^{\prime}\end{smallmatrix}\right]$
is a cokernel of $\left[\begin{smallmatrix}i\\\ -f\end{smallmatrix}\right]$,
so it suffices to prove that the latter is an admissible monic. But this
follows from [E1] since $\left[\begin{smallmatrix}i\\\
-f\end{smallmatrix}\right]$ is equal to the composition of the morphisms
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]}$$\textstyle{A\oplus
A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}1&0\\\
-f&1\end{smallmatrix}\right]}$$\scriptstyle{\cong}$$\textstyle{A\oplus
A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}i&0\\\
0&1\end{smallmatrix}\right]}$$\textstyle{B\oplus A^{\prime}}$
which are all admissible monics by Lemma 2.7, Remark 2.3 and Proposition 2.9,
respectively.
(ii) $\Rightarrow$ (iii) and (iii) $\Rightarrow$ (i): obvious.
(i) $\Rightarrow$ (iv): Let $p:B\operatorname{\twoheadrightarrow}C$ be a
cokernel of $i$. The push-out property of the square yields that there is a
unique morphism $p^{\prime}:B^{\prime}\to C$ such that
$p^{\prime}f^{\prime}=p$ and $p^{\prime}i^{\prime}=0$. Observe that
$p^{\prime}f^{\prime}=p$ implies that $p^{\prime}$ is epic. In order to see
that $p^{\prime}$ is a cokernel of $i^{\prime}$, let $g:B^{\prime}\to X$ be
such that $gi^{\prime}=0$. Then $gf^{\prime}i=gi^{\prime}f=0$, so
$gf^{\prime}$ factors uniquely over a morphism $h:C\to X$ such that
$gf^{\prime}=hp$. We claim that $hp^{\prime}=g$: this follows from the push-
out property of the square because $hp^{\prime}f^{\prime}=hp=gf^{\prime}$ and
$hp^{\prime}i^{\prime}=0=gi^{\prime}$. Since $p^{\prime}$ is epic, the
factorization $h$ of $g$ is unique, so $p^{\prime}$ is a cokernel of
$i^{\prime}$.
(iv) $\Rightarrow$ (ii): Form the pull-back over $p$ and $p^{\prime}$ in order
to obtain the commutative diagram
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{\prime}}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$PB$\scriptstyle{q^{\prime}}$$\scriptstyle{q}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$$\textstyle{C}$
with exact rows and columns using the dual of the implication (i)
$\Rightarrow$ (iv). Since the square
$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$$\textstyle{C}$
is commutative, there is a unique morphism $k:B\to D$ such that
$q^{\prime}k=1_{B}$ and $qk=f^{\prime}$. Since
$q^{\prime}(1_{D}-kq^{\prime})=0$, there is a unique morphism $l:D\to
A^{\prime}$ such that $j^{\prime}l=1_{D}-kq^{\prime}$. Note that $lk=0$
because $j^{\prime}lk=(1_{D}-kq^{\prime})k=0$ and $j^{\prime}$ is monic, while
the calculation
$j^{\prime}lj^{\prime}=(1_{D}-kq^{\prime})j^{\prime}=j^{\prime}$ implies
$lj^{\prime}=1_{A^{\prime}}$, again because $j^{\prime}$ is monic. Furthermore
$i^{\prime}lj=(qj^{\prime})lj=q(1_{D}-kq^{\prime})j=-(qk)(q^{\prime}j)=-f^{\prime}i=-i^{\prime}f$
implies $lj=-f$ since $i^{\prime}$ is monic.
The morphisms
$\left[\begin{smallmatrix}k&j^{\prime}\end{smallmatrix}\right]:B\oplus
A^{\prime}\to D\qquad\text{and}\qquad\left[\begin{smallmatrix}q^{\prime}\\\
l\end{smallmatrix}\right]:D\to B\oplus A^{\prime}$
are mutually inverse since
$\left[\begin{smallmatrix}k&j^{\prime}\end{smallmatrix}\right]\left[\begin{smallmatrix}q^{\prime}\\\
l\end{smallmatrix}\right]=kq^{\prime}+j^{\prime}l=1_{D}\qquad\text{and}\qquad\left[\begin{smallmatrix}q^{\prime}\\\
l\end{smallmatrix}\right]\left[\begin{smallmatrix}k&j^{\prime}\end{smallmatrix}\right]=\left[\begin{smallmatrix}q^{\prime}k&q^{\prime}j^{\prime}\\\
lk&lj^{\prime}\end{smallmatrix}\right]=\left[\begin{smallmatrix}1_{B}&0\\\
0&1_{A^{\prime}}\end{smallmatrix}\right].$
Now
$\left[\begin{smallmatrix}f^{\prime}&i^{\prime}\end{smallmatrix}\right]=q\left[\begin{smallmatrix}k&j^{\prime}\end{smallmatrix}\right]\qquad\text{and}\qquad\left[\begin{smallmatrix}i\\\
-f\end{smallmatrix}\right]=\left[\begin{smallmatrix}q^{\prime}\\\
l\end{smallmatrix}\right]j$
show that $A\xrightarrow{\left[\begin{smallmatrix}i\\\
-f\end{smallmatrix}\right]}B\oplus
A^{\prime}\xrightarrow{\left[\begin{smallmatrix}f^{\prime}&i^{\prime}\end{smallmatrix}\right]}B^{\prime}$
is isomorphic to $A\xrightarrow{j}D\xrightarrow{q}B^{\prime}$. ∎
###### 2.13 Remark.
Consider the push-out diagram
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$$\scriptstyle{a}$PO$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{B.}$
If $j^{\prime}:B^{\prime}\operatorname{\twoheadrightarrow}C^{\prime}$ is a
cokernel of $i^{\prime}$ then the unique morphism $j:B\to C^{\prime}$ such
that $ji=0$ and $jb=j^{\prime}$ is a cokernel of $i$. If
$j:B\operatorname{\twoheadrightarrow}C$ is a cokernel of $i$ then
$j^{\prime}=jb$ is a cokernel of $i^{\prime}$.
The first statement was established in the proof of the implication (i)
$\Rightarrow$ (iv) of Proposition 2.12 and we leave the easy verification of
the second statement as an exercise for the reader.
The following simple observation will only be used in the proof of Lemma 10.3.
We state it here for ease of reference.
###### 2.14 Corollary.
The surrounding rectangle in a diagram of the form
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$PB$\scriptstyle{f}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$PO$\scriptstyle{g}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\textstyle{C^{\prime}}$
is bicartesian and
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}-a\\\
gf\end{smallmatrix}\right]}$$\textstyle{A^{\prime}\oplus
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}g^{\prime}\\!f^{\prime}&c\end{smallmatrix}\right]}$$\textstyle{C^{\prime}}$
is short exact.
* Proof.
It follows from Proposition 2.12 and its dual that both squares are
bicartesian. Gluing two bicartesian squares along a common arrow yields
another bicartesian square, which entails the first part and the fact that the
sequence of the second part is a kernel-cokernel pair. The equation
$\left[\begin{smallmatrix}g^{\prime}\\!f^{\prime}&&c\end{smallmatrix}\right]=\left[\begin{smallmatrix}g^{\prime}&c\end{smallmatrix}\right]\left[\begin{smallmatrix}f^{\prime}&0\\\
0&1_{C}\end{smallmatrix}\right]$ exhibits
$\left[\begin{smallmatrix}g^{\prime}f^{\prime}&&c\end{smallmatrix}\right]$ as
a composition of admissible epics by Proposition 2.9 and Proposition 2.12. ∎
###### 2.15 Proposition.
The pull-back of an admissible monic along an admissible epic yields an
admissible monic.
* Proof.
Consider the diagram
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e^{\prime}}$PB$\scriptstyle{i^{\prime}}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\scriptstyle{pe}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{C.}$
The pull-back square exists by axiom [E$2^{\operatorname{op}}$]. Let $p$ be a
cokernel of $i$, so it is an admissible epic and $pe$ is an admissible epic by
axiom [E$1^{\operatorname{op}}$]. In any category, the pull-back of a monic is
a monic (if it exists). In order to see that $i^{\prime}$ is an admissible
monic, it suffices to prove that $i^{\prime}$ is a kernel of $pe$. Suppose
that $g^{\prime}:X\to B^{\prime}$ is such that $peg^{\prime}=0$. Since $i$ is
a kernel of $p$, there exists a unique $f:X\to A$ such that $eg^{\prime}=if$.
Applying the universal property of the pull-back square, we find a unique
$f^{\prime}:X\to A^{\prime}$ such that $e^{\prime}f^{\prime}=f$ and
$i^{\prime}f^{\prime}=g^{\prime}$. Since $i^{\prime}$ is monic, $f^{\prime}$
is the unique morphism such that $i^{\prime}f^{\prime}=g^{\prime}$ and we are
done. ∎
###### 2.16 Proposition (Obscure Axiom).
Suppose that $i:A\to B$ is a morphism in $\operatorname{\mathscr{A}}$
admitting a cokernel. If there exists a morphism $j:B\to C$ in
$\operatorname{\mathscr{A}}$ such that the composite
$ji:A\operatorname{\rightarrowtail}C$ is an admissible monic then $i$ is an
admissible monic.
###### 2.17 Remark.
The statement of the previous proposition is given as axiom c) in Quillen’s
definition of an exact category [50, § 2]. At that time, it was already proved
to be a consequence of the other axioms by Yoneda [60, Corollary, p. 525]. The
redundancy of the obscure axiom was rediscovered by Keller [36, A.1]. Thomason
baptized axiom c) the “obscure axiom” in [57, A.1.1].
A convenient and quite powerful strengthening of the obscure axiom holds under
the rather mild additional hypothesis of weak idempotent completeness, see
Proposition 7.6.
* Proof of Proposition 2.16 (Keller).
Let $k:B\to D$ be a cokernel of $i$. From the push-out diagram
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ji}$$\scriptstyle{i}$PO$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E}$
and Proposition 2.12 we conclude that $\left[\begin{smallmatrix}i\\\
ji\end{smallmatrix}\right]:A\operatorname{\rightarrowtail}A\oplus B$ is an
admissible monic. Because $\left[\begin{smallmatrix}1_{B}&0\\\
-j&1_{C}\end{smallmatrix}\right]:B\oplus C\to B\oplus C$ is an isomorphism it
is in particular an admissible monic, hence $\left[\begin{smallmatrix}i\\\
0\end{smallmatrix}\right]=\left[\begin{smallmatrix}1_{B}&0\\\
-j&1_{C}\end{smallmatrix}\right]\left[\begin{smallmatrix}i\\\
ji\end{smallmatrix}\right]$ is an admissible monic as well. Because
$\left[\begin{smallmatrix}k&0\\\ 0&1_{C}\end{smallmatrix}\right]$ is a
cokernel of $\left[\begin{smallmatrix}i\\\ 0\end{smallmatrix}\right]$, it is
an admissible epic. Consider the following diagram
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]}$$\scriptstyle{k}$PB$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}i\\\
0\end{smallmatrix}\right]}$$\textstyle{{B\oplus
C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}k&0\\\
0&1_{C}\end{smallmatrix}\right]}$$\textstyle{{D\oplus C.}}$
Since the right hand square is a pull-back, it follows that $k$ is an
admissible epic and that $i$ is a kernel of $k$, so $i$ is an admissible
monic. ∎
###### 2.18 Corollary.
Let $(i,p)$ and $(i^{\prime},p^{\prime})$ be two pairs of composable
morphisms. If the direct sum $(i\oplus i^{\prime},p\oplus p^{\prime})$ is
exact then $(i,p)$ and $(i^{\prime},p^{\prime})$ are both exact.
* Proof.
It is clear that $(i,p)$ and $(i^{\prime},p^{\prime})$ are kernel-cokernel
pairs. Since $i$ has $p$ as a cokernel and since
$\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]i=\left[\begin{smallmatrix}i&0\\\
0&i^{\prime}\end{smallmatrix}\right]\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]$
is an admissible monic, the obscure axiom implies that $i$ is an admissible
monic. ∎
###### 2.19 Exercise.
Suppose that the commutative square
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{a}$PO$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B}$
is a push-out. Prove that $a$ is an admissible monic.
_Hint:_ Let $b^{\prime}:B\operatorname{\twoheadrightarrow}B^{\prime\prime}$ be
a cokernel of $b:B^{\prime}\operatorname{\rightarrowtail}B$. Prove that
$a^{\prime}=b^{\prime}f:A\to B^{\prime\prime}$ is a cokernel of $a$, then
apply the obscure axiom.
## 3 Some Diagram Lemmas
In this section we will prove variants of diagram lemmas which are well-known
in the context of abelian categories, in particular we will prove the five
lemma and the $3\times 3$-lemma. Further familiar diagram lemmas will be
proved in section 8. The proofs will be based on the following simple
observation:
###### 3.1 Proposition.
Let $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ be an exact
category. A morphism from a short exact sequence
$A^{\prime}\operatorname{\rightarrowtail}B^{\prime}\operatorname{\twoheadrightarrow}C^{\prime}$
to another short exact sequence
$A\operatorname{\rightarrowtail}B\operatorname{\twoheadrightarrow}C$ factors
over a short exact sequence
$A\operatorname{\rightarrowtail}D\operatorname{\twoheadrightarrow}C^{\prime}$
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{a}$BC$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\scriptstyle{b^{\prime}}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\scriptstyle{b^{\prime\prime}}$BC$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{C}$
in such a way that the two squares marked BC are bicartesian. In particular
there is a canonical isomorphism of the push-out
$A\cup_{A^{\prime}}B^{\prime}$ with the pull-back $B\times_{C}C^{\prime}$.
* Proof.
Form the push-out under $f^{\prime}$ and $a$ in order to obtain the object $D$
and the morphisms $m$ and $b^{\prime}$. Let $e:D\to C^{\prime}$ be the unique
morphism such that $eb^{\prime}=g^{\prime}$ and $em=0$ and let
$b^{\prime\prime}:D\to B$ be the unique morphism $D\to B$ such that
$b^{\prime\prime}b^{\prime}=b:B^{\prime}\to B$ and $b^{\prime\prime}m=f$. It
is easy to see that $e$ is a cokernel of $m$ (Remark 2.13) and hence the
result follows from Proposition 2.12 since the square $DC^{\prime}BC$ is
commutative [this is because $a$ and $b^{\prime\prime}b^{\prime}$ determine
$c$ uniquely]. ∎
###### 3.2 Corollary (Five Lemma, I).
Consider a morphism of short exact sequences
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C.}$
If $a$ and $c$ are isomorphisms (or admissible monics, or admissible epics)
then so is $b$.
* Proof.
Assume first that $a$ and $c$ are isomorphisms. Because isomorphisms are
preserved by push-outs and pull-backs, it follows from the diagram of
Proposition 3.1 that $b$ is the composition of two isomorphisms $B^{\prime}\to
D\to B$. If $a$ and $c$ are both admissible monics, it follows from the
diagram of Proposition 3.1 together with [E$2$] and Proposition 2.15 that $b$
is the composition of two admissible monics. The case of admissible epics is
dual. ∎
###### 3.3 Exercise.
If in a morphism
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C.}$
of short exact sequences as in the five lemma 3.2 two out of $a,b,c$ are
isomorphisms then so is the third.
_Hint:_ Use e.g. that $c$ is uniquely determined by $a$ and $b$.
###### 3.4 Remark.
The reader insisting that Corollary 3.2 should be called “three lemma” rather
than “five lemma” is cordially invited to give the details of the proof of
Lemma 8.9 and to solve Exercise 8.10. We will however use the more customary
name five lemma.
###### 3.5 Lemma (“Noether Isomorphism $C/B\cong(C/A)/(B/A)$”).
Consider the diagram
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z}$
in which the first two horizontal rows and the middle column are short exact.
Then the third column exists, is short exact, and is uniquely determined by
the requirement that it makes the diagram commutative. Moreover, the upper
right hand square is bicartesian.
* Proof.
The morphism $X\to Y$ exists since the first row is exact and the composition
$A\to C\to Y$ is zero while the morphism $Y\to Z$ exists since the second row
is exact and the composition $B\to C\to Z$ vanishes. By Proposition 2.12 the
square containing $X\to Y$ is bicartesian. It follows that $X\to Y$ is an
admissible monic and that $Y\to Z$ is its cokernel. The uniqueness assertion
is obvious. ∎
###### 3.6 Corollary ($3\times 3$-Lemma).
Consider a commutative diagram
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{a}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\scriptstyle{b}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{a^{\prime}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{b^{\prime}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c^{\prime}}$$\textstyle{A^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime\prime}}$$\textstyle{B^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime\prime}}$$\textstyle{C^{\prime\prime}}$
in which the columns are exact and assume in addition that one of the
following conditions holds:
1. (i)
the middle row and either one of the outer rows is short exact;
2. (ii)
the two outer rows are short exact and $gf=0$.
Then the remaining row is short exact as well.
* Proof.
Let us prove (i). The two possibilities are dual to each other, so we need
only consider the case that the first two rows are exact. Apply Proposition
3.1 to the first two rows so as to obtain the commutative diagram
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{a}$BC$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\scriptstyle{i}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{f}}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{g}}$$\scriptstyle{j}$BC$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{C}$
where $ji=b$—notice that $i$ and $j$ are admissible monics by axiom [E$2$] and
Proposition 2.15, respectively. By Remark 2.13 the morphism $i^{\prime}:D\to
A^{\prime\prime}$ determined by $i^{\prime}i=0$ and
$i^{\prime}\bar{f}=a^{\prime}$ is a cokernel of $i$ and the morphism
$j^{\prime}:B\operatorname{\twoheadrightarrow}C^{\prime\prime}$ given by
$j^{\prime}=c^{\prime}g=g^{\prime\prime}b^{\prime}$ is a cokernel of $j$.
If we knew that the diagram
$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$$\scriptstyle{j}$$\textstyle{A^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime\prime}}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b^{\prime}}$$\scriptstyle{j^{\prime}}$$\textstyle{B^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime\prime}}$$\textstyle{C^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\prime\prime}}$
is commutative then we would conclude from the Noether isomorphism 3.5 that
$(f^{\prime\prime},g^{\prime\prime})$ is a short exact sequence. It therefore
remains to prove that $f^{\prime\prime}i^{\prime}=b^{\prime}j$ since the other
commutativity relations $b=ji$ and $g^{\prime\prime}b^{\prime}=j^{\prime}$
hold by construction. We are going to apply the push-out property of the
square $A^{\prime}B^{\prime}AD$. We have
$(f^{\prime\prime}i^{\prime})i=0=b^{\prime}b=(b^{\prime}j)i\qquad\text{and}\qquad(b^{\prime}j)\bar{f}=b^{\prime}f=f^{\prime\prime}a^{\prime}=(f^{\prime\prime}i^{\prime})\bar{f}$
which together with
$(f^{\prime\prime}i^{\prime}\bar{f})a=(f^{\prime\prime}i^{\prime}i)f^{\prime}=0\qquad\text{and}\qquad(b^{\prime}j\bar{f})a=f^{\prime\prime}a^{\prime}a=0=b^{\prime}bf^{\prime}=(b^{\prime}ji)f^{\prime}$
show that both $f^{\prime\prime}i^{\prime}$ and $b^{\prime}j$ are solutions to
the same push-out problem, hence they are equal. This settles case (i).
In order to prove (ii) we start by forming the push-out under $g^{\prime}$ and
$b$ so that we have the following commutative diagram with exact rows and
columns
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\scriptstyle{b}$PO$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\scriptstyle{b^{\prime}}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k^{\prime}}$$\textstyle{B^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime\prime}}$
in which the cokernel $k^{\prime}$ of $k$ is determined by
$k^{\prime}j=b^{\prime}$ and $k^{\prime}k=0$, while $i=bf^{\prime}$ is a
kernel of the admissible epic $j$, see Remark 2.13 and Proposition 2.15. The
push-out property of the square $B^{\prime}C^{\prime}BD$ applied to the square
$B^{\prime}C^{\prime}BC$ yields a unique morphism $d^{\prime}:D\to C$ such
that $d^{\prime}k=c$ and $d^{\prime}j=g$. The diagram
$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{\prime}}$$\scriptstyle{k^{\prime}}$$\textstyle{B^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime\prime}}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c^{\prime}}$$\textstyle{C^{\prime\prime}}$
has exact rows and it is commutative: Indeed, $c=d^{\prime}k$ holds by
construction of $d^{\prime}$, while
$c^{\prime}d^{\prime}=g^{\prime\prime}k^{\prime}$ follows from
$c^{\prime}d^{\prime}j=c^{\prime}g=g^{\prime\prime}b^{\prime}=g^{\prime\prime}k^{\prime}j$
and the fact that $j$ is epic. We conclude from Proposition 2.12 that
$DCB^{\prime\prime}C^{\prime\prime}$ is a pull-back, so $d^{\prime}$ is an
admissible epic and so is $g=d^{\prime}j$. The unique morphism
$d:A^{\prime\prime}\to D$ such that $k^{\prime}d=f^{\prime\prime}$ and
$d^{\prime}d=0$ is a kernel of $d^{\prime}$. By the pull-back property of
$DCB^{\prime\prime}C^{\prime\prime}$ the diagram
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{a^{\prime}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{j}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{\prime}}$$\textstyle{C}$
is commutative as
$k^{\prime}(da^{\prime})=f^{\prime\prime}a^{\prime}=b^{\prime}f=k^{\prime}(jf)$
and $d^{\prime}(da^{\prime})=0=gf=d^{\prime}(jf)$. Notice that the hypothesis
that $gf=0$ enters at this point of the argument. It follows from the dual of
Proposition 2.12 that $ABA^{\prime\prime}D$ is bicartesian, so $f$ is a kernel
of $g$ by Proposition 2.15. ∎
###### 3.7 Exercise.
Consider the solid arrow diagram
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\prime\prime}}$
with exact rows and columns. Strengthen the Noether isomorphism 3.5 to the
statement that there exist unique maps $C^{\prime}\to C$ and $C\to
C^{\prime\prime}$ making the diagram commutative and the sequence
$C^{\prime}\operatorname{\rightarrowtail}C\operatorname{\twoheadrightarrow}C^{\prime\prime}$
is short exact.
###### 3.8 Exercise.
In the situation of the $3\times 3$-lemma prove that there are two exact
sequences $A^{\prime}\operatorname{\rightarrowtail}A\oplus B^{\prime}\to
B\operatorname{\twoheadrightarrow}C^{\prime\prime}$ and
$A^{\prime}\operatorname{\rightarrowtail}B\to B^{\prime\prime}\oplus
C\operatorname{\twoheadrightarrow}C^{\prime\prime}$ in the sense that the
morphisms $\to$ factor as
$\operatorname{\twoheadrightarrow}\operatorname{\rightarrowtail}$ in such a
way that consecutive
$\operatorname{\rightarrowtail}\operatorname{\twoheadrightarrow}$ are short
exact [compare also with Definition 8.8].
_Hint:_ Apply Proposition 3.1 to the first two rows in order to obtain a short
exact sequence $A^{\prime}\operatorname{\rightarrowtail}A\oplus
B^{\prime}\operatorname{\twoheadrightarrow}D$ using Proposition 2.12. Conclude
from the push-out property of $DC^{\prime}BC$ that
$D\operatorname{\rightarrowtail}B$ has $C^{\prime\prime}$ as cokernel.
###### 3.9 Exercise (Heller [26, 6.2]).
Let $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ be an exact
category and consider $\operatorname{\mathscr{E}}$ as a full subcategory of
$\operatorname{\mathscr{A}}^{\to\to}$. We have shown that
$\operatorname{\mathscr{E}}$ is additive in Corollary 2.10. Let
$\operatorname{\mathscr{F}}$ be the class of short sequences
$\textstyle{(A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(A^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\prime\prime})}$
over $\operatorname{\mathscr{E}}$ with short exact columns [we write
$(A\operatorname{\rightarrowtail}B\operatorname{\twoheadrightarrow}C)$ to
indicate that we think of the sequence as an object of
$\operatorname{\mathscr{E}}$]. Prove that
$(\operatorname{\mathscr{E}},\operatorname{\mathscr{F}})$ is an exact
category.
###### 3.10 Remark.
The category of short exact sequences in a nonzero abelian category is _not_
abelian, see [43, XII.6, p. 375].
###### 3.11 Exercise (Künzer’s Axiom, cf. e.g. [41]).
1. (i)
If $f:A\to B$ is a morphism and $g:B\operatorname{\twoheadrightarrow}C$ is an
admissible epic such that $h=gf:A\operatorname{\rightarrowtail}C$ is an
admissible monic then $f$ is an admissible monic and the morphism
$\operatorname{Ker}{g}\to\operatorname{Coker}{f}$ is an admissible monic as
well.
_Hint:_ Form the pull-back $P$ over $h$ and $g$, use Proposition 2.15 and
factor $f$ over $P$ to see the first part (see also Remark 7.4). For the
second part use the Noether isomorphism 3.5.
2. (ii)
Let $\operatorname{\mathscr{E}}$ be a class of kernel-cokernel pairs in the
additive category $\operatorname{\mathscr{A}}$. Assume that
$\operatorname{\mathscr{E}}$ is closed under isomorphisms and contains the
split exact sequences. If $\operatorname{\mathscr{E}}$ enjoys the property of
point (i) and its dual then it is an exact structure.
_Hint:_ Let $f:A\operatorname{\rightarrowtail}B$ be an admissible monic and
let $a:A\to A^{\prime}$ be arbitrary. The push-out axiom follows from the
commutative diagram
---
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\left[\begin{smallmatrix}a\\\
f\end{smallmatrix}\right]}$$\textstyle{B}$$\textstyle{A^{\prime}\oplus
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}0&1\end{smallmatrix}\right]}$$\scriptstyle{\left[\begin{smallmatrix}f^{\prime}&-b\end{smallmatrix}\right]}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]}$$\textstyle{B^{\prime}}$
in which $\left[\begin{smallmatrix}a\\\ f\end{smallmatrix}\right]$ and
$f^{\prime}$ are admissible monics by (i) and
$\left[\begin{smallmatrix}f^{\prime}&-b\end{smallmatrix}\right]$ is a cokernel
of $\left[\begin{smallmatrix}a\\\ f\end{smallmatrix}\right]$. Next observe
that the dual of (i) implies that if in addition $a$ is an admissible epic
then so is $b$. In order to prove the composition axiom, let $f$ and $g$ be
admissible monics and choose a cokernel $f^{\prime}$ of $f$. Form the push-out
under $g$ and $f^{\prime}$ and verify that $gf$ is a kernel of the push-out of
$f^{\prime}$.
## 4 Quasi-Abelian Categories
###### 4.1 Definition.
An additive category $\operatorname{\mathscr{A}}$ is called _quasi-abelian_ if
1. (i)
Every morphism has a kernel and a cokernel.
2. (ii)
The class of kernels is stable under push-out along arbitrary morphisms and
the class of cokernels is stable under pull-back along arbitrary morphisms.
###### 4.2 Remark.
The concept of a quasi-abelian category is self-dual, that is to say
$\operatorname{\mathscr{A}}$ is quasi-abelian if and only if
$\operatorname{\mathscr{A}}^{\operatorname{op}}$ is quasi-abelian.
###### 4.3 Exercise.
Let $\operatorname{\mathscr{A}}$ be an additive category with kernels. Prove
that every pull-back of a kernel is a kernel.
###### 4.4 Proposition (Schneiders [54, 1.1.7]).
The class $\operatorname{\mathscr{E}}_{\max}$ of all kernel-cokernel pairs in
a quasi-abelian category is an exact structure.
* Proof.
It is clear that $\operatorname{\mathscr{E}}_{\max}$ is closed under
isomorphisms and that the classes of kernels and cokernels contain the
identity morphisms. The pull-back and push-out axioms are part of the
definition of quasi-abelian categories. By duality it only remains to show
that the class of cokernels is closed under composition. So let
$f:A\operatorname{\twoheadrightarrow}B$ and
$g:B\operatorname{\twoheadrightarrow}C$ be cokernels and put $h=gf$. In the
diagram
$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v}$$\scriptstyle{\ker{h}}$$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ker{g}}$$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ker{f}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{h}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C}$
there exist unique morphisms $u$ and $v$ making it commutative. The upper
right hand square is a pull-back, so $v$ is a cokernel and $u$ is its kernel.
But then it follows by duality that the upper right hand square is also a
push-out and this together with the fact that $h$ is epic implies that $h$ is
a cokernel of $\ker{h}$. ∎
###### 4.5 Remark.
Note that we have just re-proved the Noether isomorphism 3.5 in the special
case of quasi-abelian categories.
###### 4.6 Definition.
The _coimage_ of a morphism $f$ in a category with kernels and cokernels is
$\operatorname{Coker}{(\ker{f})}$, while the _image_ is defined to be
$\operatorname{Ker}{(\operatorname{coker}{f})}$. The _analysis_ (cf. [43,
IX.2]) of $f$ is the commutative diagram
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\operatorname{coim}{f}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{coker}{f}}$$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ker{f}}$$\textstyle{\operatorname{Coim}@wrapper{\operatorname{Coim}@presentation}{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{f}}$$\textstyle{\operatorname{Im}@wrapper{\operatorname{Im}@presentation}{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{im}{f}}$$\textstyle{\operatorname{Coker}@wrapper{\operatorname{Coker}@presentation}{f}}$
in which $\hat{f}$ is uniquely determined by requiring the diagram to be
commutative.
###### 4.7 Remark.
The difference between quasi-abelian categories and abelian categories is that
in the quasi-abelian case the canonical morphism $\hat{f}$ in the analysis $f$
is not in general an isomorphism. Indeed, it is easy to see that a quasi-
abelian category is abelian _provided_ that $\hat{f}$ is always an
isomorphism. Equivalently, not every monic is a kernel and not every epic is a
cokernel.
###### 4.8 Proposition ([54, 1.1.5]).
Let $f$ be a morphism in the quasi-abelian category
$\operatorname{\mathscr{A}}$. The canonical morphism
$\hat{f}:\operatorname{Coim}{f}\to\operatorname{Im}{f}$ is monic and epic.
* Proof.
By duality it suffices to check that the morphism $\bar{f}$ in the diagram
---
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{j}$$\textstyle{B}$$\textstyle{\operatorname{Coim}@wrapper{\operatorname{Coim}@presentation}{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{f}}$
is monic. Let $x:X\to\operatorname{Coim}{f}$ be a morphism such that
$\bar{f}x=0$. The pull-back $y:Y\to A$ of $x$ along $j$ satisfies $fy=0$, so
$y$ factors over $\operatorname{Ker}{f}$ and hence $jy=0$. But then the map
$Y\operatorname{\twoheadrightarrow}X\to\operatorname{Coim}{f}$ is zero as
well, so $x=0$. ∎
###### 4.9 Remark.
Every morphism $f$ in a quasi-abelian category $\operatorname{\mathscr{A}}$
has two epic-monic factorizations, one over $\operatorname{Coim}{f}$ and one
over $\operatorname{Im}{f}$. The quasi-abelian category
$\operatorname{\mathscr{A}}$ is abelian if and only if the two factorizations
coincide for all morphisms $f$.
###### 4.10 Remark.
An additive category with kernels and cokernels is called _semi-abelian_ if
the canonical morphism $\operatorname{Coim}{f}\to\operatorname{Im}{f}$ is
always monic and epic. We have just proved that quasi-abelian categories are
semi-abelian. It may seem obvious that the concept of semi-abelian categories
is strictly weaker than the concept of a quasi-abelian category. However, it
is surprisingly delicate to come up with an explicit example. This led Raĭkov
to conjecture that every semi-abelian category is quasi-abelian. A
counterexample to this conjecture was recently found by Rump [52].
###### 4.11 Remark.
We do not develop the theory of quasi-abelian categories any further. The
interested reader may consult Schneiders [54], Rump [51] and the references
therein.
## 5 Exact Functors
###### 5.1 Definition.
Let $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ and
$(\operatorname{\mathscr{A}}^{\prime},\operatorname{\mathscr{E}}^{\prime})$ be
exact categories. An (additive) functor
$F:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{A}}^{\prime}$ is called
_exact_ if
$F(\operatorname{\mathscr{E}})\subset\operatorname{\mathscr{E}}^{\prime}$. The
functor $F$ _reflects exactness_ if
$F(\sigma)\in\operatorname{\mathscr{E}}^{\prime}$ implies
$\sigma\in\operatorname{\mathscr{E}}$ for all
$\sigma\in\operatorname{\mathscr{A}}^{\to\to}$.
###### 5.2 Proposition.
An exact functor preserves push-outs along admissible monics and pull-backs
along admissible epics.
* Proof.
An exact functor preserves admissible monics and admissible epics, in
particular it preserves diagrams of type
and
so the result follows immediately from Proposition 2.12 and its dual. ∎
The following exercises show how one can induce new exact structures using
functors satisfying certain exactness properties.
###### 5.3 Exercise (Heller [26, 7.3]).
Let
$F:(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})\to(\operatorname{\mathscr{A}}^{\prime},\operatorname{\mathscr{E}}^{\prime})$
be an exact functor and let $\operatorname{\mathscr{F}}^{\prime}$ be another
exact structure on $\operatorname{\mathscr{A}}^{\prime}$. Then
$\operatorname{\mathscr{F}}=\\{\sigma\in\operatorname{\mathscr{E}}\,:\,F(\sigma)\in\operatorname{\mathscr{F}}^{\prime}\\}$
is an exact structure on $\operatorname{\mathscr{A}}$.
###### 5.4 Remark (Heller).
The prototypical application of the previous exercise is the following: A
(unital) ring homomorphism $\varphi:R^{\prime}\to R$ yields an exact functor
$\varphi^{\ast}:\leftidx_{{R}}{\operatorname{\mathbf{Mod}}}\to\leftidx_{{R^{\prime}}}{\operatorname{\mathbf{Mod}}}$
of the associated module categories. Let $\operatorname{\mathscr{F}}^{\prime}$
be the class of split exact sequences on
$\leftidx_{{R^{\prime}}}{\operatorname{\mathbf{Mod}}}$. The induced structure
$\operatorname{\mathscr{F}}$ on $\leftidx_{R}{\operatorname{\mathbf{Mod}}}$
consisting of sequences $\sigma$ such that $\varphi^{\ast}(\sigma)$ is split
exact is the _relative exact structure_ with respect to $\varphi$. This
structure is used in particular to define the relative derived functors such
as the relative $\operatorname{Tor}$ and $\operatorname{Ext}$ functors.
###### 5.5 Exercise (Künzer).
Let
$F:(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})\to(\operatorname{\mathscr{A}}^{\prime},\operatorname{\mathscr{E}}^{\prime})$
be a functor which preserves admissible kernels, i.e., for every morphism
$f:B\to C$ with an admissible monic $k:A\operatorname{\rightarrowtail}B$ as
kernel, the morphism $F(k)$ is an admissible monic and a kernel of $F(f)$. Let
$\operatorname{\mathscr{F}}=\\{\sigma\in\operatorname{\mathscr{E}}\,:\,F(\sigma)\in\operatorname{\mathscr{E}}^{\prime}\\}$
be the largest subclass of $\operatorname{\mathscr{E}}$ on which $F$ is exact.
Prove that $\operatorname{\mathscr{F}}$ is an exact structure.
_Hint:_ Axioms [E$0$], [E$0^{\operatorname{op}}$] and
[E$1^{\operatorname{op}}$] are easy. To check axiom [E$1$] use the obscure
axiom 2.16 and the $3\times 3$-lemma 3.6. Axiom [E$2$] follows from the
obscure axiom 2.16 and Proposition 2.12 (iv), while axiom
[E$2^{\operatorname{op}}$] follows from the fact that $F$ preserves certain
pull-back squares.
###### 5.6 Exercise.
Let $\operatorname{\mathscr{P}}$ be a set of objects in the exact category
$(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$. Consider the class
$\operatorname{\mathscr{E}}_{\operatorname{\mathscr{P}}}$ consisting of the
sequences
$A^{\prime}\operatorname{\rightarrowtail}A\operatorname{\twoheadrightarrow}A^{\prime\prime}$
in $\operatorname{\mathscr{E}}$ such that
$\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(P,A^{\prime})}\operatorname{\rightarrowtail}\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(P,A)}\operatorname{\twoheadrightarrow}\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(P,A^{\prime\prime})}$
is an exact sequence of abelian groups for all
$P\in\operatorname{\mathscr{P}}$. Prove that
$\operatorname{\mathscr{E}}_{\operatorname{\mathscr{P}}}$ is an exact
structure on $\operatorname{\mathscr{A}}$.
_Hint:_ Use Exercise 5.5.
## 6 Idempotent Completion
In this section we discuss Karoubi’s construction of ‘the’ idempotent
completion of an additive category, see [33, 1.2] and show how to extend it to
exact categories. Admittedly, the constructions and arguments presented here
are quite obvious (once the definitions are given) and thus rather boring, but
as the author is unaware of a reasonably complete exposition it seems
worthwhile to outline the details. A different account for small categories
(not necessarily additive) is given in Borceux [5, Proposition 6.5.9, p. 274].
It appears that the latter approach is due to M. Bunge [12].
###### 6.1 Definition.
An additive category $\operatorname{\mathscr{A}}$ is _idempotent complete_
[33, 1.2.1, 1.2.2] if for every idempotent $p:A\to A$, _i.e._ $p^{2}=p$, there
is a decomposition $A\cong K\oplus I$ of $A$ such that
$p\cong\left[\begin{smallmatrix}0&0\\\ 0&1\end{smallmatrix}\right]$.
###### 6.2 Remark.
The additive category $\operatorname{\mathscr{A}}$ is idempotent complete if
and only if every idempotent has a kernel.
Indeed, suppose that every idempotent has a kernel. Let $k:K\to A$ be a kernel
of the idempotent $p:A\to A$ and let $i:I\to A$ be a kernel of the idempotent
$1-p$. Because $p(1-p)=0$, we have $(1-p)=kl$ for a unique $l:A\to K$ and
because $(1-p)p=0$ we have $p=ij$ for a unique $j:A\to I$. Since $k$ is monic
and $kli=(1-p)i=0$ we have $li=0$ and because $klk=(1-p)k=pk+(1-p)k=k$ we have
$lk=1_{K}$. Similarly, $jk=0$ and $ji=1_{I}$. Therefore
$\left[\begin{smallmatrix}k&i\end{smallmatrix}\right]:K\oplus I\to A$ and
$\left[\begin{smallmatrix}l\\\ j\end{smallmatrix}\right]:A\to K\oplus I$ are
inverse to each other and $\left[\begin{smallmatrix}l\\\
j\end{smallmatrix}\right]p\left[\begin{smallmatrix}k&i\end{smallmatrix}\right]=\left[\begin{smallmatrix}l\\\
j\end{smallmatrix}\right]ij\left[\begin{smallmatrix}k&i\end{smallmatrix}\right]=\left[\begin{smallmatrix}0&0\\\
0&1_{I}\end{smallmatrix}\right]$ as desired. Notice that we have constructed
an analysis of $p$:
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{j}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{K,}$
in particular $k:K\operatorname{\rightarrowtail}A$ is a kernel of $p$ and
$i:I\operatorname{\rightarrowtail}A$ is an image of $p$. The converse
direction is even more obvious.
###### 6.3 Remark.
Every additive category $\operatorname{\mathscr{A}}$ can be fully faithfully
embedded into an idempotent complete additive category
$\operatorname{\mathscr{A}}^{\wedge}$.
The objects of $\operatorname{\mathscr{A}}^{\wedge}$ are the pairs $(A,p)$
consisting of an object $A$ of $\operatorname{\mathscr{A}}$ and an idempotent
$p:A\to A$ while the sets of morphisms are
$\operatorname{Hom}_{\operatorname{\mathscr{A}}^{\wedge}}{((A,p),(B,q))}=q\circ\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(A,B)}\circ
p$
with the composition induced by the composition in
$\operatorname{\mathscr{A}}$. It is easy to see that
$\operatorname{\mathscr{A}}^{\wedge}$ is additive with biproduct
$(A,p)\oplus(A^{\prime},p^{\prime})=(A\oplus A^{\prime},p\oplus p^{\prime})$
and obviously the functor
$i_{\operatorname{\mathscr{A}}}:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{A}}^{\wedge}$
given by $i_{\operatorname{\mathscr{A}}}{(A)}=(A,1_{A})$ and
$i_{\operatorname{\mathscr{A}}}(f)=f$ is fully faithful. In order to see that
$\operatorname{\mathscr{A}}^{\wedge}$ is idempotent complete, suppose $pfp$ is
an idempotent of $(A,p)$ in $\operatorname{\mathscr{A}}^{\wedge}$. _A
fortiori_ $pfp$ is an idempotent of $\operatorname{\mathscr{A}}$ and the
object $(A,p)$ is isomorphic to the direct sum $(A,p-pfp)\oplus(A,pfp)$ via
the morphisms $\left[\begin{smallmatrix}p-pfp\\\ pfp\end{smallmatrix}\right]$
and $\left[\begin{smallmatrix}p-pfp&&pfp\end{smallmatrix}\right]$. The
equation $\left[\begin{smallmatrix}p-pfp\\\
pfp\end{smallmatrix}\right]pfp\left[\begin{smallmatrix}p-pfp&&pfp\end{smallmatrix}\right]=\left[\begin{smallmatrix}0&0\\\
0&pfp\end{smallmatrix}\right]$ proves $\operatorname{\mathscr{A}}^{\wedge}$ to
be idempotent complete.
###### 6.4 Definition.
The pair
$(\operatorname{\mathscr{A}}^{\wedge},i_{\operatorname{\mathscr{A}}})$
constructed in Remark 6.3 is called the _idempotent completion_ of
$\operatorname{\mathscr{A}}$.
The next goal is to characterize the pair
$(\operatorname{\mathscr{A}}^{\wedge},i_{\operatorname{\mathscr{A}}})$ by a
universal property (Proposition 6.10). We first work out some nice properties
of the explicit construction.
###### 6.5 Remark.
If $\operatorname{\mathscr{A}}$ is idempotent complete then
$i_{\operatorname{\mathscr{A}}}:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{A}}^{\wedge}$
is an equivalence of categories. In order to construct a quasi-inverse functor
of
$i_{\operatorname{\mathscr{A}}}:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{A}}^{\wedge}$,
choose for each idempotent $p:A\to A$ a kernel $K_{p}$, an image $I_{p}$ and
morphisms $i_{p},j_{p},k_{p},l_{p}$ as in Remark 6.2 and map the object
$(A,p)$ of $\operatorname{\mathscr{A}}^{\wedge}$ to $I_{p}$. A morphism
$(A,p)\to(B,q)$ of $\operatorname{\mathscr{A}}^{\wedge}$ can be written as
$qfp$ and map it to $j_{q}qfpi_{p}=j_{q}fi_{p}$. Obviously, this yields a
quasi-inverse functor of
$i_{\operatorname{\mathscr{A}}}:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{A}}^{\wedge}$.
###### 6.6 Remark.
A functor $F:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{B}}$ yields a
functor
$F^{\wedge}:\operatorname{\mathscr{A}}^{\wedge}\to\operatorname{\mathscr{B}}^{\wedge}$,
simply by setting $F^{\wedge}(A,p)=(F(A),F(p))$ and
$F^{\wedge}(qfp)=F(q)F(f)F(p)$. Obviously,
$F^{\wedge}i_{\operatorname{\mathscr{A}}}=i_{\operatorname{\mathscr{B}}}F$ and
$(GF)^{\wedge}=G^{\wedge}F^{\wedge}$.
###### 6.7 Remark.
A natural transformation $\alpha:F\Rightarrow G$ of functors
$\operatorname{\mathscr{A}}\to\operatorname{\mathscr{B}}$ yields a unique
natural transformation $\alpha^{\wedge}:F^{\wedge}\Rightarrow G^{\wedge}$.
Observe first that a natural transformation
$\alpha^{\prime}:F^{\prime}\Rightarrow G^{\prime}$ of functors
$\operatorname{\mathscr{A}}^{\wedge}\to\operatorname{\mathscr{B}}^{\wedge}$ is
completely determined by its values on
$i_{\operatorname{\mathscr{A}}}{(\operatorname{\mathscr{A}})}$ by the
following argument. Every object $(A,p)$ of
$\operatorname{\mathscr{A}}^{\wedge}$ is canonically a retract of $(A,1_{A})$
via the morphisms $s:(A,p)\to(A,1_{A})$ and $r:(A,1_{A})\to(A,p)$ given by
$p\in\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(A,A)}$. Therefore, by
naturality, we must have
$\alpha^{\prime}_{(A,p)}=\alpha^{\prime}_{(A,p)}F^{\prime}(r)F^{\prime}(s)=G^{\prime}(r)\alpha^{\prime}_{(A,1_{A})}F^{\prime}(s),$
so $\alpha^{\prime}$ is completely determined by its values on
$i_{\operatorname{\mathscr{A}}}{(\operatorname{\mathscr{A}})}$. Now given a
natural transformation $\alpha:F\Rightarrow G$ of functors
$\operatorname{\mathscr{A}}\to\operatorname{\mathscr{B}}$ put
$\alpha^{\wedge}_{(A,p)}=G^{\wedge}(r)i_{\operatorname{\mathscr{B}}}(\alpha_{A})F^{\wedge}(s)$
which is simply the element $G(p)\alpha_{A}F(p)$ in
$\operatorname{Hom}_{\operatorname{\mathscr{B}}^{\wedge}}{(F^{\wedge}(A,p),G^{\wedge}(A,p))}=G(p)\circ\operatorname{Hom}_{\operatorname{\mathscr{B}}}(F(A),G(A))\circ
F(p).$
It is easily checked that this definition of $\alpha^{\wedge}$ indeed yields a
natural transformation $F^{\wedge}\Rightarrow G^{\wedge}$ as desired.
###### 6.8 Remark.
The assignment $\alpha\mapsto\alpha^{\wedge}$ is compatible with vertical and
horizontal composition (see e.g. [44, II.5, p. 42f]): For functors
$F,G,H:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{B}}$ and natural
transformations $\alpha:F\Rightarrow G$ and $\beta:G\Rightarrow H$ we have
$(\beta\circ\alpha)^{\wedge}=\beta^{\wedge}\circ\alpha^{\wedge}$ while for
functors $F,G:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{B}}$ and
$H,K:\operatorname{\mathscr{B}}\to\operatorname{\mathscr{C}}$ with natural
transformations $\alpha:F\Rightarrow G$ and $\beta:H\Rightarrow K$ we have
$(\beta\ast\alpha)^{\wedge}=\beta^{\wedge}\ast\alpha^{\wedge}$.
###### 6.9 Remark.
A functor $F:\operatorname{\mathscr{A}}^{\wedge}\to\operatorname{\mathscr{I}}$
to an idempotent complete category is determined up to unique isomorphism by
its values on $i_{\operatorname{\mathscr{A}}}{(\operatorname{\mathscr{A}})}$.
A natural transformation $\alpha:F\Rightarrow G$ of functors
$\operatorname{\mathscr{A}}^{\wedge}\to\operatorname{\mathscr{I}}$ is
determined by its values on
$i_{\operatorname{\mathscr{A}}}{(\operatorname{\mathscr{A}})}$.
Indeed, exhibit each $(A,p)$ as a retract of $(A,1)$ as in Remark 6.7.
Consider $p$ as an idempotent of $(A,1)$, so $F(p)$ is an idempotent of
$F(A,1)$. Choosing an image $I_{F(p)}$ of $F(p)$ as in Remark 6.2, it is clear
that the functor $F$ must map $(A,p)$ to $I_{F(p)}$ and is thus determined up
to a unique isomorphism. The claim about natural transformations is analogous
to the argument in Remark 6.7.
###### 6.10 Proposition.
The functor
$i_{\operatorname{\mathscr{A}}}:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{A}}^{\wedge}$
is $2$-universal among functors from $\operatorname{\mathscr{A}}$ to
idempotent complete categories:
1. (i)
For every functor $F:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{I}}$
to an idempotent complete category $\operatorname{\mathscr{I}}$ there exists a
functor
$\widetilde{F}:\operatorname{\mathscr{A}}^{\wedge}\to\operatorname{\mathscr{I}}$
and a natural isomorphism
$\alpha:F\Rightarrow\widetilde{F}i_{\operatorname{\mathscr{A}}}$.
2. (ii)
Given a functor
$G:\operatorname{\mathscr{A}}^{\wedge}\to\operatorname{\mathscr{I}}$ and a
natural transformation $\gamma:F\Rightarrow Gi_{\operatorname{\mathscr{A}}}$
there is a unique natural transformation $\beta:\widetilde{F}\Rightarrow G$
such that $\gamma=\beta\ast\alpha$.
* Sketch of the Proof.
Given a functor $F:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{I}}$,
the construction outlined in Remark 6.9 yields candidates for
$\widetilde{F}:\operatorname{\mathscr{A}}^{\wedge}\to\operatorname{\mathscr{I}}$
and $\alpha:F\Rightarrow\widetilde{F}i_{\operatorname{\mathscr{A}}}$ and we
leave it to the reader to check that this works. Once $\widetilde{F}$ and
$\alpha$ are fixed, $\widetilde{\beta}:=\gamma\ast\alpha^{-1}$ yields a
natural transformation $\widetilde{F}i_{\operatorname{\mathscr{A}}}\Rightarrow
Gi_{\operatorname{\mathscr{A}}}$ and the procedure in Remark 6.9 shows what an
extension $\beta:\widetilde{F}\Rightarrow G$ of $\widetilde{\beta}$ must look
like and again we leave it to the reader to check that this works. ∎
###### 6.11 Corollary.
Let $\operatorname{\mathscr{A}}$ be a small additive category. The functor
$i_{\operatorname{\mathscr{A}}}:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{A}}^{\wedge}$
induces an equivalence of functor categories
$(i_{\operatorname{\mathscr{A}}})^{\ast}:\operatorname{Hom}{(\operatorname{\mathscr{A}}^{\wedge},\operatorname{\mathscr{I}})}\xrightarrow{\simeq}\operatorname{Hom}{(\operatorname{\mathscr{A}},\operatorname{\mathscr{I}})}$
for every idempotent complete category $\operatorname{\mathscr{I}}$.
* Proof.
Point (i) of Proposition 6.10 states that
$(i_{\operatorname{\mathscr{A}}})^{\ast}$ is essentially surjective and it
follows from point (ii) that it is fully faithful, hence it is an equivalence
of categories. ∎
###### 6.12 Example.
Let $\operatorname{\mathscr{F}}$ be the category of (finitely generated) free
modules over a ring $R$. Its idempotent completion
$\operatorname{\mathscr{F}}^{\wedge}$ is equivalent to the category of
(finitely generated) projective modules over $R$.
Let now $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ be an exact
category. Call a sequence in $\operatorname{\mathscr{A}}^{\wedge}$ short exact
if it is a direct summand in $(\operatorname{\mathscr{A}}^{\wedge})^{\to\to}$
of a sequence in $\operatorname{\mathscr{E}}$ and denote the class of short
exact sequences in $\operatorname{\mathscr{A}}^{\wedge}$ by
$\operatorname{\mathscr{E}}^{\wedge}$.
###### 6.13 Proposition.
The class $\operatorname{\mathscr{E}}^{\wedge}$ is an exact structure on
$\operatorname{\mathscr{A}}^{\wedge}$. The inclusion functor
$i_{\operatorname{\mathscr{A}}}:(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})\to(\operatorname{\mathscr{A}}^{\wedge},\operatorname{\mathscr{E}}^{\wedge})$
preserves and reflects exactness and is $2$-universal among exact functors to
idempotent complete exact categories:
1. (i)
Let $F:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{I}}$ be an exact
functor to an idempotent complete exact category $\operatorname{\mathscr{I}}$.
There exists an exact functor
$\widetilde{F}:\operatorname{\mathscr{A}}^{\wedge}\to\operatorname{\mathscr{I}}$
together with a natural isomorphism
$\alpha:F\Rightarrow\widetilde{F}i_{\operatorname{\mathscr{A}}}$.
2. (ii)
Given another exact functor
$G:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{I}}$ together with a
natural transformation $\gamma:F\Rightarrow Gi_{\operatorname{\mathscr{A}}}$,
there is a unique natural transformation $\beta:\widetilde{F}\Rightarrow G$
such that $\gamma=\beta\ast\alpha$.
* Proof.
To prove that $\operatorname{\mathscr{E}}^{\wedge}$ is an exact structure is
straightforward but rather tedious, so we skip it.111Thomason [57, A.9.1 (b)]
gives a short argument relying on the embedding into an abelian category, but
it can be done by completely elementary means as well. Given this, it is clear
that the functor
$\operatorname{\mathscr{A}}\to\operatorname{\mathscr{A}}^{\wedge}$ is exact
and reflects exactness. If
$F:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{I}}$ is an exact
functor to an idempotent complete exact category then, as every sequence in
$\operatorname{\mathscr{E}}^{\wedge}$ is a direct summand of a sequence in
$\operatorname{\mathscr{E}}$, an extension $\widetilde{F}$ of $F$ as provided
by Proposition 6.10 must carry exact sequences in
$\operatorname{\mathscr{A}}^{\wedge}$ to exact sequences in
$\operatorname{\mathscr{I}}$. Thus statements (i) and (ii) follow from the
corresponding statements in Proposition 6.10. ∎
###### 6.14 Corollary.
For a small exact category
$(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$, the exact functor
$i_{\operatorname{\mathscr{A}}}$ induces an equivalence of the categories of
exact functors
$(i_{\operatorname{\mathscr{A}}})^{\ast}:\operatorname{Hom}_{\operatorname{Ex}}{((\operatorname{\mathscr{A}}^{\wedge},\operatorname{\mathscr{E}}^{\wedge}),\operatorname{\mathscr{I}})}\xrightarrow{\simeq}\operatorname{Hom}_{\operatorname{Ex}}{((\operatorname{\mathscr{A}},\operatorname{\mathscr{E}}),\operatorname{\mathscr{I}})}$
to every idempotent complete exact category $\operatorname{\mathscr{I}}$. ∎
## 7 Weak Idempotent Completeness
Thomason introduced in [57, A.5.1] the notion of an exact category with
“weakly split idempotents”. It turns out that this is a property of the
underlying additive category rather than the exact structure.
Recall that in an arbitrary category a morphism $r:B\to C$ is called a
_retraction_ if there exists a _section_ $s:C\to B$ of $r$ in the sense that
$rs=1_{C}$. Dually, a morphism $c:A\to B$ is a _coretraction_ if it admits a
section $s:B\to A$, i.e., $sc=1_{A}$. Observe that retractions are epics and
coretractions are monics. Moreover, a section of a retraction is a
coretraction and a section of a coretraction is a retraction.
###### 7.1 Lemma.
In an additive category $\operatorname{\mathscr{A}}$ the following are
equivalent:
1. (i)
Every coretraction has a cokernel.
2. (ii)
Every retraction has a kernel.
###### 7.2 Definition.
If the conditions of the previous lemma hold then $\operatorname{\mathscr{A}}$
is said to be _weakly idempotent complete_.
###### 7.3 Remark.
Freyd [19] uses the more descriptive terminology _retracts have complements_
for weakly idempotent complete categories. He proves in particular that a
weakly idempotent complete category with countable coproducts is idempotent
complete.
###### 7.4 Remark.
Assume that $r:B\to C$ is a retraction with section $s:C\to B$. Then $sr:B\to
B$ is an idempotent. Let us prove that this idempotent gives rise to a
splitting of $B$ if $r$ admits a kernel $k:A\to B$.
Indeed, since $r(1_{B}-sr)=0$, there is a unique morphism $t:B\to A$ such that
$kt=1_{B}-sr$. It follows that $k$ is a coretraction because
$ktk=(1_{B}-sr)k=k$ implies that $tk=1_{A}$. Moreover $kts=0$, so $ts=0$,
hence $\left[\begin{smallmatrix}k&s\end{smallmatrix}\right]:A\oplus C\to B$ is
an isomorphism with inverse $\left[\begin{smallmatrix}t\\\
r\end{smallmatrix}\right]$. In particular, the sequences $A\to B\to C$ and
$A\to A\oplus C\to C$ are isomorphic.
* Proof of Lemma 7.1.
By duality it suffices to prove that (ii) implies (i).
Let $c:C\to B$ be a coretraction with section $s$. Then $s$ is a retraction
and, assuming (ii), it admits a kernel $k:A\to B$. By the discussion in Remark
7.4, $k$ is a coretraction with section $t:B\to A$ and it is obvious that $t$
is a cokernel of $c$. ∎
###### 7.5 Corollary.
Let $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ be an exact
category. The following are equivalent:
1. (i)
The additive category $\operatorname{\mathscr{A}}$ is weakly idempotent
complete.
2. (ii)
Every coretraction is an admissible monic.
3. (iii)
Every retraction is an admissible epic.
* Proof.
It follows from Remark 7.4 that every retraction $r:B\to C$ admitting a kernel
gives rise to a sequence $A\to B\to C$ which is isomorphic to the split exact
sequence $A\operatorname{\rightarrowtail}A\oplus
C\operatorname{\twoheadrightarrow}C$, hence $r$ is an admissible epic by Lemma
2.7, whence (i) implies (iii). By duality (i) implies (ii) as well.
Conversely, every admissible monic has a cokernel and every admissible epic
has a kernel, hence (ii) and (iii) both imply (i). ∎
In a weakly idempotent complete exact category the obscure axiom (Proposition
2.16) has an easier statement—this is Heller’s cancellation axiom [26, (P2),
p. 492]:
###### 7.6 Proposition.
Let $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ be an exact
category. The following are equivalent:
1. (i)
The additive category $\operatorname{\mathscr{A}}$ is weakly idempotent
complete.
2. (ii)
Consider two morphisms $g:B\to C$ and $f:A\to B$. If
$gf:A\operatorname{\twoheadrightarrow}C$ is an admissible epic then $g$ is an
admissible epic.
* Proof.
(i) $\Rightarrow$ (ii): Form the pull-back over $g$ and $gf$ and consider the
diagram
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{A}}$$\scriptstyle{f}$$\scriptstyle{\exists!}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$PB$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{gf}$$\textstyle{C}$
which proves $g^{\prime}$ to be a retraction, so $g^{\prime}$ has a kernel
$K^{\prime}\to B^{\prime}$. Because the diagram is a pull-back, the composite
$K^{\prime}\to B^{\prime}\to B$ is a kernel of $g$ and now the dual of
Proposition 2.16 applies to yield that $g$ is an admissible epic.
For the implication (ii) $\Rightarrow$ (i) simply observe that (ii) implies
that retractions are admissible epics. ∎
###### 7.7 Corollary.
An exact category is weakly idempotent complete if and only if it has the
following property: all morphisms $g:B\to C$ for which there is a commutative
diagram
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{gf}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{C}$
are admissible epics.
* Proof.
A weakly idempotent complete exact category enjoys the stated property by
Proposition 7.6.
Conversely, let $r:B\to C$ and $s:C\to B$ be such that $rs=1_{C}$. We want to
show that $r$ is an admissible epic. The sequences
$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}1\\\
-r\end{smallmatrix}\right]}$$\textstyle{B\oplus
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}r&1\end{smallmatrix}\right]}$$\textstyle{C}$
and
$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}-s\\\
1\end{smallmatrix}\right]}$$\textstyle{B\oplus
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}1&s\end{smallmatrix}\right]}$$\textstyle{B}$
are split exact, so $\left[\begin{smallmatrix}r&1\end{smallmatrix}\right]$ and
$\left[\begin{smallmatrix}1&s\end{smallmatrix}\right]$ are admissible epics.
But the diagram
$\textstyle{B\oplus
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}1&s\end{smallmatrix}\right]}$$\scriptstyle{\left[\begin{smallmatrix}r&1\end{smallmatrix}\right]}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\textstyle{C}$
is commutative, hence $r$ is an admissible epic. ∎
###### 7.8 Remark ([47, 1.12]).
Every small additive category $\operatorname{\mathscr{A}}$ has a _weak
idempotent completion_ $\operatorname{\mathscr{A}}^{\prime}$. Objects of
$\operatorname{\mathscr{A}}^{\prime}$ are the pairs $(A,p)$, where $p:A\to A$
is an idempotent factoring as $p=cr$ for some retraction $r:A\to X$ and
coretraction $c:X\to A$ with $rc=1_{B}$, while the morphisms are given by
$\operatorname{Hom}_{\operatorname{\mathscr{A}}^{\prime}}{((A,p),(B,q))}=q\circ\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(A,B)}\circ
p.$
It is easy to see that the functor
$\operatorname{\mathscr{A}}\to\operatorname{\mathscr{A}}^{\prime}$ given on
objects by $A\mapsto(A,1_{A})$ is $2$-universal among functors from
$\operatorname{\mathscr{A}}$ to a weakly idempotent complete category.
Moreover, if $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ is
exact then so is
$(\operatorname{\mathscr{A}}^{\prime},\operatorname{\mathscr{E}}^{\prime})$,
where the sequences in $\operatorname{\mathscr{E}}^{\prime}$ are the direct
summands in $\operatorname{\mathscr{A}}^{\prime}$ of sequences in
$\operatorname{\mathscr{E}}$, and the functor
$\operatorname{\mathscr{A}}\to\operatorname{\mathscr{A}}^{\prime}$ preserves
and reflects exactness and is $2$-universal among exact functors to weakly
idempotent complete categories.
###### 7.9 Remark.
Contrary to the construction of the idempotent completion, there is the set-
theoretic subtlety that the weak idempotent completion might not be well-
defined if $\operatorname{\mathscr{A}}$ is not small: it is _not_ clear a
priori that the objects $(A,p)$ form a class—essentially for the same reason
that the monics in a category need not form a class, see e.g. the discussion
in Borceux [6, p. 373f].
## 8 Admissible Morphisms and the Snake Lemma
###### 8.1 Definition.
A morphism $f:A\to B$ in an exact category
---
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{e}$$\textstyle{B}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$
is called _admissible_ if it factors as a composition of an admissible monic
with an admissible epic. Admissible morphisms will sometimes be displayed as
in diagrams.
###### 8.2 Remark.
Let $f$ be an admissible morphism. If $e^{\prime}$ is an admissible epic and
$m^{\prime}$ is an admissible monic then $m^{\prime}fe^{\prime}$ is admissible
if the composition is defined. However, admissible morphisms are _not_ closed
under composition in general. Indeed, consider a morphism $g:A\to B$ which is
not admissible. The morphisms $\left[\begin{smallmatrix}1\\\
g\end{smallmatrix}\right]:A\to A\oplus B$ and
$\left[\begin{smallmatrix}0&1\end{smallmatrix}\right]:A\oplus B\to B$ are
admissible since they are part of split exact sequences. But
$g=\left[\begin{smallmatrix}0&1\end{smallmatrix}\right]\left[\begin{smallmatrix}1\\\
g\end{smallmatrix}\right]$ is not admissible by hypothesis.
###### 8.3 Remark.
We choose the terminology _admissible morphism_ even though _strict morphism_
seems to be more standard (see e.g. [51, 54]). By Exercise 2.6 an admissible
monic is the same thing as an admissible morphism which happens to be monic.
###### 8.4 Lemma ([26, 3.4]).
The factorization of an admissible morphism is unique up to unique
isomorphism. More precisely: In a commutative diagram of the form
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\scriptstyle{e^{\prime}}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\scriptstyle{i}$$\textstyle{I^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m^{\prime}}$$\scriptstyle{i^{\prime}}$$\textstyle{B}$
there exist unique morphisms $i$, $i^{\prime}$ making the diagram commutative.
In particular, $i$ and $i^{\prime}$ are mutually inverse isomorphisms.
* Proof.
Let $k$ be a kernel of $e$. Since $m^{\prime}e^{\prime}k=mek=0$ and
$m^{\prime}$ is monic we have $e^{\prime}k=0$, hence there exists a unique
morphism $i:I\to I^{\prime}$ such that $e^{\prime}=ie$. Moreover,
$m^{\prime}ie=m^{\prime}e^{\prime}=me$ and $e$ epic imply $m^{\prime}i=m$.
Dually for $i^{\prime}$. ∎
###### 8.5 Remark.
An admissible morphism has an _analysis_ (cf. [43, IX.2])
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{e}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{C}$
where $k$ is a kernel, $c$ is a cokernel, $e$ is a coimage and $m$ is an image
of $f$ and the isomorphism classes of $K$, $I$ and $C$ are well-defined by
Lemma 8.4.
###### 8.6 Exercise.
If $\operatorname{\mathscr{A}}$ is an exact category in which every morphism
is admissible then $\operatorname{\mathscr{A}}$ is abelian. [A solution is
given by Freyd in [18, Proposition 3.1]].
###### 8.7 Lemma.
Admissible morphisms are stable under push-out along admissible monics and
pull-back along admissible epics.
* Proof.
Let $A\operatorname{\twoheadrightarrow}I\operatorname{\rightarrowtail}B$ be an
admissible epic-admissible monic factorization of an admissible morphism. To
prove the claim about push-outs construct the diagram
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$PO$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$PO$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime}.}$
Proposition 2.15 yields that $A^{\prime}\to I^{\prime}$ is an admissible epic
and the rest is clear. ∎
###### 8.8 Definition.
A sequence of admissible morphisms
---
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{e}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{e^{\prime}}$$\textstyle{A^{\prime\prime}}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{I^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m^{\prime}}$
is _exact_ if
$I\operatorname{\rightarrowtail}A\operatorname{\twoheadrightarrow}I^{\prime}$
is short exact. Longer sequences of admissible morphisms are exact if the
sequence given by any two consecutive morphisms is exact. Since the term
“exact” is heavily overloaded, we also use the synonym _“acyclic”_ , in
particular in connection with chain complexes.
###### 8.9 Lemma (Five Lemma, II).
If the commutative diagram
$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{A_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{A_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{A_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{5}}$
has exact rows then $f$ is an isomorphism.
* Sketch of the Proof.
By hypothesis we may choose factorizations
$A_{i}\operatorname{\twoheadrightarrow}I_{i}\operatorname{\rightarrowtail}A_{i+1}$
of $A_{i}\to A_{i+1}$ and
$B_{i}\operatorname{\twoheadrightarrow}J_{i}\operatorname{\rightarrowtail}B_{i+1}$
of $B_{i}\to B_{i+1}$ for $i=1,\ldots,4$. Using Lemma 8.4 and Exercise 3.3
there are isomorphisms $I_{1}\cong J_{1}$ and $I_{2}\cong J_{2}$ which one may
insert into the diagram without destroying its commutativity. Dually for
$I_{4}\cong J_{4}$ and $I_{3}\cong J_{3}$. The five lemma 3.2 then implies
that $f$ is an isomorphism. ∎
###### 8.10 Exercise.
Assume that $\operatorname{\mathscr{A}}$ is weakly idempotent complete
(Definition 7.2).
1. (i)
(Sharp Four Lemma) Consider a commutative diagram
$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{A_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{A_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{4}}$
with exact rows. Prove that $f$ is an admissible monic. Dualize.
2. (ii)
(Sharp Five Lemma) If the commutative diagram
$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{A_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{A_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{A_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{5}}$
has exact rows then $f$ is an isomorphism.
_Hint:_ Use Proposition 7.6, Exercise 2.6, Exercise 3.3 as well as Corollary
3.2.
###### 8.11 Proposition
($\operatorname{Ker}$–$\operatorname{Coker}$–Sequence).
Assume that $\operatorname{\mathscr{A}}$ is a weakly idempotent complete exact
category. Let $f:A\to B$ and $g:B\to C$ be admissible morphisms such that
$h=gf:A\to C$ is admissible as well. There is an exact sequence
$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Coker}@wrapper{\operatorname{Coker}@presentation}{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Coker}@wrapper{\operatorname{Coker}@presentation}{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Coker}@wrapper{\operatorname{Coker}@presentation}{g}}$
depending naturally on the diagram $h=gf$.
* Proof.
Observe that the morphism
$\operatorname{Ker}{f}\operatorname{\rightarrowtail}A$ factors over
$\operatorname{Ker}{h}\operatorname{\rightarrowtail}A$ via a unique morphism
$\operatorname{Ker}{f}\to\operatorname{Ker}{h}$ which is an admissible monic
by Proposition 7.6. Let
$\operatorname{Ker}{h}\operatorname{\twoheadrightarrow}X$ be a cokernel of
$\operatorname{Ker}{f}\operatorname{\rightarrowtail}\operatorname{Ker}{h}$.
Dually, there is an admissible epic
$\operatorname{Coker}{h}\operatorname{\twoheadrightarrow}\operatorname{Coker}{g}$
with $Z\operatorname{\rightarrowtail}\operatorname{Coker}{h}$ as kernel. The
Noether isomorphism 3.5 implies that there are two commutative diagrams with
exact rows and columns
$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Im}@wrapper{\operatorname{Im}@presentation}{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Im}@wrapper{\operatorname{Im}@presentation}{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Im}@wrapper{\operatorname{Im}@presentation}{h}}$
and
$\textstyle{\operatorname{Im}@wrapper{\operatorname{Im}@presentation}{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Im}@wrapper{\operatorname{Im}@presentation}{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Im}@wrapper{\operatorname{Im}@presentation}{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Coker}@wrapper{\operatorname{Coker}@presentation}{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Coker}@wrapper{\operatorname{Coker}@presentation}{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Coker}@wrapper{\operatorname{Coker}@presentation}{g}.}$
It is easy to see that there is an admissible monic
$X\operatorname{\rightarrowtail}\operatorname{Ker}{g}$ whose cokernel we
denote by $\operatorname{Ker}{g}\operatorname{\twoheadrightarrow}Y$. Therefore
the $3\times 3$-lemma yields a commutative diagram with exact rows and columns
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Im}@wrapper{\operatorname{Im}@presentation}{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Coker}@wrapper{\operatorname{Coker}@presentation}{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Im}@wrapper{\operatorname{Im}@presentation}{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Im}@wrapper{\operatorname{Im}@presentation}{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z.}$
The desired $\operatorname{Ker}$–$\operatorname{Coker}$–sequence is now
obtained by splicing: Start with the first row of the first diagram, splice it
with the first row of the third diagram, and continue with the third row of
the third diagram and the third row of the second diagram. The naturality
assertion is obvious. ∎
###### 8.12 Lemma.
Let $\operatorname{\mathscr{A}}$ be an exact category in which each
commutative triangle of admissible morphisms yields an exact
$\operatorname{Ker}$–$\operatorname{Coker}$–sequence where exactness is
understood in the sense of admissible morphisms. Then
$\operatorname{\mathscr{A}}$ is weakly idempotent complete.
* Proof.
We check the criterion in Corollary 7.7. We need to show that in every
commutative diagram of the form
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{h}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{C}$
the morphism $g$ is an admissible epic. Given such a diagram, consider the
diagram
$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{C}$
whose associated $\operatorname{Ker}$–$\operatorname{Coker}$–sequence is
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
so that $g$ is an admissible epic. ∎
The $\operatorname{Ker}$–$\operatorname{Coker}$–sequence immediately yields
the following version of the snake lemma, the neat proof of which was pointed
out to the author by Matthias Künzer.
###### 8.13 Corollary (Snake Lemma, I).
Let $\operatorname{\mathscr{A}}$ be weakly idempotent complete. Consider a
morphism of short exact sequences
$A^{\prime}\operatorname{\rightarrowtail}A\operatorname{\twoheadrightarrow}A^{\prime\prime}$
and
$B^{\prime}\operatorname{\rightarrowtail}B\operatorname{\twoheadrightarrow}B^{\prime\prime}$
with admissible components. There is a commutative diagram
$\textstyle{K^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k^{\prime}}$$\textstyle{K^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c^{\prime}}$$\textstyle{C^{\prime\prime}}$
with exact rows and columns and there is a connecting morphism
$\delta:K^{\prime\prime}\to C^{\prime}$ fitting into an exact sequence
$\textstyle{K^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k^{\prime}}$$\textstyle{K^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c^{\prime}}$$\textstyle{C^{\prime\prime}}$
depending naturally on the morphism of short exact sequences.
###### 8.14 Remark.
Using the notations of the proof of Lemma 8.12 consider the diagram
$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
The sequence
$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
provided by the snake lemma shows that $g$ is an admissible epic. It follows
that the snake lemma can only hold if the category is weakly idempotent
complete.
* Proof of Corollary 8.13.
By Proposition 3.1 and Lemma 8.7 we get the commutative diagram
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$BC$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$BC$\textstyle{A^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime\prime}}$
[more explicitly, the diagram is obtained by forming the push-out
$A^{\prime}AB^{\prime}D$]. The
$\operatorname{Ker}$–$\operatorname{Coker}$–sequence of the commutative
triangle of admissible morphisms
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B}$
yields the desired result by Remark 2.13. ∎
###### 8.15 Exercise (Snake Lemma, II, cf. [26, 4.3]).
Formulate and prove a snake lemma for a diagram of the form
|
---|---
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime\prime}}$
in weakly idempotent complete categories. Prove
$\operatorname{Ker}{(A^{\prime}\to A)}=\operatorname{Ker}{(K^{\prime}\to K)}$
and $\operatorname{Coker}{(B\to B^{\prime\prime})}=\operatorname{Coker}{(C\to
C^{\prime\prime})}$.
_Hint:_ Reduce to Corollary 8.13 by using Proposition 7.6 and the Noether
isomorphism 3.5.
###### 8.16 Remark.
Heller [26, 4.3] gives a direct proof of the snake lemma starting from his
axioms. Using the Noether isomorphism 3.5 and the $3\times 3$-lemma 3.6 as
well as Proposition 7.6, Heller’s proof is easily adapted to a proof from
Quillen’s axioms.
## 9 Chain Complexes and Chain Homotopy
The notion of chain complexes makes sense in every additive category
$\operatorname{\mathscr{A}}$. A _(chain) complex_ is a diagram
$(A^{\bullet},d_{A}^{\bullet})$
$\cdots\xrightarrow{}A^{n-1}\xrightarrow{d_{A}^{n-1}}A^{n}\xrightarrow{d_{A}^{n}}A^{n+1}\xrightarrow{}\cdots$
subject to the condition that $d^{n}d^{n-1}=0$ for all $n$ and a _chain map_
is a morphism of such diagrams. The category of complexes and chain maps is
denoted by $\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$.
Obviously, the category
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$ is additive.
###### 9.1 Lemma.
If $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ is an exact
category then $\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$ is an
exact category with respect to the class
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{E}})}$ of short sequences
of chain maps which are exact in each degree. If $\operatorname{\mathscr{A}}$
is abelian then so is
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$.
* Proof.
The point is that (as in every functor category) limits and colimits of
diagrams in $\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$ are
obtained by taking the limits and colimits pointwise (in each degree), in
particular push-outs under admissible monics and pull-backs over admissible
epics exist and yield admissible monics and epics. The rest is obvious. ∎
###### 9.2 Definition.
The _mapping cone_ of a chain map $f:A\to B$ is the complex
$\operatorname{cone}{(f)}^{n}=A^{n+1}\oplus B^{n}\qquad\text{with
differential}\qquad d_{f}^{n}=\left[\begin{smallmatrix}-d_{A}^{n+1}&0\\\
f^{n+1}&d_{B}^{n}\end{smallmatrix}\right].$
Notice that $d_{f}^{n+1}d_{f}^{n}=0$ precisely because $f$ is a chain map. It
is plain that the mapping cone defines a functor from the category of
morphisms in $\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$ to
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$.
The _translation functor_ on
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$ is defined to be
$\Sigma A=\operatorname{cone}{(A\to 0)}$. More explicitly, $\Sigma A$ is the
complex with components $(\Sigma A)^{n}=A^{n+1}$ and differentials $d_{\Sigma
A}^{n}=-d_{A}^{n+1}$. If $f$ is a chain map, its translate is given by
$(\Sigma f)^{n}=f^{n+1}$. Clearly, $\Sigma$ is an additive automorphism of
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$.
The _strict triangle_ over the chain map $f:A\to B$ is the $3$-periodic (or
rather $3$-helicoidal, if you insist) sequence
$A\xrightarrow{f}B\xrightarrow{i_{f}}\operatorname{cone}{(f)}\xrightarrow{j_{f}}\Sigma
A\xrightarrow{\Sigma f}\Sigma B\xrightarrow{\Sigma
i_{f}}\Sigma\operatorname{cone}{(f)}\xrightarrow{\Sigma j_{f}}\cdots,$
where the chain map $i_{f}$ has components $\left[\begin{smallmatrix}0\\\
1\end{smallmatrix}\right]$ and $j_{f}$ has components
$\left[\begin{smallmatrix}1&0\end{smallmatrix}\right]$.
###### 9.3 Remark.
Let $f:A\to B$ be a chain map. Observe that the sequence of chain maps
$B\xrightarrow{i_{f}}\operatorname{cone}{(f)}\xrightarrow{j_{f}}\Sigma A$
splits in each degree, however it need not be a split exact sequence in
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$, because the
degreewise splitting maps need not assemble to chain maps. In fact, it is
straightforward to verify that the above sequence is split exact in
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$ if and only if $f$
is chain homotopic to zero in the sense of Definition 9.5.
###### 9.4 Exercise.
Assume that $\operatorname{\mathscr{A}}$ is an abelian category. Prove that
the strict triangle over the chain map $f:A\to B$ gives rise to a long exact
homology sequence
$\cdots\xrightarrow{}H^{n}(A)\xrightarrow{H^{n}(f)}H^{n}(B)\xrightarrow{H^{n}(i_{f})}H^{n}{(\operatorname{cone}{(f)})}\xrightarrow{H^{n}{(j_{f})}}H^{n+1}{(A)}\xrightarrow{}\cdots.$
Deduce that $f$ induces an isomorphism of $H^{\ast}(A)$ with $H^{\ast}(B)$ if
and only if $\operatorname{cone}{(f)}$ is acyclic.
###### 9.5 Definition.
A chain map $f:A\to B$ is _chain homotopic to zero_ if there exist morphisms
$h^{n}:A^{n}\to B^{n-1}$ such that $f^{n}=d_{B}^{n-1}h^{n}+h^{n+1}d_{A}^{n}$.
A chain complex $A$ is called _null-homotopic_ if $1_{A}$ is chain homotopic
to zero.
###### 9.6 Remark.
The maps which are chain homotopic to zero form an ideal in
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$, that is to say if
$h:B\to C$ is chain homotopic to zero then so are $hf$ and $gh$ for all
morphisms $f:A\to B$ and $g:C\to D$, if $h_{1}$ and $h_{2}$ are chain
homotopic to zero then so is $h_{1}\oplus h_{2}$. The set $N{(A,B)}$ of chain
maps $A\to B$ which are chain homotopic to zero is a subgroup of the abelian
group
$\operatorname{Hom}_{\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}}{(A,B)}$.
###### 9.7 Definition.
The _homotopy category_
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$ is the category with
the chain complexes over $\operatorname{\mathscr{A}}$ as objects and
$\operatorname{Hom}_{\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}}{(A,B)}:=\operatorname{Hom}_{\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}}{(A,B)}/N{(A,B)}$
as morphisms.
###### 9.8 Remark.
Notice that every null-homotopic complex is isomorphic to the zero object in
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$. It turns out that
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$ is additive, but it
is very rarely abelian or exact with respect to a non-trivial exact structure
(see Verdier [58, Ch.II, 1.3.6]). However,
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$ has the structure of
a _triangulated category_ induced by the _strict triangles_ in
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$, see e.g. Verdier
[58], Beĭlinson-Bernstein-Deligne [4], Gelfand-Manin [23], Grivel [8, Chapter
I], Kashiwara-Schapira [34], Keller [38], Neeman [48] or Weibel [59].
###### 9.9 Remark.
For each object $A\in\operatorname{\mathscr{A}}$, define
$\operatorname{cone}{(A)}=\operatorname{cone}{(1_{A})}$. Notice that
$\operatorname{cone}{(A)}$ is null-homotopic with
$\left[\begin{smallmatrix}0&1\\\ 0&0\end{smallmatrix}\right]$ as contracting
homotopy.
###### 9.10 Remark.
If $f$ and $g$ are chain homotopy equivalent, i.e., $f-g$ is chain homotopic
to zero, then $\operatorname{cone}{(f)}$ and $\operatorname{cone}{(g)}$ are
isomorphic in $\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$ but
the isomorphism and its homotopy class will generally depend on the choice of
a chain homotopy. In particular, the mapping cone construction does not yield
a functor defined on morphisms of
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$.
###### 9.11 Remark.
A chain map $f:A\to B$ is chain homotopic to zero if and only if it factors as
$hi_{A}=f$ over $h:\operatorname{cone}{(A)}\to B$, where
$i_{A}=i_{1_{A}}:A\to\operatorname{cone}{(A)}$. Moreover, $h$ has components
$\left[\begin{smallmatrix}h^{n+1}&f^{n}\end{smallmatrix}\right]$, where the
family of morphisms $\\{h^{n}\\}$ is a chain homotopy of $f$ to zero.
Similarly, $f$ is chain homotopic to zero if and only if $f$ factors through
$j_{\Sigma^{-1}B}=j_{1_{\Sigma^{-1}B}}:\operatorname{cone}{(\Sigma^{-1}B)}\to
B$.
###### 9.12 Remark.
The mapping cone construction yields the push-out diagram
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{i_{A}}$PO$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{f}}$$\textstyle{\operatorname{cone}@wrapper{\operatorname{cone}@presentation}{(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}1&0\\\
0&f\end{smallmatrix}\right]}$$\textstyle{\operatorname{cone}@wrapper{\operatorname{cone}@presentation}{(f)}}$
in $\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$. Now suppose
that $g:B\to C$ is a chain map such that $gf$ is chain homotopic to zero. By
Remark 9.11, $gf$ factors over $i_{A}$ and using the push-out property of the
above diagram it follows that $g$ factors over $i_{f}$. This construction will
depend on the choice of an explicit chain homotopy $gf\simeq 0$ in general. In
particular, $\operatorname{cone}(f)$ is a _weak cokernel_ in
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$ of the homotopy
class of $f$ in that it has the factorization property of a cokernel but
without uniqueness. Similarly, $\Sigma^{-1}\operatorname{cone}{(f)}$ is a
_weak kernel_ of $f$ in
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$.
## 10 Acyclic Complexes and Quasi-Isomorphisms
The present section is probably only of interest to readers acquainted with
triangulated categories or at least with the construction of the derived
category of an abelian category. After giving the fundamental definition of
acyclicity of a complex over an exact category, we may formulate the
intimately connected notion of quasi-isomorphisms.
We will give an elementary proof of the fact that the homotopy category
$\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})}$ of acyclic
complexes over an exact category $\operatorname{\mathscr{A}}$ is a
triangulated category. It turns out that
$\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})}$ is a strictly full
subcategory of the homotopy category of chain complexes
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$ if and only if
$\operatorname{\mathscr{A}}$ is idempotent complete, and in this case
$\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})}$ is even thick in
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$. Since thick
subcategories are strictly full by definition,
$\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})}$ is thick if and
only if $\operatorname{\mathscr{A}}$ is idempotent complete.
By [48, Chapter 2], the Verdier quotient
$\operatorname{\mathbf{K}}/\operatorname{\mathscr{T}}$ is defined for any
(strictly full) triangulated subcategory $\operatorname{\mathscr{T}}$ of a
triangulated category $\operatorname{\mathbf{K}}$ and it coincides with the
Verdier quotient $\operatorname{\mathbf{K}}/\bar{\operatorname{\mathscr{T}}}$,
where $\bar{\operatorname{\mathscr{T}}}$ is the _thick closure_ of
$\operatorname{\mathscr{T}}$. The case we are interested in is
$\operatorname{\mathbf{K}}=\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$
and
$\operatorname{\mathscr{T}}=\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})}$.
The Verdier quotient
$\operatorname{\mathbf{D}}{(\operatorname{\mathscr{A}})}=\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}/\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})}$
is the _derived category_ of $\operatorname{\mathscr{A}}$. If
$\operatorname{\mathscr{A}}$ is idempotent complete then
$\overline{\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})}}=\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})}$
and it is clear that quasi-isomorphisms are then precisely the chain maps with
acyclic mapping cone. If $\operatorname{\mathscr{A}}$ fails to be idempotent
complete, it turns out that the thick closure
$\overline{\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})}}$ of
$\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})}$ is the same as the
closure of $\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})}$ under
isomorphisms in $\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$, so
a chain map $f$ is a quasi-isomorphism if and only if
$\operatorname{cone}{(f)}$ is homotopy equivalent to an acyclic complex.
Similarly, the derived categories of bounded, left bounded or right bounded
complexes are constructed as in the abelian setting. It is useful to notice
that for $\ast\in\\{+,-,b\\}$ the category
$\operatorname{\mathbf{Ac}}^{\ast}{(\operatorname{\mathscr{A}})}$ is thick in
$\operatorname{\mathbf{K}}^{\ast}{(\operatorname{\mathscr{A}})}$ if and only
if $\operatorname{\mathscr{A}}$ is weakly idempotent complete, which leads to
an easier description of quasi-isomorphisms.
If $\operatorname{\mathscr{B}}$ is a fully exact subcategory of
$\operatorname{\mathscr{A}}$, the inclusion
$\operatorname{\mathscr{B}}\to\operatorname{\mathscr{A}}$ yields a canonical
functor
$\operatorname{\mathbf{D}}{(\operatorname{\mathscr{B}})}\to\operatorname{\mathbf{D}}{(\operatorname{\mathscr{A}})}$
and we state conditions which ensure that this functor is essentially
surjective or fully faithful.
We end the section with a short discussion of Deligne’s approach to total
derived functors.
### 10.1 The Homotopy Category of Acyclic Complexes
###### 10.1 Definition.
A chain complex $A$ over an exact category is called _acyclic_ (or _exact_) if
each differential factors as
$A^{n}\operatorname{\twoheadrightarrow}Z^{n+1}A\operatorname{\rightarrowtail}A^{n+1}$
in such a way that each sequence
$Z^{n}A\operatorname{\rightarrowtail}A^{n}\operatorname{\twoheadrightarrow}Z^{n+1}A$
is exact.
###### 10.2 Remark.
An acyclic complex is a complex with admissible differentials (Definition 8.1)
which is exact in the sense of Definition 8.8. In particular, $Z^{n}A$ is a
kernel of $A^{n}\to A^{n+1}$, an image and coimage of $A^{n-1}\to A^{n}$ and a
cokernel of $A^{n-2}\to A^{n-1}$.
###### 10.3 Lemma (Neeman [47, 1.1]).
The mapping cone of a chain map $f:A\to B$ between acyclic complexes is
acyclic.
* Proof.
An easy diagram chase shows that the dotted morphisms in the diagram
$\textstyle{A^{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{A}^{n-1}}$$\scriptstyle{j_{A}^{n-1}}$$\scriptstyle{f^{n-1}}$$\textstyle{A^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{A}^{n}}$$\scriptstyle{j_{A}^{n}}$$\scriptstyle{f^{n}}$$\textstyle{A^{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{n+1}}$$\textstyle{Z^{n}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{A}^{n}}$$\scriptstyle{\exists!g^{n}}$$\textstyle{Z^{n+1}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{A}^{n+1}}$$\scriptstyle{\exists!g^{n+1}}$$\textstyle{Z^{n}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{B}^{n}}$$\textstyle{Z^{n+1}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{B}^{n+1}}$$\textstyle{B^{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{B}^{n-1}}$$\scriptstyle{j_{B}^{n-1}}$$\textstyle{B^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{B}^{n}}$$\scriptstyle{j_{B}^{n}}$$\textstyle{B^{n+1}}$
exist and are the unique morphisms $g^{n}$ making the diagram commutative.
By Proposition 3.1 we find objects $Z^{n}C$ fitting into a commutative diagram
$\textstyle{A^{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{A}^{n-1}}$$\scriptstyle{j_{A}^{n-1}}$$\scriptstyle{f^{\prime
n-1}}$$\textstyle{A^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{A}^{n}}$$\scriptstyle{j_{A}^{n}}$$\scriptstyle{f^{\prime
n}}$$\textstyle{A^{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime
n+1}}$$\textstyle{Z^{n}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{A}^{n}}$$\scriptstyle{g^{n}}$BC$\textstyle{Z^{n+1}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{A}^{n+1}}$$\scriptstyle{g^{n+1}}$BC$\textstyle{Z^{n-1}C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{n-1}}$$\scriptstyle{f^{\prime\prime
n-1}}$BC$\textstyle{Z^{n}C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{n}}$$\scriptstyle{f^{\prime\prime
n}}$BC$\textstyle{Z^{n+1}C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime\prime
n+1}}$$\textstyle{Z^{n}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k^{n}}$$\scriptstyle{i_{B}^{n}}$$\textstyle{Z^{n+1}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k^{n+1}}$$\scriptstyle{i_{B}^{n+1}}$$\textstyle{B^{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{B}^{n-1}}$$\scriptstyle{j_{B}^{n-1}}$$\textstyle{B^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{B}^{n}}$$\scriptstyle{j_{B}^{n}}$$\textstyle{B^{n+1}}$
where $f^{n}=f^{\prime\prime n}f^{\prime n}$ and the quadrilaterals marked BC
are bicartesian. Recall that the objects $Z^{n}C$ are obtained by forming the
push-outs under $i_{A}^{n}$ and $g^{n}$ (or the pull-backs over $j_{B}^{n}$
and $g^{n+1}$) and that
$Z^{n}B\operatorname{\rightarrowtail}Z^{n}C\operatorname{\twoheadrightarrow}Z^{n+1}A$
is short exact.
It follows from Corollary 2.14 that for each $n$ the sequence
$\textstyle{Z^{n-1}C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}-i_{A}^{n}h^{n-1}\\\
f^{\prime\prime n-1}\end{smallmatrix}\right]}$$\textstyle{A^{n}\oplus
B^{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}f^{\prime
n}&&k^{n}j_{B}^{n-1}\end{smallmatrix}\right]}$$\textstyle{Z^{n}C}$
is short exact and the commutative diagram
---
proves that $\operatorname{cone}{(f)}$ is acyclic. ∎
###### 10.4 Remark.
Retaining the notations of the proof we have a short exact sequence
$Z^{n}B\operatorname{\rightarrowtail}Z^{n}C\operatorname{\twoheadrightarrow}Z^{n+1}A.$
This sequence exhibits
$Z^{n}C=\operatorname{Ker}{\left[\begin{smallmatrix}-d_{A}^{n+1}&0\\\
f^{n+1}&d_{B}^{n}\end{smallmatrix}\right]}$ as an extension of
$Z^{n+1}A=\operatorname{Ker}{d_{A}^{n+1}}$ by
$Z^{n}B=\operatorname{Ker}{d_{B}^{n}}$.
Let $\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})}$ be the full
subcategory of the homotopy category
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$ consisting of
acyclic complexes over the exact category $\operatorname{\mathscr{A}}$. It
follows from Proposition 2.9 that the direct sum of two acyclic complexes is
acyclic. Thus $\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})}$ is a
full additive subcategory of
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$. The previous lemma
implies that even more is true:
###### 10.5 Corollary.
The homotopy category of acyclic complexes
$\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})}$ is a triangulated
subcategory of $\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$. ∎
###### 10.6 Remark.
For reasons of convenience, many authors assume that triangulated
subcategories are not only full but _strictly full_. We do not do so because
$\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})}$ is closed under
isomorphisms in $\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$ if
and only if $\operatorname{\mathscr{A}}$ is idempotent complete, see
Proposition 10.9.
###### 10.7 Lemma.
Assume that $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ is
idempotent complete. Every retract in
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$ of an acyclic
complex $A$ is acyclic.
* Proof (cf. [36, 2.3 a)]).
Let the chain map $f:X\to A$ be a coretraction, i.e., there is a chain map
$s:A\to X$ such that $s^{n}f^{n}-1_{X^{n}}=d_{X}^{n-1}h^{n}+h^{n+1}d_{X}^{n}$
for some morphisms $h^{n}:X^{n}\to X^{n-1}$. Obviously, the complex $IX$ with
components
$(IX)^{n}=X^{n}\oplus X^{n+1}\qquad\text{and
differential}\qquad\left[\begin{smallmatrix}0&1\\\
0&0\end{smallmatrix}\right]$
is acyclic. There is a chain map $i_{X}:X\to IX$ given by
$i_{X}^{n}=\left[\begin{smallmatrix}1_{X^{n}}\\\
d_{X}^{n}\end{smallmatrix}\right]:X^{n}\to X^{n}\oplus X^{n+1}$
and the chain map
$\left[\begin{smallmatrix}f\\\ i_{X}\end{smallmatrix}\right]:X\to A\oplus IX$
has the chain map
$\left[\begin{smallmatrix}s^{n}&-d_{X}^{n-1}h^{n}&-h^{n+1}\end{smallmatrix}\right]:A^{n}\oplus
X^{n}\oplus X^{n+1}\to X^{n}$
as a left inverse. Hence, on replacing the acyclic complex $A$ by the acyclic
complex $A\oplus IX$, we may assume that $f:X\to A$ has $s$ as a left inverse
in $\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$. But then
$e=fs:A\to A$ is an idempotent in
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$ and it induces an
idempotent on the exact sequences
$Z^{n}A\operatorname{\rightarrowtail}A^{n}\operatorname{\twoheadrightarrow}Z^{n+1}A$
witnessing that $A$ is acyclic as in the first diagram of the proof of Lemma
10.3. This means that
$Z^{n}A\operatorname{\rightarrowtail}A^{n}\operatorname{\twoheadrightarrow}Z^{n+1}A$
decomposes as a direct sum of two short exact sequences (Corollary 2.18) since
$\operatorname{\mathscr{A}}$ is idempotent complete. Therefore the acyclic
complex $A=X^{\prime}\oplus Y^{\prime}$ is a direct sum of the acyclic
complexes $X^{\prime}$ and $Y^{\prime}$, and $f$ induces an isomorphism from
$X$ to $X^{\prime}$ in
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$. The details are
left to the reader. ∎
###### 10.8 Exercise.
Prove that the sequence $X\to\operatorname{cone}{(X)}\to\Sigma X$ from Remark
9.3 is isomorphic to a sequence $X\to IX\to\Sigma X$ in
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$.
###### 10.9 Proposition ([38, 11.2]).
The following are equivalent:
1. (i)
Every null-homotopic complex in
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$ is acyclic.
2. (ii)
The category $\operatorname{\mathscr{A}}$ is idempotent complete.
3. (iii)
The class of acyclic complexes is closed under isomorphisms in
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$.
* Proof (Keller).
Let us prove that (i) implies (ii). Let $e:A\to A$ be an idempotent of
$\operatorname{\mathscr{A}}$. Consider the complex
$\cdots\xrightarrow{1-e}A\xrightarrow{e}A\xrightarrow{1-e}A\xrightarrow{e}\cdots$
which is null-homotopic. By (i) this complex is acyclic. This means by
definition that $e$ has a kernel and hence $\operatorname{\mathscr{A}}$ is
idempotent complete.
Let us prove that (ii) implies (iii). Assume that $X$ is isomorphic in
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$ to an acyclic
complex $A$. Using the construction in the proof of Lemma 10.7 one shows that
$X$ is a direct summand in
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$ of the acyclic
complex $A\oplus IX$ and we conclude by Lemma 10.7.
That (iii) implies (i) follows from the fact that a null-homotopic complex $X$
is isomorphic in $\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$ to
the (acyclic) zero complex and hence $X$ is acyclic. ∎
###### 10.10 Remark.
Recall that a subcategory $\operatorname{\mathscr{T}}$ of a triangulated
category $\operatorname{\mathbf{K}}$ is called _thick_ if it is strictly full
and $X\oplus Y\in\operatorname{\mathscr{T}}$ implies
$X,Y\in\operatorname{\mathscr{T}}$.
###### 10.11 Corollary.
The triangulated subcategory
$\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})}$ of
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$ is thick if and only
if $\operatorname{\mathscr{A}}$ is idempotent complete. ∎
### 10.2 Boundedness Conditions
A complex $A$ is called _left bounded_ if $A^{n}=0$ for $n\ll 0$, _right
bounded_ if $A^{n}=0$ for $n\gg 0$ and _bounded_ if $A^{n}=0$ for $|n|\gg 0$.
###### 10.12 Definition.
Denote by $\operatorname{\mathbf{K}}^{+}{(\operatorname{\mathscr{A}})}$,
$\operatorname{\mathbf{K}}^{-}{(\operatorname{\mathscr{A}})}$ and
$\operatorname{\mathbf{K}}^{b}{(\operatorname{\mathscr{A}})}$ the full
subcategories of $\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$
generated by the left bounded complexes, right bounded complexes and bounded
complexes over $\operatorname{\mathscr{A}}$.
Observe that
$\operatorname{\mathbf{K}}^{b}{(\operatorname{\mathscr{A}})}=\operatorname{\mathbf{K}}^{+}{(\operatorname{\mathscr{A}})}\cap\operatorname{\mathbf{K}}^{-}{(\operatorname{\mathscr{A}})}$.
Note further that
$\operatorname{\mathbf{K}}^{\ast}{(\operatorname{\mathscr{A}})}$ is _not_
closed under isomorphisms in
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$ for
$\ast\in\\{+,-,b\\}$ unless $\operatorname{\mathscr{A}}=0$.
###### 10.13 Definition.
For $\ast\in\\{+,-,b\\}$ we define
$\operatorname{\mathbf{Ac}}^{\ast}{(\operatorname{\mathscr{A}})}=\operatorname{\mathbf{K}}^{\ast}{(\operatorname{\mathscr{A}})}\cap\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})}$.
Plainly, $\operatorname{\mathbf{K}}^{\ast}{(\operatorname{\mathscr{A}})}$ is a
full triangulated subcategory of
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$ and
$\operatorname{\mathbf{Ac}}^{\ast}{(\operatorname{\mathscr{A}})}$ is a full
triangulated subcategory of
$\operatorname{\mathbf{K}}^{\ast}{(\operatorname{\mathscr{A}})}$ by Lemma
10.3.
###### 10.14 Proposition.
The following assertions are equivalent:
1. (i)
The subcategories
$\operatorname{\mathbf{Ac}}^{+}{(\operatorname{\mathscr{A}})}$ and
$\operatorname{\mathbf{Ac}}^{-}{(\operatorname{\mathscr{A}})}$ of
$\operatorname{\mathbf{K}}^{+}{(\operatorname{\mathscr{A}})}$ and
$\operatorname{\mathbf{K}}^{-}{(\operatorname{\mathscr{A}})}$ are thick.
2. (ii)
The subcategory $\operatorname{\mathbf{Ac}}^{b}{(\operatorname{\mathscr{A}})}$
of $\operatorname{\mathbf{K}}^{b}{(\operatorname{\mathscr{A}})}$ is thick.
3. (iii)
The category $\operatorname{\mathscr{A}}$ is weakly idempotent complete.
* Proof.
Since
$\operatorname{\mathbf{Ac}}^{b}{(\operatorname{\mathscr{A}})}=\operatorname{\mathbf{Ac}}^{+}{(\operatorname{\mathscr{A}})}\cap\operatorname{\mathbf{Ac}}^{-}{(\operatorname{\mathscr{A}})}$,
we see that (i) implies (ii). Let us prove that (ii) implies (iii). Let
$s:B\to A$ and $t:A\to B$ be morphisms of $\operatorname{\mathscr{A}}$ such
that $ts=1_{B}$. We need to prove that $s$ has a cokernel and $t$ has a
kernel. The complex $X$ given by
$\cdots\xrightarrow{}0\xrightarrow{}B\xrightarrow{s}A\xrightarrow{1-st}A\xrightarrow{t}B\xrightarrow{}0\xrightarrow{}\cdots$
is a direct summand of $X\oplus\Sigma X$ and the latter complex is acyclic
since there is an isomorphism in
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$
$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]}$$\textstyle{B\oplus
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}0&-t\\\
s&1-st\end{smallmatrix}\right]}$$\scriptstyle{\left[\begin{smallmatrix}0&0\\\
0&1\end{smallmatrix}\right]}$$\textstyle{A\oplus
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}-1+st&st\\\
st&1-st\end{smallmatrix}\right]}$$\scriptstyle{\left[\begin{smallmatrix}1&0\\\
0&0\end{smallmatrix}\right]}$$\textstyle{A\oplus
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}1-st&-s\\\
t&0\end{smallmatrix}\right]}$$\scriptstyle{\left[\begin{smallmatrix}0&1\end{smallmatrix}\right]}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}0\\\
s\end{smallmatrix}\right]}$$\textstyle{B\oplus
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}-s&0\\\
0&1-st\end{smallmatrix}\right]}$$\textstyle{A\oplus
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}-1+st&0\\\
0&t\end{smallmatrix}\right]}$$\textstyle{A\oplus
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}-t&0\end{smallmatrix}\right]}$$\textstyle{B}$
where the upper row is obviously acyclic and the lower row is $X\oplus\Sigma
X$. Since $\operatorname{\mathbf{Ac}}^{b}{(\operatorname{\mathscr{A}})}$ is
thick, we conclude that $X$ is acyclic, so that $s$ has a cokernel and $t$ has
a kernel. Therefore $\operatorname{\mathscr{A}}$ is weakly idempotent
complete.
Let us prove that (iii) implies (i). Assume that $X$ is a direct summand in
$\operatorname{\mathbf{K}}^{+}{(\operatorname{\mathscr{A}})}$ of a complex
$A\in\operatorname{\mathbf{Ac}}^{+}{(\operatorname{\mathscr{A}})}$. This means
that we are given a chain map $f:X\to A$ for which there exists a chain map
$s:A\to X$ and morphisms $h^{n}:X^{n}\to X^{n-1}$ such that
$s^{n}f^{n}-1_{X^{n}}=d_{X}^{n-1}h^{n}+h^{n+1}d_{X}^{n}$. On replacing $A$ by
the acyclic complex $A\oplus IX$ as in the proof of Lemma 10.7, we may assume
that $s$ is a left inverse of $f$ in
$\operatorname{\mathbf{Ch}}^{+}{(\operatorname{\mathscr{A}})}$. In particular,
since $\operatorname{\mathscr{A}}$ is assumed to be weakly idempotent
complete, Proposition 7.6 implies that each $f^{n}$ is an admissible monic and
that each $s^{n}$ is an admissible epic. Moreover, as both complexes $X$ and
$A$ are left bounded, we may assume that $A^{n}=0=X^{n}$ for $n<0$. It follows
that $d_{A}^{0}:A^{0}\operatorname{\rightarrowtail}A^{1}$ is an admissible
monic since $A$ is acyclic. But then $d_{A}^{0}f^{0}=f^{1}d_{X}^{0}$ is an
admissible monic, hence Proposition 7.6 implies that $d_{X}^{0}$ is an
admissible monic as well. Let
$e^{1}_{X}:X^{1}\operatorname{\twoheadrightarrow}Z^{2}X$ be a cokernel of
$d_{X}^{0}$ and let $e^{1}_{A}:A^{1}\operatorname{\twoheadrightarrow}Z^{2}A$
be a cokernel of $d_{A}^{0}$. The dotted morphisms in the diagram
$\textstyle{X^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{X}^{0}}$$\scriptstyle{f^{0}}$$\textstyle{X^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{X}^{1}}$$\scriptstyle{f^{1}}$$\textstyle{Z^{2}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{2}}$$\textstyle{A^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{A}^{0}}$$\scriptstyle{s^{0}}$$\textstyle{A^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{A}^{1}}$$\scriptstyle{s^{1}}$$\textstyle{Z^{2}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{2}}$$\textstyle{X^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{X}^{0}}$$\textstyle{X^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{X}^{1}}$$\textstyle{Z^{2}X}$
are uniquely determined by requiring the resulting diagram to be commutative.
Since $s^{0}f^{0}=1_{X^{0}}$ and $s^{1}f^{1}=1_{X^{1}}$ it follows that
$t^{2}g^{2}=1_{Z^{2}X}$, so $t^{2}$ is an admissible epic and $g^{2}$ is an
admissible monic by Proposition 7.6.
Now since $A$ and $X$ are complexes, there are unique maps
$m^{2}_{X}:Z^{2}X\to X^{2}$ and $m^{2}_{A}:Z^{2}A\to A^{2}$ such that
$d^{1}_{X}=m^{2}_{X}e^{1}_{X}$ and $d^{1}_{A}=m^{2}_{A}e^{1}_{A}$. Note that
$m^{2}_{A}$ is an admissible monic since $A$ is acyclic. The upper square in
the diagram
$\textstyle{Z^{2}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m^{2}_{X}}$$\scriptstyle{g^{2}}$$\textstyle{X^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{2}}$$\textstyle{Z^{2}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m^{2}_{A}}$$\scriptstyle{t^{2}}$$\textstyle{A^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s^{2}}$$\textstyle{Z^{2}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m^{2}_{X}}$$\textstyle{X^{2}}$
is commutative because $e^{1}_{X}$ is epic and the lower square is commutative
because $e^{1}_{A}$ is epic. From the commutativity of the upper square it
follows in particular that $m^{2}_{X}$ is an admissible monic by Proposition
7.6. An easy induction now shows that $X$ is acyclic. The assertion about
$\operatorname{\mathbf{Ac}}^{-}{(\operatorname{\mathscr{A}})}$ follows by
duality. ∎
###### 10.15 Remark.
The isomorphism of complexes in the proof that (ii) implies (iii) appears in
Neeman [47, 1.9].
### 10.3 Quasi-Isomorphisms
In abelian categories, quasi-isomorphisms are defined to be chain maps
inducing an isomorphism in homology. Taking the observation in Exercise 9.4
and Proposition 10.9 into account, one arrives at the following generalization
for general exact categories:
###### 10.16 Definition.
A chain map $f:A\to B$ is called a _quasi-isomorphism_ if its mapping cone is
homotopy equivalent to an acyclic complex.
###### 10.17 Remark.
Assume that $\operatorname{\mathscr{A}}$ is idempotent complete. By
Proposition 10.9, a chain map $f$ is a quasi-isomorphism if and only if
$\operatorname{cone}{(f)}$ is acyclic. In particular, for abelian categories,
a quasi-isomorphism is the same thing as a chain map inducing an isomorphism
on homology.
###### 10.18 Remark.
If $p:A\to A$ is an idempotent in $\operatorname{\mathscr{A}}$ which does not
split, then the complex $C$ given by
$\cdots\xrightarrow{1-p}A\xrightarrow{p}A\xrightarrow{1-p}A\xrightarrow{p}\cdots$
is null-homotopic but _not_ acyclic. However, $f:0\to C$ is a chain homotopy
equivalence, hence it should be a quasi-isomorphism, but
$\operatorname{cone}{(f)}=C$ fails to be acyclic.
### 10.4 The Definition of the Derived Category
The _derived category_ of the exact category $\operatorname{\mathscr{A}}$ is
defined to be the _Verdier quotient_
$\operatorname{\mathbf{D}}{(\operatorname{\mathscr{A}})}=\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}/\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})}$
as described e.g. in Neeman [48, Chapter 2] or Keller [38, §§ 10, 11]. For the
description of derived functors given in section 10.6 it is useful to recall
that the Verdier quotient can be explicitly described by a calculus of
fractions. A morphism $A\to B$ in
$\operatorname{\mathbf{D}}{(\operatorname{\mathscr{A}})}$ can be represented
by a _fraction_ $(f,s)$
$A\xrightarrow{f}B^{\prime}\xleftarrow{s}B$
where $f:A\to B$ is a morphism in
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$ and $s:B\to
B^{\prime}$ is a quasi-isomorphism in
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$. Two fractions
$(f,s)$ and $(g,t)$ are equivalent if there exists a fraction $(h,u)$ and a
commutative diagram
---
$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{h}$$\scriptstyle{g}$$\textstyle{B^{\prime\prime\prime}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{u}$$\scriptstyle{t}$$\textstyle{B^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
or, in words, if the fractions $(f,s)$ and $(g,t)$ have a common expansion
$(h,u)$. We refer to Keller [38, §§ 9, 10] for further details.
When dealing with the boundedness condition $\ast\in\\{+,-,b\\}$ we define
$\operatorname{\mathbf{D}}^{\ast}{(\operatorname{\mathscr{A}})}=\operatorname{\mathbf{K}}^{\ast}{(\operatorname{\mathscr{A}})}/\operatorname{\mathbf{Ac}}^{\ast}{(\operatorname{\mathscr{A}})}.$
It is not difficult to prove that the canonical functor
$\operatorname{\mathbf{D}}^{\ast}{(\operatorname{\mathscr{A}})}\to\operatorname{\mathbf{D}}{(\operatorname{\mathscr{A}})}$
is an equivalence between
$\operatorname{\mathbf{D}}^{\ast}{(\operatorname{\mathscr{A}})}$ and the full
subcategory of $\operatorname{\mathbf{D}}{(\operatorname{\mathscr{A}})}$
generated by the complexes satisfying the boundedness condition $\ast$, see
Keller [38, 11.7].
###### 10.19 Remark.
If $\operatorname{\mathscr{A}}$ is idempotent complete then a chain map
becomes an isomorphism in
$\operatorname{\mathbf{D}}{(\operatorname{\mathscr{A}})}$ if and only if its
cone is acyclic by Corollary 10.11. If $\operatorname{\mathscr{A}}$ is weakly
idempotent complete then a chain map in
$\operatorname{\mathbf{Ch}}^{\ast}{(\operatorname{\mathscr{A}})}$ becomes an
isomorphism in
$\operatorname{\mathbf{D}}^{\ast}{(\operatorname{\mathscr{A}})}$ if and only
if its cone is acyclic by Proposition 10.14.
### 10.5 Derived Categories of Fully Exact Subcategories
The proof of the following lemma is straightforward and left to the reader as
an exercise. That admissible monics and epics are closed under composition
follows from the Noether isomorphism 3.5.
###### 10.20 Lemma.
Let $\operatorname{\mathscr{A}}$ be an exact category and suppose that
$\operatorname{\mathscr{B}}$ is a full additive subcategory of
$\operatorname{\mathscr{A}}$ which is closed under extensions in the sense
that the existence of a short exact sequence
$B^{\prime}\operatorname{\rightarrowtail}A\operatorname{\twoheadrightarrow}B^{\prime\prime}$
with $B^{\prime},B^{\prime\prime}\in\operatorname{\mathscr{B}}$ implies that
$A$ is isomorphic to an object of $\operatorname{\mathscr{B}}$. The sequences
in $\operatorname{\mathscr{B}}$ which are exact in
$\operatorname{\mathscr{A}}$ form an exact structure on
$\operatorname{\mathscr{B}}$. ∎
###### 10.21 Definition.
A _fully exact subcategory_ $\operatorname{\mathscr{B}}$ of an exact category
$\operatorname{\mathscr{A}}$ is a full additive subcategory which is closed
under extensions and equipped with the exact structure from the previous
lemma.
###### 10.22 Theorem ([38, 12.1]).
Let $\operatorname{\mathscr{B}}$ be a fully exact subcategory of
$\operatorname{\mathscr{A}}$ and consider the functor
$\operatorname{\mathbf{D}}^{+}{(\operatorname{\mathscr{B}})}\to\operatorname{\mathbf{D}}^{+}{(\operatorname{\mathscr{A}})}$
induced by the inclusion
$\operatorname{\mathscr{B}}\subset\operatorname{\mathscr{A}}$.
1. (i)
Assume that for every object $A\in\operatorname{\mathscr{A}}$ there exists an
admissible monic $A\operatorname{\rightarrowtail}B$ with
$B\in\operatorname{\mathscr{B}}$. For every left bounded complex $A$ over
$\operatorname{\mathscr{A}}$ there exists a left bounded complex $B$ over
$\operatorname{\mathscr{B}}$ and a quasi-isomorphism $A\to B$. In particular
$\operatorname{\mathbf{D}}^{+}{(\operatorname{\mathscr{B}})}\to\operatorname{\mathbf{D}}^{+}{(\operatorname{\mathscr{A}})}$
is essentially surjective.
2. (ii)
Assume that for every short exact sequence
$B^{\prime}\operatorname{\rightarrowtail}A\to A^{\prime\prime}$ of
$\operatorname{\mathscr{A}}$ with $B^{\prime}\in\operatorname{\mathscr{B}}$
there exists a commutative diagram with exact rows
$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime\prime}.}$
For every quasi-isomorphism $s:B\to A$ in
$\operatorname{\mathbf{K}}^{+}{(\operatorname{\mathscr{A}})}$ with $B$ a
complex over $\operatorname{\mathscr{B}}$ there exists a morphism $t:A\to
B^{\prime}$ in $\operatorname{\mathbf{K}}^{+}{(\operatorname{\mathscr{A}})}$
such that $ts:B\to B^{\prime}$ is a quasi-isomorphism. In particular,
$\operatorname{\mathbf{D}}^{+}{(\operatorname{\mathscr{B}})}\to\operatorname{\mathbf{D}}^{+}{(\operatorname{\mathscr{A}})}$
is fully faithful.
###### 10.23 Remark.
The condition in (ii) holds if condition (i) holds and, moreover, for all
short exact sequences
$B^{\prime}\operatorname{\rightarrowtail}B\operatorname{\twoheadrightarrow}A^{\prime\prime}$
with $B^{\prime},B\in\operatorname{\mathscr{B}}$ it follows that
$A^{\prime\prime}$ is isomorphic to an object in $\operatorname{\mathscr{B}}$.
To see this, start with a short exact sequence
$B^{\prime}\operatorname{\rightarrowtail}A\operatorname{\twoheadrightarrow}A^{\prime\prime}$,
then choose an admissible monic $A\operatorname{\rightarrowtail}B$, form the
push-out $AA^{\prime\prime}BB^{\prime\prime}$ and apply Proposition 2.12 and
Proposition 2.15.
###### 10.24 Example.
Let $\operatorname{\mathscr{I}}$ be the full subcategory spanned by the
_injective_ objects of the exact category
$(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$, see Definition
11.1. Clearly, $\operatorname{\mathscr{I}}$ is a fully exact subcategory of
$\operatorname{\mathscr{A}}$ (the induced exact structure consists of the
split exact sequences) and it satisfies condition (ii) of Theorem 10.22. If
$\operatorname{\mathscr{I}}$ satisfies condition (i) then there are _enough
injectives_ in $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$, see
Definition 11.9. A quasi-isomorphism of left bounded complexes of injectives
is a chain homotopy equivalence, hence
$\operatorname{\mathbf{K}}^{+}{(\operatorname{\mathscr{I}})}$ is equivalent to
$\operatorname{\mathbf{D}}^{+}{(\operatorname{\mathscr{I}})}$. By Theorem
10.22 $\operatorname{\mathbf{K}}^{+}{(\operatorname{\mathscr{I}})}$ is
equivalent to the full subcategory of
$\operatorname{\mathbf{D}}{(\operatorname{\mathscr{A}})}$ spanned by the left
bounded complexes with injective components. Moreover, if
$(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ has enough
injectives, then the functor
$\operatorname{\mathbf{K}}^{+}{(\operatorname{\mathscr{I}})}\to\operatorname{\mathbf{D}}^{+}{(\operatorname{\mathscr{A}})}$
is an equivalence of triangulated categories.
### 10.6 Total Derived Functors
With these constructions at hand one can now introduce (total) derived
functors in the sense of Grothendieck-Verdier and Deligne, see e.g. Keller
[38, §§13-15] or any one of the references given in Remark 9.8. We follow
Keller’s exposition of the Deligne approach.
The problem is the following: An additive functor
$F:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{B}}$ from an exact
category to another induces functors
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}\to\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{B}})}$
and
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}\to\operatorname{\mathbf{K}}{(\operatorname{\mathscr{B}})}$
in an obvious way. By abuse of notation we still denote these functors by $F$.
The next question to ask is whether the functor descends to a functor of
derived categories, i.e., we look for a commutative diagram
$\textstyle{\operatorname{\mathbf{K}}@wrapper{\operatorname{\mathbf{K}}@presentation}{(\operatorname{\mathscr{A}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q_{\operatorname{\mathscr{A}}}}$$\scriptstyle{F}$$\textstyle{\operatorname{\mathbf{K}}@wrapper{\operatorname{\mathbf{K}}@presentation}{(\operatorname{\mathscr{B}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q_{\operatorname{\mathscr{B}}}}$$\textstyle{\operatorname{\mathbf{D}}@wrapper{\operatorname{\mathbf{D}}@presentation}{(\operatorname{\mathscr{A}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exists?}$$\textstyle{\operatorname{\mathbf{D}}@wrapper{\operatorname{\mathbf{D}}@presentation}{(\operatorname{\mathscr{B}})}.}$
If the functor $F:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{B}}$ is
exact, this problem has a solution by the universal property of the derived
category since then
$F(\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{A}})})\subset\operatorname{\mathbf{Ac}}{(\operatorname{\mathscr{B}})}$.
However, if $F$ fails to be exact, it will not send acyclic complexes to
acyclic complexes, or, in other words, it will not send quasi-isomorphisms to
quasi-isomorphisms and our naïve question will have a negative answer.
Deligne’s solution consists in constructing for each
$A\in\operatorname{\mathbf{D}}{(\operatorname{\mathscr{A}})}$ a functor
$\operatorname{\mathbf{r}}\\!{F}({-},A):(\operatorname{\mathbf{D}}{(\operatorname{\mathscr{B}})})^{\operatorname{op}}\to\operatorname{\mathbf{Ab}}.$
If the functor $\operatorname{\mathbf{r}}\\!{F}({-},A)$ is representable, a
representing object will be denoted by $\operatorname{\mathbf{R}}\\!{F}(A)$
and $\operatorname{\mathbf{R}}\\!{F}$ is said to be _defined at $A$_. To be a
little more specific, for
$B\in\operatorname{\mathbf{D}}{(\operatorname{\mathscr{B}})}$ we define the
abelian group $\operatorname{\mathbf{r}}\\!{F}(B,A)$ by the equivalence
classes of diagrams
$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{F(A^{\prime})}$$\textstyle{A^{\prime}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$
where $f:B\to F(A^{\prime})$ is a morphism of
$\operatorname{\mathbf{D}}{(\operatorname{\mathscr{B}})}$ and $s:A\to
A^{\prime}$ is a quasi-isomorphism in
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$. Observe the analogy
to the description of morphisms in
$\operatorname{\mathbf{D}}{(\operatorname{\mathscr{A}})}$; it is useful to
think of the diagram as “$F$-fractions”. Accordingly, two $F$-fractions
$(f,s)$ and $(g,t)$ are said to be _equivalent_ if there exist commutative
diagrams
---
$\textstyle{F(A^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F(v)}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{h}$$\scriptstyle{g}$$\textstyle{F(A^{\prime\prime\prime})}$$\textstyle{A^{\prime\prime\prime}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{u}$$\scriptstyle{t}$$\textstyle{F(A^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F(w)}$$\textstyle{A^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w}$
in $\operatorname{\mathbf{D}}{(\operatorname{\mathscr{B}})}$ and
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$, where $(h,u)$ is
another $F$-fraction. On morphisms of
$\operatorname{\mathbf{D}}{(\operatorname{\mathscr{B}})}$ define
$\operatorname{\mathbf{r}}\\!F{(-,A)}$ by pre-composition. By defining
$\operatorname{\mathbf{r}}\\!F$ on morphisms of
$\operatorname{\mathbf{D}}{(\operatorname{\mathscr{A}})}$ one obtains a
_functor_ from $\operatorname{\mathbf{D}}{(\operatorname{\mathscr{A}})}$ to
the category of functors
$(\operatorname{\mathbf{D}}{(\operatorname{\mathscr{B}})})^{\operatorname{op}}\to\operatorname{\mathbf{Ab}}$.
Let
$\operatorname{\mathscr{T}}\subset\operatorname{\mathbf{D}}{(\operatorname{\mathscr{A}})}$
be the full subcategory of objects at which $\operatorname{\mathbf{R}}\\!{F}$
is defined and choose for each $A\in\operatorname{\mathscr{T}}$ a representing
object $\operatorname{\mathbf{R}}\\!F(A)$ and an isomorphism
$\operatorname{Hom}_{\operatorname{\mathbf{D}}{(\operatorname{\mathscr{B}})}}{({-},\operatorname{\mathbf{R}}\\!F(A))}\xrightarrow{\sim}\operatorname{\mathbf{r}}\\!F{({-},A)}.$
These choices force the definition of $\operatorname{\mathbf{R}}\\!F$ on
morphisms and thus
$\operatorname{\mathbf{R}}\\!F:\operatorname{\mathscr{T}}\to\operatorname{\mathbf{D}}{(\operatorname{\mathscr{B}})}$
is a functor. Even more is true:
###### 10.25 Theorem (Deligne).
Let
$F:\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}\to\operatorname{\mathbf{K}}{(\operatorname{\mathscr{B}})}$
be a functor and let $\operatorname{\mathscr{T}}$ be the full subcategory of
$\operatorname{\mathbf{D}}{(\operatorname{\mathscr{A}})}$ at which
$\operatorname{\mathbf{R}}\\!F$ is defined. Let $\operatorname{\mathscr{S}}$
be the full subcategory of
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$ spanned by the
objects of $\operatorname{\mathscr{T}}$. Denote by
$I:\operatorname{\mathscr{S}}\to\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$
the inclusion.
1. (i)
The category $\operatorname{\mathscr{T}}$ is a triangulated subcategory of
$\operatorname{\mathbf{D}}{(\operatorname{\mathscr{A}})}$ and
$\operatorname{\mathscr{S}}$ is a triangulated subcategory of
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$.
2. (ii)
The functor
$\operatorname{\mathbf{R}}\\!F:\operatorname{\mathscr{T}}\to\operatorname{\mathbf{D}}{(\operatorname{\mathscr{B}})}$
is a triangle functor and there is a morphism of triangle functors
$Q_{\operatorname{\mathscr{B}}}FI\Rightarrow\operatorname{\mathbf{R}}\\!FQ_{\operatorname{\mathscr{A}}}I$.
3. (iii)
For every morphism $\mu:F\Rightarrow F^{\prime}$ of triangle functors
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}\to\operatorname{\mathbf{K}}{(\operatorname{\mathscr{B}})}$
there is an induced morphism of triangle functors
$\operatorname{\mathbf{R}}\\!\mu:\operatorname{\mathbf{R}}\\!F\Rightarrow\operatorname{\mathbf{R}}\\!F^{\prime}$
on the intersection of the domains of $\operatorname{\mathbf{R}}\\!F$ and
$\operatorname{\mathbf{R}}\\!F^{\prime}$.
The only subtle part of the previous theorem is the fact that
$\operatorname{\mathscr{T}}$ is triangulated. The rest is a straightforward
but rather tedious verification. The essential details and references are
given in Keller [38, §13].
The next question that arises is whether one can get some information on
$\operatorname{\mathscr{T}}$. A complex $A$ is said to be _$F$ -split_ if
$\operatorname{\mathbf{R}}\\!F$ is defined at $A$ and the canonical morphism
$F(A)\to\operatorname{\mathbf{R}}\\!F(A)$ is invertible. An object $A$ of
$\operatorname{\mathscr{A}}$ is said to be _$F$ -acyclic_ if it is $F$-split
when considered as complex concentrated in degree zero.
###### 10.26 Lemma ([38, 15.1, 15.3]).
Let $\operatorname{\mathscr{C}}$ be a fully exact subcategory of
$\operatorname{\mathscr{A}}$ satisfying hypothesis (ii) of Theorem 10.22.
Assume that the restriction of
$F:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{B}}$ to
$\operatorname{\mathscr{C}}$ is exact. Then each object of
$\operatorname{\mathscr{C}}$ is $F$-acyclic.
Conversely, let $\operatorname{\mathscr{C}}$ be the full subcategory of
$\operatorname{\mathscr{A}}$ consisting of the $F$-acyclic objects. Then
$\operatorname{\mathscr{C}}$ is a fully exact subcategory of
$\operatorname{\mathscr{A}}$, it satisfies condition (ii) of Theorem 10.22 and
the restriction of $F$ to $\operatorname{\mathscr{C}}$ is exact.
Now let $\operatorname{\mathscr{C}}$ be a fully exact subcategory of
$\operatorname{\mathscr{A}}$ consisting of $F$-acyclic objects and suppose
that $\operatorname{\mathscr{C}}$ satisfies conditions (i) and (ii) of Theorem
10.22. By these assumptions, the inclusion
$\operatorname{\mathscr{C}}\to\operatorname{\mathscr{A}}$ induces an
equivalence
$\operatorname{\mathbf{D}}^{+}{(\operatorname{\mathscr{C}})}\to\operatorname{\mathbf{D}}^{+}{(\operatorname{\mathscr{A}})}$.
As the restriction of $F$ to $\operatorname{\mathscr{C}}$ is exact, it yields
a triangle functor
$F:\operatorname{\mathbf{D}}^{+}{(\operatorname{\mathscr{C}})}\to\operatorname{\mathbf{D}}^{+}{(\operatorname{\mathscr{B}})}$.
To choose a quasi-inverse for the canonical functor
$\operatorname{\mathbf{D}}^{+}{(\operatorname{\mathscr{C}})}\to\operatorname{\mathbf{D}}^{+}{(\operatorname{\mathscr{A}})}$
amounts to choosing for each complex
$A\in\operatorname{\mathbf{K}}^{+}{(\operatorname{\mathscr{A}})}$ a quasi-
isomorphism $s:A\to C$ with
$C\in\operatorname{\mathbf{K}}^{+}{(\operatorname{\mathscr{C}})}$ by [40,
1.6], a proof of which is given in [37, 6.7]. As we have just seen, $C$ is
$F$-split, hence $s$ yields an isomorphism
$\operatorname{\mathbf{R}}\\!F(A)\to
F(C)\cong\operatorname{\mathbf{R}}\\!F(C)$. Such a quasi-isomorphism $A\to C$
exists by the construction in the proof of Theorem 12.7 and our assumptions.
The admittedly concise _résumé_ given here provides the basic toolkit for
treating derived functors. We refer to Keller [38, §§13–15] for a much more
thorough and general discussion and precise statements of the composition
formula
$\operatorname{\mathbf{R}}\\!F\circ\operatorname{\mathbf{R}}\\!G\cong\operatorname{\mathbf{R}}\\!(FG)$
and adjunction formulæ of left and right derived functors of adjoint pairs of
functors,
## 11 Projective and Injective Objects
###### 11.1 Definition.
An object $P$ of an exact category $\operatorname{\mathscr{A}}$ is called
_projective_ if the represented functor
$\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(P,{-})}:\operatorname{\mathscr{A}}\to\operatorname{\mathbf{Ab}}$
is exact. An object $I$ of an exact category is called _injective_ if the
corepresented functor
$\operatorname{Hom}_{\operatorname{\mathscr{A}}}{({-},I)}:\operatorname{\mathscr{A}}^{\operatorname{op}}\to\operatorname{\mathbf{Ab}}$
is exact.
###### 11.2 Remark.
The concepts of projectivity and injectivity are dual to each other in the
sense that $P$ is projective in $\operatorname{\mathscr{A}}$ if and only if
$P$ is injective in $\operatorname{\mathscr{A}}^{\operatorname{op}}$. For our
purposes it is therefore sufficient to deal with projective objects.
###### 11.3 Proposition.
An object $P$ of an exact category is projective if and only if any one of the
following conditions holds:
1. (i)
For all admissible epics $A\operatorname{\twoheadrightarrow}A^{\prime\prime}$
and all morphisms $P\to A^{\prime\prime}$ there exists a solution to the
lifting problem
$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exists}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime\prime}}$
making the diagram above commutative.
2. (ii)
The functor
$\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(P,{-})}:\operatorname{\mathscr{A}}\to\operatorname{\mathbf{Ab}}$
sends admissible epics to surjections.
3. (iii)
Every admissible epic $A\operatorname{\twoheadrightarrow}P$ splits (has a
right inverse).
* Proof.
Since $\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(P,{-})}$ transforms
exact sequences to left exact sequences in $\operatorname{\mathbf{Ab}}$ for
all objects $P$ (see the proof of Corollary A.8), it is clear that conditions
(i) and (ii) are equivalent to the projectivity of $P$. If $P$ is projective
and $A\operatorname{\twoheadrightarrow}P$ is an admissible epic then
$\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(P,A)}\operatorname{\twoheadrightarrow}\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(P,P)}$
is surjective, and every pre-image of $1_{P}$ is a splitting map of
$A\operatorname{\twoheadrightarrow}P$. Conversely, let us prove that condition
(iii) implies condition (i): given a lifting problem as in (i), form the
following pull-back diagram
$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{a^{\prime}}$PB$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{A^{\prime\prime}.}$
By hypothesis, there exists a right inverse $b^{\prime}$ of $a^{\prime}$ and
$f^{\prime}b^{\prime}$ solves the lifting problem because
$af^{\prime}b^{\prime}=fa^{\prime}b^{\prime}=f$. ∎
###### 11.4 Corollary.
If $P$ is projective and $P\to A$ has a right inverse then $A$ is projective.
* Proof.
This is a trivial consequence of condition (i) in Proposition 11.3. ∎
###### 11.5 Remark.
If $\operatorname{\mathscr{A}}$ is weakly idempotent complete, the above
corollary amounts to the familiar “direct summands of projective objects are
projective” in abelian categories.
###### 11.6 Corollary.
A sum $P=P^{\prime}\oplus P^{\prime\prime}$ is projective if and only if both
$P^{\prime}$ and $P^{\prime\prime}$ are projective. ∎
More generally:
###### 11.7 Corollary.
Let $\\{P_{i}\\}_{i\in I}$ be a family of objects for which the coproduct
$P=\coprod_{i\in I}P_{i}$ exists in $\operatorname{\mathscr{A}}$. The object
$P$ is projective if and only if each $P_{i}$ is projective. ∎
###### 11.8 Remark.
The dual of the previous result is that a product (if it exists) is injective
if and only if each of its factors is injective.
###### 11.9 Definition.
An exact category $\operatorname{\mathscr{A}}$ is said to have _enough
projectives_ if for every object $A\in\operatorname{\mathscr{A}}$ there exists
a projective object $P$ and an admissible epic
$P\operatorname{\twoheadrightarrow}A$.
###### 11.10 Exercise (Heller [26, 5.6]).
Assume that $\operatorname{\mathscr{A}}$ has enough projectives. Prove that
$A^{\prime}\to A\to A^{\prime\prime}$ is short exact if and only if
$\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(P,A^{\prime})}\operatorname{\rightarrowtail}\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(P,A)}\operatorname{\twoheadrightarrow}\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(P,A^{\prime\prime})}$
is short exact for all projective objects $P$.
_Hint:_ For sufficiency prove first that $A^{\prime}\to A$ is a monomorphism,
then prove that it is a kernel of $A\to A^{\prime\prime}$ and finally apply
the obscure axiom 2.16. In all three steps use that there are enough
projectives.
###### 11.11 Exercise (Heller [26, 5.6]).
Assume that $\operatorname{\mathscr{A}}$ is weakly idempotent complete and has
enough projectives. Prove that the sequence
$A_{n}\to A_{n-1}\to\cdots\to A_{1}\to A_{0}\to 0$
is an exact sequence of admissible morphisms if and only if for all
projectives $P$ the sequence
$\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(P,A_{n})}\to\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(P,A_{n-1})}\to\cdots\to\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(P,A_{1})}\to\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(P,A_{0})}\to
0$
is an exact sequence of abelian groups.
## 12 Resolutions and Classical Derived Functors
###### 12.1 Definition.
A _projective resolution_ of the object $A$ is a positive complex
$P_{\bullet}$ with projective components together with a morphism $P_{0}\to A$
such that the _augmented complex_
$\cdots\to P_{n+1}\to P_{n}\to\cdots\to P_{1}\to P_{0}\to A$
is exact.
###### 12.2 Proposition (Resolution Lemma).
If $\operatorname{\mathscr{A}}$ has enough projectives then every object
$A\in\operatorname{\mathscr{A}}$ has a projective resolution.
* Proof.
This is an easy induction. Because $\operatorname{\mathscr{A}}$ has enough
projectives, there exists a projective object $P_{0}$ and an admissible epic
$P_{0}\operatorname{\twoheadrightarrow}A$ with $P_{0}$. Choose an admissible
monic $A_{0}\operatorname{\rightarrowtail}P_{0}$ such that
$A_{0}\operatorname{\rightarrowtail}P_{0}\operatorname{\twoheadrightarrow}A$
is exact. Now choose a projective $P_{1}$ and an admissible epic
$P_{1}\operatorname{\twoheadrightarrow}A_{0}$. Continue with an admissible
monic $A_{1}\operatorname{\rightarrowtail}P_{1}$ such that
$A_{1}\operatorname{\rightarrowtail}P_{1}\operatorname{\twoheadrightarrow}A_{0}$
is exact, and so on. One thus obtains a sequence
|
---|---
$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{P_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A}$$\textstyle{A_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
which is exact by construction, so $P_{\bullet}\to A$ is a projective
resolution. ∎
###### 12.3 Remark.
The defining concept of projectivity is not used in the previous proof. That
is, we have proved: If $\operatorname{\mathscr{P}}$ is a class in
$\operatorname{\mathscr{A}}$ such that for each object
$A\in\operatorname{\mathscr{A}}$ there is an admissible epic
$P\operatorname{\twoheadrightarrow}A$ with $P\in\operatorname{\mathscr{P}}$
then each object of $\operatorname{\mathscr{A}}$ has a
$\operatorname{\mathscr{P}}$-resolution
$P_{\bullet}\operatorname{\twoheadrightarrow}A$.
Consider a morphism $f:A\to B$ in $\operatorname{\mathscr{A}}$. Let
$P_{\bullet}$ be a complex of projectives with $P_{n}=0$ for $n<0$ and let
$\alpha:P_{0}\to A$ be a morphism such that the composition $P_{1}\to P_{0}\to
A$ is zero [e.g. $P_{\bullet}\to A$ is a projective resolution of $A$]. Let
$Q_{\bullet}\xrightarrow{\beta}B$ be a resolution (not necessarily
projective).
###### 12.4 Theorem (Comparison Theorem).
Under the above hypotheses there exists a chain map
$f_{\bullet}:P_{\bullet}\to Q_{\bullet}$ such that the following diagram
commutes:
$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exists
f_{2}}$$\textstyle{P_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exists
f_{1}}$$\textstyle{P_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\exists
f_{0}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{B.}$
Moreover, the _lift_ $f_{\bullet}$ of $f$ is unique up to homotopy
equivalence.
* Proof.
It is convenient to put $P_{-1}=A$, $Q_{0}^{\prime}=Q_{-1}=B$ and $f_{-1}=f$.
_Existence:_ The question of existence of $f_{0}$ is the lifting problem given
by the map $f\alpha:P_{0}\to B$ and the admissible epic
$\beta:Q_{0}\operatorname{\twoheadrightarrow}B$. This problem has a solution
by projectivity of $P_{0}$.
Let $n\geq 0$ and suppose by induction that there are morphisms
$f_{n}:P_{n}\to Q_{n}$ and $f_{n-1}:P_{n-1}\to Q_{n-1}$ such that
$df_{n}=f_{n-1}d$. Consider the following diagram:
| |
---|---|---
| |
$\textstyle{P_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exists
f_{n+1}}$$\scriptstyle{\exists!f_{n+1}^{\prime}}$$\scriptstyle{d}$$\textstyle{P_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\scriptstyle{f_{n}}$$\textstyle{P_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{n-1}}$$\textstyle{{Q_{n+1}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{Q_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{Q_{n-1}}$$\textstyle{{Q_{n}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
By induction the right hand square is commutative, so the morphism $P_{n+1}\to
Q_{n-1}$ is zero because the morphism $P_{n+1}\to P_{n-1}$ is zero. The
morphism $P_{n+1}\to Q_{n}^{\prime}$ is zero as well because
$Q_{n}^{\prime}\operatorname{\rightarrowtail}Q_{n-1}$ is monic. Since
$Q_{n+1}^{\prime}\operatorname{\rightarrowtail}Q_{n}\operatorname{\twoheadrightarrow}Q_{n}^{\prime}$
is exact, there exists a unique morphism $f_{n+1}^{\prime}:P_{n+1}\to
Q_{n+1}^{\prime}$ making the upper right triangle in the left hand square
commute. Because $P_{n+1}$ is projective and
$Q_{n+1}\operatorname{\twoheadrightarrow}Q_{n+1}^{\prime}$ is an admissible
epic, there is a morphism $f_{n+1}:P_{n+1}\to Q_{n+1}$ such that the left hand
square commutes. This settles the existence of $f_{\bullet}$.
_Uniqueness:_ Let $g_{\bullet}:P_{\bullet}\to Q_{\bullet}$ be another lift of
$f$ and put $h_{\bullet}=f_{\bullet}-g_{\bullet}$. We will construct by
induction a chain contraction $s_{n}:P_{n-1}\to Q_{n}$ for $h$. For $n\leq 0$
we put $s_{n}=0$. For $n\geq 0$ assume by induction that there are morphisms
$s_{n-1},s_{n}$ such that $h_{n-1}=ds_{n}+s_{n-1}d$. Because of this
assumption and the fact that $h$ is a chain map, we have
$d(h_{n}-s_{n}d)=h_{n-1}d-(h_{n-1}-s_{n-1}d)d=0$ so the following diagram
commutes
---
|
$\textstyle{P_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{n}-s_{n}d}$$\scriptstyle{\exists!s_{n+1}^{\prime}}$$\scriptstyle{\exists
s_{n+1}}$$\scriptstyle{0}$$\textstyle{{Q_{n+1}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{Q_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{Q_{n-1}}$$\textstyle{{Q_{n}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
and we get a morphism $s_{n+1}:P_{n}\to Q_{n+1}$ such that
$ds_{n+1}=h_{n}-s_{n}d$ as in the existence proof. ∎
###### 12.5 Corollary.
Any two projective resolutions of an object $A$ are chain homotopy equivalent.
∎
###### 12.6 Corollary.
Let $P_{\bullet}$ be a right bounded complex of projectives and let
$A_{\bullet}$ be an acyclic complex. Then
$\operatorname{Hom}_{\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}}{(P_{\bullet},A_{\bullet})}=0$.
∎
In order to deal with derived functors on the level of the derived category,
one needs to sharpen both the resolution lemma and the comparison theorem.
###### 12.7 Theorem ([36, 4.1, Lemma, b)]).
Let $\operatorname{\mathscr{A}}$ be an exact category with enough projectives.
For every right bounded complex $A$ over $\operatorname{\mathscr{A}}$ exists a
right bounded complex with projective components $P$ and a quasi-isomorphism
$P\xrightarrow{\alpha}A$.
* Proof.
Renumbering if necessary, we may suppose $A_{n}=0$ for $n<0$. The complex $P$
will be constructed by induction. For the inductive formulation it is
convenient to define $P_{n}=B_{n}=0$ for $n<0$. Put $B_{0}=A_{0}$, choose an
admissible epic $p_{0}^{\prime}:P_{0}\operatorname{\twoheadrightarrow}B_{0}$
from a projective $P_{0}$ and define $p_{0}^{\prime\prime}=d^{A}_{0}$. Let
$B_{1}$ be the pull-back over $p_{0}^{\prime}$ and $p_{0}^{\prime\prime}$.
Consider the following commutative diagram:
---
$\textstyle{P_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{0}^{\prime}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{0}^{\prime\prime}}$$\scriptstyle{i_{0}^{\prime}}$PB$\textstyle{B_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$PB$\textstyle{0}$$\textstyle{A_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{A}_{2}}$$\scriptstyle{0}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{A}_{1}}$$\scriptstyle{0}$$\scriptstyle{\exists!p_{1}^{\prime\prime}}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{A}_{0}}$$\scriptstyle{p_{0}^{\prime\prime}}$$\textstyle{A_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
The morphism $p_{1}^{\prime\prime}$ exists by the universal property of the
pull-back and moreover $p_{1}^{\prime\prime}d^{A}_{2}=0$ because
$d^{A}_{1}d^{A}_{2}=0$.
Suppose by induction that in the following diagram everything is constructed
except $B_{n+1}$ and the morphisms terminating or issuing from there. Assume
further that $P_{n}$ is projective and that
$p_{n}^{\prime\prime}d^{A}_{n+1}=0$.
---
$\textstyle{P_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{n}^{\prime}}$$\textstyle{P_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{n-1}^{\prime}}$$\textstyle{B_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{n}^{\prime\prime}}$$\scriptstyle{i_{n}^{\prime}}$PB$\textstyle{B_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{n-1}^{\prime\prime}}$$\scriptstyle{i_{n-1}^{\prime}}$PB$\textstyle{B_{n-1}}$$\textstyle{A_{n+3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{A}_{n+2}}$$\textstyle{A_{n+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{A}_{n+1}}$$\scriptstyle{\exists!p_{n+1}^{\prime\prime}}$$\textstyle{A_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{A}_{n}}$$\scriptstyle{p_{n}^{\prime\prime}}$$\textstyle{A_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{n-1}^{\prime\prime}}$
As indicated in the diagram, we obtain $B_{n+1}$ by forming the pull-back over
$p_{n}^{\prime}$ and $p_{n}^{\prime\prime}$. We complete the induction by
choosing an admissible epic
$p_{n+1}^{\prime}:P_{n+1}\operatorname{\twoheadrightarrow}B_{n+1}$ from a
projective $P_{n+1}$, constructing $p_{n+1}^{\prime\prime}$ as in the first
paragraph and finally noticing that $p_{n+1}^{\prime\prime}d^{A}_{n+2}=0$.
The projective complex is given by the $P_{n}$’s and the differential
$d^{P}_{n-1}=i_{n-1}^{\prime}p_{n}^{\prime}$, which satisfies $(d^{P})^{2}=0$
by construction.
Let $\alpha$ be given by $\alpha_{n}=i_{n-1}^{\prime\prime}p_{n}^{\prime}$ in
degree $n$, manifestly a chain map. We claim that $\alpha$ is a quasi-
isomorphism. The mapping cone of $\alpha$ is seen to be exact using
Proposition 2.12: For each $n$ there is an exact sequence
$B_{n+1}\xrightarrow{i_{n}=\left[\begin{smallmatrix}-i_{n}^{\prime}\\\
i_{n}^{\prime\prime}\end{smallmatrix}\right]}P_{n}\oplus
A_{n+1}\xrightarrow{p_{n}=\left[\begin{smallmatrix}p_{n}^{\prime}&p_{n}^{\prime\prime}\end{smallmatrix}\right]}B_{n}.$
We thus obtain an exact complex $C$ with $C_{n}=P_{n}\oplus A_{n+1}$ in degree
$n$ and differential
$d^{C}_{n-1}=i_{n-1}p_{n}=\left[\begin{smallmatrix}-i_{n-1}^{\prime}p_{n}^{\prime}&-i_{n-1}^{\prime}p_{n}^{\prime\prime}\\\
i_{n-1}^{\prime\prime}p_{n}^{\prime}&i_{n-1}^{\prime\prime}p_{n}^{\prime\prime}\end{smallmatrix}\right]=\left[\begin{smallmatrix}-d^{P}_{n-1}&0\\\
\alpha_{n}&d^{A}_{n}\end{smallmatrix}\right]$
which shows that $C=\operatorname{cone}{(\alpha)}$. ∎
###### 12.8 Theorem (Horseshoe Lemma).
A horseshoe can be filled in: Suppose we are given a horseshoe diagram
$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{2}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{1}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{0}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{2}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{1}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{0}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime\prime},}$
that is to say, the column is short exact and the horizontal rows are
projective resolutions of $A^{\prime}$ and $A^{\prime\prime}$. Then the direct
sums $P_{n}=P_{n}^{\prime}\oplus P_{n}^{\prime\prime}$ assemble to a
projective resolution of $A$ in such a way that the horseshoe can be embedded
into a commutative diagram with exact rows and columns
$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{2}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{1}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{0}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{2}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{1}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{0}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime\prime}.}$
###### 12.9 Remark.
All the columns except the rightmost one are split exact. However, the
morphisms $P_{n+1}\to P_{n}$ are _not_ the sums of the morphisms
$P_{n+1}^{\prime}\to P_{n}^{\prime}$ and $P_{n+1}^{\prime\prime}\to
P_{n}^{\prime\prime}$. This only happens in the trivial case that the sequence
$A^{\prime}\operatorname{\rightarrowtail}A\operatorname{\twoheadrightarrow}A^{\prime\prime}$
is already split exact.
* Proof.
This is an easy application of the five lemma 3.2 and the $3\times 3$-lemma
3.6. By lifting the morphism
$\varepsilon^{\prime\prime}:P_{0}^{\prime\prime}\to A^{\prime\prime}$ over the
admissible epic $A\operatorname{\twoheadrightarrow}A^{\prime\prime}$ we obtain
a morphism $\varepsilon:P_{0}\to A$ and a commutative diagram
$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{\varepsilon^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{0}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon^{\prime}}$$\scriptstyle{\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{\varepsilon}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\scriptstyle{\left[\begin{smallmatrix}0&1\end{smallmatrix}\right]}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{\varepsilon^{\prime\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{0}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon^{\prime\prime}}$$\textstyle{A^{\prime\prime}.}$
It follows from the five lemma that $\varepsilon$ is actually an admissible
epic, so its kernel exists. The two vertical dotted morphisms exist since the
second and the third column are short exact. Now the $3\times 3$-lemma implies
that the dotted column is short exact. Finally note that $P_{1}^{\prime}\to
P_{0}^{\prime}$ and $P_{1}^{\prime\prime}\to P_{0}^{\prime\prime}$ factor over
admissible epics to $\operatorname{Ker}{\varepsilon^{\prime}}$ and
$\operatorname{Ker}{\varepsilon^{\prime\prime}}$ and proceed by induction. ∎
###### 12.10 Remark.
In concrete situations it may be useful to remember that only the projectivity
of $P_{n}^{\prime\prime}$ is used in the proof.
###### 12.11 Remark (Classical Derived Functors).
Using the results of this section, the theory of classical derived functors,
see e.g. Cartan-Eilenberg [14], Mac Lane [43], Hilton-Stammbach [28] or Weibel
[59], is easily adapted to the following situation:
Let $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ be an exact
category with enough projectives and let
$F:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{B}}$ be an additive
functor to an abelian category. By the resolution lemma 12.2 a projective
resolution $P_{\bullet}\operatorname{\twoheadrightarrow}A$ exists for every
object $A\in\operatorname{\mathscr{A}}$ and is well-defined up to homotopy
equivalence by the comparison theorem (Corollary 12.5). It follows that for
two projective resolutions $P_{\bullet}\operatorname{\twoheadrightarrow}A$ and
$Q_{\bullet}\operatorname{\twoheadrightarrow}A$ the complexes $F(P_{\bullet})$
and $F(Q_{\bullet})$ are chain homotopy equivalent. Therefore it makes sense
to define the _left derived functors_ of $F$ as
$L_{i}F(A):=H_{i}(F(P_{\bullet})).$
Let us indicate why $L_{i}F(A)$ is a functor. First observe that a morphism
$f:A\to A^{\prime}$ extends uniquely up to chain homotopy equivalence to a
chain map $f_{\bullet}:P_{\bullet}\to P_{\bullet}^{\prime}$ if
$P_{\bullet}\operatorname{\twoheadrightarrow}A$ and
$P_{\bullet}^{\prime}\operatorname{\twoheadrightarrow}A^{\prime}$ are
projective resolutions of $A$ and $A^{\prime}$. From this uniqueness it
follows easily that $L_{i}F(fg)=L_{i}F(f)L_{i}F(g)$ and
$L_{i}F(1_{A})=1_{L_{i}F(A)}$ as desired. Using the horseshoe lemma 12.8 one
proves that a short exact sequence
$A^{\prime}\operatorname{\rightarrowtail}A\operatorname{\twoheadrightarrow}A^{\prime\prime}$
yields a long exact sequence
$\cdots\to L_{i+1}F(A^{\prime\prime})\to L_{i}F(A^{\prime})\to L_{i}F(A)\to
L_{i}F(A^{\prime\prime})\to L_{i-1}F(A^{\prime})\to\cdots$
and that $L_{0}F$ sends exact sequences to right exact sequences in
$\operatorname{\mathscr{B}}$ so that the $L_{i}F$ are a universal
$\delta$-functor. Moreover, $L_{0}F$ is characterized by being the best right
exact approximation to $F$ and the $L_{i}F$ measure the failure of $L_{0}F$ to
be exact. In particular, if $F$ sends exact sequences to right exact sequences
then $L_{0}F\cong F$ and if $F$ is exact, then in addition $L_{i}F=0$ if
$i>0$.
###### 12.12 Remark.
By the discussion in section 10.6, the assumption that
$(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ has enough
projectives is unnecessarily restrictive. In order for the classical left
derived functor of $F:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{B}}$
to exist, it suffices to assume that there is a fully exact subcategory
$\operatorname{\mathscr{C}}\subset\operatorname{\mathscr{A}}$ satisfying the
duals of the conditions in Theorem 10.22 with the additional property that $F$
restricted to $\operatorname{\mathscr{C}}$ is exact (see Lemma 10.26). These
conditions ensure that the total derived functor
$\operatorname{\mathbf{L}}\\!F:\operatorname{\mathbf{D}}^{-}{(\operatorname{\mathscr{A}})}\to\operatorname{\mathbf{D}}^{-}{(\operatorname{\mathscr{B}})}$
exists and thus it makes sense to define
$L_{i}F(A)=H_{i}{(\operatorname{\mathbf{L}}\\!F(A))}$, where the object
$A\in\operatorname{\mathscr{A}}$ is considered as a complex concentrated in
degree zero. More explicitly, choose a $\operatorname{\mathscr{C}}$-resolution
$C_{\bullet}\to A$ and let $L_{i}F(A):=H_{i}{(F(C_{\bullet}))}$. It is not
difficult to check that the $L_{i}F$ are a universal $\delta$-functor: They
form a $\delta$-functor as $\operatorname{\mathbf{L}}\\!F$ is a triangle
functor and
$H_{\ast}:\operatorname{\mathbf{D}}^{-}{(\operatorname{\mathscr{B}})}\to\operatorname{\mathscr{B}}$
sends distinguished triangles to long exact sequences; this $\delta$-functor
is universal because it is effaçable, as $L_{i}F(C)=0$ for $i>0$.
###### 12.13 Exercise (Heller [26, 6.3, 6.5]).
Let $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ be an exact
category and consider the exact category
$(\operatorname{\mathscr{E}},\operatorname{\mathscr{F}})$ as described in
Exercise 3.9. Prove that an exact sequence
$P^{\prime}\operatorname{\rightarrowtail}P\operatorname{\twoheadrightarrow}P^{\prime\prime}$
is projective in $(\operatorname{\mathscr{E}},\operatorname{\mathscr{F}})$ if
$P^{\prime}$ and $P^{\prime\prime}$ (and hence $P$) are projective in
$(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$. If
$(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ has enough
projectives then so has
$(\operatorname{\mathscr{E}},\operatorname{\mathscr{F}})$ and every projective
is of the form described before.
## 13 Examples and Applications
It is of course impossible to give an exhaustive list of examples. We simply
list some of the popular ones.
### 13.1 Additive Categories
Every additive category $\operatorname{\mathscr{A}}$ is exact with respect to
the class $\operatorname{\mathscr{E}}_{\min}$ of split exact sequences, i.e.,
the sequences isomorphic to
$A\xrightarrow{\left[\begin{smallmatrix}1\\\ 0\end{smallmatrix}\right]}A\oplus
B\xrightarrow{\left[\begin{smallmatrix}0&1\end{smallmatrix}\right]}B$
for $A,B\in\operatorname{\mathscr{A}}$. Every object
$A\in\operatorname{\mathscr{A}}$ is both projective and injective with respect
to this exact structure.
### 13.2 Quasi-Abelian Categories
We have seen in Section 4 that quasi-abelian categories are exact with respect
to the class $\operatorname{\mathscr{E}}_{\max}$ of all kernel-cokernel pairs.
Evidently, this class of examples includes in particular all abelian
categories. There is an abundance of non-abelian quasi-abelian categories
arising in functional analysis:
###### 13.1 Example (Cf. e.g. [11, IV.2]).
Let $\operatorname{\mathbf{Ban}}$ be the category of Banach spaces and bounded
linear maps over the field $k$ of real or complex numbers. It has kernels and
cokernels—the cokernel of a morphism $f:A\to B$ is given by
$B/\overline{\operatorname{Im}{f}}$. It is an easy consequence of the open
mapping theorem that $\operatorname{\mathbf{Ban}}$ is quasi-abelian. Notice
that the forgetful functor
$\operatorname{\mathbf{Ban}}\to\operatorname{\mathbf{Ab}}$ is exact and
reflects exactness, it preserves monics but fails to preserve epics (morphisms
with dense range). The ground field $k$ is projective and by Hahn-Banach it
also is injective. More generally, it is easy to see that for each set $S$ the
space $\ell^{1}{(S)}$ is projective and $\ell^{\infty}{(S)}$ is injective.
Since every Banach space $A$ is isometrically isomorphic to a quotient of
$\ell^{1}{(B_{\leq 1}{A})}$ and to a subspace of $\ell^{\infty}{(B_{\leq
1}A^{\ast})}$ there are enough of both, projective and injective objects in
$\operatorname{\mathbf{Ban}}$.
###### 13.2 Example.
Let $\operatorname{\mathbf{Fre}}$ be the category of completely metrizable
topological vector spaces (Fréchet spaces) and continuous linear maps. Again,
$\operatorname{\mathbf{Fre}}$ is quasi-abelian by the open mapping theorem
(the proof of Theorem 2.3.3 in Chapter IV.2 of [11] applies _mutatis
mutandis_), and there are exact functors
$\operatorname{\mathbf{Ban}}\to\operatorname{\mathbf{Fre}}$ and
$\operatorname{\mathbf{Fre}}\to\operatorname{\mathbf{Ab}}$. It is still true
that $k$ is projective, but $k$ fails to be injective (Hahn-Banach breaks
down).
###### 13.3 Example.
Consider the category $\operatorname{\mathbf{Pol}}$ of polish abelian groups
(i.e., second countable and completely metrizable topological groups) and
continuous homomorphisms. From the open mapping theorem—which is a standard
consequence of Pettis’ theorem (cf. e.g. [35, (9.9), p. 61]) stating that for
a non-meager set $A$ in $G$ the set $A^{-1}A$ is a neighborhood of the
identity—it follows that $\operatorname{\mathbf{Pol}}$ is quasi-abelian (again
one easily adapts the proof of Theorem 2.3.3 in Chapter IV.2 of [11]).
Further functional analytic examples are discussed in detail e.g. in Rump [49]
and Schneiders [54]. Rump [52] gives a rather long list of examples.
### 13.3 Fully Exact Subcategories
Recall from section 10.5 that a _fully exact subcategory_
$\operatorname{\mathscr{B}}$ of an exact category $\operatorname{\mathscr{A}}$
is a full subcategory $\operatorname{\mathscr{B}}$ which is closed under
extensions and equipped with the exact structure formed by the sequences which
are exact in $\operatorname{\mathscr{A}}$ (see Lemma 10.20).
###### 13.4 Example.
By the embedding theorem A.1, every small exact category is a fully exact
subcategory of an abelian category.
###### 13.5 Example.
The full subcategories of projective or injective objects of an exact category
$\operatorname{\mathscr{A}}$ are fully exact. The induced exact structures are
the split exact structures.
###### 13.6 Example.
Let $\operatorname{\widehat{\otimes}}$ be the projective tensor product of
Banach spaces. A Banach space $F$ is _flat_ if
$F\operatorname{\widehat{\otimes}}{-}$ is exact. It is well-known that the
flat Banach spaces are precisely the $\operatorname{\mathscr{L}}_{1}$-spaces
of Lindenstrauss-Pełczyński. The category of flat Banach spaces is a fully
exact subcategory of $\operatorname{\mathbf{Ban}}$. The exact structure is the
pure exact structure consisting of the short sequences whose Banach dual
sequences are split exact, see [11, Ch. IV.2] for further information and
references.
### 13.4 Frobenius Categories
An exact category is said to be _Frobenius_ provided that it has enough
projectives and injectives and, moreover, the classes of projectives and
injectives coincide [27]. Frobenius categories $\operatorname{\mathscr{A}}$
give rise to _algebraic triangulated categories_ (see [39, 3.6]) by passing to
the _stable category_ $\underline{\operatorname{\mathscr{A}}}$ of
$\operatorname{\mathscr{A}}$. By definition,
$\underline{\operatorname{\mathscr{A}}}$ is the category consisting of the
same objects as $\operatorname{\mathscr{A}}$ and in which a morphism of
$\operatorname{\mathscr{A}}$ is identified with zero if it factors over an
injective object. It is not hard to prove that
$\underline{\operatorname{\mathscr{A}}}$ is additive and it has the structure
of a triangulated category as follows:
The translation functor is obtained by choosing for each object $A$ a short
exact sequence
$A\operatorname{\rightarrowtail}I(A)\operatorname{\twoheadrightarrow}\Sigma(A)$
where $I(A)$ is injective. The assignment $A\mapsto\Sigma(A)$ induces an auto-
equivalence of $\underline{\operatorname{\mathscr{A}}}$. Given a morphism
$f:A\to B$ in $\operatorname{\mathscr{A}}$ consider the push-out diagram
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$PO$\textstyle{I(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C(f)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma(A)}$
and call the sequence $A\to B\to C(f)\to\Sigma(A)$ a _standard triangle_. The
class $\Delta$ of _distinguished triangles_ consists of the triangles which
are isomorphic in $\underline{\operatorname{\mathscr{A}}}$ to (the image of) a
standard triangle.
###### 13.7 Theorem (Happel [24, 2.6, p.16]).
The triple $(\underline{\operatorname{\mathscr{A}}},\Sigma,\Delta)$ is a
triangulated category.
###### 13.8 Example.
Consider the category
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$ of complexes over
the additive category $\operatorname{\mathscr{A}}$ equipped with the
degreewise split exact sequences. It turns out that
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$ is a _Frobenius
category_. The complex $I(A)$ introduced in the proof of Lemma 10.7 is
injective. It is not hard to verify that the stable category
$\underline{\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}}$
coincides with the homotopy category
$\operatorname{\mathbf{K}}{(\operatorname{\mathscr{A}})}$ and that the
triangulated structure provided by Happel’s theorem 13.7 is the same as the
one mentioned in Remark 9.8 (see also Exercise 10.8).
The reader may consult Happel [24] for further information, examples and
applications.
### 13.5 Further Examples
###### 13.9 Example (Vector bundles).
Let $X$ be a scheme. The category of algebraic vector bundles over $X$, i.e.,
the category of locally free and coherent
$\operatorname{\mathscr{O}}_{X}$-modules, is an exact category with the exact
structure consisting of the locally split short exact sequences.
###### 13.10 Example (Chain complexes).
If $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ is an exact
category then the category of chain complexes
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{A}})}$ is an exact
category with respect to the exact structure
$\operatorname{\mathbf{Ch}}{(\operatorname{\mathscr{E}})}$ of short sequences
of complexes which are exact in each degree, see Lemma 9.1.
###### 13.11 Example (Diagram Categories).
Let $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ be an exact
category and let $\operatorname{\mathscr{D}}$ be a small category. The
category $\operatorname{\mathscr{A}}^{\operatorname{\mathscr{D}}}$ of functors
$\operatorname{\mathscr{D}}\to\operatorname{\mathscr{A}}$ is an exact category
with the exact structure
$\operatorname{\mathscr{E}}^{\operatorname{\mathscr{D}}}$. The verification of
the axioms of an exact category for
$(\operatorname{\mathscr{A}}^{\operatorname{\mathscr{D}}},\operatorname{\mathscr{E}}^{\operatorname{\mathscr{D}}})$
is straightforward, as limits and colimits in
$\operatorname{\mathscr{A}}^{\operatorname{\mathscr{D}}}$ are formed
pointwise, see e.g. Borceux [5, 2.15.1, p. 87].
###### 13.12 Example (Filtered Objects).
Let $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ be an exact
category. A (bounded) _filtered object_ $A$ in $\operatorname{\mathscr{A}}$ is
a sequence of admissible monics in $\operatorname{\mathscr{A}}$
$A=(\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.75pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}\crcr}{\immediate}}}{\immediate}{\immediate}\ignorespaces{\immediate}{\immediate}\ignorespaces}\immediate\immediate\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-6.75pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
4.25pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\hbox{{}{\hbox{\kern 5.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}}}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
30.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{A^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
45.11133pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\hbox{{}{\hbox{\kern 5.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}}}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
54.20506pt\raise 6.42223pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.05556pt\hbox{$\scriptstyle{i^{n}_{A}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 71.61133pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
71.61133pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{A^{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
93.12822pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\hbox{{}{\hbox{\kern 5.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}}}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
119.6282pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
119.6282pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\cdots}$}}}}}}}}}}}\ignorespaces)$
such that $A^{n}=0$ for $n\ll 0$ and that $i^{n}_{A}$ is an isomorphism for
$n\gg 0$. A _morphism_ $f$ from the filtered object $A$ to the filtered object
$B$ is a collection of morphisms $f^{n}:A^{n}\to B^{n}$ in
$\operatorname{\mathscr{A}}$ satisfying $f^{n+1}i^{n}_{A}=i^{n}_{B}f^{n}$.
Thus there is a category
$\operatorname{\mathscr{F}}\\!\operatorname{\mathscr{A}}$ of filtered objects.
It follows from Proposition 2.9 that
$\operatorname{\mathscr{F}}\\!\operatorname{\mathscr{A}}$ is additive. The
$3\times 3$-lemma 3.6 implies that the class
$\operatorname{\mathscr{F}}\\!\operatorname{\mathscr{E}}$ consisting of the
pairs of morphisms $(i,p)$ of
$\operatorname{\mathscr{F}}\\!\operatorname{\mathscr{A}}$ such that
$(i^{n},p^{n})$ is in $\operatorname{\mathscr{E}}$ for each $n$ is an exact
structure on $\operatorname{\mathscr{F}}\\!\operatorname{\mathscr{A}}$. Notice
that for a nonzero abelian category $\operatorname{\mathscr{A}}$ the category
of filtered objects $\operatorname{\mathscr{F}}\\!\operatorname{\mathscr{A}}$
is _not_ abelian.
###### 13.13 Example.
Paul Balmer [2] (following Knebusch) gives the following definition: An _exact
category with duality_ is a triple $(\operatorname{\mathscr{A}},\ast,\varpi)$
consisting of an exact category $\operatorname{\mathscr{A}}$, a contravariant
and exact endofunctor $\ast$ on $\operatorname{\mathscr{A}}$ together with a
natural isomorphism
$\varpi:\operatorname{id}_{\operatorname{\mathscr{A}}}\Rightarrow\ast\circ\ast$
satisfying $\varpi_{M}^{\ast}\varpi_{M^{\ast}}=\operatorname{id}_{M^{\ast}}$
for all $M\in\operatorname{\mathscr{A}}$. There are natural notions of
symmetric spaces and isometries of symmetric spaces, (admissible) Lagrangians
of a symmetric space and hence of metabolic spaces If
$\operatorname{\mathscr{A}}$ is essentially small it makes sense to speak of
the set ${\rm MW}(\operatorname{\mathscr{A}},\ast,\varpi)$ of isometry classes
of symmetric spaces and the subset ${\rm
NW}{(\operatorname{\mathscr{A}},\ast,\varpi)}$ of isometry classes of
metabolic spaces and both turn out to be abelian monoids with respect to the
_orthogonal sum_ of symmetric spaces. The _Witt group_ is ${\rm
W}{(\operatorname{\mathscr{A}},\ast,\varpi)}={\rm
MW}{(\operatorname{\mathscr{A}},\ast,\varpi)}/{\rm
NW}{(\operatorname{\mathscr{A}},\ast,\varpi)}$. In case
$\operatorname{\mathscr{A}}$ is the category of vector bundles over a scheme
$(X,\operatorname{\mathscr{O}}_{X})$ and
$\ast=\operatorname{Hom}_{\operatorname{\mathscr{O}}_{X}}({-},\operatorname{\mathscr{O}}_{X})$
is the usual duality functor, one obtains the classical Witt group of a
scheme.
Extending these considerations to the level of the derived category leads to
_Balmer’s triangular Witt groups_ which had a number of striking applications
to the theory of quadratic forms and $K$-theory, we refer the interested
reader to Balmer’s survey [2]. For a beautiful link to algebraic $K$-theory we
refer to Schlichting [53].
### 13.6 Higher Algebraic $K$-Theory
Let $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ be a small exact
category. The _Grothendieck group_
$K_{0}(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ is defined to
be the quotient of the free (abelian) group generated by the isomorphism
classes of objects of $\operatorname{\mathscr{A}}$ modulo the relations
$[A]=[A^{\prime}][A^{\prime\prime}]$ for each short exact sequence
$A^{\prime}\operatorname{\rightarrowtail}A\operatorname{\twoheadrightarrow}A^{\prime\prime}$
in $\operatorname{\mathscr{E}}$. This generalizes the $K$-theory of a ring,
where $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ is taken to be
the category of finitely generated projective modules over $R$ with the split
exact structure. If $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$
is the category of algebraic vector bundles over a scheme $X$ then by
definition $K_{0}{(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})}$ is
the (naïve) Grothendieck group $K_{0}{(X)}$ of the scheme (cf. [57, 3.2, p.
313]).
Quillen’s landmark paper [50] introduces today’s definition of higher
algebraic $K$-theory and proves its basic properties. Exact categories enter
via the $Q$-construction, which we outline briefly. Given a small exact
category $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ one forms a
new category $Q\operatorname{\mathscr{A}}$: The objects of
$Q\operatorname{\mathscr{A}}$ are the objects of $\operatorname{\mathscr{A}}$
and $\operatorname{Hom}_{Q\operatorname{\mathscr{A}}}(A,A^{\prime})$ is
defined to be the set of equivalence classes of diagrams
$\textstyle{A}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{i}$$\textstyle{A^{\prime},}$
in which $p$ is an admissible epic and $i$ is an admissible monic, where two
diagrams are considered equivalent if there is an isomorphism of such diagrams
inducing the identity on $A$ and $A^{\prime}$. The composition of two
morphisms $(p,i)$, $(p^{\prime},i^{\prime})$ is given by the following
construction: form the pull-back over $p^{\prime}$ and $i$ so that by
Proposition 2.15 there is a diagram
$\textstyle{B^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{j^{\prime}}$PB$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{i}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$$\scriptstyle{i^{\prime}}$$\textstyle{A}$$\textstyle{A^{\prime}}$$\textstyle{A^{\prime\prime}}$
and put $(p^{\prime},i^{\prime})\circ(p,i)=(pq,i^{\prime}j^{\prime})$. This is
easily checked to yield a category and it is not hard to make sense of the
statement that the morphisms $A\to A^{\prime}$ in
$Q\operatorname{\mathscr{A}}$ correspond to the different ways that $A$ arises
as an admissible subquotient of $A^{\prime}$.
Now any small category $\operatorname{\mathscr{C}}$ gives rise to a simplicial
set $N\operatorname{\mathscr{C}}$, called the _nerve of
$\operatorname{\mathscr{C}}$_ whose $n$ simplices are given by sequences of
composable morphisms
$C_{0}\to C_{1}\to\cdots\to C_{n},$
where the $i$-th face map is obtained by deleting the object $C_{i}$ and the
$i$-th degeneracy map is obtained by replacing $C_{i}$ by $1_{C_{i}}:C_{i}\to
C_{i}$. The _classifying space_ $B\operatorname{\mathscr{C}}$ of
$\operatorname{\mathscr{C}}$ is the geometric realization of the nerve
$N\operatorname{\mathscr{C}}$.
Quillen proves the fundamental result that
$K_{0}(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})\cong\pi_{1}(B(Q\operatorname{\mathscr{A}}),0)$
which motivates the definition
$K_{n}{(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})}:=\pi_{n+1}(B(Q\operatorname{\mathscr{A}}),0).$
Obviously, an exact functor
$F:(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})\to(\operatorname{\mathscr{A}}^{\prime},\operatorname{\mathscr{E}}^{\prime})$
yields a functor $Q\operatorname{\mathscr{A}}\to
Q\operatorname{\mathscr{A}}^{\prime}$ and hence a homomorphism
$F_{\ast}:K_{n}{(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})}\to
K_{n}(\operatorname{\mathscr{A}}^{\prime},\operatorname{\mathscr{E}}^{\prime})$
which is easily seen to depend only on the isomorphism class of $F$.
We do not discuss $K$-theory any further and recommend the lecture of
Quillen’s original article [50] and Srinivas’s book [56] expanding on
Quillen’s article. For a good overview over many topics of current interest we
refer to the handbook of $K$-theory [20].
## Appendix A The Embedding Theorem
For abelian categories, one has the Freyd-Mitchell embedding theorem, see [17]
and [46], allowing one to prove diagram lemmas in abelian categories “by
chasing elements”. In order to prove diagram lemmas in exact categories, a
similar technique works. More precisely, one has:
###### A.1 Theorem ([57, A.7.1, A.7.16]).
Let $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ be a small exact
category.
1. (i)
There is an abelian category $\operatorname{\mathscr{B}}$ and a fully faithful
exact functor $i:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{B}}$ that
reflects exactness. Moreover, $\operatorname{\mathscr{A}}$ is closed under
extensions in $\operatorname{\mathscr{B}}$.
2. (ii)
The category $\operatorname{\mathscr{B}}$ may canonically be chosen to be the
category of left exact functors
$\operatorname{\mathscr{A}}^{\operatorname{op}}\to\operatorname{\mathbf{Ab}}$
and $i$ to be the Yoneda embedding
$i(A)=\operatorname{Hom}_{\operatorname{\mathscr{A}}}{({-},A)}$.
3. (iii)
Assume moreover that $\operatorname{\mathscr{A}}$ is weakly idempotent
complete. If $f$ is a morphism in $\operatorname{\mathscr{A}}$ and $i(f)$ is
epic in $\operatorname{\mathscr{B}}$ then $f$ is an admissible epic.
###### A.2 Remark.
In order for (iii) to hold it is necessary to assume weak idempotent
completeness of $\operatorname{\mathscr{A}}$. Indeed, if
$\operatorname{\mathscr{A}}$ fails to be weakly idempotent complete, there
must be a retraction $r$ without kernel. By definition there exists $s$ such
that $rs=1$, but then $i(rs)$ is epic, so $i(r)$ is epic as well.
###### A.3 Remark.
Let $\operatorname{\mathscr{B}}$ be an abelian category and assume that
$\operatorname{\mathscr{A}}$ is a full subcategory which is closed under
extensions, i.e., $\operatorname{\mathscr{A}}$ is fully exact subcategory of
$\operatorname{\mathscr{B}}$ in the sense of Definition 10.21. Then, by Lemma
10.20, $\operatorname{\mathscr{A}}$ is an exact category with respect to the
class $\operatorname{\mathscr{E}}$ of short sequences in
$\operatorname{\mathscr{A}}$ which are exact in $\operatorname{\mathscr{B}}$.
This is a basic recognition principle of exact categories, for many examples
arise in this way. The embedding theorem provides a partial converse to this
recognition principle.
###### A.4 Remark.
Quillen states in [50, p. “92/16/100”]:
> Now suppose given an exact category $\operatorname{\mathscr{M}}$. Let
> $\operatorname{\mathscr{A}}$ be the additive category of additive
> contravariant functors from $\operatorname{\mathscr{M}}$ to abelian groups
> which are left exact, i.e. carry [an exact sequence
> $M^{\prime}\operatorname{\rightarrowtail}M\operatorname{\twoheadrightarrow}M^{\prime\prime}$]
> to an exact sequence
>
> $0\to F(M^{\prime\prime})\to F(M)\to F(M^{\prime}).$
>
> (Precisely, choose a universe containing $\operatorname{\mathscr{M}}$, and
> let $\operatorname{\mathscr{A}}$ be the category of left exact functors
> whose values are abelian groups in the universe.) Following well-known ideas
> (e.g. [22]), one can prove $\operatorname{\mathscr{A}}$ is an abelian
> category, that the Yoneda functor $h$ embeds $\operatorname{\mathscr{M}}$ as
> a full subcategory of $\operatorname{\mathscr{A}}$ closed under extensions,
> and finally that a [short] sequence […] is in $\operatorname{\mathscr{E}}$
> if and only if $h$ carries it into an exact sequence in
> $\operatorname{\mathscr{A}}$. The details will be omitted, as they are not
> really important for the sequel.
Freyd stated a similar theorem in [16], again without proof, and with the
additional assumption of idempotents completeness, since he uses Heller’s
axioms. The first proof published is in Laumon [42, 1.0.3], relying on the
Grothendieck-Verdier theory of sheafification [55]. The sheafification
approach was also used and further refined by Thomason [57, Appendix A]. A
quite detailed sketch of the proof alluded to by Quillen is given in Keller
[36, A.3].
The proof given here is due to Thomason [57, A.7] amalgamated with the proof
in Laumon [42, 1.0.3]. We also take the opportunity to fix a slight gap in
Thomason’s argument (our Lemma A.10, compare with the first sentence after
[57, (A.7.10)]). Since Thomason fails to spell out the nice sheaf-theoretic
interpretations of his construction and since referring to SGA 4 seems rather
brutal, we use the terminology of the more lightweight Mac Lane-Moerdijk [45,
Chapter III]. Other good introductions to the theory of sheaves may be found
in Artin [1] or Borceux [7], for example.
### A.1 Separated Presheaves and Sheaves
Let $(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ be a small exact
category. For each object $A\in\operatorname{\mathscr{A}}$, let
$\operatorname{\mathscr{C}}_{A}=\\{(p^{\prime}:A^{\prime}\operatorname{\twoheadrightarrow}A)\,:\,A^{\prime}\in\operatorname{\mathscr{A}}\\}$
be the set of admissible epics onto $A$. The elements of
$\operatorname{\mathscr{C}}_{A}$ are the _coverings_ of $A$.
###### A.5 Lemma.
The family
$\\{\operatorname{\mathscr{C}}_{A}\\}_{A\in\operatorname{\mathscr{A}}}$ is a
_basis for a Grothendieck topology_ $J$ on $\operatorname{\mathscr{A}}$:
1. (i)
If $f:A\to B$ is an isomorphism then $f\in\operatorname{\mathscr{C}}_{B}$.
2. (ii)
If $g:A\to B$ is arbitrary and
$(q^{\prime}:B^{\prime}\operatorname{\twoheadrightarrow}B)\in\operatorname{\mathscr{C}}_{B}$
then the pull-back
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$PB$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q^{\prime}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{B}$
yields a morphism $p^{\prime}\in\operatorname{\mathscr{C}}_{A}$. _(“Stability
under base-change”)_
3. (iii)
If $(p:B\operatorname{\twoheadrightarrow}A)\in\operatorname{\mathscr{C}}_{A}$
and $(q:C\operatorname{\twoheadrightarrow}B)\in\operatorname{\mathscr{C}}_{B}$
then $pq\in\operatorname{\mathscr{C}}_{A}$. _(“Transitivity”)_
In particular, $(\operatorname{\mathscr{A}},J)$ is a _site_.
* Proof.
This is obvious from the definition, see [45, Definition 2, p. 111]. ∎
The Yoneda functor
$y:\operatorname{\mathscr{A}}\to\operatorname{\mathbf{Ab}}^{\operatorname{\mathscr{A}}^{\operatorname{op}}}$
associates to each object $A\in\operatorname{\mathscr{A}}$ the presheaf (of
abelian groups)
$y(A)=\operatorname{Hom}_{\operatorname{\mathscr{A}}}{({-},A)}$. In general, a
_presheaf_ is the same thing as a functor
$G:\operatorname{\mathscr{A}}^{\operatorname{op}}\to\operatorname{\mathbf{Ab}}$,
which we will assume to be additive except in the next lemma. We will see
shortly that $y(A)$ is in fact a _sheaf_ on the site
$(\operatorname{\mathscr{A}},J)$.
###### A.6 Lemma.
Consider the site $(\operatorname{\mathscr{A}},J)$ and let
$G:\operatorname{\mathscr{A}}^{\operatorname{op}}\to\operatorname{\mathbf{Ab}}$
be a functor.
1. (i)
The presheaf $G$ is _separated_ if and only if for each admissible epic $p$
the morphism $G(p)$ is monic.
2. (ii)
The presheaf $G$ is a _sheaf_ if and only if for each admissible epic
$p:A\operatorname{\twoheadrightarrow}B$ the diagram
$\textstyle{G(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G(p)}$$\textstyle{G(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{0}=G(p_{0})}$$\scriptstyle{d^{1}=G(p_{1})}$$\textstyle{G(A\times_{B}A)}$
is an _equalizer_ (difference kernel), where
$p_{0},p_{1}:A\times_{B}A\operatorname{\twoheadrightarrow}A$ denote the two
projections. In other words, the presheaf $G$ is a sheaf if and only if for
all admissible epics $p:A\operatorname{\twoheadrightarrow}B$ the diagram
$\textstyle{G(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G(p)}$$\scriptstyle{G(p)}$$\textstyle{G(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{1}}$$\textstyle{G(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{0}}$$\textstyle{G(A\times_{B}A)}$
is a pull-back.
* Proof.
Again, this is obtained by making the definitions explicit. Point (i) is the
definition, [45, p. 129], and point (ii) is [45, Proposition 1[bis], p. 123].
∎
The following lemma shows that the sheaves on the site
$(\operatorname{\mathscr{A}},J)$ are quite familiar gadgets.
###### A.7 Lemma.
Let
$G:\operatorname{\mathscr{A}}^{\operatorname{op}}\to\operatorname{\mathbf{Ab}}$
be an additive functor. The following are equivalent:
1. (i)
The presheaf $G$ is a sheaf on the site $(\operatorname{\mathscr{A}},J)$.
2. (ii)
For each admissible epic $p:B\operatorname{\twoheadrightarrow}C$ the sequence
$0\xrightarrow{}G(C)\xrightarrow{G(p)}G(B)\xrightarrow{d^{0}-d^{1}}G(B\times_{C}B)$
is exact.
3. (iii)
For each short exact sequence
$A\operatorname{\rightarrowtail}B\operatorname{\twoheadrightarrow}C$ in
$\operatorname{\mathscr{A}}$ the sequence
$0\xrightarrow{}G(C)\xrightarrow{}G(B)\xrightarrow{}G(A)$
is exact, i.e., $G$ is _left exact_.
* Proof.
By Lemma A.6 (ii) we have that $G$ is a sheaf if and only if the sequence
$0\xrightarrow{}G(C)\xrightarrow{\left[\begin{smallmatrix}G(p)\\\
G(p)\end{smallmatrix}\right]}G(B)\oplus
G(B)\xrightarrow{\left[\begin{smallmatrix}G(p_{0})&&-G(p_{1})\end{smallmatrix}\right]}G(B\times_{C}B)$
is exact. Since $p_{1}:B\times_{C}B\operatorname{\twoheadrightarrow}B$ is a
split epic with kernel $A$, there is an isomorphism $B\times_{C}B\to A\oplus
B$ and it is easy to check that the above sequence is isomorphic to
$0\xrightarrow{}G(C)\xrightarrow{}G(B)\oplus G(B)\xrightarrow{}G(A)\oplus
G(B).$
Because left exact sequences are stable under taking direct sums and passing
to direct summands, (i) is equivalent to (iii). That (i) is equivalent to (ii)
is obvious by Lemma A.6 (ii). ∎
###### A.8 Corollary ([57, A.7.6]).
The represented functor
$y(A)=\operatorname{Hom}_{\operatorname{\mathscr{A}}}{({-},A)}$ is a sheaf for
every object $A$ of $\operatorname{\mathscr{A}}$.
* Proof.
Given an exact sequence
$B^{\prime}\operatorname{\rightarrowtail}B\operatorname{\twoheadrightarrow}B^{\prime\prime}$
we need to prove that
$0\xrightarrow{}\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(B^{\prime\prime},A)}\xrightarrow{}\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(B,A)}\xrightarrow{}\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(B^{\prime},A)}$
is exact. That the sequence is exact at
$\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(B,A)}$ follows from the fact
that $B\operatorname{\twoheadrightarrow}B^{\prime\prime}$ is a cokernel of
$B^{\prime}\operatorname{\rightarrowtail}B$. That the sequence is exact at
$\operatorname{Hom}_{\operatorname{\mathscr{A}}}{(B^{\prime\prime},A)}$
follows from the fact that
$B\operatorname{\twoheadrightarrow}B^{\prime\prime}$ is epic. ∎
### A.2 Outline of the Proof
Let now $\operatorname{\mathscr{Y}}$ be the category of additive functors
$\operatorname{\mathscr{A}}^{\operatorname{op}}\to\operatorname{\mathbf{Ab}}$
and let $\operatorname{\mathscr{B}}$ be the category of (additive) sheaves on
the site $(\operatorname{\mathscr{A}},J)$. Let
$j_{\ast}:\operatorname{\mathscr{B}}\to\operatorname{\mathscr{A}}$ be the
inclusion. By Corollary A.8, the Yoneda functor $y$ factors as
$\textstyle{\operatorname{\mathscr{A}}@wrapper{\operatorname{\mathscr{A}}@presentation}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y}$$\scriptstyle{i}$$\textstyle{\operatorname{\mathscr{B}}@wrapper{\operatorname{\mathscr{B}}@presentation}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\ast}}$$\textstyle{\operatorname{\mathscr{Y}}@wrapper{\operatorname{\mathscr{Y}}@presentation}}$
via a functor $i:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{B}}$. We
will prove that the category
$\operatorname{\mathscr{B}}=\operatorname{Sheaves}{(A,J)}$ is abelian and we
will check that the functor $i$ has the properties asserted in the embedding
theorem.
The category $\operatorname{\mathscr{Y}}$ is a Grothendieck abelian category
(there is a generator, small products and coproducts exist and filtered
colimits are exact)—as a functor category, these properties are inherited from
$\operatorname{\mathbf{Ab}}$, as limits and colimits are taken pointwise. The
crux of the proof of the embedding theorem is to show that $j_{\ast}$ has a
left adjoint $j^{\ast}$ such that
$j^{\ast}j_{\ast}=\operatorname{id}_{\operatorname{\mathscr{B}}}$, namely
sheafification. As soon as this is established, the rest will be relatively
painless.
### A.3 Sheafification
The goal of this section is to construct the sheafification functor on the
site $(\operatorname{\mathscr{A}},J)$ and to prove its basic properties. We
will construct an endofunctor
$L:\operatorname{\mathscr{Y}}\to\operatorname{\mathscr{Y}}$ which associates
to each presheaf a separated presheaf and to each separated presheaf a sheaf.
The sheafification functor will then be given by $j^{\ast}=LL$.
We need one more concept from the theory of sites:
###### A.9 Lemma.
Let $A\in\operatorname{\mathscr{A}}$. A covering
$p^{\prime\prime}:A^{\prime\prime}\operatorname{\twoheadrightarrow}A$ is a
_refinement_ of the covering
$p^{\prime}:A^{\prime}\operatorname{\twoheadrightarrow}A$ if and only if there
exists a morphism $a:A^{\prime\prime}\to A^{\prime}$ such that
$p^{\prime}a=p^{\prime\prime}$.
* Proof.
This is the specialization of a _matching family_ as given in [45, p. 121] in
the present context. ∎
By definition, refinement gives the structure of a filtered category on
$\operatorname{\mathscr{C}}_{A}$ for each $A\in\operatorname{\mathscr{A}}$.
More precisely, let $\operatorname{\mathscr{D}}_{A}$ be the following
category: the objects are the coverings
$(p^{\prime}:A^{\prime}\operatorname{\twoheadrightarrow}A)$ and there exists
at most one morphism between any two objects of
$\operatorname{\mathscr{D}}_{A}$: there exists a morphism
$(p^{\prime}:A^{\prime}\operatorname{\twoheadrightarrow}A)\to(p^{\prime\prime}:A^{\prime\prime}\operatorname{\twoheadrightarrow}A)$
in $\operatorname{\mathscr{D}}_{A}$ if and only if there exists
$a:A^{\prime\prime}\to A^{\prime}$ such that $p^{\prime}a=p^{\prime\prime}$.
To see that $\operatorname{\mathscr{D}}_{A}$ is filtered, let
$(p^{\prime}:A^{\prime}\operatorname{\twoheadrightarrow}A)$ and
$(p^{\prime\prime}:A^{\prime\prime}\operatorname{\twoheadrightarrow}A)$ be two
objects and put $A^{\prime\prime\prime}=A^{\prime}\times_{A}A^{\prime\prime}$,
so there is a pull-back diagram
$\textstyle{A^{\prime\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{a^{\prime}}$PB$\textstyle{A^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime\prime}}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$$\textstyle{A.}$
Put $p^{\prime\prime\prime}=p^{\prime}a=p^{\prime\prime}a^{\prime}$, so the
object
$(p^{\prime\prime\prime}:A^{\prime\prime\prime}\operatorname{\twoheadrightarrow}A)$
of $\operatorname{\mathscr{D}}_{A}$ is a common refinement of
$(p^{\prime}:A^{\prime}\operatorname{\twoheadrightarrow}A)$ and
$(p^{\prime\prime}:A^{\prime\prime}\operatorname{\twoheadrightarrow}A)$.
###### A.10 Lemma.
Let $A_{1},A_{2}\in\operatorname{\mathscr{A}}$ be any two objects.
1. (i)
There is a functor
$Q:\operatorname{\mathscr{D}}_{A_{1}}\times\operatorname{\mathscr{D}}_{A_{2}}\to\operatorname{\mathscr{D}}_{A_{1}\oplus
A_{2}},\;(p_{1}^{\prime},p_{2}^{\prime})\mapsto(p_{1}^{\prime}\oplus
p_{2}^{\prime})$.
2. (ii)
Let $(p^{\prime}:A^{\prime}\operatorname{\twoheadrightarrow}A_{1}\oplus
A_{2})$ be an object of $\operatorname{\mathscr{D}}_{A_{1}\oplus A_{2}}$ and
for $i=1,2$ let
$\textstyle{A_{i}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$PB$\scriptstyle{p_{i}^{\prime}}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$$\textstyle{A_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{1}\oplus
A_{2}}$
be a pull-back diagram in which the bottom arrow is the inclusion. This
construction defines a functor
$P:\operatorname{\mathscr{D}}_{A_{1}\oplus
A_{2}}\longrightarrow\operatorname{\mathscr{D}}_{A_{1}}\times\operatorname{\mathscr{D}}_{A_{2}},\quad
p^{\prime}\longmapsto(p_{1}^{\prime},p_{2}^{\prime}).$
3. (iii)
There are a natural transformation
$\operatorname{id}_{\operatorname{\mathscr{D}}_{A_{1}\oplus A_{2}}}\Rightarrow
PQ$ and a natural isomorphism
$QP\cong\operatorname{id}_{\operatorname{\mathscr{D}}_{A_{1}}\times\operatorname{\mathscr{D}}_{A_{2}}}$.
In particular, the images of $P$ and $Q$ are cofinal.
* Proof.
That $P$ is a functor follows from its construction and the universal property
of pull-back diagrams in conjunction with axiom [E2${}^{\operatorname{op}}$].
That $Q$ is well-defined follows from Proposition 2.9 and that
$PQ\cong\operatorname{id}_{\operatorname{\mathscr{D}}_{A_{1}}\times\operatorname{\mathscr{D}}_{A_{2}}}$
is easy to check. That there is a natural transformation
$\operatorname{id}_{\operatorname{\mathscr{D}}_{A_{1}\oplus A_{2}}}\Rightarrow
QP$ follows from the universal property of products. ∎
Let $(p^{\prime\prime}:A^{\prime\prime}\operatorname{\twoheadrightarrow}A)$ be
a refinement of $(p^{\prime}:A^{\prime}\operatorname{\twoheadrightarrow}A)$,
and let $a:A^{\prime\prime}\to A^{\prime}$ be such that
$p^{\prime}a=p^{\prime\prime}$. By the universal property of pull-backs, $a$
yields a unique morphism $A^{\prime\prime}\times_{A}A^{\prime\prime}\to
A^{\prime}\times_{A}A^{\prime}$ which we denote by $a\times_{A}a$. Hence, for
every additive functor
$G:\operatorname{\mathscr{A}}^{\operatorname{op}}\to\operatorname{\mathbf{Ab}}$,
we obtain a commutative diagram in $\operatorname{\mathbf{Ab}}$:
$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{(d^{0}-d^{1})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exists!}$$\textstyle{G(A^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{0}-d^{1}}$$\scriptstyle{G{(a)}}$$\textstyle{G(A^{\prime}\times_{A}A^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G(a\times_{A}a)}$$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{(d^{0}-d^{1})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G(A^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{0}-d^{1}}$$\textstyle{G(A^{\prime\prime}\times_{A}A^{\prime\prime}).}$
The next thing to observe is that the dotted morphism does not depend on the
choice of $a$. Indeed, if $\tilde{a}$ is another morphism such that
$p^{\prime}\tilde{a}=p^{\prime\prime}$, consider the diagram
$\textstyle{A^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{\tilde{a}}$$\scriptstyle{\exists!b}$$\textstyle{A^{\prime}\times_{A}A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}^{\prime}}$$\scriptstyle{p_{0}^{\prime}}$PB$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$$\textstyle{A}$
and $b:A^{\prime\prime}\to A^{\prime}\times_{A}A^{\prime}$ is such that
$G(b)(d^{0}-d^{1})=G(b)G(p_{0}^{\prime})-G(b)G(p_{1}^{\prime})=G(a)-G(\tilde{a}),$
so $G(a)-G(\tilde{a})=0$ on $\operatorname{Ker}{(d^{0}-d^{1})}$.
For
$G:\operatorname{\mathscr{A}}^{\operatorname{op}}\to\operatorname{\mathbf{Ab}}$,
we put $\ell
G(p^{\prime}:A^{\prime}\operatorname{\twoheadrightarrow}A):=\operatorname{Ker}{(G(A^{\prime})\xrightarrow{d^{0}-d^{1}}G(A^{\prime}\times_{A}A^{\prime}))}$
and we have just seen that this defines a functor $\ell
G:\operatorname{\mathscr{D}}_{A}\to\operatorname{\mathbf{Ab}}$.
###### A.11 Lemma.
Define
$LG(A)=\varinjlim_{\operatorname{\mathscr{D}}_{A}}\ell
G(p^{\prime}:A^{\prime}\operatorname{\twoheadrightarrow}A).$
1. (i)
$LG$ is an additive contravariant functor in $A$.
2. (ii)
$L$ is a covariant functor in $G$.
* Proof.
This is immediate from going through the definitions:
To prove (i), let $f:A\to B$ be an arbitrary morphism. Lemma A.5 (ii) shows
that by taking pull-backs we obtain a functor
$\operatorname{\mathscr{D}}_{B}\xrightarrow{f^{\ast}}\operatorname{\mathscr{D}}_{A}$
which, by passing to the colimit, induces a unique morphism
$LG(B)\xrightarrow{LG(f)}LG(A)$ compatible with $f^{\ast}$. From this
uniqueness, we deduce $LG(fg)=LG(g)LG(f)$. The additivity of $LG$ is a
consequence of Lemma A.10.
To prove (ii), let $\alpha:F\Rightarrow G$ be a natural transformation between
two (additive) presheaves. Given an object $A\in\operatorname{\mathscr{A}}$,
we obtain a morphism between the colimit diagrams defining $LF(A)$ and $LG(A)$
and we denote the unique resulting map by $L(\alpha)_{A}$. Given a morphism
$f:A\to B$, there is a commutative diagram
$\textstyle{LF(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L(\alpha)_{B}}$$\scriptstyle{LF(f)}$$\textstyle{LF(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L(\alpha)_{A}}$$\textstyle{LG(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{LG(f)}$$\textstyle{LG(A),}$
as is easily checked. The uniqueness in the definition of $L(\alpha)_{A}$
implies that for each $A\in\operatorname{\mathscr{A}}$ the equation
$L(\alpha\circ\beta)_{A}=L(\alpha)_{A}\circ L(\beta)_{A}$
holds. The reader in need of more details may consult [7, p. 206f]. ∎
###### A.12 Lemma ([57, A.7.8]).
The functor $L:\operatorname{\mathscr{Y}}\to\operatorname{\mathscr{Y}}$ has
the following properties:
1. (i)
It is additive and preserves finite limits.
2. (ii)
There is a natural transformation
$\eta:\operatorname{id}_{\operatorname{\mathscr{Y}}}\Rightarrow L$.
* Proof.
That $L$ preserves finite limits follows from the fact that filtered colimits
and kernels in $\operatorname{\mathbf{Ab}}$ commute with finite limits, as
limits in $\operatorname{\mathscr{Y}}$ are formed pointwise, see also [7,
Lemma 3.3.1]. Since $L$ preserves finite limits, it preserves in particular
finite products, hence it is additive. This settles point (i).
For each
$(p^{\prime}:A^{\prime}\operatorname{\twoheadrightarrow}A)\in\operatorname{\mathscr{D}}_{A}$
the morphism $G(p^{\prime}):G(A)\to G(A^{\prime})$ factors uniquely over
$\tilde{\eta}_{p^{\prime}}:G(A)\to\operatorname{Ker}{(G(A^{\prime})\to
G(A^{\prime}\times_{A}A^{\prime}))}.$
By passing to the colimit over $\operatorname{\mathscr{D}}_{A}$, this induces
a morphism $\tilde{\eta}_{A}:G(A)\to LG(A)$ which is clearly natural in $A$.
In other words, the $\tilde{\eta}_{A}$ yield a natural transformation
$\eta_{G}:G\Rightarrow LG$, i.e., a morphism in $\operatorname{\mathscr{Y}}$.
We leave it to the reader to check that the construction of $\eta_{G}$ is
compatible with natural transformations $\alpha:G\Rightarrow F$ so that the
$\eta_{G}$ assemble to yield a natural transformation
$\eta:\operatorname{id}_{\operatorname{\mathscr{Y}}}\Rightarrow L$, as claimed
in point (ii). ∎
###### A.13 Lemma ([57, A.7.11, (a), (b), (c)]).
Let $G\in\operatorname{\mathscr{Y}}$ and let $A\in\operatorname{\mathscr{A}}$.
1. (i)
For all $x\in LG(A)$ there exists an admissible epic
$p^{\prime}:A^{\prime}\operatorname{\twoheadrightarrow}A$ and $y\in
G(A^{\prime})$ such that $\eta(y)=LG(p^{\prime})(x)$ in $LG(A^{\prime})$.
2. (ii)
For all $x\in G(A)$, we have $\eta(x)=0$ in $LG(A)$ if and only if there
exists an admissible epic
$p^{\prime}:A^{\prime}\operatorname{\twoheadrightarrow}A$ such that
$G(p^{\prime})(x)=0$ in $G(A^{\prime})$.
3. (iii)
We have $LG=0$ if and only if for all $A\in\operatorname{\mathscr{A}}$ and all
$x\in G(A)$ there exists an admissible epic
$p^{\prime}:A^{\prime}\operatorname{\twoheadrightarrow}A$ such that
$G(p^{\prime})(x)=0$.
* Proof.
Points (i) and (ii) are immediate from the definitions. Point (iii) follows
from (i) and (ii). ∎
###### A.14 Lemma ([45, Lemma 2, p. 131], [57, A.7.11, (d), (e)]).
Let $G\in\operatorname{\mathscr{Y}}$.
1. (i)
The presheaf $G$ is separated if and only if $\eta_{G}:G\to LG$ is monic.
2. (ii)
The presheaf $G$ is a sheaf if and only if $\eta_{G}:G\to LG$ is an
isomorphism.
* Proof.
Point (i) follows from Lemma A.13 (ii) and point (ii) follows from the
definitions. ∎
###### A.15 Proposition ([57, A.7.12]).
Let $G\in\operatorname{\mathscr{Y}}$.
1. (i)
The presheaf $LG$ is separated.
2. (ii)
If $G$ is separated then $LG$ is a sheaf.
* Proof.
Let us prove (i) by applying Lemma A.6 (i), so let $x\in LG(A)$ and let
$b:B\operatorname{\twoheadrightarrow}A$ be an admissible epic for which
$LG(b)(x)=0$. We have to prove that then $x=0$ in $LG(A)$. By the definition
of $LG(A)$, we know that $x$ is represented by some
$y\in\operatorname{Ker}{(G(A^{\prime})\xrightarrow{d^{0}-d^{1}}G(A^{\prime}\times_{A}A^{\prime}))}$
for some admissible epic
$(p^{\prime}:A^{\prime}\operatorname{\twoheadrightarrow}A)$ in
$\operatorname{\mathscr{D}}_{A}$. Since $LG(b)(x)=0$ in $LG(B)$, we know that
the image of $y$ in
$\operatorname{Ker}{(G(A^{\prime}\times_{A}B)\xrightarrow{d^{0}-d^{1}}G((A^{\prime}\times_{A}B)\times_{B}(A^{\prime}\times_{A}B)))}$
is equivalent to zero in the filtered colimit over
$\operatorname{\mathscr{D}}_{B}$ defining $LG(B)$. Therefore there exists a
morphism $D\to A^{\prime}\times_{A}B$ in $\operatorname{\mathscr{A}}$ such
that its composite with the projection onto $B$ is an admissible epic
$D\operatorname{\twoheadrightarrow}B$. By Lemma A.13 (ii), it follows that $y$
maps to zero in $G(D)$. Now the composite
$D\operatorname{\twoheadrightarrow}B\operatorname{\twoheadrightarrow}A$ is in
$\operatorname{\mathscr{D}}_{A}$ and hence $y$ is equivalent to zero in the
filtered colimit over $\operatorname{\mathscr{D}}_{A}$ defining $LG(A)$. Thus,
$x=0$ in $LG(A)$ as required.
Let us prove (ii). If $G$ is a separated presheaf, we have to check that for
every admissible epic $B\operatorname{\twoheadrightarrow}A$ the diagram
$\textstyle{LG(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{LG(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{0}=G(p_{0})}$$\scriptstyle{d^{1}=G(p_{1})}$$\textstyle{LG(B\times_{A}B)}$
is a difference kernel. By (i) $LG$ is separated, so $LG(A)\to LG(B)$ is
monic, and it remains to prove that every element $x\in LG(B)$ with
$(d^{0}-d^{1})x=0$ is in the image of $LG(A)$. By Lemma A.13 (i) there is an
admissible epic $q:C\operatorname{\twoheadrightarrow}B$ and $y\in G(C)$ such
that $\eta(y)=LG(q)(x)$. It follows that $\eta G(p_{0})(y)=\eta G(p_{1})(y)$
in $LG(C\times_{A}C)$. Now, $G$ is separated, so $\eta:G\Rightarrow LG$ is
monic by Lemma A.14, and we conclude from this that $G(p_{0})(y)=G(p_{1})(y)$
in $G(C\times_{A}C)$. In other words,
$y\in\operatorname{Ker}{(G(C)\xrightarrow{d^{0}-d^{1}}G(C\times_{A}C))}$
yields a class in $LG(A)$ representing $x$. ∎
###### A.16 Corollary.
For a presheaf $G\in\operatorname{\mathscr{Y}}$ we have $LG=0$ if and only if
$LLG=0$.
* Proof.
Obviously $LG=0$ entails $LLG=0$ as $L$ is additive by Lemma A.12. Conversely,
as $LG$ is separated by Proposition A.15, it follows that the morphism
$\eta_{LG}:LG\to LLG$ is monic by Lemma A.14 (i), so if $LLG=0$ we must have
$LG=0$. ∎
###### A.17 Definition.
The _sheaf ification functor_ is
$j^{\ast}=LL:\operatorname{\mathscr{Y}}\to\operatorname{\mathscr{B}}$.
###### A.18 Lemma.
The sheafification functor
$j^{\ast}:\operatorname{\mathscr{Y}}\to\operatorname{\mathscr{B}}$ is left
adjoint to the inclusion functor
$j_{\ast}:\operatorname{\mathscr{B}}\to\operatorname{\mathscr{Y}}$ and
satisfies
$j^{\ast}j_{\ast}\cong\operatorname{id}_{\operatorname{\mathscr{B}}}$.
Moreover, sheafification is exact.
* Proof.
By Lemma A.14 (ii) the morphism $\eta_{G}:G\to LG$ is an isomorphism if and
only if $G$ is a sheaf, so it follows that
$j^{\ast}j_{\ast}\cong\operatorname{id}_{\operatorname{\mathscr{B}}}$.
Let $Y\in\operatorname{\mathscr{Y}}$ be a presheaf and let
$B\in\operatorname{\mathscr{B}}$ be a sheaf. The natural transformation
$\eta:\operatorname{id}_{\operatorname{\mathscr{Y}}}\Rightarrow L$ gives us on
the one hand a natural transformation
$\varrho_{Y}=\eta_{LY}\eta_{Y}:Y\longrightarrow LLY=j_{\ast}j^{\ast}Y$
and on the other hand a natural isomorphism
$\lambda_{B}=(\eta_{LB}\eta_{B})^{-1}:j^{\ast}j_{\ast}B=LLB\longrightarrow B.$
Now the compositions
$j_{\ast}B\xrightarrow{\varrho_{j_{\ast}B}}j_{\ast}j^{\ast}j_{\ast}B\xrightarrow{j_{\ast}\lambda_{B}}j_{\ast}B\qquad\text{and}\qquad
j^{\ast}Y\xrightarrow{j^{\ast}\varrho_{Y}}j^{\ast}j_{\ast}j^{\ast}Y\xrightarrow{\lambda_{j^{\ast}Y}}j^{\ast}Y$
are manifestly equal to $\operatorname{id}_{j_{\ast}B}$ and
$\operatorname{id}_{j^{\ast}Y}$ so that $j^{\ast}$ is indeed left adjoint to
$j_{\ast}$. In particular $j^{\ast}$ preserves cokernels. That $j^{\ast}$
preserves kernels follows from the fact that
$L:\operatorname{\mathscr{Y}}\to\operatorname{\mathscr{Y}}$ has this property
by Lemma A.12 (i) and the fact that $\operatorname{\mathscr{B}}$ is a full
subcategory of $\operatorname{\mathscr{Y}}$. Therefore $j^{\ast}$ is exact. ∎
###### A.19 Remark.
It is an illuminating exercise to prove exactness of $j^{\ast}$ directly by
going through the definitions.
###### A.20 Lemma.
The category $\operatorname{\mathscr{B}}$ is abelian.
* Proof.
It is clear that $\operatorname{\mathscr{B}}$ is additive. The sheafification
functor $j^{\ast}=LL$ preserves kernels by Lemma A.12 (i) and as a left
adjoint it preserves cokernels. To prove $\operatorname{\mathscr{B}}$ abelian,
it suffices to check that every morphism $f:A\to B$ has an analysis
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}@wrapper{\operatorname{Ker}@presentation}{(f)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Coim}@wrapper{\operatorname{Coim}@presentation}{(f)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\operatorname{Im}@wrapper{\operatorname{Im}@presentation}{(f)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Coker}@wrapper{\operatorname{Coker}@presentation}{(f)}.}$
Since $j^{\ast}$ preserves kernels and cokernels and
$j^{\ast}j_{\ast}\cong\operatorname{id}_{\operatorname{\mathscr{B}}}$ such an
analysis can be obtained by applying $j^{\ast}$ to an analysis of $j_{\ast}f$
in $\operatorname{\mathscr{Y}}$. ∎
### A.4 Proof of the Embedding Theorem
Let us recapitulate: one half of the axioms of an exact structure yields that
a small exact category $\operatorname{\mathscr{A}}$ becomes a _site_
$(\operatorname{\mathscr{A}},J)$. We denoted the Yoneda category of
contravariant functors
$\operatorname{\mathscr{A}}\to\operatorname{\mathbf{Ab}}$ by
$\operatorname{\mathscr{Y}}$ and the Yoneda embedding
$A\mapsto\operatorname{Hom}{({-},A)}$ by
$y:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{Y}}$. We have shown
that the category $\operatorname{\mathscr{B}}$ of sheaves on the site $(A,J)$
is abelian, being a full reflective subcategory of
$\operatorname{\mathscr{Y}}$ with sheafification
$j^{\ast}:\operatorname{\mathscr{Y}}\to\operatorname{\mathscr{B}}$ as
reflector (left adjoint). Following Thomason, we denoted the inclusion
$\operatorname{\mathscr{B}}\to\operatorname{\mathscr{Y}}$ by $j_{\ast}$.
Moreover, we have shown that the Yoneda embedding takes its image in
$\operatorname{\mathscr{B}}$, so we obtained a commutative diagram of
categories
$\textstyle{\operatorname{\mathscr{A}}@wrapper{\operatorname{\mathscr{A}}@presentation}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{y}$$\textstyle{\operatorname{\mathscr{B}}@wrapper{\operatorname{\mathscr{B}}@presentation}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\ast}}$$\textstyle{\operatorname{\mathscr{Y}}@wrapper{\operatorname{\mathscr{Y}}@presentation},}$
in other words $y=j_{\ast}i$. By the Yoneda lemma, $y$ is fully faithful and
$j_{\ast}$ is fully faithful, hence $i$ is fully faithful as well. This
settles the first part of the following lemma:
###### A.21 Lemma.
The functor $i:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{B}}$ is
fully faithful and exact.
* Proof.
By the above discussion, it remains to prove exactness.
Clearly, the Yoneda embedding sends exact sequences in
$\operatorname{\mathscr{A}}$ to left exact sequences in
$\operatorname{\mathscr{Y}}$. Sheafification $j^{\ast}$ is exact and since
$j^{\ast}j_{\ast}\cong\operatorname{id}_{\operatorname{\mathscr{B}}}$, we have
that $j^{\ast}y=j^{\ast}j_{\ast}i\cong i$ is left exact as well. It remains to
prove that for each admissible epic $p:B\operatorname{\twoheadrightarrow}C$
the morphism $i(p)$ is epic. By Corollary A.16, it suffices to prove that
$G=\operatorname{Coker}{y(p)}$ satisfies $LG=0$, because
$\operatorname{Coker}{i(p)}=j^{\ast}\operatorname{Coker}{y(p)}=LLG=0$ then
implies that $i(p)$ is epic. To this end we use the criterion in Lemma A.13
(iii), so let $A\in\operatorname{\mathscr{A}}$ be any object and $x\in G(A)$.
We have an exact sequence
$\operatorname{Hom}{(A,B)}\xrightarrow{y(p)_{A}}\operatorname{Hom}{(A,C)}\xrightarrow{q_{A}}G(A)\xrightarrow{}0$,
so $x=q_{A}(f)$ for some morphism $f:A\to C$. Now form the pull-back
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$PB$\scriptstyle{f^{\prime}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{C}$
and observe that
$G(p^{\prime})(x)=G(p^{\prime})(q_{A}(f))=q_{A^{\prime}}(fp^{\prime})=q_{A^{\prime}}(pf^{\prime})=0$.
∎
###### A.22 Lemma ([57, A.7.15]).
Let $A\in\operatorname{\mathscr{A}}$ and $B\in\operatorname{\mathscr{B}}$ and
suppose there is an epic $e:B\operatorname{\twoheadrightarrow}i(A)$. There
exist $A^{\prime}\in\operatorname{\mathscr{A}}$ and $k:i(A^{\prime})\to B$
such that $ek:A^{\prime}\to A$ is an admissible epic.
* Proof.
Let $G$ be the cokernel of $j_{\ast}e$ in $\operatorname{\mathscr{Y}}$. Then
we have $0=j^{\ast}G=LLG$ because $j^{\ast}j_{\ast}e\cong e$ is epic. By
Corollary A.16 it follows that $LG=0$ as well. Now observe that
$G(A)\cong\operatorname{Hom}{(A,A)}/\operatorname{Hom}{(i(A),B)}$ and let
$x\in G(A)$ be the class of $1_{A}$. From Lemma A.13 (iii) we conclude that
there is an admissible epic
$p^{\prime}:A^{\prime}\operatorname{\twoheadrightarrow}A$ such that
$G(p^{\prime})(x)=0$ in
$G(A^{\prime})\cong\operatorname{Hom}{(A^{\prime},A)}/\operatorname{Hom}{(i(A^{\prime}),B)}$.
But this means that the admissible epic $p^{\prime}$ factors as $ek$ for some
$k\in\operatorname{Hom}{(i(A^{\prime}),B)}$ as claimed. ∎
###### A.23 Lemma.
The functor $i$ reflects exactness.
* Proof.
Suppose $A\xrightarrow{m}B\xrightarrow{e}C$ is a sequence in
$\operatorname{\mathscr{A}}$ such that
$i(A)\xrightarrow{i(m)}i(B)\xrightarrow{i(e)}i(C)$ is short exact in
$\operatorname{\mathscr{B}}$. In particular, $i(m)$ is a kernel of $i(e)$.
Since $i$ is fully faithful, it follows that $m$ is a kernel of $e$ in
$\operatorname{\mathscr{A}}$, hence we are done as soon as we can show that
$e$ is an admissible epic. Because $i(e)$ is epic, Lemma A.22 allows us to
find $A^{\prime}\in\operatorname{\mathscr{A}}$ and $k:i(A^{\prime})\to i(B)$
such that $ek$ is an admissible epic and since $e$ has a kernel we conclude by
the dual of Proposition 2.16. ∎
###### A.24 Lemma.
The essential image of
$i:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{B}}$ is closed under
extensions.
* Proof.
Consider a short exact sequence
$i(A)\operatorname{\rightarrowtail}G\operatorname{\twoheadrightarrow}i(B)$ in
$\operatorname{\mathscr{B}}$, where $A,B\in\operatorname{\mathscr{A}}$. By
Lemma A.22 we find an admissible epic $p:C\operatorname{\twoheadrightarrow}B$
such that $i(p)$ factors over $G$. Now consider the pull-back diagram
$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$PB$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{i(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i(p)}$$\textstyle{i(B)}$
and observe that $D\operatorname{\twoheadrightarrow}i(C)$ is a split epic
because $i(p)$ factors over $G$. Therefore we have isomorphisms $D\cong
i(A)\oplus i(C)\cong i(A\oplus C)$. If $K$ is a kernel of $p$ then $i(K)$ is a
kernel of $D\operatorname{\twoheadrightarrow}G$, so we obtain an exact
sequence
$\textstyle{i(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}i(a)\\\
i(c)\end{smallmatrix}\right]}$$\textstyle{i(A)\oplus
i(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G,}$
where $c=\ker{p}$, which shows that $G$ is the push-out
$\textstyle{i(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i(c)}$$\scriptstyle{i(a)}$PO$\textstyle{i(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{i(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G.}$
Now $i$ is exact by Lemma A.21 and hence preserves push-outs along admissible
monics by Proposition 5.2, so $i$ preserves the push-out
$G^{\prime}=A\cup_{K}C$ of $a$ along the admissible monic $c$ and thus $G$ is
isomorphic to $i(G^{\prime})$. ∎
* Proof of the Embedding Theorem A.1.
Let us summarize what we know: the embedding
$i:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{B}}$ is fully faithful
and exact by Lemma A.21. It reflects exactness by Lemma A.23 and its image is
closed under extensions in $\operatorname{\mathscr{B}}$ by Lemma A.24. This
settles point (i) of the theorem.
Point (ii) is taken care of by Lemma A.7 and Corollary A.8.
It remains to prove (iii). Assume that $\operatorname{\mathscr{A}}$ is weakly
idempotent complete. We claim that every morphism $f:B\to C$ such that $i(f)$
is epic is in fact an admissible epic. Indeed, by Lemma A.22 we find a
morphism $k:A\to B$ such that $fk:A\operatorname{\twoheadrightarrow}C$ is an
admissible epic and we conclude by Proposition 7.6. ∎
## Appendix B Heller’s Axioms
###### B.1 Proposition (Quillen).
Let $\operatorname{\mathscr{A}}$ be an additive category and let
$\operatorname{\mathscr{E}}$ be a class of kernel-cokernel pairs in
$\operatorname{\mathscr{A}}$. The pair
$(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ is a weakly
idempotent complete exact category if and only if $\operatorname{\mathscr{E}}$
satisfies Heller’s axioms:
1. (i)
Identity morphisms are both admissible monics and admissible epics;
2. (ii)
The class of admissible monics and the class of admissible epics are closed
under composition;
3. (iii)
Let $f$ and $g$ be composable morphisms. If $gf$ is an admissible monic then
so is $f$ and if $gf$ is an admissible epic then so is $g$;
4. (iv)
Assume that all rows and the second two columns of the commutative diagram
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{a}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\scriptstyle{b}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{a^{\prime}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{b^{\prime}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c^{\prime}}$$\textstyle{A^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime\prime}}$$\textstyle{B^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime\prime}}$$\textstyle{C^{\prime\prime}}$
are in $\operatorname{\mathscr{E}}$ then the first column is also in
$\operatorname{\mathscr{E}}$.
* Proof.
Note that (i) and (ii) are just axioms [E$0$], [E$1$] and their duals.
For a weakly idempotent complete exact category
$(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$, point (iii) is
proved in Proposition 7.6 and point (iv) follows from the $3\times 3$-lemma
3.6.
Conversely, assume that $\operatorname{\mathscr{E}}$ has properties (i)–(iv)
and let us check that $\operatorname{\mathscr{E}}$ is an exact structure.
By properties (i) and (iii) an isomorphism is both an admissible monic and an
admissible epic since by definition $f^{-1}f=1$ and $ff^{-1}=1$. If the short
sequence $\sigma=(A^{\prime}\to A\to A^{\prime\prime})$ is isomorphic to the
short exact sequence
$B^{\prime}\operatorname{\rightarrowtail}B\operatorname{\twoheadrightarrow}B^{\prime\prime}$
then property (iv) tells us that $\sigma$ is short exact. Thus,
$\operatorname{\mathscr{E}}$ is closed under isomorphisms.
Heller proves [26, Proposition 4.1] that (iv) implies its dual, that is: if
the commutative diagram in (iv) has exact rows and both $(a,a^{\prime})$ and
$(b,b^{\prime})$ belong to $\operatorname{\mathscr{E}}$ then so does
$(c,c^{\prime})$.222Indeed, by (iii) $c^{\prime}$ is an admissible epic and so
it has a kernel $D$. Because $c^{\prime}gb=0$, there is a morphism
$B^{\prime}\to D$ and replacing $C^{\prime}$ by $D$ in the diagram of (iv) we
see that
$A^{\prime}\operatorname{\rightarrowtail}B^{\prime}\operatorname{\twoheadrightarrow}D$
is short exact. Therefore $C^{\prime}\cong D$ and we conclude by the fact that
$\operatorname{\mathscr{E}}$ is closed under isomorphisms. It follows that
Heller’s axioms are self-dual.
Let us prove that [E$2$] holds—the remaining axiom [E$2^{\operatorname{op}}$]
will follow from the dual argument. Given the diagram
---
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{a}$$\textstyle{B^{\prime}}$$\textstyle{A}$
we want to construct its push-out $B$ and prove that the morphism $A\to B$ is
an admissible monic. Observe that $\left[\begin{smallmatrix}a\\\
f^{\prime}\end{smallmatrix}\right]:A^{\prime}\to A\oplus B^{\prime}$ is the
composition
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}0\\\
1\end{smallmatrix}\right]}$$\textstyle{A\oplus
A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\left[\begin{smallmatrix}1&a\\\
0&1\end{smallmatrix}\right]}$$\textstyle{A\oplus
A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}1&0\\\
0&f^{\prime}\end{smallmatrix}\right]}$$\textstyle{A\oplus B^{\prime}.}$
By (iii) split exact sequences belong to $\operatorname{\mathscr{E}}$, and the
proof of Proposition 2.9 shows that the direct sum of two sequences in
$\operatorname{\mathscr{E}}$ also belongs to $\operatorname{\mathscr{E}}$.
Therefore $\left[\begin{smallmatrix}a\\\ f^{\prime}\end{smallmatrix}\right]$
is an admissible monic and it has a cokernel
$\left[\begin{smallmatrix}-f&b\end{smallmatrix}\right]:A\oplus
B^{\prime}\operatorname{\twoheadrightarrow}B$. It follows that the left hand
square in the diagram
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{a}$BC$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\scriptstyle{g^{\prime}}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{C^{\prime}}$
is bicartesian. Let
$g^{\prime}:B^{\prime}\operatorname{\twoheadrightarrow}C^{\prime}$ be a
cokernel of $f^{\prime}$ and let $g$ be the morphism $B\to C^{\prime}$ such
that $gf=0$ and $gb=g^{\prime}$. Now consider the commutative diagram
$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}0\\\
1\end{smallmatrix}\right]}$$\textstyle{A\oplus
A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}-1&0\end{smallmatrix}\right]}$$\scriptstyle{\left[\begin{smallmatrix}1&a\\\
0&f^{\prime}\end{smallmatrix}\right]}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}a\\\
f\end{smallmatrix}\right]}$$\textstyle{A\oplus
B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[\begin{smallmatrix}-f&b\end{smallmatrix}\right]}$$\scriptstyle{\left[\begin{smallmatrix}0&g^{\prime}\end{smallmatrix}\right]}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\prime}}$
in which the rows are exact and the first two columns are exact. It follows
that the third column is exact and hence $f$ is an admissible monic.
Now that we know that
$(\operatorname{\mathscr{A}},\operatorname{\mathscr{E}})$ is an exact
category, we conclude from (iii) and Proposition 7.6 that
$\operatorname{\mathscr{A}}$ must be weakly idempotent complete. ∎
## Acknowledgments
I would like to thank Paul Balmer for introducing me to exact categories and
Bernhard Keller for answering numerous questions via email or via his
excellent articles. Part of the paper was written in November 2008 during a
conference at the Erwin–Schrödinger–Institut in Vienna. The final version of
this paper was prepared at the Forschungsinstitut für Mathematik of the ETH
Zürich. I would like to thank Marc Burger and his team for their support and
for providing excellent working conditions. Matthias Künzer and an anonymous
referee made numerous valuable suggestions which led to substantial
improvements of the text. Finally, I am very grateful to Ivo Dell’Ambrogio for
his detailed comments on various preliminary versions of this paper.
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|
arxiv-papers
| 2008-11-10T14:57:27 |
2024-09-04T02:48:58.694248
|
{
"license": "Public Domain",
"authors": "Theo Buehler",
"submitter": "Theo Buehler",
"url": "https://arxiv.org/abs/0811.1480"
}
|
0811.1514
|
# Scalar Tetraquark Currents with application to the QCD sum rule
Hua-Xing Chen Research Center for Nuclear Physics, Osaka University, Ibaraki
567–0047, Japan
Department of Physics, Peking University, Beijing 100871, China
hxchen@rcnp.osaka-u.ac.jp Atsushi Hosaka Research Center for Nuclear
Physics, Osaka University, Ibaraki 567–0047, Japan Shi-Lin Zhu Department of
Physics, Peking University, Beijing 100871, China
###### Abstract
We study the light scalar mesons in the QCD sum rule. We construct both the
diquark-antidiquark currents $(qq)(\bar{q}\bar{q})$ and the meson-meson
currents $(\bar{q}q)(\bar{q}q)$. We find that there are five independent
currents for both cases, and derive the relations between them. For the meson-
meson currents, five independent currents are formed by products of color
singlet $\bar{q}q$ pairs or color octet pairs. However, they can be related to
each other, and the relations are derived. We obtain the masses of the light
scalar mesons which are consistent with the experiments.
###### keywords:
tetraquark; QCD sum rule.
Received (Day Month Year)Revised (Day Month Year)
PACS Nos.: 12.39.Mk, 12.38.Lg, 12.40.Yx.
## 1 Introduction
Most of hadron states, including mesons and baryons, can be well classified by
the quark content $\bar{q}q$ and $qqq$ in the quark model. However, there are
still many observed states, which can not be explained or difficult to be
explained by just using $\bar{q}q$ and $qqq$. One important example is the
pentaquark $\Theta^{+}$. After three years of intense study, the status of
$\Theta^{+}$ is still controversial. The charm-strange mesons $D_{sJ}(2317)$,
$D_{sJ}(2460)$ and the charmonium state $X(3872)$, $Y(4260)$ are also
difficult to be explained by the conventional picture of $\bar{q}q$ in the
quark model.
Besides them, the light scalar mesons with masses below $1$ GeV have been
discussed for thirties year, but are still controversial [1, 2]. By using the
conventional picture of $\bar{q}q$, the mass ordering are expected be
$m_{\sigma}\sim m_{a_{0}}<m_{\kappa}<m_{f_{0}}$. However, from the
experiments, the mass ordering are $m_{\sigma}<m_{\kappa}<m_{f_{0}}\sim
m_{a_{0}}$ [3], which can be explained well assuming that they were tetraquark
states.
In this paper, we study the light scalar mesons in the QCD sum rule employing
tetraquark currents for the interpolating fields. We construct both the
diquark-antidiquark currents $(qq)(\bar{q}\bar{q})$ and the meson-meson
currents $(\bar{q}q)(\bar{q}q)$. We find that there are five independent
currents for each scalar tetraquark state, and derive the relations between
them. We also find that there are five meson-meson currents where “mesons”
inside are color singlets, and five ones where “mesons” inside are color
octets. However, they can be related to each other, and the relations are
obtained. The masses of the light scalar mesons are calculated in the QCD sum
rule, which are consistent with the experiments.
## 2 Diquark-Antidiquark Currents
In this section, we construct the diquark-antidiquark currents
$(qq)(\bar{q}\bar{q})$ for the state $\sigma(600)$. The currents for other
scalar mesons are similar. In order to make a scalar tetraquark current, the
diquark and antidiquark fields should have the same color, spin and orbital
symmetries. Therefore, they must have the same flavor symmetry, which is
either antisymmetric ($\mathbf{\bar{3}_{f}}\otimes\mathbf{3_{f}}$) or
symmetric ($\mathbf{6_{f}}\otimes\mathbf{\bar{6}_{f}}$). In this paper we
choose the antisymmetric one. The details about their flavor structure can be
found in the reference [4].
Using the antisymmetric combination for diquark flavor structure, we arrive at
the following five independent currents
$\displaystyle S^{\sigma}_{3}$ $\displaystyle=$
$\displaystyle(u_{a}^{T}C\gamma_{5}d_{b})(\bar{u}_{a}\gamma_{5}C\bar{d}_{b}^{T}-\bar{u}_{b}\gamma_{5}C\bar{d}_{a}^{T})\,,$
$\displaystyle V^{\sigma}_{3}$ $\displaystyle=$
$\displaystyle(u_{a}^{T}C\gamma_{\mu}\gamma_{5}d_{b})(\bar{u}_{a}\gamma^{\mu}\gamma_{5}C\bar{d}_{b}^{T}-\bar{u}_{b}\gamma^{\mu}\gamma_{5}C\bar{d}_{a}^{T})\,,$
$\displaystyle T^{\sigma}_{6}$ $\displaystyle=$
$\displaystyle(u_{a}^{T}C\sigma_{\mu\nu}d_{b})(\bar{u}_{a}\sigma^{\mu\nu}C\bar{d}_{b}^{T}+\bar{u}_{b}\sigma^{\mu\nu}C\bar{d}_{a}^{T})\,,$
(1) $\displaystyle A^{\sigma}_{6}$ $\displaystyle=$
$\displaystyle(u_{a}^{T}C\gamma_{\mu}d_{b})(\bar{u}_{a}\gamma^{\mu}C\bar{d}_{b}^{T}+\bar{u}_{b}\gamma^{\mu}C\bar{d}_{a}^{T})\,,$
$\displaystyle P^{\sigma}_{3}$ $\displaystyle=$
$\displaystyle(u_{a}^{T}Cd_{b})(\bar{u}_{a}C\bar{d}_{b}^{T}-\bar{u}_{b}C\bar{d}_{a}^{T})\,,$
where the sum over repeated indices ($\mu$, $\nu,\cdots$ for Dirac, and
$a,b,\cdots$ for color indices) is taken. Either plus or minus sign in the
second parentheses ensures that the diquarks form the antisymmetric
combination in the flavor space. The currents $S$, $V$, $T$, $A$ and $P$ are
constructed by scalar, vector, tensor, axial-vector, pseudoscalar diquark and
antidiquark fields, respectively. The subscripts $3$ and $6$ show that the
diquarks (antidiquark) are combined into the color representation
$\mathbf{\bar{3}_{c}}$ and $\mathbf{6_{c}}$ ($\mathbf{3_{c}}$ or
$\mathbf{\bar{6}_{c}}$), respectively.
## 3 Meson-Meson Currents
In this section, we construct the meson-meson currents $(\bar{q}q)(\bar{q}q)$
for the state $\sigma(600)$. We find that there are five currents where
“mesons” inside are color singlets
$\displaystyle S^{\sigma}_{1}$ $\displaystyle=$
$\displaystyle(\bar{u}_{a}u_{a})(\bar{d}_{b}d_{b})-(\bar{u}_{a}d_{a})(\bar{d}_{b}u_{b})\,,$
$\displaystyle V^{\sigma}_{1}$ $\displaystyle=$
$\displaystyle(\bar{u}_{a}\gamma_{\mu}u_{a})(\bar{d}_{b}\gamma^{\mu}d_{b})-(\bar{u}_{a}\gamma_{\mu}d_{a})(\bar{d}_{b}\gamma^{\mu}u_{b})\,,$
$\displaystyle T^{\sigma}_{1}$ $\displaystyle=$
$\displaystyle(\bar{u}_{a}\sigma_{\mu\nu}u_{a})(\bar{d}_{b}\sigma^{\mu\nu}d_{b})-(\bar{u}_{a}\sigma_{\mu\nu}d_{a})(\bar{d}_{b}\sigma^{\mu\nu}u_{b})\,,$
(2) $\displaystyle A^{\sigma}_{1}$ $\displaystyle=$
$\displaystyle(\bar{u}_{a}\gamma_{\mu}\gamma_{5}u_{a})(\bar{d}_{b}\gamma^{\mu}\gamma_{5}d_{b})-(\bar{u}_{a}\gamma_{\mu}\gamma_{5}d_{a})(\bar{d}_{b}\gamma^{\mu}\gamma_{5}u_{b})\,,$
$\displaystyle P^{\sigma}_{1}$ $\displaystyle=$
$\displaystyle(\bar{u}_{a}\gamma_{5}u_{a})(\bar{d}_{b}\gamma_{5}d_{b})-(\bar{u}_{a}\gamma_{5}d_{a})(\bar{d}_{b}\gamma_{5}u_{b})\,.$
The minus sign ensures that the diquarks (anti-diquarks) form the
antisymmetric combination in the flavor space. These five currents are
independent, and can be related to the five diquark-antidiquark currents
$\displaystyle 8S^{\sigma}_{3}$ $\displaystyle=$
$\displaystyle-2S^{\sigma}_{1}-2V^{\sigma}_{1}+T^{\sigma}_{1}-2A^{\sigma}_{1}-2P^{\sigma}_{1}\,,$
$\displaystyle 2V^{\sigma}_{3}$ $\displaystyle=$ $\displaystyle
2S^{\sigma}_{1}-V^{\sigma}_{1}+A^{\sigma}_{1}-2P^{\sigma}_{1}\,,$
$\displaystyle 2T^{\sigma}_{6}$ $\displaystyle=$ $\displaystyle
6S^{\sigma}_{1}+T^{\sigma}_{1}+6P^{\sigma}_{1}\,,$ (3) $\displaystyle
2A^{\sigma}_{6}$ $\displaystyle=$ $\displaystyle
2S^{\sigma}_{1}+V^{\sigma}_{1}-A^{\sigma}_{1}-2P^{\sigma}_{1}\,,$
$\displaystyle 8P^{\sigma}_{3}$ $\displaystyle=$
$\displaystyle-2S^{\sigma}_{1}+2V^{\sigma}_{1}+T^{\sigma}_{1}+2A^{\sigma}_{1}-2P^{\sigma}_{1}\,.$
We find the other five currents where “mesons” inside are color octets
$\displaystyle S_{8}$ $\displaystyle=$
$\displaystyle(\bar{u}_{a}{\lambda^{n}_{ab}}u_{b})(\bar{d}_{c}{\lambda^{n}_{cd}}d_{d})-(\bar{u}_{a}{\lambda^{n}_{ab}}d_{b})(\bar{d}_{c}{\lambda^{n}_{cd}}u_{d})\,,$
$\displaystyle V_{8}$ $\displaystyle=$
$\displaystyle(\bar{u}_{a}\gamma_{\mu}{\lambda^{n}_{ab}}u_{b})(\bar{d}_{c}\gamma^{\mu}{\lambda^{n}_{cd}}d_{d})-(\bar{u}_{a}\gamma_{\mu}{\lambda^{n}_{ab}}d_{b})(\bar{d}_{c}\gamma^{\mu}{\lambda^{n}_{cd}}u_{d})\,,$
$\displaystyle T_{8}$ $\displaystyle=$
$\displaystyle(\bar{u}_{a}\sigma_{\mu\nu}{\lambda^{n}_{ab}}u_{b})(\bar{d}_{c}\sigma^{\mu\nu}{\lambda^{n}_{cd}}d_{d})-(\bar{u}_{a}\sigma_{\mu\nu}{\lambda^{n}_{ab}}d_{b})(\bar{d}_{c}\sigma^{\mu\nu}{\lambda^{n}_{cd}}u_{d})\,,$
(4) $\displaystyle A_{8}$ $\displaystyle=$
$\displaystyle(\bar{u}_{a}\gamma_{\mu}\gamma_{5}{\lambda^{n}_{ab}}u_{b})(\bar{d}_{c}\gamma^{\mu}\gamma_{5}{\lambda^{n}_{cd}}d_{d})-(\bar{u}_{a}\gamma_{\mu}\gamma_{5}{\lambda^{n}_{ab}}d_{b})(\bar{d}_{c}\gamma^{\mu}\gamma_{5}{\lambda^{n}_{cd}}u_{d})\,,$
$\displaystyle P_{8}$ $\displaystyle=$
$\displaystyle(\bar{u}_{a}\gamma_{5}{\lambda^{n}_{ab}}u_{b})(\bar{d}_{c}\gamma_{5}{\lambda^{n}_{cd}}d_{d})-(\bar{u}_{a}\gamma_{5}{\lambda^{n}_{ab}}d_{b})(\bar{d}_{c}\gamma_{5}{\lambda^{n}_{cd}}u_{d})\,.$
They are also independent, and can be related to the five diquark-antidiquark
currents, as well as to the five meson-meson currents $S^{\sigma}_{1}$,
$V^{\sigma}_{1}$, $T^{\sigma}_{1}$, $A^{\sigma}_{1}$ and $P^{\sigma}_{1}$
$\displaystyle 12S^{\sigma}_{8}$ $\displaystyle=$
$\displaystyle-2S^{\sigma}_{1}+6V^{\sigma}_{1}+3T^{\sigma}_{1}-6A^{\sigma}_{1}-6P^{\sigma}_{1}\,,$
$\displaystyle 3V^{\sigma}_{3}$ $\displaystyle=$ $\displaystyle
6S^{\sigma}_{1}-5V^{\sigma}_{1}-3A^{\sigma}_{1}-6P^{\sigma}_{1}\,,$
$\displaystyle 3T^{\sigma}_{6}$ $\displaystyle=$ $\displaystyle
18S^{\sigma}_{1}-5T^{\sigma}_{1}+18P^{\sigma}_{1}\,,$ (5) $\displaystyle
3A^{\sigma}_{6}$ $\displaystyle=$
$\displaystyle-6S^{\sigma}_{1}-3V^{\sigma}_{1}-5A^{\sigma}_{1}+6P^{\sigma}_{1}\,,$
$\displaystyle 12P^{\sigma}_{3}$ $\displaystyle=$ $\displaystyle
6S^{\sigma}_{1}-6V^{\sigma}_{1}+3T^{\sigma}_{1}+6A^{\sigma}_{1}-2P^{\sigma}_{1}\,.$
## 4 QCD sum rule analysis
We have performed the QCD sum rule analysis for each single current and their
linear combinations. We have performed the OPE calculation up to dimension
eight, which contains the four-quark condensates. We find that the results for
single currents are not always reliable, while a good sum rule is achieved by
a linear combination of $A_{6}^{\sigma}$ and $V_{3}^{\sigma}$
$\displaystyle\eta^{\sigma}$ $\displaystyle=$ $\displaystyle\cos\theta
A^{\sigma}_{6}+\sin\theta V^{\sigma}_{3}\,,$ (6)
where $\theta$ is the mixing angle. The best choice of the mixing angle turns
out to be $\cot\theta=1/\sqrt{2}$. The mixed currents for $\kappa$, $a_{0}$
and $f_{0}$ can be found in the similar way.
By using this mixed current $\eta^{\sigma}$, we studied Borel mass $M_{B}$ and
threshold value $s_{0}$ dependences, which are quite stable. The convergence
of the OPE is also good with the positivity of the spectral densities being
maintained, and with sufficient pole contribution. Therefore, we have achieved
a good QCD sum rule within the present calculation of OPE. We also considered
the finite decay width by using the Gaussian distribution instead of the pole
term in the phenomenological side, where the predicted masses do not change
much as far as the Borel mass is within a reasonable range. Then we can still
reproduce the experimental data.
We have also performed the QCD sum rule analysis with the conventional
$\bar{q}q$ currents. Their masses are calculated to be around 1.2 GeV as in
the previous work [5]. This indicates that the tetraquark currents are more
suitable for the description of the light scalar mesons than the conventional
ones.
In summary, our QCD sum rule analysis supports a tetraquark structure for low-
lying scalar mesons. We construct both the diquark-antidiquark currents and
the meson-meson currents. We find that there are five independent currents in
both constructions. However, currents in different constructions can be
related to each other. Therefore, all the scalar tetraquark currents can be
written as a combination of five meson-meson currents where “mesons” inside
are color singlets. This conclusion can be extended to other tetraquark
currents of different quantum numbers, as well as pentaquark currents.
## Acknowledgments
H.X.C. is grateful for Monkasho support for his stay at the Research Center
for Nuclear Physics where this work is done. A.H. is supported in part by the
Grant for Scientific Research ((C) No.19540297) from the Ministry of
Education, Culture, Science and Technology, Japan. S.L.Z. was supported by the
National Natural Science Foundation of China under Grants 10625521 and
10721063 and Ministry of Education of China.
## References
* [1] W. M. Yao et al. [Particle Data Group], J. Phys. G 33, 1 (2006).
* [2] R. L. Jaffe, Phys. Rev. D 15, 267 (1977); R. L. Jaffe, Phys. Rev. D 15, 281 (1977).
* [3] E. M. Aitala et al., Phys. Rev. Lett. 89, 121801 (2002); M. Ablikim et al., Phys. Lett. B 633, 681 (2006).
* [4] H. X. Chen, A. Hosaka and S. L. Zhu, Phys. Rev. D 76, 094025 (2007).
* [5] L. J. Reinders, S. Yazaki and H. R. Rubinstein, Nucl. Phys. B 196, 125 (1982).
|
arxiv-papers
| 2008-11-10T16:42:37 |
2024-09-04T02:48:58.718178
|
{
"license": "Public Domain",
"authors": "Hua-Xing Chen, Atsushi Hosaka, and Shi-Lin Zhu",
"submitter": "Hua-Xing Chen",
"url": "https://arxiv.org/abs/0811.1514"
}
|
0811.1683
|
EXCESS SPECIFIC HEAT OF PTFE AND PCTFE AT LOW TEMPERATURES: APPROXIMATION
DETAILS
Nina B. Bogdanova$\,{}^{a)}$, B.M. Terziyska$\,{}^{b)}$
$\,{}^{a)}$Inst.of Nuclear Research,BAS, 72 Tzarigradsko choussee, 1784 Sofia,
Bulgaria, Email:nibogd@inrne.bas.bg
$\,{}^{b)}$Inst.of Solid State Physics,BAS, 72 Tzarigradsko choussee, 1784
Sofia, Bulgaria, Email:terzyska@issp.bas.bg
###### Abstract
Approximation of the previously estimated excess specific heat
$C^{excess}/T^{5}$ of two fluoropolymers, $PTFE$ and $PCTFE$, is presented
using Orthonormal Polynomial Expansion Method (OPEM). The new type of
weighting functions in OPEM involves the experimental errors in every point of
the studied thermal characteristic. The investigated temperature dependence of
the function $C^{excess}/T^{5}$ is described in the whole temperature ranges
$0.4\div 8K$ and $2.5\div 7K$ respectively for PTFE and PCTFE as well as in
two subintervals $(0.4\div 2)K$, $(2.5\div 8)K$ for PTFE. Numerical results of
the deviations between the evaluated $C^{excess}/T^{5}$ data and their
approximating values are given. The usual polynomial coefficients obtained by
orthonormal ones in our OPEM approach and the calculated in every point
absolute, relative and specific sensitivities of the studied thermal
characteristic are proposed too. The approximation parameters of this type
thermal characteristic are shown in Figures and Tables.
Key words: orthonormal and usual polynomial approximation, low-temperature
excess specific heat of two fluoropolymers - PTFE and PCTFE
I.INTRODUCTION
The unusual thermal properties concerning the low-temperature specific
features of the heat capacities of the polytetrafluoroethylene (PTFE) and
polyclorothrifluoroethylene (PCTFE) were considered in an earlier paper [1].
The estimated there excess specific heat over Debye contribution below 10 K of
these fluoroplasts was discussed in the frame of the recently developed Soft-
Potential Model (SPM).
The present study is devoted to the mathematical description of the low-
temperature excess specific heat of these semi-crystalline polymers applying
Orthonormal Polynomial Expansion Method (OPEM) [2].
II.EXCESS SPECIFIC HEAT DATA
The previous work [1] clarifies several points concerning the specific heat
peculiarities observed in $C_{p}/T^{3}$ temperature dependencies of PTFE and
PCTFE, a maximum appearing around $T_{max}$= 4 K for both polymers and a found
shallow minimum for PTFE centered at $T_{min}=1~{}K$. Following the Soft-
Potential Model (SPM) [3, 4, 5, 6], supposing a coexistence of acoustic
phonons with quasi-localized low-frequency (soft) modes in glasses and
successfully applied by us to some chalcogenide glasses [7, 8], the specific
heat data, taken from the available calorimetric measurements at low-
temperatures ($T~{}<10$ K) for the studied fluoroplasts [9, 10], were
described in a paper [1]. The $C_{p}$ components are:
i) $C_{p}^{TLS}$ \- a linear contribution, described by double-well
potentials, conditioned by the thermal exitations of the tunneling state
(TLS); for PTFE it was established to predominate at $T~{}\leq 0.2~{}K$.
ii) $(C_{p}^{D})^{acou}$ \- a cubic Debye contribution, evaluated and
discussed in details in a paper [1].
iii) $C_{p}^{exc(SM)}$ \- an excess specific heat (a soft mode) contribution
of the quasi-harmonic exitations, described by single-well potentials.
$C_{p}^{exc(SM)}$ component was evaluated [1] by difference between the
measured specific heat $C_{p}$ [9, 10] and the sum ($C_{p}^{TLS}$ \+
$(C_{p}^{D})^{acou}$) as follows.
$C_{p}^{exc(SM)}=C_{p}-[C_{p}^{TLS}+(C_{p}^{D})^{acou}],$ (1)
The temperature dependencies of the three specific heat components in Eq. 1
are:
$C_{p}^{TLS}=C_{TLS}T,\hskip 8.5359ptC_{p}^{D}=(C_{D})^{acou}T^{3},\hskip
8.5359ptC_{p}^{exc}=C^{exc(SM)}T^{5}.$ (2)
The abbreviations of Two Level State, Debye and excess specific heat (Soft
Modes) are marked with $TLS$, $D$, and $exc(SM)$, respectively. Here,
$C_{TLS}$ was determined by the $C_{p}$ experimental data of Nittke et al [9];
$(C_{D})^{acou},$ the true elastic coefficient, was calculated [1] by the
macroscopic parameters of the investigated materials and the average sound
velocity $v_{s}$, evaluated [1] for PTFE (PCTFE) by available measurements of
transversely and longitudinally polarized 10 (5) MHz ordinary sound waves
between 4.2 and 140 (180) K. Note that the average sound velocity $v_{s}$ for
both polymers changes its value about 0.3 % within 0 and 10 K. This fact
allows us to accept up to 10 K a constant Debye coefficient of the Debye
specific heat contribution by acoustic measurements .
The quantities $C_{D}$ up to 10 K and $C_{TLS}$ are independent of the
temperature. In accordance with the SPM, the softening of the lattice
vibrations leads to increasing of the density of states of the harmonic
oscillators with rising of their energy $E$. This increase appears to be
proportional to the energy as $E^{4}$ leading to proportional to $T^{5}$ term
in $C_{p}(T)$ at enough low-temperatures, i.e. the quantity $C^{exc(SM)}$ is
temperature independent only in a narrow low-temperature range,
$0<T<1.5T_{min}$. It is worth to mention that the quantity $C^{exc(SM)}$
evaluated for some glasses [11, 12, 13] confirms this prediction of the SPM.
For the studied polymers [1] $T_{min}$ was estimated to be very close to 1 K.
But in wider low-temperature range the temperature dependencies of the
quantities $C_{p}^{exc(SM)}/T^{5}$ and the excess specific heat
$C_{p}^{exc(SM)}$ presented, respectively in Fig.1 and in the inset of the
Fig.1, are temperature dependent. As it can be seen in the inset of Fig.1, the
Soft Modes (SM) delocalized at T = 8 K for PTFE and at T = 7 K for PCTFE.
Figure 1: Temperature dependencies of the excess specific heat presented as
$C^{exc}/T^{5}$; inset: Temperature dependencies of the excess specific heat
$C^{exc}$ $[mJ/gK]$ of the PTFE and PCTFE
.
It is important to note that $C^{exc}/T^{5}$ function changes its temperature
behavior just at $T=(3/2)T_{min}$.
III. MATHEMATICAL APPROACH
Our Orthonormal Polynomial Expansion Method(OPEM) and its applications in
cryogenic thermometry are presented in papers [2, 14, 15, 16]. Some important
features of OPEM concerning cryogenic thermometry at the approximation of
thermometric characteristics of different type low- temperature sensors are
protected by a patent for an invention [15]. Our OPEM is defined on Forsythe
[16] three-term relation for constructing orthogonal polynomials over discrete
point set with arbitrary weights in the term of the least square method. The
one-dimensional recurrence for generation of orthonormal polynomials
$\\{\Psi_{k}^{(0)}$, $k=1,2,\ldots\\}$ and their derivatives
$\\{\Psi_{k}^{(m)},m=0,1,2,\ldots\\}$, in OPEM is:
$\Psi_{k+1}^{(m)}(q)=\gamma_{k+1}[(q-\alpha_{k+1})\Psi_{k}^{(m)}(q)-\\\
(1-\delta_{k0})\beta_{k}\Psi_{k-1}^{(m)}(q)+m\Psi_{k}^{(m-1)}(q)]$
OPEM is a development of the Forsythe approach for receiving derivatives and
integrals with fourth term in the Eq. (3). The polynomials
$\\{\Psi_{k}^{(0)}\\}$ satisfy the orthogonality relations:
$\sum_{i=1}^{M}w_{i}\Psi_{k}^{(0)}(q_{i})\Psi_{l}^{(0)}(q_{i})=\delta_{kl}$
over the point set $\\{q_{i},i=1,2,\ldots M\\}$ with weights
$w_{i}=1/\sigma_{i}^{2}$ , depending on errors $\sigma_{i}$ in every point.
The approximating values $f^{appr}$ of the function and its $m$-th derivative
$f^{(m)appr}$,$\\{m=0,1,..\\}$ are calculated by
$f^{(m)appr}(q)=\sum_{k=0}^{N}a_{k}\Psi_{k}^{(m)}(q)=\sum_{k=0}^{N}c_{k}q^{k}.$
(3)
The optimal degree $N$ of the approximating polynomials in Eq.(4) is selected
by the algorithm, combining the following two criteria. First, the fitting
curve should lie in the error corridor of the dependent variable
$(q_{j},f_{j}^{exp}\pm{\sigma_{j}},j=1,...M)$.
$(f_{j}^{appr}-f_{j}^{exp})^{2}w_{j}\leq{1}.$ (4)
Second, the minimum $\chi^{2}$ should be reached.
$\sum_{j=1}^{M}{w_{j}(f_{j}^{appr}-f_{j}^{exp})^{2}}/(M-N-1)\rightarrow{min}.$
(5)
When the first criterion is satisfied, the search of the minimum $\chi^{2}$
stops. The development of the algorithm in the biophysics with the total
variance formula for involving the errors in independent variable was
published in a paper [17]. The last version with obtaining of usual $c_{k}$
coefficients from orthogonal ones $a_{k}$ from Eq. 3 is developed in our work,
RSI-2005 [2].
IV. APPROXIMATION RESULTS
A. Orthonormal expansion
The temperature dependence of the $C^{exc}/T^{5}$ function is described by
orthonormal polynomials in the whole temperature range $0.4\div 8$ $[K]$ or in
two subintervals $0.4\div 2$ $[K]$, $2.5\div 8$ $[K]$ for PTFE, and in the
temperature range $2.5\div 7$ $[K]$ for PCTFE, using the new type of weights,
$W^{C^{exc}/T^{5}}$. The studied thermal characteristic $C^{exc}/T^{5}$ and
its sensitivities, absolute $d(C^{exc}/T^{5})/dT$, relative
$[1/(C^{exc}/T^{5})]d(C^{exc}/T^{5})/dT$ and specific,
$[T/(C^{exc}/T^{5})]d(C^{exc}/T^{5})/dT$, evaluated in every point, are shown
in linear-linear plot in Figs.2 and 3 for PTFE and PCTFE, respectively. It is
worth to note that the subintervals for PTFE are chosen by the temperature
behavior of the specific sensitivity of this polymer (see Fig.2).
Figure 2: Temperature dependencies of the relative
$[1/(C^{exc}/T^{5})]d(C^{exc}/T^{5})/dT$ $[K^{-1}]$ and specific
$[T/(C^{exc}/T^{5})]d(C^{exc}/T^{5})/dT$ $[-]$ sensitivities of the PTFE;
inset: Temperature dependencies of the absolute sensitivity
$d(C^{exc}/T^{5})/dT$.
Figure 3: Temperature dependencies of the relative
$[1/(C^{exc}/T^{5})]d(C^{exc}/T^{5})/dT$ $[K^{-1}]$ and specific
$[T/(C^{exc}/T^{5})]d(C^{exc}/T^{5})/dT$ $[-]$ sensitivities of the PCTFE;
inset: Temperature dependencies of the absolute sensitivity
$d(C^{exc}/T^{5})/dT$ $[mJ/gK^{7}]$.
By definition the weighting function $W^{C^{exc}/T^{5}}$ is $1/\sigma^{2}$,
where $\sigma^{2}$ is a variance of the thermal characteristic $C^{exc}/T^{5}$
versus temperature $T$. In our investigation this variance is accepted to be,
correspondingly square of the absolute heat capacity resolution $\Delta
C^{exc}_{acr}$, determined by the experimental specific heat accuracy as
follows: $(\Delta C^{exc}_{acr})_{i}=0.025(C^{exc}/T^{5})_{i}$ $[mJ/gK^{6}]$
for the first approximating interval of PTFE and $(\Delta
C^{exc}_{acr})_{i}=0.1(C^{exc}/T^{5})_{i}$ $[mJ/gK^{6}]$ for both, the whole
PCTFE approximating interval and the second approximating interval of PTFE.
Here the weights, $W^{C^{exc}/T^{5}}$ are expressed by the relations:
$(W^{C^{exc}/T^{5}})_{i}=1/(\Delta
C^{exc}/T^{5}_{acr})_{i}=1600/(C^{exc}/T^{5})^{2}_{i}~{}~{}[mJ/gK^{6}]^{-2}$
(6)
for the first approximating interval of PTFE, or
$(W^{C^{exc}/T^{5}})_{i}=1/(\Delta
C^{exc}_{acr})_{i}=100/(C^{exc}/T^{5})^{2}_{i}~{}~{}[mJ/gK^{6}]^{-2}$ (7)
for the PCTFE and the second approximating interval of PTFE. The deviations
$\Delta(C^{exc}/T^{5})_{i}$ between experimental and approximating values of
the excess specific heat are estimated in every point by the expression:
$\Delta(C^{exc}/T^{5})_{i}=(C^{exc}/T^{5})_{i}^{exp}-(C^{exc}/T^{5})_{i}^{appr}$
(8)
The temperature behavior of the calculated differences
$\Delta(C^{exc}/T^{5})_{i}$, the root mean square deviations $RMS^{C}$ and the
mean absolute deviations $MAD^{C}$ respectively for PTFE and PCTFE excess
specific heat approximations are shown in the Figs.4,5 and 6.
Figure 4: Temperature dependencies of $\Delta(C^{exc}/T^{5})$,
$(\Delta(C^{exc}/T^{5})_{acr}$, $MAD^{C^{exc}/T^{5}}$ and
$RMS^{C^{exc}/T^{5}}$ of the PTFE for the whole approximated temperature
range;inset:Temperature dependence of the weighting function $w(T)$ for FTPE .
Figure 5: Temperature dependencies of $\Delta(C^{exc}/T^{5})$,
$(\Delta(C^{exc}/T^{5})_{acr}$, $MAD^{C^{exc}/T^{5}}$ and
$RMS^{C^{exc}/T^{5}}$ of the PTFE for two interval approximation.
Figure 6: Temperature dependencies of $\Delta(C^{exc}/T^{5})$,
$\Delta(C^{exc})_{acr}$, $MAD^{C^{exc}/T^{5}}$ and $RMS^{C^{exc}/T^{5}}$ of
the PCTFE for the whole approximated temperature range.
The RMS deviation is more popular. It is given in another our paper [2]. The
characteristics MAD is defined as follows:
$MAD^{C}=\frac{1}{M}\sum_{i}|{\Delta(C^{exc}/T^{5})_{i}-\overline{\Delta(C^{exc}/T^{5})}|},$
(9)
where
$\overline{\Delta(C^{exc}/T^{5})}=\frac{1}{M}\sum_{i}{\Delta(C^{exc}/T^{5})_{i}}$
and $\Delta(C^{exc}/T^{5})_{i}$ are calculated from Eq.(9).
Following the cited criteria in Eqs.(5,6) the deviations
$\Delta(C^{exc}/T^{5})_{i}$ are in the error corridor (see Eq.5). As a result
of our approximation, the polynomial degree for temperature range of PTFE
subintervals is lower than for the whole temperature range. For
$C^{exc}/T^{5}$ vs. $T$ approximation the optimal degree $N$ as well as some
main characteristics including the overall approximation characteristics: RMS
and MAD of $\Delta C^{exc}/T^{5}$ vs. $T$ approximation, the weighting
function $W^{C}$ and a goodness of fit $\chi^{2}$ are presented in the TABLE
1.
B.Usual expansion obtained by orhonormal one
The next step in our approximation is done here. The OPEM is extended by
calculation of usual coefficients $\\{{c}_{k}\\}$ from orthonormal ones
$\\{{a}_{k}\\}$ using Eq.(4). This is carried out for the investigated
intervals. The calculations are made for the $C^{exc}/T^{5}=f(T)$
descriptions, in two runs: first - in an interval [-1,1], and second - in the
input intervals. Optimal values of usual polynomial degrees are chosen using
two new criteria.
Table 1: OPEM approximations of $C^{excess}/T^{5}$ for $PTFE$ and $PCTFE$ T | C | N | $W^{C}$ | $RMS^{C}$ | $MAD^{C}$ | $\chi^{2}$
---|---|---|---|---|---|---
$[K]$ | $[\mu J/gK^{6}]$ | - | $[mJ/gK^{6}]$ | $[mJ/gK^{6}]$ | - |
$0.4\div 2$ | $152\div 1$ | 6 | $6.91\times{10}^{4}\div 1.60\times{10}^{9}$ | $0.121\times 10^{-2}$ | $0.828\times 10^{-3}$ | 1.38
$2.5\div 8$ | $1\div 0.015$ | 4 | $2.65\times 10^{8}\div 4.44\times 10^{11}$ | $0.265\times 10^{-4}$ | $0.208\times 10^{-4}$ | 0.57
$0.4\div 8$ | $152\div 0.015$ | 8 | $6.91\times 10^{4}\div 4.44\times 10^{11}$ | $0.109\times 10^{-2}$ | $0.326\times 10^{-3}$ | 0.95
$2.5\div 7$ | $1.75\div 0.031$ | 6 | $3.26\times 10^{7}\div 1.04\times 10^{11}$ | $0.428\times 10^{-4}$ | $0.306\times 10^{-4}$ | 0.58
Table 2: Usual coefficients for $C^{excess}/T^{5}$ $[10^{-3}mJ/gK^{6}]$ vs. $T$ approximation $material$ | $PTFE$ | $PCTFE$
---|---|---
$RangeT(K)$ | $0.4\div 2$ | $2.5\div 8$ | $0.4\div 8$ | $2.5\div 7$
$Range(C^{exc.}/T^{5})$ | $152\div 1$ | $1\div 0.015$ | $152\div 0.015$ | $1.75\div 0.031$
$c_{0}$ | .022854 | .003118 | .016067 | .182459
$c_{1}$ | -.001420 | -.009158 | -.063937 | -.667996
$c_{2}$ | .105876 | .011825 | .103755 | .997805
$c_{3}$ | -.582978 | -.006389 | -.070355 | -.769121
$c_{4}$ | -.390617 | .001190 | -.008661 | .320591
$c_{5}$ | .845833 | - | .048421 | -.068068
$c_{6}$ | .696453 | - | -.033867 | .005687
$c_{7}$ | - | - | .010411 | -
$c_{8}$ | - | - | -.001230 | -
The first criterion is:
$\max|f_{i}^{exp}-f_{i}^{appr,u}|/f_{i}^{exp}\rightarrow{min},$ (10)
where $f^{appr,u}$ is the approximating function defined with usual expansion.
The second criterion is:
$\Delta\\{c_{k}\\}_{1}^{N}\rightarrow{min},$ (11)
where $\\{\Delta c_{k}\\}$ are inherited errors in usual coefficients, defined
in OPEM, discussed in a paper[18]. In the TABLE 2 the usual coefficients for
$C^{exc}/T^{5}$ vs. $T$ approximation are presented.
In conclusion, the heat capacity components of the studied fluoroplasts are
well defined earlier [1] using the SPM. In this study, the temperature
dependencies of the excess specific heat of PTFE and PCTFE below 10 K
presented as $C^{exc(SM)}/T^{5}$ are described by an Orthonormal Polynomial
Expansion Method (OPEM). An approximation of the above mentioned thermal
characteristic by usual polynomials, obtained by orthonormal ones, is given
too. The numerical results for approximation characteristics as root mean
square deviation, mean absolute deviation and normalized $\chi$ square assure
good accuracy for calculated values of physical quantity in the input
temperature intervals. Our approach for description of this characteristic
proposes possibility for the future analyzing the low-temperature excess
specific heat of the materials with glass-like behavior.
## References
* [1] B. Terziyska, H. Madge, Some special feature of low-temperature specific heat of PTFE and PCTFE analyzed within the Soft Potential Model, to be published.
* [2] N.Bogdanova, B. Terzijska, Thermometric characteristics approximation of Germanium film temperature microsensors by orthonormal polynomials, Rev. Sci. Instrum. 68 (10), 3766-3771 (2005).
* [3] M.A.Il’in, Karpov V.G., and Parshin D.A., Parameters of soft atomic potentials in glasses, Zh. Exper. Teor. Fiz. 92, 291 (1987).
* [4] V.G.Karpov , Klinger M.I., and Ignat’ev F.N., Theory of low-temperature anomalies in the thermal properties of amorphic structures, Zh. Exper. Teor. Fiz. 84 (2), 760-775 (1983).
* [5] D.A.Parshin, Interactions of soft atomic potentials and universality of low- temperature properties of glasses, Phys. Rev. B 49 (14), 9400-9418 (1994).
* [6] D.A.Parshin, Liu X., Brand O., and Lohneysen H.v., Analysis of the low-temperature specific heat of amorphous AsxSe1-x within the soft potential model, Z. Phys. B 93 (1), 57-62 (1993).
* [7] B. Terziyska, H. Misiorek, E. Vateva, A. Jezovski, and D.Arsova, Low-temperature thermal conductivity of GexAs40-xS60 glasses, SSC 134, 349 (2005).
* [8] B. Terziyska, A. Czopnik, E. Vateva, D. Arsova, and R. Czopnik, Low-temperature specific heat of Ge-As-S glasses, Phil. Mag. Letters, 85, 145-150 (2005).
* [9] A. Nittke, P. Esquinazi, H.C. Semmelhack, A.L. Burin, A.Z. Patashinskii, Eur. Phys. J. B8, 19 (1999).
* [10] B. Terziyska, H. Madge, and V. Lovtchinov, Specific heat of PTFE and PCTFE within the temperatute range 2.5-20 K, Journal of Thermal Analysis 20, 33 (1981).
* [11] M.A.Ramos, Talon C, and Vieira S.,The Boson peak in structural and oriental glasses of simple alcohols: specific heat at low temperatures, J. Non-Cryst. Sol. 307-310, 80-86 (2002) and references therein.
* [12] C.Talon , Ramos M.A., and Vieira S.,Low-temperature specific heat of amorphous, orientational glass, and crystal phases of ethanol, Physical Review B 66, 012201(4)(2002).
* [13] M.A.Ramos, Talon C., Jimenez-Rioboo RJ, and Vieira S. Low-temperature specific heat of structural and orientational glasses of simple alcohols, J. Phys.:Condens. Matter 15, S1007-S1018 (2003).
* [14] N.Bogdanova, B. Terzijska, A novel approach to the Rh-Fe thermometric characteristic approximation, Commun. JINR, Dubna, E11–97-396 (1997).
* [15] B. Terzijska, N.Bogdanova, Weighted Orthonormal Polynomial Method in Cryogenic Thermometry, A Patent for an Invention, Bulgaria, No. 62582 (2000).
* [16] G.Forsythe, Generation and Use of Orthogonal Polynomials, J. Soc. Ind. Appl. Math. 5, 74-88 (1957).
* [17] N.Bogdanova, S. Todorov, Fitting of water Hydrogen bond energy data with uncertainties in both variables by help of orthogonal polynomials, Int. J. Modern Physics C 12, 1 (2001).
* [18] N.Bogdanova, Orthonormal Polynomial Expansion Method with errors in variables, E11-98-3, Communication JINR, Dubna (1998).
|
arxiv-papers
| 2008-11-11T11:00:16 |
2024-09-04T02:48:58.726341
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nina B. Bogdanova, B.M.Terziyska",
"submitter": "Nina Bogdanova Bogdanova",
"url": "https://arxiv.org/abs/0811.1683"
}
|
0811.1723
|
# Finite-temperature Screening and the Specific Heat of Doped Graphene Sheets
M.R. Ramezanali Department of Physics, Sharif University of Technology,
Tehran 11155-9161, Iran M.M. Vazifeh Department of Physics, Sharif
University of Technology, Tehran 11155-9161, Iran Reza Asgari
asgari@theory.ipm.ac.ir School of Physics, Institute for research in
fundamental sciences, IPM 19395-5531 Tehran, Iran Marco Polini
m.polini@sns.it NEST-CNR-INFM and Scuola Normale Superiore, I-56126 Pisa,
Italy A.H. MacDonald Department of Physics, The University of Texas at
Austin, Austin, Texas 78712, USA
###### Abstract
At low energies, electrons in doped graphene sheets are described by a
massless Dirac fermion Hamiltonian. In this work we present a semi-analytical
expression for the dynamical density-density linear-response function of
noninteracting massless Dirac fermions (the so-called “Lindhard” function) at
finite temperature. This result is crucial to describe finite-temperature
screening of interacting massless Dirac fermions within the Random Phase
Approximation. In particular, we use it to make quantitative predictions for
the specific heat and the compressibility of doped graphene sheets. We find
that, at low temperatures, the specific heat has the usual normal-Fermi-liquid
linear-in-temperature behavior, with a slope that is solely controlled by the
renormalized quasiparticle velocity.
††: J. Phys. A: Math. Gen.
## 1 Introduction
Graphene is a newly realized two-dimensional (2D) electron system that has
attracted a great deal of interest in the scientific community because of the
new physics which it exhibits and because of its potential as a new material
for electronic technology [1, 2]. The agent responsible for many of the
interesting electronic properties of graphene sheets is the non-Bravais
honeycomb-lattice arrangement of Carbon atoms, which leads to a gapless
semiconductor with valence and conduction $\pi$-bands. States near the Fermi
energy of a graphene sheet are described by a spin-independent massless Dirac
Hamiltonian [3]
${\cal H}_{\rm D}=v_{\rm F}{\bm{\sigma}}\cdot{\bm{p}}~{},$ (1)
where $v_{\rm F}$ is the Fermi velocity, which is density-independent and
roughly three-hundred times smaller that the velocity of light in vacuum, and
${\bm{\sigma}}=(\sigma^{x},\sigma^{y})$ is a vector constructed with two Pauli
matrices $\\{\sigma^{i},i=x,y\\}$, which operate on pseudospin (sublattice)
degrees of freedom. Note that the eigenstates of ${\cal H}_{\rm D}$ have a
definite chirality rather than a definite pseudospin, i.e. they have a
definite projection of the honeycomb-sublattice pseudospin onto the momentum
${\bm{p}}$.
When non-relativistic Coulombic electron-electron interactions are added to
the kinetic Hamiltonian (1), graphene represents a new type of many-electron
problem, distinct from both an ordinary 2D electron gas (EG) and from quantum
electrodynamics. The Dirac-like wave equation and the chirality of its
eigenstates lead indeed to both unusual electron-electron interaction effects
[4, 5, 6, 7, 8] and to unusual response to external potentials [9, 10].
Within this low energy description, the properties of doped graphene sheets
depend on the dimensionless coupling constant
$\alpha_{\rm gr}=g\frac{e^{2}}{\epsilon\hbar v_{\rm F}}~{},$ (2)
and on an ultraviolet cut-off $\Lambda=k_{\rm c}/k_{\rm F}$. Here $g=g_{\rm
s}g_{\rm v}=4$ accounts for spin and valley degeneracy, $k_{\rm F}=(4\pi
n/g)^{1/2}$ is the Fermi wave number with $n$ the electron density, and
$k_{\rm c}$ should be assigned a value corresponding to the wavevector range
over which the continuum model (1) describes graphene. For definiteness we
take $k_{\rm c}$ to be such that $\pi k^{2}_{\rm c}=(2\pi)^{2}/{\cal A}_{0}$,
where ${\cal A}_{0}=3\sqrt{3}a^{2}_{0}/2$ is the area of the unit cell in the
honeycomb lattice, with $a_{0}\simeq 1.42$ Å the Carbon-Carbon distance. With
this choice
$\Lambda=\frac{\sqrt{g}}{\sqrt{n{\cal A}_{0}}}~{}.$ (3)
The continuum model is useful when $k_{\rm c}\gg k_{\rm F}$, i.e. when
$\Lambda\gg 1$.
Vafek [11] has recently shown that the specific heat of undoped graphene
sheets presents an anomalous low-temperature behavior showing a logarithmic
suppression with respect to its noninteracting counterpart, $C_{V}(T\to
0)/C_{V}(0)\propto T/\ln(T)$. On the other hand, in Refs. [6, 7] we have
demonstrated (see also Ref. [8]) that doped graphene sheets are normal
(pseudochiral) Fermi liquids, with Landau parameters that possess, however, a
quite distinct behavior from those of conventional 2D EGs. In this work we
calculate the Helmholtz free energy ${\cal F}(T)$ of doped graphene sheets
within the Random Phase Approximation (RPA) [12, 13]. This allows us to access
important thermodynamic quantities, such as the compressibility and the
specific heat, which can be calculated by taking appropriate derivatives of
the free energy. We show that, at low temperatures, the specific heat of doped
graphene, contrary to the one of the undoped system [11], has the usual
linear-in-temperature behavior, which is solely controlled by the renormalized
velocity of quasiparticles as in a normal Fermi liquid.
## 2 The Helmoltz free energy and the Lindhard response function at finite
temperature
The free energy ${\cal F}={\cal F}_{0}+{\cal F}_{\rm int}$ is usually
decomposed into the sum of a noninteracting term ${\cal F}_{0}$ and an
interaction contribution ${\cal F}_{\rm int}$. To evaluate the interaction
contribution to the Helmholtz free energy we follow a familiar strategy [13]
by combining a coupling constant integration expression for ${\cal F}_{\rm
int}$ valid for uniform continuum models ($\hbar=1$ from now on),
${\cal F}_{\rm
int}(T)=\frac{N}{2}\int_{0}^{1}d\lambda\int\frac{d^{2}{\bm{q}}}{(2\pi)^{2}}v_{q}\left[S^{(\lambda)}(q,T)-1\right]~{},$
(4)
with a fluctuation-dissipation-theorem (FDT) expression [13] for the static
structure factor,
$S^{(\lambda)}(q,T)=-\frac{1}{\pi
n}\int_{0}^{+\infty}d\omega~{}\coth{(\beta\omega/2)}\Im
m\chi^{(\lambda)}_{\rho\rho}(q,\omega,T)~{}.$ (5)
Here $v_{q}=2\pi e^{2}/(\epsilon q)$ is the 2D Fourier transform of the
Coulomb potential and $\beta=(k_{\rm B}T)^{-1}$. We anticipate that this
version of the FDT (in which the frequency integration has to be performed
over the real-frequency axis) requires care in handling the plasmon
contribution to ${\cal F}_{\rm int}(T)$ (see discussion below).
The RPA approximation for ${\cal F}_{\rm int}$ then follows from the RPA
approximation for $\chi^{(\lambda)}_{\rho\rho}(q,\omega)$:
$\chi^{(\lambda)}_{\rho\rho}(q,\omega,T)=\frac{\chi^{(0)}(q,\omega,T)}{1-\lambda
v_{q}\chi^{(0)}(q,\omega,T)}$ (6)
where $\chi^{(0)}(q,\omega,T)$ is the noninteracting density-density response-
function,
$\displaystyle\chi^{(0)}(q,\omega,T)$ $\displaystyle=$ $\displaystyle
g\lim_{\eta\to
0^{+}}\sum_{s,s^{\prime}=\pm}\int\frac{d^{2}{\bm{k}}}{(2\pi)^{2}}\frac{1+ss^{\prime}\cos(\theta_{{\bm{k}},{\bm{k}}+{\bm{q}}})}{2}$
(7) $\displaystyle\times$ $\displaystyle\frac{n_{\rm
F}(\varepsilon_{{\bm{k}},s})-n_{\rm
F}(\varepsilon_{{\bm{k}}+{\bm{q}},s^{\prime}})}{\omega+\varepsilon_{{\bm{k}},s}-\varepsilon_{{\bm{k}}+{\bm{q}},s^{\prime}}+i\eta}~{}.$
Here $\varepsilon_{{\bm{k}},s}=sv_{\rm F}k$ are the Dirac band energies and
$n_{\rm F}(\varepsilon)=\\{\exp[\beta(\varepsilon-\mu_{0})]+1\\}^{-1}$ is the
usual Fermi-Dirac distribution function, $\mu_{0}=\mu_{0}(T)$ being the
noninteracting chemical potential. As usual, this is determined by the
normalization condition
$\mu_{0}(T)=\int_{-\infty}^{+\infty}d\varepsilon~{}\nu(\varepsilon)n_{\rm
F}(\varepsilon)~{},$ (8)
where $\nu(\varepsilon)=g\varepsilon/(2\pi v^{2}_{\rm F})$ is the
noninteracting density of states. For $T\to 0$ one finds
$\mu_{0}(T)=\varepsilon_{\rm F}-\pi^{2}(T/T_{\rm F})^{2}/6$, where $T_{\rm
F}=\varepsilon_{\rm F}/k_{B}$ is the Fermi temperature. The factor in the
first line of Eq. (7), which depends on the angle
$\theta_{{\bm{k}},{\bm{k}}+{\bm{q}}}$ between ${\bm{k}}$ and
${\bm{k}}+{\bm{q}}$, describes the dependence of Coulomb scattering on the
relative chirality $ss^{\prime}$ of the interacting electrons.
After some straightforward algebraic manipulations we arrive at the following
expressions for the imaginary [$\Im m~{}\chi^{(0)}(q,\omega,T)$] and the real
[$\Re e~{}\chi^{(0)}(q,\omega,T)$] parts of the noninteracting density-density
response function for $\omega>0$:
$\displaystyle\Im m~{}\chi^{(0)}(q,\omega,T)$ $\displaystyle=$
$\displaystyle\frac{g}{4\pi}\sum_{\alpha=\pm}\Bigg{\\{}\Theta(v_{\rm
F}q-\omega)q^{2}f(v_{\rm F}q,\omega)$ $\displaystyle\times$
$\displaystyle\left[G^{(\alpha)}_{+}(q,\omega,T)-G^{(\alpha)}_{-}(q,\omega,T)\right]$
$\displaystyle+$ $\displaystyle\Theta(\omega-v_{\rm F}q)q^{2}f(\omega,v_{\rm
F}q)\left[-\frac{\pi}{2}\delta_{\alpha,-}+H^{(\alpha)}_{+}(q,\omega,T)\right]\Bigg{\\}}$
and
$\displaystyle\Re e~{}\chi^{(0)}(q,\omega,T)$ $\displaystyle=$
$\displaystyle\frac{g}{4\pi}\sum_{\alpha=\pm}\Bigg{\\{}\frac{-2k_{\rm
B}T\ln[1+e^{\alpha\mu_{0}/(k_{\rm B}T)}]}{v^{2}_{\rm F}}+\Theta(\omega-v_{\rm
F}q)$ $\displaystyle\times$ $\displaystyle q^{2}f(\omega,v_{\rm
F}q)\left[G^{(\alpha)}_{-}(q,\omega,T)-G^{(\alpha)}_{+}(q,\omega,T)\right]$
$\displaystyle+$ $\displaystyle\Theta(v_{\rm F}q-\omega)q^{2}f(v_{\rm
F}q,\omega)\left[-\frac{\pi}{2}\delta_{\alpha,-}+H^{(\alpha)}_{-}(q,\omega,T)\right]\Bigg{\\}}~{}.$
Here
$f(x,y)=\frac{1}{2\sqrt{x^{2}-y^{2}}}~{},$ (11)
$G^{(\alpha)}_{\pm}(q,\omega,T)=\int_{1}^{\infty}du~{}\frac{\sqrt{u^{2}-1}}{\exp\left({\displaystyle\frac{|v_{\rm
F}qu\pm\omega|-2\alpha\mu_{0}}{2k_{\rm B}T}}\right)+1}~{},$ (12)
and
$H^{(\alpha)}_{\pm}(q,\omega,T)=\int_{-1}^{1}du~{}\frac{\sqrt{1-u^{2}}}{\exp\left({\displaystyle\frac{|v_{\rm
F}qu\pm\omega|-2\alpha\mu_{0}}{2k_{\rm B}T}}\right)+1}~{}.$ (13)
These semi-analytical expressions for $\Re e~{}\chi^{(0)}(q,\omega,T)$ and
$\Im m~{}\chi^{(0)}(q,\omega,T)$ constitute the first important result of this
work. In Fig. 1 we have plotted the static response, $\Re
e~{}\chi^{(0)}(q,0,T)$, as a function of $q/k_{\rm F}$ for different values of
$T/T_{\rm F}$. The temperature dependence of the Lindhard function at finite
frequency is instead presented in Fig. 2. An illustrative plot of the
imaginary part of the inverse RPA dielectric function
$\varepsilon(q,\omega,T)=1-v_{q}\chi^{(0)}(q,\omega,T)$ is reported in Fig. 3.
Figure 1: The static response function $\Re e~{}\chi^{(0)}(q,0,T)$ [in units
of $-\nu(\varepsilon_{\rm F})$] as a function of $q/k_{\rm F}$ for three
values of $0\leq T/T_{\rm F}\leq 1$.
Figure 2: Left panel: the real part of the dynamical response function $\Re
e~{}\chi^{(0)}(q,\omega,T)$ [in units of $-\nu(\varepsilon_{\rm F})$] as a
function of $q/k_{\rm F}$ for $\omega=2\varepsilon_{\rm F}$ and three values
of $0\leq T/T_{\rm F}\leq 1$. Right panel: same as in the left panel but for
the imaginary part. Figure 3: Left panel: $\Im
m~{}[\varepsilon^{-1}(q,\omega,T)]$ as a function of $q/k_{\rm F}$ and
$\omega/\varepsilon_{\rm F}$ for $\alpha_{\rm gr}=2$ and $T=0$. The red solid
line is the plasmon dispersion relation. Right panel: same as in the left
panel but for $T=0.2~{}T_{\rm F}$ (corresponding roughly to room temperature).
The coupling constant integration in Eq. (4) can be carried out partly
analytically due to the simple RPA expression (6). We find that the
interaction contribution to the free energy per particle $f_{\rm int}(T)$ is
given by
$\displaystyle f_{\rm int}(T)$ $\displaystyle\equiv$ $\displaystyle\frac{{\cal
F}_{\rm
int}(T)}{N}=\frac{1}{2}\int\frac{d^{2}{\bm{q}}}{(2\pi)^{2}}\left\\{-\frac{1}{\pi
n}\int_{0}^{+\infty}d\omega\coth{(\beta\omega/2)}\right.$ (14)
$\displaystyle\times$ $\displaystyle\left.\arctan\left[\frac{v_{q}\Im
m~{}\chi^{(0)}(q,\omega,T)}{1-v_{q}\Re
e~{}\chi^{(0)}(q,\omega,T)}\right]-v_{q}\right\\}$ $\displaystyle+$
$\displaystyle\frac{1}{2n}\int\frac{d^{2}{\bm{q}}}{(2\pi)^{2}}\int_{0}^{1}\frac{d\lambda}{\lambda}\coth{(\beta\omega_{\rm
pl}/2)}\Re e~{}\chi^{(0)}(q,\omega_{\rm pl},T)$ $\displaystyle\times$
$\displaystyle\left|{\frac{\partial[\Re
e~{}\chi^{(0)}(q,\omega,T)]}{\partial\omega}}\right|^{-1}_{\omega=\omega_{\rm
pl}}\,.$
In this equation the first term comes from the smooth electron-hole
contribution to $\Im m~{}\chi^{(\lambda)}_{\rho\rho}$, while the second term
comes from the plasmon contribution; $\omega_{\rm pl}=\omega_{\rm
pl}(q,T,\lambda)$ is the plasmon dispersion relation at coupling constant
$\lambda$ which can be found numerically by solving the equation $1-\lambda
v_{q}\Re e~{}\chi^{(0)}(q,\omega,T)=0$. Note that in a usual 2D EG the
exchange energy starts to matter little for $T$ of order $T_{\rm F}$ because
all the occupation numbers are small and the Pauli exclusion principle matters
little. In the graphene case however exchange interactions with the negative
energy sea remain important as long as $T$ is small compared to $v_{\rm
F}k_{\rm c}/k_{\rm B}=T_{\rm F}\Lambda$.
The free energy calculated according to Eq. (14) is divergent since it
includes the interaction energy of the model’s infinite sea of negative energy
particles. Following Vafek [11], we choose the free energy at $T=0$, $f(T=0)$,
as our “reference” free energy, and thus introduce the regularized quantity
$\delta f\equiv f(T)-f(T=0)$. This again can be decomposed into the sum of a
noninteracting contribution, $\delta f_{0}(T\to 0)=-g\varepsilon_{\rm
F}\pi^{2}(T/T_{\rm F})^{2}/12$, and an interaction-induced contribution
$\delta f_{\rm int}(T)=f_{\rm int}(T)-f_{\rm int}(T=0)$, which can be
calculated from Eq. (14). Numerical results for $\delta f_{\rm int}(T)$ as a
function of the reduced temperature $T/T_{\rm F}$ are presented in the left
panel of Fig. 4.
The low-temperature behavior of the interaction contribution to the free
energy can be extracted analytically with some patience. After some lenghty
but straightfoward algebra we find, to leading order in $\Lambda$,
$\displaystyle\delta f_{\rm int}(T\to 0)$ $\displaystyle=$
$\displaystyle\varepsilon_{\rm F}\frac{\pi^{2}}{3}\left(\frac{T}{T_{\rm
F}}\right)^{2}\frac{\alpha_{\rm gr}[1-\alpha_{\rm gr}\xi(\alpha_{\rm
gr})]}{4g}~{}\ln{\Lambda}+{\rm R.~{}T.}~{},$ (15)
where the function $\xi(x)$, defined as in Eq. (14) of Ref. [6], is given by
$\xi(x)=128/(\pi^{2}x^{3})-32/(\pi^{2}x^{2})+1/x-h(\pi x/8)$, with
$h(x)=\left\\{\begin{array}[]{ll}{\displaystyle\frac{1}{2x^{3}\sqrt{1-x^{2}}}\arctan{\left(\frac{\sqrt{1-x^{2}}}{x}\right)}}&{\displaystyle{\rm
for}~{}x<1\vspace{0.1 cm}}\\\
{\displaystyle\frac{1}{4x^{3}\sqrt{x^{2}-1}}\ln{\left(\frac{x+\sqrt{x^{2}-1}}{x-\sqrt{x^{2}-1}}\right)}}&{\displaystyle{\rm
for}~{}x>1}\end{array}\right.\,.$ (16)
The symbol “${\rm R.~{}T.}$” in the l.f.s. of Eq. (15) indicates “regular
terms”, i.e. terms that, by definition, are finite in the limit
$\Lambda\to\infty$. Eq. (15) represents the second important result of this
work.
Befor concluding this Section, we remind the reader that in Ref. [6] it has
been proven that the renormalized RPA quasiparticle velocity $v^{\star}$ is
given, at weak coupling and to leading order in $\Lambda$, by
$\frac{v^{\star}}{v_{\rm F}}=1+\frac{\alpha_{\rm gr}[1-\alpha_{\rm
gr}\xi(\alpha_{\rm gr})]}{4g}~{}\ln{\Lambda}~{}.$ (17)
## 3 The specific heat and the compressibility
The specific heat can be calculated from the second derivative of the
Helmholtz free energy, $C_{V}=-T\partial^{2}[n\delta f(T)]/\partial T^{2}$
111The second derivative is calculated using the full temperature dependent
free-energy of the non-interacting system, $\delta f_{0}(T)$, and not its
analytical expression reported above that is valid only for $T\ll T_{\rm F}$..
Numerical results for $C_{V}(T)$ as a function of temperature are reported in
Fig. 4.
Figure 4: Left panel: the (regularized) interaction contribution to the free
energy $\delta f_{\rm int}(T)$ (in units of the Fermi energy $\varepsilon_{\rm
F}$) as a function of $T/T_{\rm F}$ for $\Lambda=10^{2}$. Right panel: the
specific heat $C_{V}(T)$ (in units of $k_{\rm B}$) as a function of $T/T_{\rm
F}$.
We thus see that $\delta f_{\rm int}(T\to 0)\propto T^{2}$ in Eq. (15) implies
a conventional Fermi-liquid behavior with a linear-in-$T$ specific heat.
Moreover, comparing Eq. (15) with Eq. (17) we find that the ratio between
$C_{V}$ and its noninteracting value $C^{(0)}_{V}$ is given by
$\lim_{T\to 0}\frac{C_{V}}{C^{(0)}_{V}}=\frac{v_{\rm F}}{v^{\star}}~{},$ (18)
a well-known property of normal Fermi liquids [12, 13]. We are thus led to
conclude, in full agreement with the zero-temperature calculations of the
quasiparticle energy and lifetime performed in Refs. [6, 7], that doped
graphene sheets are normal Fermi liquids. Note that the fact that interactions
enhance the quasiparticle velocity [see Eq. (17)] implies that the specific
heat of doped graphene sheets is suppressed with respect to its noninteracting
value.
The compressibility can be calculated from the following equation
$\frac{1}{n^{2}\kappa(T)}=\frac{1}{n^{2}\kappa_{0}(T)}+\frac{\partial^{2}[n\delta
f_{\rm int}(T)]}{\partial n^{2}}~{},$ (19)
where $\kappa^{-1}_{0}(T)$ is the inverse compressibility of the
noninteracting system at finite temperature. In the low-temperature limit
$1/[n^{2}\kappa_{0}(T\to 0)]=n\varepsilon_{\rm F}/2+gn\varepsilon_{\rm
F}\pi^{2}(T/T_{\rm F})^{2}/48$. The dependence of the ratio
$\kappa(T)/\kappa_{0}(T)$ on $\alpha_{\rm gr}$ and $T/T_{\rm F}$ is shown in
Fig. 5.
Figure 5: The dimensionless ratio $\kappa(T)/\kappa_{0}(T)$ as a function of
graphene’s coupling constant $\alpha_{\rm gr}$ for three values of $0\leq
T/T_{\rm F}\leq 0.2$.
## 4 Conclusions
In this work we have presented semi-analytical expressions for the real and
the imaginary parts of the density-density linear-response function of
noninteracting massless Dirac fermions at finite temperature. These results
are very useful to study finite-temperature screening within the Random Phase
Approximation. For example they can be used to calculate the conductivity at
finite temperature within Boltzmann transport theory and make quantitative
comparisons with recent experimental results in unsuspended [14, 15] and
suspended graphene sheets [16, 17].
The Lindhard function at finite temperature is also extremely useful to
calculate finite-temperature equilibrium properties of interacting massless
Dirac fermions, such as the specific heat and the compressibility. For
example, in this work we have been able to show that, at low temperatures, the
specific heat of interacting massless Dirac fermions has the usual normal-
Fermi-liquid linear-in-temperature behavior, with a slope that is solely
controlled by the renormalized quasiparticle velocity.
M.P. was partly supported by the CNR-INFM “Seed Projects”.
## References
## References
* [1] Geim A K and Novoselov K S 2007 The rise of graphene Nature Mater. 6 183; Exploring graphene — Recent research advances, Solid State Commun.143 (2007), edited by Das Sarma S, Geim A K, Kim P, and MacDonald A H; Castro Neto A H, Guinea F, Peres N M R, Novoselov K S and Geim A K 2007 The electronic properties of graphene arXiv:0709.1163 [to appear in Rev. Mod. Phys.].
* [2] For a recent popular review see Geim A K and MacDonald A H (2007) Graphene: exploring carbon flatland Phys. Today 60 35\.
* [3] Slonczewski J C and Weiss P R (1958) Band structure of graphite Phys. Rev.109 272; Ando T, Nakanishi T and Saito R (1998) Berry’s phase and absence of back scattering in carbon nanotubes J. Phys. Soc. Jpn. 67 2857\.
* [4] González J, Guinea F and Vozmediano M A (1996) Unconventional quasiparticle lifetime in graphite Phys. Rev. Lett.77 3589; Gonzáles J, Guinea F and Vozmediano M A (1999) Marginal-Fermi-liquid behavior from two-dimensional Coulomb interaction Phys. Rev. B 59 R2474 .
* [5] Barlas Y, Pereg-Barnea T, Polini M, Asgari R and MacDonald A H (2007) Chirality and correlations in graphene Phys. Rev. Lett.98 236601\.
* [6] Polini M, Asgari R, Barlas Y, Pereg-Barnea T and MacDonald A H (2007) Graphene: a pseudochiral Fermi liquid Solid State Commun.143 58\.
* [7] Polini M, Asgari R, Borghi G, Barlas Y, Pereg-Barnea T and MacDonald A H (2008) Plasmons and the spectral function of graphene Phys. Rev. B 77 081411(R).
* [8] Hwang E H and Das Sarma S (2007) Dielectric function, screening, and plasmons in two-dimensional graphene Phys. Rev. B 75 205418; Das Sarma S, Hwang E H and Tse W-K (2007) Many-body interaction effects in doped and undoped graphene: Fermi liquid versus non-Fermi liquid Phys. Rev. B 75 121406(R); Hwang E H, Hu B Y-K and Das Sarma S (2007) Inelastic carrier lifetime in graphene Phys. Rev. B 76 115434; Hwang E H, Hu B Y-K and Das Sarma S (2007) Density dependent exchange contribution to $\partial\mu/\partial n$ and compressibility in graphene Phys. Rev. Lett.99, 226801.
* [9] Polini M, Tomadin A, Asgari R and MacDonald A H (2008) Density functional theory of graphene sheets Phys. Rev. B 78, 115426.
* [10] Rossi E and Das Sarma S (2008) Ground State of graphene in the presence of random charged impurities Phys. Rev. Lett.101 166803\.
* [11] Vafek O (2007) Anomalous thermodynamics of Coulomb-interacting massless Dirac fermions in two spatial dimensions Phys. Rev. Lett.98 216401\.
* [12] Pines D and Noziéres P (1966) The Theory of Quantum Liquids (Addison-Wesley: Menlo Park).
* [13] Giuliani G F and Vignale G (2005) Quantum Theory of the Electron Liquid (Cambridge University Press: Cambridge).
* [14] Morozov S V, Novoselov K S, Katsnelson M I, Schedin F, Elias D C, Jaszczak J A and Geim A K (2008) Giant intrinsic carrier mobilities in graphene and its bilayer Phys. Rev. Lett.100 016602\.
* [15] Chen J H, Jang C, Xiao S, Ishigami M and Fuhrer M S (2008) Intrinsic and extrinsic performance limits of graphene devices on ${\rm SiO}_{2}$ Nature Nanotechnology 3 206\.
* [16] Bolotin K I, Sikes K J, Hone J, Stormer H L and Kim P (2008) Temperature-dependent transport in suspended graphene Phys. Rev. Lett.101 096802\.
* [17] Du X, Skachko I, Barker A and Andrei E Y (2008) Approaching ballistic transport in suspended graphene Nature Nanotechnology 3 491\.
|
arxiv-papers
| 2008-11-11T15:42:21 |
2024-09-04T02:48:58.732688
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M.R. Ramezanali, M.M. Vazifeh, Reza Asgari, Marco Polini, A.H.\n MacDonald",
"submitter": "Reza Asgari",
"url": "https://arxiv.org/abs/0811.1723"
}
|
0811.1921
|
# Symmetry-Breaking and Symmetry-Restoring Dynamics of a Mixture of Bose-
Einstein Condensates in a Double Well
Indubala I Satija and Philip Naudus (George Mason University, Fairfax, VA),
Radha Balakrishnan ( Institute of Mathematical Sciences, Chennai, India),
Jeffrey Heward ,Mark Edwards( Department of Physics, Georgia Southern
University, Statesboro, GA ) and and Charles W Clark (National Institute of
Standards and Technology, Gaithersburg, Maryland )
###### Abstract
We study the coherent nonlinear tunneling dynamics of a binary mixture of
Bose-Einstein condensates in a double-well potential. We demonstrate the
existence of a new type of mode associated with the ”swapping” of the two
species in the two wells of the potential. In contrast to the symmetry
breaking macroscopic quantum self-trapping (MQST) solutions, the swapping
modes correspond to the tunneling dynamics that preserves the symmetry of the
double well potential. As a consequence of two distinct types of broken
symmetry MQST phases where the two species localize in the different potential
welils or coexist in the same well, the corresponding symmetry restoring
swapping modes result in dynamics where the the two species either avoid or
chase each other. In view of the possibility to control the interaction
between the species, the binary mixture offers a very robust system to observe
these novel effects as well as the phenomena of Josephson oscillations and pi-
modes.
## I Introduction
Ultracold laboratories have had great success in creating Bose-Einstein
condensates (BECs) ande in a variety of atomic gases such as Rubidium (Rb),
Lithium (Li), Sodium (Na) and Ytterbium (Yb). These quantum fluids exist in
various isotopic forms as well as in different hyperfine states. The rapid
pace of development in this field has led to condensates which are robust and
relatively easy to manipulate experimentally. In particular, the tunability of
inter-species and intra-species interactions thal via magnetic and optical
Feshbach resonances makes the BEC mixture a very attractive candidate for
exploring new phenomena involving quantum coherence and nonlinearity in a
multicomponent system.
The subject of this paper is to investigate the tunneling dynamics of a binary
mixture of BECs in a double well potential. A single species of BEC in a
double well is called a bosonic Josephson junction (BJJ), since it is a
bosonic analog of the well known superconducting Josephson junction. In
addition to Josephson oscillations (JO), the BJJ exhibits various novel
phenomena such as $\pi$-modes and macroscopic quantum self-trapping (MQST), as
predicted theoretically smer ; milb . In the JO and the $\pi$-modes, the
condensate oscillates symmetrically about the two wells of the potential. In
contrast to this, the MQST dynamics represents a broken symmetry phase as the
tunneling solutions exhibit population imbalance between the two wells of the
potential. These various features have been observed experimentally exptbjj .
Our motivation is to explore whether new phenomena arise when there are two
interacting condensates trapped in a symmetric double well.
Although our formulation and results are valid for a variety of BEC mixtures,
our main focus here is the Rb family of two isotopes, namely the mixture of
87Rb and 85Rb, motivated by the experimental setup at JILAthesis . The
scattering length of 87Rb is known to be $100$ atomic units while the
interspecies scattering length is $213$ atomic units. In experiments, the
scattering length of 85Rb can be tuned using the Feshbach resonance method
feshbach_reference .
The ability to tune the scattering length of one of the species makes this
mixture of isotopes an ideal candidate for studying the coupled BJJ system.
First, it opens up the possibility of exploring the parameter space where the
Rb 85–85 scattering length is equal to the Rb 87–87 scattering length. As will
be discussed below, this symmetric parameter regime simplifies the theoretical
analysis of the system and also captures most of the new phenomena that
underlie the dynamics of the binary mixture. Furthermore, the tunability of
the 85Rb scattering length can be exploited to study a unique possibility
where one of the species has a negative scattering length, a case which
strongly favors the $\pi$-mode oscillations that have not been observed so
far.
In our exploration of nonlinear tunneling dynamics of coupled BJJ systems, the
MQST states are found to be of two types. In the broken–symmetry MQST state,
the two components may localize in different wells resulting in a phase
separation or they may localize in the same well and hence coexist. By varying
the parameters such as initial conditions, the phase–separated broken–symmetry
MQST states can be transformed to a symmetry–restoring phase where the species
continually “avoid” each other by swapping places between the two wells. In
other words, if the dynamics is initiated with both species in the same
potential well, the sustained tunneling oscillations are seen where the two
species swap places between the well one and the well two. From the coexisting
MQST phase, one can achieve symmetry restoring swapping dynamics by initiating
the dynamics with two species in the separate wells. In this case, the
emergence of the swapping modes can be interpreted as a phase where the two
species “chase” each other.
The paper is organized as follows. In section II, we discuss the model and use
the two–mode approximation to the Gross–Pitaevskii (GP) equation to map it to
a system of two coupled pendulums with momentum–dependent lengths and
coupling. Section III discusses the stationary solutions and their stability.
These results enable us to look for various qualitatively different effects
without actually solving the GP equations. Section IV describes the numerical
solutions of the GP equations as various parameters of the system are tuned.
Although we have explored the multi-dimensional parameter space, the novelties
attributed to the binary mixture in a double well trap are presented in a
restricted parameter space where the scattering lengths of the two species are
equal. Additionally, in our numerical results described here, we fix the ratio
of Rb 87–87 interaction to Rb 85-87 interaction to be $2.13$. This restricted
parameter space is accessible in the JILA setup and provides a simple means to
describe various highlights of the mixture dynamics. Section V provides
additional details of the JILA setup relevant for our investigation. A summary
is given in section VI.
## II Two-mode GP Equation for the Binary Mixture
In the semiclassical regime where the fluctuations around the mean values are
small, the two-component BEC is described by the following coupled GP
equations for the two condensate wave functions $\Phi_{l}(x,t)$, with $l=a,b$
representing the two species in the mixture.
$\displaystyle i\hbar\dot{\Phi}_{a}$ $\displaystyle=$
$\displaystyle(-\frac{\hbar^{2}}{2m_{a}}\nabla^{2}+V_{a})\Phi_{a}+(g_{a}|\Phi_{a}|^{2}+g_{ab}|\Phi_{b}|^{2})\Phi_{a}$
$\displaystyle i\hbar\dot{\Phi}_{b}$ $\displaystyle=$
$\displaystyle(-\frac{\hbar^{2}}{2m_{b}}\nabla^{2}+V_{b})\Phi_{b}+(g_{b}|\Phi_{b}|^{2}+g_{ab}|\Phi_{a}|^{2})\Phi_{b}.$
Here, $m_{l}$, $V_{l}$ and $g_{l}=4\pi\hbar^{2}a_{l}/m_{l}$, denote
respectively, the mass, the trapping potential and the intra-atomic
interaction of each species, with $a_{l}$ as the corresponding scattering
length. $g_{ab}=2\pi\hbar^{2}(1/m_{a}+1/m_{b})a_{ab}$ is the inter-species
interaction, where $a_{ab}$ is the corresponding scattering length. For the
JILA experiment, in view of the tight confinement of the condensate transverse
to the trap, it is sufficient to consider the corresponding one-dimensional
GPE equations.
The condensate wave functions satisfy the normalization conditions,
$\int d^{3}r|\Phi_{l}|^{2}=N_{l}$ (1)
The total number of atoms in the mixture is $N=N_{a}+N_{b}$. In the weakly
linked limit, the dynamical oscillations of the two-component BEC can be
described by two wave functions representing the condensate in each trap
labeled by $k=1,2$, with the spatial and the temporal contribution factored as
follows:
$\displaystyle\left(\begin{array}[]{cc}\Phi_{a}\\\
\Phi_{b}\end{array}\right)=\left(\begin{array}[]{cc}\chi^{a}_{1}(x)\psi^{a}_{1}(t)\\\
\chi^{b}_{1}(x)\psi^{b}_{1}(t)\end{array}\right)+\left(\begin{array}[]{cc}\chi^{a}_{2}(x)\psi^{a}_{2}(t)\\\
\chi^{b}_{2}(x)\psi^{b}_{2}(t)\end{array}\right)$ (8)
The localized spatial modes $\chi^{(l)}_{k}(x)$ are computed as sums and
differences of the symmetric and antisymmetric solutions of the
time–independent, coupled GP equations.
To derive the equations of motion in the two–mode approximation, we introduce
$z_{l}(t)$, the population imbalance, and $\phi_{l}(t)$, the relative phase of
of species $l$ between the left and right sides of the double well potential,
$z_{l}\left(t\right)=(|\psi^{l}_{1}|^{2}-|\psi^{l}_{2}|^{2})/N,$ (9)
$\phi_{l}\left(t\right)=(\theta^{l}_{1}-\theta^{l}_{2}).$ (10)
where $\psi^{(l)}_{k}(t)=|\psi^{(l)}_{k}(t)|\exp{i\theta^{(l)}_{k}}$ are the
time–dependent coefficients in the two–mode equations .Substituting equations
(8) into the coupled GP equations and integrating over the spatial degrees of
freedom, we obtain the following four coupled, nonlinear, ordinary
differential equations which we refer to as the “two–mode” model.
$\displaystyle\dot{Z}_{a}$ $\displaystyle=$
$\displaystyle-\bar{K}_{a}\sqrt{1-Z_{a}^{2}}\,\,\sin\phi_{a}$ (11)
$\displaystyle\dot{Z}_{b}$ $\displaystyle=$
$\displaystyle-\bar{K}_{b}\sqrt{1-Z_{b}^{2}}\,\,\sin\phi_{b}$ (12)
$\displaystyle\dot{\phi}_{a}$ $\displaystyle=$
$\displaystyle\bar{\Lambda}_{a}f_{a}Z_{a}+\Lambda_{ab}f_{b}Z_{b}+\bar{K}_{a}\frac{Z_{a}}{\sqrt{1-Z_{a}^{2}}}\,\,\cos\phi_{a}$
(13) $\displaystyle\dot{\phi}_{b}$ $\displaystyle=$
$\displaystyle\bar{\Lambda}_{b}f_{b}Z_{b}+\Lambda_{ab}f_{a}Z_{a}+\bar{K}_{b}\frac{Z_{b}}{\sqrt{1-Z_{b}^{2}}}\,\,\cos\phi_{b}.$
(14)
where $Z_{l}=z_{l}/f_{l}$. In the above equations, $f_{l}=N_{l}/N$ denotes the
fraction of atoms of species $l$, while $\bar{K}_{l}$ and $\bar{\Lambda}_{l}$
are given by
$\displaystyle\bar{K}_{a}$ $\displaystyle=$ $\displaystyle
K_{a}-2f_{a}C_{a}\sqrt{1-Z_{a}^{2}}\cos\phi_{a}+f_{b}D_{ab}\sqrt{1-Z_{b}^{2}}\cos\phi_{b}$
$\displaystyle\bar{K}_{b}$ $\displaystyle=$ $\displaystyle
K_{b}-2f_{b}C_{b}\sqrt{1-Z_{b}^{2}}\cos\phi_{b}+f_{a}D_{ba}\sqrt{1-Z_{a}^{2}}\cos\phi_{a}$
$\displaystyle\bar{\Lambda}_{a}$ $\displaystyle=$
$\displaystyle\Lambda_{a}+C_{a}$ $\displaystyle\bar{\Lambda}_{b}$
$\displaystyle=$ $\displaystyle\Lambda_{b}+C_{b}$
In the above, the space and time-independent parameters $K_{l}$,
$\Lambda_{l}$, $\Lambda_{ab}$, $C_{l}$ and $D_{ab}$ can be expressed in terms
of various microscopic parameters that appear in GP equation and the localized
modes, $\chi^{l}_{k}(x)$ and their overlap (integrated over spatial degrees of
freedom). The explicit expressions for these parameters are given in the
Appendix.
The parameters $K_{l}$ describe the tunneling amplitude while $\Lambda_{l}$ is
related to the corresponding scattering length of the species. The parameters
$C_{l}$ and $D_{ab}$ have their origin in the overlaps between the spatial
modes $\chi_{1}$ and $\chi_{2}$, and are expected to be small in the weak
tunneling limit. These overlaps modify the bare parameters denoted by the
interaction $\Lambda_{l}$ and the tunneling $K_{l}$. Consequently, we have a
variable tunneling model, since the tunneling parameters $\bar{K}_{l}$ depend
explicitly on the dynamical variables $Z_{l}$ and $\phi_{l}$.
In our analysis, we will mostly restrict ourselves to the case where the two
species are equally populated, namely $f_{a}=f_{b}=1/2$. In this case, the
above system of equations (14) can be viewed as the Hamilton equations in
terms of the canonical variables $Z_{l}$ (momenta) and $\phi_{l}$ (co-
ordinates), with the Hamiltonian given by the following form:
$H=\frac{1}{2}[\bar{\Lambda}_{a}Z_{a}^{2}+\bar{\Lambda}_{b}Z_{b}^{2}+2\Lambda_{ab}Z_{a}Z_{b}]-\sum_{l=a,b}\bar{K}_{l}\sqrt{1-Z_{l}^{2}}\,\,\cos\phi_{l}.$
(15)
For the case where the overlap between the spatial modes $\chi_{1}$ and
$\chi_{2}$ can be neglected, and the effective tunneling $\bar{K}_{l}$ can be
replaced by its bare value $K_{l}$ the above system can be viewed as a coupled
pair of non-rigid pendulums, with momentum–dependent lengths. The coupling
between the pendulums is also momentum dependent.
We parenthetically remark that this system can also be mapped to a pair of
classical spins with Cartesian components
$\displaystyle S^{l}_{x}$ $\displaystyle=$
$\displaystyle\sqrt{1-Z_{l}^{2}}\,\,\cos\,\phi_{l}$ $\displaystyle S^{l}_{y}$
$\displaystyle=$ $\displaystyle\sqrt{1-Z_{l}^{2}}\,\,\sin\,\theta_{l}$
$\displaystyle S^{l}_{z}$ $\displaystyle=$ $\displaystyle Z_{l},$
so that $(S^{l})^{2}=1$. Thus the spin vector locates a point on the unit
sphere given by polar angles $\theta_{l},\phi_{l}$, with
$Z_{l}=\cos\theta_{l}$. The corresponding spin Hamiltonian, written in terms
of bare variables, can be shown to be
$\displaystyle
H=\sum_{l=a,b}[\frac{1}{2}(\Lambda_{l}+C_{l})(S^{l}_{z})^{2}+C_{l}(S^{l}_{x})^{2}-K_{l}S^{l}_{x}]$
$\displaystyle+\Lambda_{ab}(S^{a}_{z}S^{b}_{z})-D_{ab}(S^{a}_{x}S^{b}_{x}).$
The spin mapping provides an alternative means to visualize the effective
interaction between the two species during the tunneling. If we ignore the
spatial overlap integrals between the localized modes in two wells, (
$C_{l}=0$, $D_{ab}=0$ ), the binary mixture of condensates in two-mode
approximation, maps to two Ising-like spins in a transverse magnetic field.
The full two-mode variable tunneling feature induces XY-like spin interaction.
In this paper, we find it convenient to exploit mapping to the coupled
pendulums, for exploring tunneling dynamics in the coupled BJJ. Although we
have explored the full two-mode variable tunneling model, we will only discuss
the constant tunneling case ( $\bar{K}_{l}$ replaced by $K_{l}$ and
$\bar{\Lambda}_{l}$ replaced by $\Lambda_{l}$.), as the overlap integrals are
small and the various novel effects of the mixture described here are found to
be robust and unaffected by the variable tunneling parameters.
## III Stationary Solutions: Fixed Points
The solutions of the coupled system are characterized by the interactions
$\Lambda_{l}$, the ratio of the tunneling amplitude for the two species,
$K_{a}/K_{b}$ which we denote by $R$ as well as the initial phase difference
$\phi_{l}(t=0)$ and the initial population imbalance $Z_{l}(t=0)$. In the
multi– dimensional parameter space the equilibrium or fixed–point solutions,
in which the right–hand-sides of Eqs. (LABEL:ftun)-(LABEL:tun) are zero,
provide an effective tool to classify different categories of behavior of the
system.
In general, these fixed–point equations are transcendental and have to be
solved numerically. However, in the symmetric case where
$\Lambda_{a}=\Lambda_{b}=\Lambda$, , $K_{a}=K_{b}=K$, the fixed point
equations can be tackled analytically. Further, as can be seen from Eqs.
(LABEL:ftun)–(LABEL:tun), the parameter $K$ can be eliminated in this case by
rescaling $t$ ($t\rightarrow Kt$) and redefining $\Lambda_{x}$ as
$\Lambda_{x}\rightarrow\Lambda_{x}/K$. Our detailed analysis shows that this
special case captures many relevant phenomena characterizing the binary
mixture in a double well. In this case, the fixed points belong to two broad
categories as stated below, resulting in two types of small amplitude
oscillations about these two fixed points. It is important to note that this
type of MQST phase does not exist in a BJJ with a single species.
(I) Zero-mode Fixed Points ($\phi^{*}_{a}=\phi^{*}_{b}=0)$
(1) $Z^{\ast}_{a}=Z^{*}_{b}=0$
(2)
$Z^{*}_{a}=-Z^{*}_{b}=\pm\frac{\sqrt{(\Lambda_{ab}-\Lambda)^{2}-4K^{2})}}{|(\Lambda_{ab}-\Lambda)|}$
(II) $\pi$-mode Fixed Points ( $\phi^{*}_{a}=\phi^{*}_{b}=\pi$ )
(1) $Z^{*}_{a}=Z^{*}_{b}=0$
(2)
$Z^{*}_{a}=Z^{*}_{b}=\pm\frac{\sqrt{(\Lambda_{ab}+\Lambda)^{2}-4K^{2})}}{|(\Lambda_{ab}+\Lambda)|}$
It should be noted that the mixed-mode Fixed Points, ($\phi^{*}_{a}=0$ and
$\phi^{*}_{b}=\pi)$ , ($Z^{*}_{a}=Z^{*}_{b}=0$)
are unstable for the restricted parameter regime we are considering here and
hence will not be discussed.
The small oscillations about the fixed point ($Z^{\ast}=0,\phi^{\ast}=0$)
result in Zero-mode while small oscillations about
($Z^{\ast}=0,\phi^{\ast}=\pi$) lead to $\pi$-mode. The oscillation frequencies
are in the next subsection.
The non–trivial fixed points ($Z^{\ast}\neq 0$ ) result in solutions with
population imbalance and lead to tunneling dynamics with macroscopic quantum
self-trapping or the MQST. In view of the $a-b$ symmetry, we have two sets of
stationary solutions: $Z_{x}^{\ast}$ and $-Z_{x}^{\ast}$), ($x=a,b$) This
suggests the possibility of modes where each species oscillates about the
binary fixed points, going back and forth between the two wells. Unlike MQST,
these modes will preserve the symmetry of the double well. However, in
contrast to Zero-modes, these modes are non-linear and give rise to ”swapping
phase” that will be discussed later.
The emergence of fixed points with opposite signs for the two species,
($Z_{a}^{\ast}=-Z_{b}^{\ast}$) in the Zero mode phase suggests that MQST in
Zero–mode is accompanied by phase separation of the two species. In contrast,
in the $\pi$-mode MQST phase , the two species could coexist in the same
potential well as ($Z_{a}^{\ast}=Z_{b}^{\ast}$). Therefore, the fixed point
equations suggest that $\pi$-modes mimic attractive interaction between the
two species.
The onset from oscillatory to MQST phase corresponds to the values of the
parameters where the non–trivial fixed points move from the complex to the
real plane. Alternatively, the condition for the broken symmetry phase can be
obtained by linear stability analysis of the fixed–point equations. This is
discussed in the next sub-section.
In the asymmetric case when $\Lambda_{a}\neq\Lambda_{b}$ the fixed points are
obtained by solving the coupled transcendental equations:
$\displaystyle(-1)^{p}\frac{K_{a}Z^{*}_{a}}{\sqrt{1-(Z^{*}_{a})^{2}}}+\frac{1}{2}(\Lambda
Z^{*}_{a}+\Lambda_{ab}Z^{*}_{b})$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle(-1)^{p}\frac{K_{b}Z^{*}_{b}}{\sqrt{(}1-(Z^{*}_{b})^{2})}+\frac{1}{2}(\Lambda
Z^{*}_{b}+\Lambda_{ab}Z^{*}_{a})$ $\displaystyle=$ $\displaystyle 0,$
where $p=0(1)$ for $\phi^{*}_{a}=0(\pi)$ and $\phi^{*}_{b}=0(\pi)$..
Analogous to the symmetric case, both the Zero and the $\pi$-mode solutions
including those corresponding to MQST can be found numerically. As expected,
for the MQST fixed points $Z^{*}_{a}\neq-Z^{*}_{b}$ in the Zero-mode and
$Z^{*}_{a}\neq Z^{*}_{b}$ in the $\pi$-mode and we do not have the permutation
symmetry or the $a-b$ symmetry. However, unlike the symmetric case, $K_{l}$s
do not scale time $t$ and the parameters and hence the ratio
$R=\frac{K_{a}}{K_{b}}$ emerges as a new parameter.
### III.1 Normal Modes: Linear Stability Analysis of Fixed Points
Frequencies of small amplitude oscillations about about ($Z^{*}=0,\phi^{*}=0$)
and ($Z^{*}=0,\phi^{*}=\pi$) respectively referred to as the Zero-mode or the
$\pi$-modes are given by
$\displaystyle\omega^{2}=\frac{1}{2}(K_{a}\Lambda^{*}_{a}+K_{b}\Lambda^{*}_{b})+\frac{1}{2}\pm\sqrt{(K_{a}\Lambda^{*}_{a}-K_{b}\Lambda^{*}_{b})^{2}+4K_{a}K_{b}\Lambda^{2}_{ab}}]$
where
$\displaystyle\Lambda^{*}_{a}$ $\displaystyle=$
$\displaystyle(-1)^{p}K_{a}+f_{a}\Lambda_{a}$ $\displaystyle\Lambda^{*}_{b}$
$\displaystyle=$ $\displaystyle(-1)^{p}K_{b}+f_{b}\Lambda_{b},$
where $p=0$ for the Zero-mode , and $p=1$ for the $\pi$-mode. In the symmetric
case, with $\Lambda_{a}=\Lambda_{b}$ and $f_{a}=f_{b}$, the normal mode
frequencies $\omega_{0}$ and $\omega_{\pi}$ simplify to,
$\displaystyle\omega^{2}_{0}$ $\displaystyle=$ $\displaystyle
K^{2}+K(\Lambda\pm\Lambda_{ab})/2$ $\displaystyle\omega^{2}_{\pi}$
$\displaystyle=$ $\displaystyle K^{2}-K(\Lambda\pm\Lambda_{ab})/2$
The condition for the instability of the fixed point is determined when one of
the normal mode frequencies become complex. This gives rise to new fixed
points where $Z_{x}^{*}\neq 0$ resulting in MQST phase where there is a
population imbalance between the two wells of the double well potential for
each species. The condition for the existence of MQST is given by,
$\displaystyle f_{a}f_{b}\Lambda^{2}_{ab}$ $\displaystyle\geq$
$\displaystyle\Lambda^{*}_{a}\Lambda^{*}_{b}$ (16)
For the parameter values where both the Zero and the $\pi$-modes coexist,
$\pi$-mode frequencies are smaller than the Zero-mode frequencies.
Figure 1: (color online) The upper(black) and the lower(yellow) shaded regime
corresponds to the parameter values for the existence of stable Zero-mode and
$\pi$-mode with $R=1$.
Figure 1 shows the values of $\Lambda_{a},\Lambda_{b}$ where the tunneling is
governed by the Zero-mode and the $\pi$-mode. For $\Lambda_{b}>0$, the regime
where the $\pi$-modes exist is is small but finite. However, by tuning
$\Lambda_{b}$ to negative values, the $\pi$-modes that have not been seen in
earlier studies, can be observed. Variation with the parameter $R$, the
tunneling ratio for the two species, leads to similar results, with the
parameter space for the existence of $\pi$ mode increasing slightly with $R$.
The unshaded regime corresponds to MQST phase.
## IV Tunneling Dynamics with $\Lambda_{a}=\Lambda_{b}$
We now describe numerical solution of the tunneling equations. For small
population imbalance, we confirm the dynamics predicted by the fixed points as
discussed above. However, numerical solutions also illustrate nonlinear modes,
not described by the fixed point analysis. The fact that new features continue
to exist in the nonlinear regime, assures their robustness.
In our numerics, we set $\Lambda_{ab}=2.13\Lambda$ and study the dynamics for
different values of $\Lambda$. These conditions can be achieved by first
tuning the $g_{b}$ via a Feshbach resonance so that $\Lambda_{a}=\Lambda_{b}$.
The variation of $\Lambda$ corresponds to varying the number of atoms in the
double well trap. As already mentioned, $K$ can be eliminated by using
$t\rightarrow Kt$ and $\Lambda\rightarrow\Lambda/K$. The dynamics is governed
by $\Lambda$ and the initial conditions: $Z_{a}(0)$, $Z_{b}(0)$, $\phi_{a}(0)$
and $\phi_{b}(0)$.
As we discuss below, tunneling solutions belong to three broad categories:
(I) ” Zero-phase Mode ” , characterized by $<\phi_{l}>=0$
(II) ”$\pi$-phase Mode ” characterized by $<\phi_{l}>=\pi$
(III) ” Running-Phase Mode ” characterized by $<\phi_{l}>$ proportional to $t$
In the single species case, $<\phi_{l}>=0$ also corresponds to $<Z_{l}>=0$.
However, as we discuss below, in a binary mixture, we can have $<\phi_{l}>=0$
but $<Z_{l}>\neq 0$. This gives rise to a broken symmetry MQST phase in Zero-
modes as well.
### IV.1 Zero-Modes
For $\Lambda<\Lambda^{0}_{c}\approx 1.77$, and $\phi_{a}(0)=\phi_{b}(0)=0$,
and $|Z_{l}(0)|<<1$, both species execute small amplitude oscillations (like
oscillations of a non-rigid pendulum) with, $<Z_{l}(t)>=0$ and
$<\phi_{l}(t)>=0$ as shown in Fig. 2. Such modes exhibit quasiperiodic
dynamics characterized by superposition of sinusoidal modes with two competing
frequencies. As $Z_{l}(0)$ increases, we see large amplitude non-sinusoidal
oscillations. Therefore, in spite of the repulsive interaction between the two
condensates, the two species execute a coherent oscillatory dynamics as
expected from the zero-mode fixed point analysis described earlier.
Figure 2: (color online) Time series for $Z_{l}$(top), $\phi_{l}$ (middle) and
phase portrait for $\Lambda=0.6$ with initial conditions shown in the figure.
The red and blue corresponds to the $a$ and $b$ species.
### IV.2 $\pi$-modes
If the dynamics is initiated with $\phi_{a}(t=0)=\phi_{b}(t=0)=\pi$, both
species oscillate in $\pi$-mode provided $\Lambda<\Lambda^{0}_{c}\approx
0.67$, and initial population imbalance is small ( $|Z_{l}(0)|<<1$) .
Analogous to the Zero-mode, the dynamics in the $\pi$ mode is in general
quasiperiodic. As seen in the figure, the motion is in phase with the slow
mode and out of phase with the other. Comparison with the Zero and the
$\pi$-mode oscillations show that species move more sluggishly in $\pi$-mode
compared to the Zero-mode as the Zero-mode frequencies are larger than those
of the $\pi$-mode.
Figure 3: (color online) Same parameters as Fig. 2, the only exception being
that $\phi_{l}(t=0)=\pi$ here.
### IV.3 Symmetry Breaking and Phase Separation: MQST in Zero-Mode
Figure 4: (color online) Transition from Josephson oscillations (top with
$\Lambda=1.6$ ) to MQST with phase separation (bottom, with $\Lambda=1.8$ ),
obtained by varying $\Lambda$. The left and right plots show the time series
and phase portraits respectively.
Beyond a critical value of $\Lambda$, the system enters the symmetry breaking
MQST phase, as predicted by the fixed point analysis earlier. One of the novel
aspects of the binary mixture is the existence of Zero-mode MQST accompanied
by phase separation of the two species. Even with the initial conditions
corresponding to both species abundance in the same well, the two components
localize in the two different wells. In this case, transition to MQST is
accompanied by phase separation: although the two species overlap for some
time, the $<Z_{a}(t)>$ and the $<Z_{b}(t)>$ have opposite signs.
### IV.4 Symmetry Restoring and Phase Separation: Swapping-Mode
As $\Lambda$ increases further,the system exhibits ”swapping-modes” where the
two species swap places between the two wells but remain phase separated as
shown in Fig. 4. As seen in the figure (at $t=0$), the dynamics is initiated
with positive population imbalance of both species. However, the resulting
dynamics corresponds to back and forth motion where the two species swap
places between the two wells. In contrast to MQST, the swapping dynamics
restores the symmetry of the tunneling solution in the double well. However,
the two species remain mostly phase separated, avoiding each other by
swapping.
In other words, the swapping phase is characterized by
$<Z_{a}(t)>=<Z_{b}(t)=0$, but $<Z_{a}(t)Z_{b}(t)<0>$ . That is, at a given
instant of time, the two species are more likely to be found in the separate
wells. Thus in the swapping mode, the two species oscillate back and forth
between the two wells and still manage to avoid each other. The swapping is
found to occur in the nonlinear Zero-mode as well as in the running mode .
Furthermore, a transition from MQST to swapping phase can be achieved either
by varying $\Lambda$ ( Fig. 5) or by varying the initial conditions ( Fig. 6).
Figure 5: (color online) Symmetry restoring transition by varying $\Lambda$
where the upper panel with $\Lambda=2.3$ shows MQST phase with phase
separation while the lower panel with $\Lambda=2.5$ shows phase separation due
to swapping mode. Figure 6: (color online) Symmetry restoring transition
obtained by changing initial conditions ($Z_{l}(t=0)$) slightly for fixed
$\Lambda=2$. Figure shows the time series as well as the phase portraits where
the straight line and crosses show the corresponding fixed points.
### IV.5 Symmetry Breaking in $\pi$-Modes: Coexistence Phase
Figure 7: (color online) Transition to MQST in $\pi$ modes. $\Lambda=0.6$
(top) and $\Lambda=0.8$ (bottom) describe, respectively, the small amplitude
$\pi$-mode oscillations and MQST in $\pi$-mode.
For $\Lambda<\Lambda^{\pi}_{c}\approx 0.67$, and
$\phi_{a}(t=0)=\phi_{b}(t=0)=\pi$, and $|Z_{l}(t=0)|<<1$,both species execute
small amplitude oscillations with $<Z_{l}(t)>=0>$ and $<\phi_{l}(t)>=\pi$, as
shown in Fig. 7 Such modes are characterized by superposition of sinusoidal
modes with two competing frequencies and the resulting dynamics is in general
quasiperiodic. As expected from the fixed point analysis, the two species with
both inter and intra-species repulsive interaction can self-trap in the same
well. That is , we have MQST where the species coexist in the same potential
well, inspite of repulsive interaction among them.
### IV.6 Swapping in $\pi$-modes
Figure 8: (color online) Transition from broken symmetry (MQST in $\pi$ modes)
to symmetric configurations, obtained by changing the initial population
imbalance. The three plots correspond to three different initial conditions.
As illustrated in figure 8, within the $\pi$-mode phase, if the dynamics of
the two species is initiated in separate wells, that is, $Z_{a}(t=0)$ and
$Z_{b}(t=0)$ have opposite signs, the MQST phase can be destroyed when the
initial population imbalance increases beyond a critical value. The tunneling
solutions become symmetric as MQST is replaced by swapping modes. In this case
the swapping can be viewed as the two species ”chasing” each other.
It should be noted that the swapping dynamics in the Zero and the $\pi$-modes
is very similar. However, swapping in the Zero-mode corresponds to two species
avoiding each other while swapping in the $\pi$-mode corresponds to one
component chasing the other. This is because, in the Zero-mode, species prefer
residing in the separate wells while in the $\pi$-mode, they like to stay in
the same well. This unique type of coherence between the two different species
is one of the most fascinating aspect of the binary mixture dynamics in double
well potential.
## V Experimental Realization
The effects described in this paper should be realizable for condensate
mixtures that already exist in the laboratory. One example in particular is a
mixture of 85Rb and 87Rb atoms that has been created in several recent
experiments at JILA feshbach_reference ; papp_exp . This system is relevant to
the analysis in this paper because the scattering length, $a_{85-85}$, that
characterizes the interaction between 85Rb atoms is tunable by an external
magnetic field via a feshbach resonance centered at approximately 155 Gauss
burke_and_bohn . Additionally, the interspecies scattering length,
$a_{85-87}$, is also tunable with two feshbach resonances (for a
$|2,-2\rangle_{85}$/$|1,-1\rangle_{87}$ collision) located at approximately
$B=267$ Gauss and $B=356$ Gauss.
In the most recent experiment papp_exp , a 85Rb/87Rb BEC mixture was produced
by trapping a thermal–gas sample of the mixture and performing evaporative
cooling on the 87Rb which sympathetically cools the 85Rb. The cold gas mixture
is then transferred to an optical trap that provides tight confinement
transverse to the trapping beam and loose confinement along the beam. If an
additional pair of beams were applied along this direction as was done in the
Albiez experiment exptbjj , it would create a setup to which the analysis in
this paper would apply.
## VI Summary
Existence of a variety of BEC species with tunable inter and intra–species
scattering lengths makes BEC mixtures one of the most attractive candidates
for exploring novel phenomena involving quantum coherence and nonlinearity.
Our analysis, based on the two-mode GP equation for the two interacting
species of BEC in a double well trap unveils a variety of phenomena describing
broken symmetry as well as subsequent restoration of symmetry, as we change
the parameters or the initial conditions. Such coherence is found to exist
over a broad range of parameters, establishing the robustness of the effects.
To make direct comparison with experiments, we need to solve the coupled GP
equations to obtain various parameters of the effective coupled pendulum
system in terms of the microscopic parameters of the system and work in this
direction is in progress. Furthermore, by quantizing the Hamiltonian (coupled
pendulum or the spin Hamiltonian), we hope to study quantum dynamics of number
fluctuations that may code the emergence of new quantum phases in the system.
## References
* (1) M. H. Anderson et al, Science 269, 198 (1995); K. B. Davis et al Phys. Rev. Lett. 75, 3969 (1995); C. C. Bradley et al, Phys. Rev. Lett. 75, 1687 (1995).
* (2) G. Thalhammer, G. Barontini, L. Sarlo, J. Catani, F. Minardi and M. Inguscio, Phys. Rev. Lett. 100, 210402 (2008).
* (3) A. Smerzi, S. Fantoni, S. Giovanazzi and S. R. Shenoy, Phys. Rev. Lett. 79, 4950 (1997) ; S. Raghavan, A. Smerzi, S. Fantoni and S. R. Shenoy, Phys. Rev. A, 59, 620 (1999).
* (4) C. J. Milburn, J. Corney, E. M. Wright and D. F. Walls, Phys. Rev. A 55, 4318 (1997).
* (5) M. Albiez et al Phys. Rev. Lett. 95, 010402 (2005); See also, cond-mat/0604348.
* (6) Scott Papp, Ph.d thesis, University of Colorado, 2007.
* (7) S. B. Papp and C. E. Wieman, Phys. Rev. Lett. 97 180404 (2006).
* (8) D. Ananikian, and T. Bergeman, PRA, 74, 039905, 2006
* (9) Tin-Lun Ho, V.B. Shenoy, Phys. Rev. Lett. 77, 0031 (1996).
* (10) Jing Chen, Yan-Qing Guo, Hai-Jing Cao, He-Shan Song, Phys. Lett.A 360, 429 (2006).
* (11) S. Ashhab and C. Lobo, Phys. Rev. A, 66, 013609 (2002).
* (12) S. B. Papp, J. M. Pino, C. E. Wieman, Phys. Rev. Lett. 101, 040402 (2008).
* (13) J. P. Burke and J. L. Bohn, Phys. Rev. A 59, 1303 (1999); J. P. Burke, J. L. Bohn, B. D. Esry, C. H. Greene, Phys. Rev. Lett. 80 2097 (1998).
*
## Appendix A Two–mode equation parameters
With $\bar{g}_{x}=g_{x}N/\hbar$ ( $x=a,b,ab$ ), the various coupling constants
in the coupled equations can be shown to be given by,
$\displaystyle\gamma_{a(b)}^{\pm}$ $\displaystyle=$
$\displaystyle\bar{g}_{a(b)}\int[(\chi^{a(b)}_{\pm})^{4}]dr$
$\displaystyle\bar{\gamma}_{a(b)}$ $\displaystyle=$
$\displaystyle\bar{g}_{a(b)}\int[(\chi^{a(b)}_{+})^{2}(\chi^{a(b)}_{-})^{2}]dr$
$\displaystyle\Delta\gamma_{a(b)}$ $\displaystyle=$
$\displaystyle\gamma_{a(b)}^{-}-\gamma_{a(b)}^{+}$
$\displaystyle\Delta\gamma_{ab}$ $\displaystyle=$
$\displaystyle\bar{g}_{ab}\int[(\chi^{a}_{-}\chi^{b}_{-})^{2}-(\chi^{a}_{+}\chi^{b}_{+})^{2}]dr$
$\displaystyle\Delta\bar{\gamma}_{ab}$ $\displaystyle=$
$\displaystyle\bar{g}_{ab}\int[(\chi^{a}_{-}\chi^{b}_{+})^{2}-(\chi^{a}_{+}\chi^{b}_{-})^{2}]dr$
$\displaystyle\Lambda_{a}(b)$ $\displaystyle=$
$\displaystyle\bar{g}_{a(b)}\int[2(\chi^{a(b)}_{+}\chi^{a(b)}_{-})^{2}-1/4((\chi^{a(b)}_{-})^{2}-(\chi^{a(b)}_{+})^{2})^{2})]dr$
$\displaystyle\Lambda_{ab}$ $\displaystyle=$ $\displaystyle
2\bar{g}_{ab}\int(\chi^{a}_{+}\chi^{a}_{-}\chi^{b}_{+}\chi^{b}_{-})dr$
$\displaystyle\bar{K}_{a}$ $\displaystyle=$ $\displaystyle[\Delta
E-f_{a}\Delta\gamma_{a}-f_{b}D_{ab}]/\hbar$ $\displaystyle\bar{K}_{b}$
$\displaystyle=$ $\displaystyle[\Delta
E-f_{b}\Delta\gamma_{b}-f_{a}D_{ab}]/\hbar$ $\displaystyle C_{a}$
$\displaystyle=$
$\displaystyle(\gamma^{a}_{+}+\gamma^{a}_{-}-2\bar{\gamma_{a}})/2\hbar$
$\displaystyle C_{b}$ $\displaystyle=$
$\displaystyle(\gamma^{b}_{+}+\gamma^{b}_{-}-2\bar{\gamma_{b}})/2\hbar$
$\displaystyle D_{ab}$ $\displaystyle=$
$\displaystyle(\Delta\gamma_{ab}-\Delta\bar{\gamma}_{ab})/2\hbar$
For each species, the localized spatial modes $\chi^{(l)}_{\pm}(x)$ are
obtained by adding and subtracting the symmetric and the antisymmetric
solutions of the time–independent coupled GPE equations. The $\Delta E$ is the
difference in the chemical potential between the (symmetric) ground and the
(anti-symmetric) first excited state of the coupled time-independent GPE
equations.
|
arxiv-papers
| 2008-11-12T15:44:34 |
2024-09-04T02:48:58.741763
|
{
"license": "Public Domain",
"authors": "Indubala I Satija and Philip Naudus (George Mason University, Fairfax,\n VA), Radha Balakrishnan (Institute of Mathematical Sciences, Chennai, India),\n Jeffrey Heward, Mark Edwards (Department of Physics, Georgia Southern\n University, Statesboro, GA) and and Charles W Clark (National Institute of\n Standards and Technology, Gaithersburg, Maryland)",
"submitter": "Indu Satija",
"url": "https://arxiv.org/abs/0811.1921"
}
|
0811.1936
|
# Semiclassical investigation of revival phenomena in one dimensional system
Zhe-xian Wang 1 Eric J. Heller 2 1 Hefei National Laboratory for Physical
Sciences at Microscale and Department of Physics, University of Science and
Technology of China, Hefei, Anhui 230026, P. R. China
2 Department of Physics and Department of Chemistry and Chemical Biology,
Harvard University, Cambridge, Massachusetts 02138, USA
###### Abstract
In a quantum revival, a localized wavepacket re-forms or ”revives” into a
compact reincarnation of itself long after it has spread in an unruly fashion
over a region restricted only by the potential energy. This is a purely
quantum phenomenon, which has no classical analog. Quantum revival, and
Anderson localization, are members of a small class of subtle interference
effects resulting in a quantum distribution radically different from the
classical after long time evolution under classically nonlinear evolution.
However it is not clear that semiclassical methods, which start with the
classical density and add interference effects, are in fact capable of
capturing the revival phenomenon. Here we investigate two different one
dimensional systems, the infinite square well and Morse potential. In both
cases, after a long time the underlying classical manifolds are spread rather
uniformly over phase space and are correspondingly spread in coordinate space,
yet the semiclassical amplitudes are able to destructively interfere over most
of coordinate space and constructively interfere in a small region, correctly
reproducing a quantum revival. Further implications of this ability are
discussed.
###### keywords:
Quantum revival , Semiclassical , Infinite square well , Morse potential
###### PACS:
03.65.Sq , 42.50.Md
## 1 Introduction
The phenomenon of ”quantum revival” attracted much attention after it was
first studied in quantum electrodynamics [1, 2]. The evolution of a quantum
wave packet in a general smooth potential has at least three regimes. First,
an initially localized packet will evolve following classical mechanics for a
time, in the sense that the mean position and momentum of the wave packet
follow classical laws. More than that, the spreading of the wave packet
follows an analogous classical distribution with appropriate initial position
and momentum densities. This is the Ehrenfest regime.
After further evolution, after the wave packet has become delocalized,
interference effects may become important, causing the classical distribution
and the quantum wave packet to have quite different details. Semiclassical
methods however are expected to be working well. They are based solely on
classical information, but incorporate interference effects by assigning an
amplitude and phase for the multiple classical paths which connect to each
final position:
${\psi(x,t)=\sum_{n}\sqrt{P_{n}(x,t)}\ e^{i\,\phi_{n}(x,t)/\hbar}}$ (1)
where $P_{n}(x)$ is the classical probability density for the $n^{th}$ way of
reaching $x$ give the initial classical manifold and $\phi_{n}(x,t)$ is the
classical action along the $n^{th}$ path reaching $x$. The Born
interpretation, namely that $\psi(x,t)$ is a probability amplitude, dictates
that the wavefunction should go as the square root of the classical
probabilities in the correspondence limit.
After a very long period of time, many classical periods in the case of an
oscillator, the quantum wave packet will reverse its seemingly unorganized
delocalized oscillation to neatly rebuild into its initial form. This is the
known quantum revival, the third regime. Quantum revival has been widely
investigated in atomic [3, 4, 5] and molecular [6, 7, 8] wave packet evolution
and other quantum mechanics systems [9, 10, 11, 12, 13]. An excellent review
on wave packet revivals is given by Robinett [14]. Precursors to the full
revival also exist, in which other organized probability distributions develop
[14]. The question addressed in this paper is: is the third, revival regime
also semiclassical? May we think of revival in semiclassical terms after all,
i.e. classical mechanics with phase interference included? It is a tall order
for semiclassical sums to self cancel almost everywhere the classical density
is large, with the exception of one region where the revival is occurring.
Time dependent semiclassical methods are exact in the limit of short time,
being equivalent to the short time limit of the quantum propagator. Increasing
time can only degrade the results. At long times, the number of terms in the
sum, Eq. 1 can become very large, and in fact the number of terms grows
exponentially in chaotic systems. This in itself does not spell the breakdown
of semiclassics. In earlier work on chaotic systems, Tomsovic et. al. [15]
showed that semiclassical amplitudes were doing well when more than 6000 terms
were needed in the sum. Other work justified the unexpected accuracy of the
semiclassical results [16]. Later, Kaplan [17] gave an ingenious analysis of
the breakdown with time in the case of chaotic systems, which built on earlier
the analyses [16] indicating that classical chaos rather surprisingly aided
accurate semiclassical propagation. The implication was that even Anderson
localization was describable semiclassically, albeit with an astronomical
number of terms in the sum, Eq. 1. Quantum revival in a potential well does
not involve chaotic spreading in phase space, and thus it could be more
difficult to describe correctly semiclassically, give the arguments in the
above references about the benefits of chaotic flow.
The revival phenomenon has no purely classical analog. At best it is a
semiclassical effect, describable in the terms of Eq. 1. The classical analog
of a localized wave packet will be a continuous density of trajectories in
phase space, well localized but consistent with the uncertainly principle. In
an anharmonic oscillator, these trajectories occupy a distribution of energies
and hence frequencies. The distribution spreads and begins to wind itself up
on a spiral (see below), with many branches at a typical position. A smooth
distribution of trajectories with a range of velocities and positions, after
spreading evenly into the available space, will never converge again on one
locale. This seems quite contradictory to the quantum result. Semiclassical
theory can bridge the gap between classical and quantum field, and provide a
simple and intuitive way to understand the subtle issue of quantum revival.
In this paper we study the quantum revival in both infinite square well and
Morse potential system. These two cases are quite different in detail. The
square well is locally linear, interrupted by discontinuities which are due to
reflections at the walls. The Morse potential is more typical, arriving at its
nonlinear evolution smoothly. Semiclassical results are analytic whenever the
dynamics is ”linear”. Examples are the free particle, the linear ramp
potential, and the harmonic oscillator. In each case, current positions and
momenta are linear functions of initial positions and momenta. The square well
is not in fact a linear system, because of reflections at the walls. However,
locally, the classical manifolds evolve linearly, suffering truncation due to
the reflections (see Figure 1). Interestingly the square well is a case with
(globally) nonlinear time evolution clearly showing revivals, yet because of
the locally linear nature of the classical dynamics the semiclassical formula
turns out to be exact. When the semiclassical method is approximate, the
delicate cancellation of amplitudes over wide areas is in question, and we
show here by example that it is still accurate enough to give the revivals.
## 2 Theory
Time-dependent semiclassical methods face difficulties when applied to long
revival time calculations. By their very nature, revivals cannot happen until
the classical manifolds have folded over on themselves many times, which means
the dynamics is in the deeply nonlinear regime. Although nothing keeps
semiclassical methods from working under these conditions in principle, and
practice the error can only grow with time. If one is looking at a subtle
phenomenon, such as near exact cancellation of semiclassical amplitudes over a
wide area, the small errors could be a problem.
A convenient way to implement the semiclassical method is via cellular
dynamics [20], which has been proven to be accurate and efficient for longtime
implementation of semiclassical calculations. The basic idea is to linearize
the classical dynamics in zones small enough to make the linearization
classically correct. The zones are typically much smaller than Planck’s
constant in area. In the following, a brief summary of cellular dynamics is
given. In the next section we discuss the revival in both infinite square well
and Morse potential systems. Further speculations are given in the Conclusion.
The starting point of semiclassical method is the Van-Vleck-Gutzwiller (VVG)
propagator [21]
$\displaystyle G\left({x,x_{0};t}\right)\ =$
$\displaystyle\left(\frac{1}{2\pi{\rm
i}\hbar}\right)^{1/2}\sum_{j}\left|\frac{\partial^{2}S_{j}(x,x_{0})}{\partial
x\partial x_{0}}\right|^{1/2}\exp\left[\frac{{\rm
i}S_{j}(x,x_{0})}{\hbar}-\frac{{\rm i}\nu_{j}\pi}{2}\right]$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2\pi{\rm
i}\hbar}\right)^{1/2}\sum_{j}\left|\frac{\partial x}{\partial
p_{0}}\right|^{-1/2}\exp\left[\frac{{\rm i}S_{j}(x,x_{0})}{\hbar}-\frac{{\rm
i}\nu_{j}\pi}{2}\right],$ (2)
where action
$S(x,x_{0})=\int_{\rm{0}}^{t}{dt^{\prime}}\left[{p\left({t^{\prime}}\right)\dot{x}\left({t^{\prime}}\right)-H\left({p\left({t^{\prime}}\right),x\left({t^{\prime}}\right)}\right)}\right]$
is the integral of the Lagrangian along classical trajectory from $x_{0}$ to
$x$, and Maslov index $\nu$ counts the number of caustic points along this
trajectory. The sum over $j$ runs over all the trajectories connecting $x_{0}$
to $x$, in other words, it counts in contributions from all the stationary
phase points. Cellular dynamics begins with a transformation of the propagator
by applying the speciality of $\delta$ function:
$\sum{\frac{1}{{\left.{\left({\partial x_{t}/\partial
p_{0}}\right)}\right|_{x=x_{t}}}}=\int{dp_{0}\delta\left({x-x_{t}\left({x_{0},p_{0}}\right)}\right)}}.$
(3)
Here $x_{t}\left({x_{0},p_{0}}\right)$ is the final position originate from
initial point $\left({x_{0},p_{0}}\right)$. The VVG propagator can now be
written as
$G\left({x,x_{0};t}\right){\rm{=}}\left({\frac{{\rm{1}}}{{{\rm{2}}\pi
i\hbar}}}\right)^{1/2}\int{dp_{0}}\left|{\frac{{\partial x_{t}}}{{\partial
p_{0}}}}\right|_{x_{0}}^{1/2}\delta\left({x-x_{t}\left({x_{0},p_{0}}\right)}\right)\exp\left[{\frac{{iS\left({x_{0},p_{0}}\right)}}{\hbar}-\frac{{i\upsilon\pi}}{2}}\right],$
(4)
with the change of action $S$ as a function of $(x_{0},p_{0})$. Then we can
get the semiclassical wave function
$\displaystyle\psi\left({x,t}\right)=$
$\displaystyle\int{dx_{0}G\left({x,x_{0};t}\right)\psi\left({x_{0},0}\right)}$
$\displaystyle=$ $\displaystyle\left({\frac{{\rm{1}}}{{{\rm{2}}\pi
i\hbar}}}\right)^{1/2}\int{dx_{0}}\int{dp_{0}\left|{\frac{{\partial
x_{t}}}{{\partial
p_{0}}}}\right|^{1/2}\delta\left({x-x_{t}}\right)e^{iS/\hbar-i\nu\pi/2}\psi\left({x_{0},0}\right)}.$
(5)
It would be difficult evaluate the integral directly since it is highly
oscillatory. However, cellular dynamics handles this difficulty by using
integration techniques similar in spirit to Filinov methods, by dividing the
region into small cells, inserting the identities
$1\approx\eta\sum\limits_{n}{\exp[-\alpha\left({x-x_{n}}\right)^{2}]}$ within
both $x$ and $p$ space. Then we have
$\displaystyle\psi\left({x,t}\right)=\eta\eta^{\prime}\sum\limits_{n}{\sum\limits_{m}{\int{dx_{0}\int{dp_{0}\left|{\frac{{\partial
x_{t}}}{{\partial
p_{0}}}}\right|^{1/2}\delta\left({x-x_{t}}\right)}}}}e^{iS/\hbar-i\nu\pi/2}$
$\displaystyle\times
e^{-\alpha\left({x_{0}-x_{n}}\right)^{2}-\beta\left({p_{0}-p_{m}}\right)^{2}}e^{-\gamma\left({x_{0}-x_{i}}\right)^{2}+ik_{i}\left({x_{0}-x_{i}}\right)^{2}},$
(6)
where the initial wave function
$\psi\left({x_{0},0}\right)=\exp\left[{-\gamma\left({x_{0}-x_{i}}\right)^{2}+ik_{i}\left({x_{0}-x_{i}}\right)^{2}}\right]$
is used. If both $\alpha$ and $\beta$ are taken to be sufficiently large, and
for sufficiently many cells, we can linearize the classical dynamics around
the central trajectory for each cell running from the initial phase space
point $\left({x_{n},p_{m}}\right)$, obtaining its contribution to the
propagation of initial wave function.
In some ways cellular dynamics resembles Miller’s initial value representation
(IVR)[22], but there are important differences. The IVR is actually
numerically superior, in that if the integral is performed the result is not
the ”primitive semiclassical” Van Vleck result, but rather a uniformized
version which is capable of describing some classically forbidden processes
and of smoothing out some semiclassical singularities. In contrast, cellular
dynamics is a direct but numerically convenient implementation of the
primitive semiclassical Green’s function. The goal of the present paper is to
test the efficacy of the primitive semiclassical propagator, but implementing
an IVR would be an interesting study.
The linearization is implemented by approximating classical action $S$ with
second order Taylor expansion and final position $x\left({x_{0},p_{0}}\right)$
with first order [20], viz.
$\begin{array}[]{rcll}S&\approx&S_{nmt}+\left({p_{nmt}m_{22}-p_{m}}\right)\left({x_{0}-x_{n}}\right)+p_{nmt}m_{21}\left({p_{0}-p_{m}}\right)\\\
&&+\frac{1}{2}m_{12}m_{22}\left({x_{0}-x_{n}}\right)^{2}+\frac{1}{2}m_{11}m_{21}\left({p_{0}-p_{m}}\right)^{2}\\\
&&+m_{12}m_{21}\left({x_{0}-x_{n}}\right)\left({p_{0}-p_{m}}\right)\\\
\lx@intercol x_{t}(x_{0},p_{0})\approx
x_{nmt}+m_{21}(p_{0}-p_{m})+m_{22}(x_{0}-x_{n})\hfil\lx@intercol.\end{array}$
(7)
where $S_{nmt}$, $x_{nmt}$, $p_{nmt}$ are the classical action, final position
and momentum of a trajectory originate from $\left({x_{n},p_{m}}\right)$
respectively, and
$M=\left({\begin{array}[]{*{20}c}{m_{11}}&{m_{12}}\\\ {m_{21}}&{m_{22}}\\\
\end{array}}\right)=\left({\begin{array}[]{*{20}c}{\partial p_{t}/\partial
p_{0}}&{\partial p_{t}/\partial x_{0}}\\\ {\partial x_{t}/\partial
p_{0}}&{\partial x_{t}/\partial x_{0}}\\\ \end{array}}\right)$ (8)
is the Jacobian matrix of the corresponding dynamical transformation [20]. The
substitution of equation (7) into (2) will simplify the quadrature into
Gaussian integration
$\psi\left({x,t}\right)=\eta\eta^{\prime}\sum\limits_{n}{\sum\limits_{m}{\int{dx_{0}\left|{\frac{{\partial
x_{t}}}{{\partial
p_{0}}}}\right|^{-1/2}e^{-a\left({x_{0}-x_{n}}\right)^{2}+b\left({x_{0}-x_{n}}\right)+c}}}},$
(9)
with the coefficients
$\displaystyle a=$
$\displaystyle\alpha+\gamma+\beta\left({\frac{{m_{22}}}{{m_{21}}}}\right)^{2}-\frac{i}{\hbar}\left({\frac{1}{2}\frac{{m_{11}m_{22}^{2}}}{{m_{21}}}-\frac{1}{2}m_{12}m_{22}}\right),$
$\displaystyle b=$ $\displaystyle\frac{{2\beta
m_{22}}}{{m_{21}^{2}}}\left({x-x_{nmt}}\right)-2\gamma\left({x_{n}-x_{i}}\right)+ik_{i}$
$\displaystyle+\frac{i}{\hbar}[\left({m_{12}-\frac{{m_{11}m_{22}}}{{m_{21}}}}\right)\left({x-x_{nmt}}\right)-p_{m}],$
$\displaystyle c=$
$\displaystyle-\frac{\beta}{{m_{21}^{2}}}\left({x-x_{nmt}}\right)^{2}-\gamma\left({x_{n}-x_{i}}\right)^{2}-\frac{{i\nu\pi}}{2}+ik_{i}\left({x_{n}-x_{i}}\right)$
$\displaystyle+\frac{i}{\hbar}[S_{nmt}+p_{nmt}\left({x-x_{nmt}}\right)+\frac{{m_{11}}}{{2m_{21}}}\left({x-x_{nmt}}\right)^{2}].$
(10)
The equation (2) can now be analytically evaluated
$\psi\left({x,t}\right)=\eta\eta^{\prime}\sum\limits_{n}{\sum\limits_{m}{\sqrt{\frac{\pi}{{am_{21}}}}e^{b^{2}/4a+c}}},$
(11)
and it is easy to implement.
## 3 Results and discussions
Figure 1: (a) Semiclassical wave functions evolve in infinite square well at
different times. (b) Partial classical manifold of an initial $\delta$
function evolves in infinite square well at time $T_{rev}$. Figure 2: The
distribution of exponential function in complex plane for position (a)$x=50$;
(b)$x=11$. The square and triangle indicate phase terms come from the first
and second exponential function in equation (3) respectively. Figure 3: Wave
functions and classical distribution probabilities in Morse potential at time
$T_{rev}/2$ and $T_{rev}$. All the functions plot in this figure are
normalized. Solid line: Semiclassical wave functions; Dash line: Exact FFT
wave functions calculated by Split-Operator method [23];Dot line: Classical
density in coordinate space which evolves from the initial density. Figure 4:
(a)Phase space diagram for Wigner transformed Gaussian evolves in Morse
potential at time $T_{rev}$. The color indicates different value of phase
(include classical action $S$ and Maslov phase) divided by $2\pi$. (b) The
blurred version of figure (a). Solid line: semiclassical wave function at time
$T_{rev}$; Dash line: Morse potential. Figure 5: (a)Vector chain for $x=-1.6$.
(b)Vector chain for $x=3.1$
In this section we will analyze the quantum revival in the infinite square
well and Morse potential in detail. First we look at the infinite square well
system, which has been well studied at many levels and from many points of
view [11, 12, 13]. We take the initial Gaussian of the form
$\psi\left({x_{0},0}\right)=\sqrt{\gamma/\pi}\exp\left[{-\gamma\left({x_{0}-x_{i}}\right)^{2}+ik_{i}\left({x_{0}-x_{i}}\right)^{2}}\right]$
(12)
and the system Hamiltonian is
$H=p^{2}/2m+V\left(x\right),{\rm{}}V\left(x\right)=\left\\{{\begin{array}[]{*{20}c}{0,{\rm{}}0<x<L}\\\
{\infty,{\rm{}}x\leq 0,x\geq L}\\\ \end{array}}\right..$ (13)
In infinite square well system, by expanding the evolving wave function with
eigen states, the revival time $T_{rev}=4mL^{2}/\hbar\pi$ can be analytically
determined [12], and it only depends on electron mass and the width of the
well. In all the calculations $m=1,{\rm{}}\hbar=1$ are used for simplicity.
With parameters $\gamma=0.02,{\rm{}}k_{i}=2,{\rm{}}x_{i}=50$ and $L=80$ we
compute the semiclassical wave function at different times using cellular
dynamics, we take 100 cells equally spaced in $x$ from 20 to 80 and 1500 cells
in $p$ from 0.8 to 3.2. One should pay attention to the Maslov phase here, in
hard wall limit it is a multiple of $\pi$ instead of $\pi/2$. As shown in Fig
1 (a), the wave packet quickly spread over the well after first several
classical periods, and at time $t=T_{rev}/2$ the wave function is a mirror
image of initial wave function, then after revival time $T_{rev}$ the wave
function is perfectly rebuilt into initial wave packet. The reason we suggest
for this astonishing relocalize of wave packet is the interference between
different contributing classical trajectories. In the following we unfold our
discussions.
As the revival in infinite square well is independent of the shape of the wave
packet, we can take a quite narrow initial wave function, such as
$\psi\left({x_{0},0}\right)=\delta\left({x_{0}-x_{i}}\right)$, sits at
$x_{i}=50$, of course it will rebuild itself at time $t=T_{rev}$. Then from
equation (2) we see the wave function directly connect to the semiclassical
propagator,
$\displaystyle\psi\left({x,t}\right)=$
$\displaystyle\int{dx_{0}G\left({x,x_{0};t}\right)\psi\left({x_{0},t}\right)}$
$\displaystyle=\int{dx_{0}G\left({x,x_{0};t}\right)\delta\left({x_{0}-x_{i}}\right)}=G\left({x,x_{0};t}\right).$
(14)
Referring to the existing works on Feynman path integral in infinite square
well [24, 25], the semiclassical propagator can be written as a summation of
contributions from all the stationary phase points
$\displaystyle G\left({x,x_{0};t}\right)=$ $\displaystyle\sqrt{{m\over{2\pi
i\hbar
t}}}\left[{\sum\limits_{n=-\infty}^{\infty}{\exp\left({{{im\left({-x-x_{0}+2nL}\right)^{2}}\over{2\hbar
t}}-i\left|{2n-1}\right|\pi}\right)}}\right.$
$\displaystyle\left.+{\sum\limits_{n=-\infty}^{\infty}{\exp\left({{{im\left({x-x_{0}+2nL}\right)^{2}}\over{2\hbar
t}}-i\left|{2n}\right|\pi}\right)}}\right]$ $\displaystyle=$
$\displaystyle\sqrt{{m\over{2\pi i\hbar
t}}}\left[{\sum\limits_{n=-\infty}^{\infty}{\exp\left({{{im\left({x-x_{0}+2nL}\right)^{2}}\over{2\hbar
t}}}\right)}}\right.$
$\displaystyle\left.-{\sum\limits_{n=-\infty}^{\infty}{\exp\left({{{im\left({-x-x_{0}+2nL}\right)^{2}}\over{2\hbar
t}}}\right)}}\right]$ (15)
Fig 1 (b) shows part of the manifold at time $T_{rev}$ which evolves from
initial $\delta$ wave function, the intersection of $x=x_{t}$ with manifold
produces stationary phase points. The two sums in Eq. 3 correspond to
stationary phase points with classical trajectories bouncing off the wall by
even and odd times respectively. To simplify the Eq. 3, we use the Jacobi
theta function
$\vartheta_{3}\left({z,T}\right)=\sum\limits_{n=-\infty}^{\infty}{\exp\left[{i\left({\pi
n^{2}T+2nz}\right)}\right]}$ and its important property [26]
$\vartheta_{3}\left({z,T}\right)=\sqrt{i/T}\exp\left({z^{2}/i\pi
T}\right)\vartheta_{3}\left({z/T,-1/T}\right),$ (16)
the semiclassical propagator becomes
$\displaystyle G\left({x,x_{0};t}\right)=$
$\displaystyle{1\over{2L}}\left[{\vartheta_{3}\left({{{\pi\left({x-x_{0}}\right)}\over{2L}},{{-\pi\hbar
t}\over{2mL^{2}}}}\right)-\vartheta_{3}\left({{{-\pi\left({x+x_{0}}\right)}\over{2L}},{{-\pi\hbar
t}\over{2mL^{2}}}}\right)}\right]$ $\displaystyle=$
$\displaystyle{1\over{2L}}\sum\limits_{n=-\infty}^{\infty}{\exp\left({{{-in^{2}\pi^{2}\hbar
t}\over{2mL^{2}}}}\right)}\left[{\exp\left({{{in\pi\left({x-x_{0}}\right)}\over
L}}\right)}\right.$
$\displaystyle\left.-{\exp\left({{{-in\pi\left({x+x_{0}}\right)}\over
L}}\right)}\right]$ $\displaystyle=$ $\displaystyle{2\over
L}\sum\limits_{n=1}^{\infty}{\exp\left({{{-in^{2}\pi^{2}\hbar
t}\over{2mL^{2}}}}\right)}\sin\left({{{n\pi x_{0}}\over
L}}\right)\sin\left({{{n\pi x}\over L}}\right).$ (17)
This is identical to the usual quantum propagator in infinite square well. At
the revival time $T_{rev}=4mL^{2}/\hbar\pi$ the wave function can be rewritten
as
$\displaystyle\psi\left({x,T_{rev}}\right)=$ $\displaystyle
G\left({x,x_{0};T_{rev}}\right)$ $\displaystyle=$
$\displaystyle{1\over{2L}}\sum\limits_{n=-\infty}^{\infty}{e^{-i2n^{2}\pi}}\left[\exp\left({{{in\pi\left({x-x_{0}}\right)}\over
L}}\right)\right.$
$\displaystyle\left.+\exp\left({{{-in\pi\left({x+x_{0}}\right)}\over
L}+i\pi}\right)\right]$ $\displaystyle=$
$\displaystyle{1\over{2L}}\sum\limits_{j}{e^{i\phi_{j}}}.$ (18)
We can compare analytically the difference between low and high amplitude
points of wave function. Taking $x_{1}=20,x_{2}=50$ for example, we find that
for the high amplitude position at $x=50$, the exponential functions
$\exp\left({i\phi_{j}}\right)$ in the summation distribute uniformly in
complex plane. There are only eight phase terms [see Fig 2 (a)] in the sum. We
need to distinguish the phase terms come from different exponential function
in Eq. 3. The second exponential function gives out all 8 different terms
distribute symmetrically around the circle so that they will cancel each
other, whereas, the first exponential function only gives out $\phi=0$ terms,
they will build up big contributions and give out high amplitude. For the low
amplitude point $x=20$, however, both exponential functions generate 16
symmetrically distributed terms [see Fig 2 (b)] on the unit circle and
therefore the summation approaches zero. Hence the interference between part
of different classical trajectories yields the revival of wave packet.
Now we come to see a more general system, the Morse potential. It is also a
widely used model in many fields. We take
$V\left(x\right)=D\left[{1-\exp\left({-\lambda x}\right)}\right]^{2}$ with
$D=150,\lambda=0.288$, its revival time
$T_{rev}=2m\pi/\left({\hbar\lambda}\right)^{2}$ can be derived by expanding
the wave function with eigen functions of Morse potential, too [see Appendix
A]. With 300 cells in $x$ and 600 cells in $p$ been used in the calculation,
the semiclassical wave functions originate from
$\psi\left({x_{0},0}\right)=\sqrt{\gamma/\pi}\exp\left[{-\gamma\left({x_{0}-x_{i}}\right)^{2}}\right]$
$\left({\gamma=2,x_{i}=3.5}\right)$ are pictured in Fig 3. Comparing to the
FFT exact wave functions we can see semiclassical wave functions agree well
for different time scales. In Fig 3 we plot the normalized classical
coordinate space density arising from the initial classical distribution.
Since the semiclassical result consists of the square root of classical
probabilities multiplied by phase terms and added together, but it is easy to
construct its purely classical result by removing the phase terms, and
squaring and adding all the square root classical densities.
It is surprising that despite the fact that the classical trajectories are
spread all over the available phase space and coordinate space, the
semiclassical approximation can still build a localized wave packet at the
revival time $T_{rev}$.
In order to demonstrate the relationship between semiclassical wave function
and classical information carried by trajectories, we first Wigner transform
the initial Gaussian distribution
$\psi\left(x\right)=\sqrt{\gamma/\pi}\exp\left[{-\gamma\left(x-x_{i}\right)^{2}}\right]$,
$\displaystyle W\left({x,p}\right)=$
$\displaystyle\frac{1}{{\pi\hbar}}\int_{-\infty}^{\infty}{\psi^{*}\left({x-s}\right)\psi\left({x+s}\right)e^{i2ps/\hbar}ds}$
$\displaystyle=$
$\displaystyle\frac{\gamma}{{\pi^{2}\hbar}}\int_{-\infty}^{\infty}{e^{-\gamma\left({x-x_{i}-s}\right)^{2}}e^{-\gamma\left({x-x_{i}+s}\right)^{2}}e^{i2ps/\hbar}ds}$
$\displaystyle=$
$\displaystyle\frac{\gamma}{{\pi^{2}\hbar}}\int_{-\infty}^{\infty}{e^{-2\gamma\left({x-x_{i}}\right)^{2}-2\gamma
s^{2}+i2ps/\hbar}ds}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\gamma}{{2\pi^{3}\hbar^{2}}}}e^{-p^{2}/2\gamma\hbar-2\gamma\left({x-x_{i}}\right)^{2}},$
which remains a Gaussian. Then we plot the evolution of this Wigner
distribution in phase space after revival time $T_{rev}$ in Fig 4. The
starting swarm of classical trajectories emerges as an elliptical disk; as
time evolves this ellipse stretches and twists, forming a large whorl.
(Indeed, the time evolution of the phase space is that of an area preserving
twist map).
It might appear that the vertical sections of the classical manifolds on the
left and right sides of the whorl would dominate the contribution to the
semiclassical wave function for two reasons: First, the prefactor
$1/\sqrt{\left|{\partial x/\partial p_{0}}\right|}$ in VVG propagator in
equation (2) is large for this part of the manifold. This is because the
density of the distribution is proportional to the probability
$1/\sqrt{\left|{\partial x/\partial p_{0}}\right|}$ of classical particles
locating at those regimes. Second, these particles have similar classical
actions [see Fig 4 (a)]. In Appendix B we prove the classical action
difference between two points equals the enclosed area of the manifold. Near
the fold regimes small enclosure areas lead to similar classical actions. The
abrupt changes of color at the turning points indicate the change of Maslov
index at those points. The combination of these two factors yields large
result refers to equation (2). In Fig 4 (b) we blurred the phase space diagram
for an intuitive view. The whorl average out and give neutral gray colors
everywhere except where the revival is occurring. When combined along vertical
lines, those regions with monochromatic bright colors will give out the
revival wave packet.
Nevertheless, one could still doubt why we don’t get a high amplitude wave
function at positions of folds on left side, they also meet the conditions
list above. To compare the difference, we write the formula of the wave
function into a compact form:
$\displaystyle\psi\left({x,t}\right)=$
$\displaystyle\int{dx_{0}G\left({x,x_{0};t}\right)\psi\left({x_{0},0}\right)}$
$\displaystyle=$ $\displaystyle\left({\frac{1}{{2\pi
i\hbar}}}\right)^{1/2}\int{dx_{0}\sum\limits_{j}{\left|{\frac{{\partial
x}}{{\partial
p_{0}}}}\right|^{-1/2}\exp\left[{\frac{{iS_{j}\left({x,x_{i}}\right)}}{\hbar}-i\upsilon_{j}\pi}\right]}}\psi\left({x_{0},0}\right)$
$\displaystyle=$
$\displaystyle\int{dx_{0}Re^{i\phi}}=\sum\limits_{n}{R_{n}e^{i\phi_{n}}\Delta
x_{0}}.$
We can approximate the quadrature numerically by a finite sum of complex
vector. We divide $x$ space into hundreds of sections, and evaluate the vector
separately in each section. By drawing each vector from the tips of previous
vector, the summation will form a chain, and the line drawn from first point
to the end point represent the quadrature. We draw two chains respectively for
$x=-1.6$ and $x=3.1$ in Fig 5.
For the low amplitude region $x=-1.6$, in the vicinity of destructive
interference, the chain circles continuously, and results in a small total
vector [see Fig 5 (a)]. This indicates the phase of stationary phase points
changes only slightly and continuously, leading to destructive interference
between classical trajectories and a small amplitude of wave function. A
different situation applies in Fig 5 (b) for the position $x=3.1$. Here the
small phase difference between stationary points accumulates a persistent
growth of the total vector, viz. the constructive interference produces high
amplitude of wave function.
## 4 Conclusions and discussion
Whenever and wherever they apply, semiclassical methods can be extremely
useful not only in computations, but in providing an underlying intuition for
quantum phenomena. Here we have shown that something so subtle as a quantum
revival still has classical underpinnings, as seen by the successful
construction of the phenomenon using only classical mechanics as input.
Semiclassical methods are accurate enough to describe the quantum revival
phenomenon. The quantum revival phenomenon does not stem from an accumulation
of classical trajectories. Rather, the classical trajectories are rather
uniformly spread, and it is through destructive interference of the
semiclassical amplitudes that the wave function is canceled in most places.
Of course, this is a momentary phenomenon, in the sense that beyond the
revival time the way packet will again begin to spread and become quite
delocalized quantum mechanically. However, imagine the following scenario: at
the moment of a revival, with the wave packet built up on one side of the
potential, suppose a time-dependent barrier is erected, preventing the wave
packet from any immediate penetration beyond the barrier. If the barrier
remained up, and the potential were sufficiently asymmetric and designed
properly the quantum mechanical wave packet would remain on one side of the
barrier forever. This raises an even more interesting and challenging
question: When the time-dependent barrier was raised, this traps classical
manifolds on the “empty” side of the barrier. Presumably at the moment of the
trapping, the semiclassical wave function would indeed be correctly
exceedingly small at that point, but for how long could this semiclassical
result correctly describe the fact that the wave function never reappeared in
that region? One could call this the “semiclassical propagation of nothing”.
That is to say, abstracting this a little further, suppose you begin with a
quite complex set of classical manifolds, interpreted semiclassically, which
gives essentially zero semiclassical wave function everywhere. Now, the
continued semiclassical propagation of these manifolds should continue to give
a vanishingly small function. Any errors in the semiclassical propagation will
cause wave function amplitude to appear incorrectly.
While we cannot fully explain the situation here, we believe this phenomenon
may be affecting the quantum classical correspondence in branched electron
flow [27]. Branched electron flows are usually ascribed to a purely classical
effect [27, 28]; however, classical and quantum electron flows begin to
disagree, with some branches suddenly missing in the quantum result as
compared to the classical, as one moves further and further from the source of
electrons. In the future we hope to verify our conjecture that the missing
branches are an effect of destructive interference of classical trajectories,
by using semiclassical methods.
## 5 Acknowledgements
One of us (Z. X. Wang) would like to acknowledge helpful discussions with
Brian Landry. This work was supported in part by the National Natural Science
Foundation of China (Grant Nos.10574121 and 10874160), ’111’ Project, Chinese
Education Ministry and Chinese Academy of Sciences.
## Appendix A Appendix A
For Morse potential $V\left(x\right)=D\left[{1-\exp\left({-\lambda
x}\right)}\right]^{2}$, one can express the time dependent wave function in
terms of eigen functions $\varphi_{n}\left(x\right)$ , via
$\psi\left({x,t}\right)=\sum\limits_{n=0}^{\infty}{a_{n}\varphi_{n}\left(x\right)e^{-iE_{n}t/\hbar}},$
(21)
where the eigen values are
$E_{n}=\alpha\left({n+1/2}\right)-\beta\left({n+1/2}\right)^{2}$ with
$\alpha=\hbar\lambda\sqrt{2D/m}$, $\beta=\hbar^{2}\lambda^{2}/2m$. The revival
condition $\psi\left({x,T}\right)=\psi\left({x,0}\right)$ requires
$E_{n}T=\left[{\alpha\left({n+1/2}\right)-\beta\left({n+1/2}\right)^{2}}\right]T=2M_{n}\pi,$
(22)
where $M_{n}$ are integers. Make a subtraction of adjacent $n$ of equation
(22) gives
$\left({\alpha-2\beta n-2\beta}\right)T=2K_{n}\pi,$ (23)
with $K_{n}$ are also integers. Then apply the subtraction of adjacent of
equation (23) again, we get the equation for the shortest revival time
$T_{rev}$ is
$2\beta T_{rev}=2\pi.$ (24)
So we have the revival time
$T_{rev}=\pi/\beta=2m\pi/\left({\hbar\lambda}\right)^{2}$.
## Appendix B Appendix B
Figure 6: Classical manifold. The area contained between intersections of the
manifold $p(q)$ and a position state (vertical line $q=q_{t}$) is $Q_{1}$.
We ought to prove the difference of classical action $S_{A}$ and $S_{B}$
equals to the shade area $Q_{1}$. First we look at point $B$ and $C$. From
classical action formula
$S=\int{p\left(q\right)dq}+\int{H\left({p,q}\right)dt}$ we have $\partial
S/\partial q=p$, thus the action difference from $B$ to $C$ is
$S_{C}-S_{B}=\int_{B}^{C}{\frac{{\partial S}}{{\partial
q}}dq}=\int_{B}^{C}{pdq}=area{\rm{}}Q_{2}.$ (25)
Then, in a similar way
$S_{A}-S_{B}=\int_{B}^{A}{\frac{{\partial S}}{{\partial
q}}dq}=\int_{B}^{A}{pdq}=area{\rm{}}Q_{1}.$ (26)
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* [27] M. A. Topinka, B. J. LeRoy, R. M. Westervelt, S. E. J. Shaw, R. Fleischmann, E. J. Heller, K. D. Maranowski, A. C. Gossard, Nature 410 (2001) 183.
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|
arxiv-papers
| 2008-11-12T16:35:07 |
2024-09-04T02:48:58.749271
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhexian Wang, Eric J. Heller",
"submitter": "Zhexian Wang",
"url": "https://arxiv.org/abs/0811.1936"
}
|
0811.1953
|
ROM2F/2008/25
Stückelino Dark Matter in Anomalous $U(1)^{\prime}$ Models
Francesco Fucito111Francesco.Fucito@roma2.infn.it♮, Andrea
Lionetto222Andrea.Lionetto@roma2.infn.it♮, Andrea
Mammarella333Andrea.Mammarella@roma2.infn.it♮, Antonio
Racioppi444Antonio.Racioppi@kbfi.ee♭
♮ Dipartimento di Fisica dell’Università di Roma , “Tor Vergata” and
I.N.F.N. - Sezione di Roma “Tor Vergata”
Via della Ricerca Scientifica, 1 - 00133 Roma, ITALY
♭ National Institute of Chemical Physics and Biophysics,
Ravala 10, Tallinn 10143, Estonia
We study a possible dark matter candidate in the framework of a minimal
anomalous $U(1)^{\prime}$ extension of the MSSM. It turns out that in a
suitable decoupling limit the Stückelino, the fermionic degree of freedom of
the Stückelberg multiplet, is the lightest supersymmetric particle (LSP). We
compute the relic density of this particle including coannihilations with the
next to lightest supersymmetric particle (NLSP) and with the next to next to
lightest supersymmetric particle (NNLSP) which are assumed almost degenerate
in mass. This assumption is needed in order to satisfy the stringent limits
that the Wilkinson Microwave Anisotropy Probe (WMAP) puts on the relic
density. We find that the WMAP constraints can be satisifed by different NLSP
and NNLSP configurations as a function of the mass gap with the LSP. These
results hold in the parameter space region where the model remains
perturbative.
## 1 Introduction
A great deal of work has been done recently to embed the standard model of
particle physics (SM) into a brane construction [1, 2, 3, 4]. This research is
part of the effort, initiated in [5], to build a fully realistic four
dimensional vacuum out of string theory. While the original models were
formulated in the framework of the heterotic string, the most recent efforts
were formulated for type II strings in order to take advantage of the recent
work on moduli stabilization using fluxes. Such brane constructions naturally
lead to extra anomalous $U(1)$’s in the four dimensional low-energy theory
and, in turn, to the presence of possible heavy $Z^{\prime}$ particles in the
spectrum. These particles should be among the early findings of LHC and
besides for the above cited models they are also a prediction of many other
theoretical models of the unification of forces (see [6] for a recent review).
It is then of some interest to know if these $Z^{\prime}$ particles contribute
to the cancelation of the gauge anomaly in the way predicted from string
theory or not. In [7] some of the present authors have studied a
supersymmetric (SUSY) extension of the minimal supersymmetric standard model
(MSSM) in which the anomaly is canceled à la Green-Schwarz. The model is only
string-inspired and is not the low-energy sector of some brane construction.
The reason of this choice rests in our curiosity to explore the phenomenology
of these models keeping a high degree of flexibility, while avoiding the
intricacies and uncertainties connected with a string theory construction. For
previous work along these lines we refer to [8]-[15]. In this work we perform
a consistency check of our model [7] by evaluating the thermal relic density
and comparing it against the WMAP data.
The cancelation of the $U(1)^{\prime}$ anomaly in our model requires the
introduction of an extra complex scalar field whose supersymmetric partner is
called the Stückelino. We will see in the following that if the latter is the
lightest supersymmetric particle (LSP) its interactions are such to define it
as an XWIMP (i.e. a weakly interacting particle with couplings at least one
order of magnitude less than the standard weak interactions): in fact it will
also turn out to be a cold relic. If the Stückelino is the lightest
supersymmetric particle (LSP), its relic density turns out to be too high with
respect to the experimental data. This is why, following [16], we favor a next
to lightest supersymmetric particle (NLSP) with a mass close to the LSP. We
show that an interesting scenario arises also for three particle
coannihilation processes. In these two cases, the model is consistent with the
experimental data. Moreover in the three particle case we find configurations
in which the LSP and the NLSP do not need to be nearly degenerate in mass. In
this case the mass gap between the two can be of the order of $20\%$. This is
the plan of the paper: in Section 2 we describe our model. In Section 3 and 4
we find the LSP and study the Stückelino interactions. Finally in Section 5 we
compute the relic density. Section 6 is a summary of our results.
## 2 Model Setup
In this section we briefly discuss our theoretical framework. We assume an
extension of the MSSM with an additional abelian vector multiplet $V^{(0)}$
with arbitrary charges. The anomalies are canceled with the Green-Schwarz (GS)
mechanism and with the Generalized Chern-Simons (GCS) terms. All the details
can be found in [7]. All the MSSM fields are charged under the additional
vector multiplet $V^{(0)}$, with charges which are given in Table 1, where
$Q_{i},L_{i}$ are the left handed quarks and leptons respectively while
$U^{c}_{i},D^{c}_{i},E^{c}_{i}$ are the right handed up and down quarks and
the electrically charged leptons. The superscript $c$ stands for charge
conjugation. The index $i=1,2,3$ denotes the three different families.
$H_{u,d}$ are the two Higgs scalars.
| SU(3)c | SU(2)L | U(1)Y | U(1)${}^{\prime}~{}$
---|---|---|---|---
$Q_{i}$ | ${\bf 3}$ | ${\bf 2}$ | $1/6$ | $Q_{Q}$
$U^{c}_{i}$ | $\bar{\bf 3}$ | ${\bf 1}$ | $-2/3$ | $Q_{U^{c}}$
$D^{c}_{i}$ | $\bar{\bf 3}$ | ${\bf 1}$ | $1/3$ | $Q_{D^{c}}$
$L_{i}$ | ${\bf 1}$ | ${\bf 2}$ | $-1/2$ | $Q_{L}$
$E^{c}_{i}$ | ${\bf 1}$ | ${\bf 1}$ | $1$ | $Q_{E^{c}}$
$H_{u}$ | ${\bf 1}$ | ${\bf 2}$ | $1/2$ | $Q_{H_{u}}$
$H_{d}$ | ${\bf 1}$ | ${\bf 2}$ | $-1/2$ | $Q_{H_{d}}$
Table 1: Charge assignment.
The key feature of this model is the mechanism of anomaly cancelation. As it
is well known, the MSSM is anomaly free. In our MSSM extension all the
anomalies that involve only the $SU(3)$, $SU(2)$ and $U(1)_{Y}$ factors vanish
identically. However, triangles with $U(1)^{\prime}$ in the external legs in
general are potentially anomalous. These anomalies are555We are working in an
effective field theory framework and we ignore throughout the paper all the
gravitational effects. In particular, we do not consider the gravitational
anomalies which, however, could be canceled by the Green-Schwarz mechanism.
$\displaystyle U(1)^{\prime}-U(1)^{\prime}-U(1)^{\prime}:$ $\displaystyle\
\mathcal{A}^{(0)}$ (1) $\displaystyle U(1)^{\prime}-U(1)_{Y}-U(1)_{Y}:$
$\displaystyle\ \mathcal{A}^{(1)}$ (2) $\displaystyle
U(1)^{\prime}-SU(2)-SU(2):$ $\displaystyle\ \mathcal{A}^{(2)}$ (3)
$\displaystyle U(1)^{\prime}-SU(3)-SU(3):$ $\displaystyle\ \mathcal{A}^{(3)}$
(4) $\displaystyle U(1)^{\prime}-U(1)^{\prime}-U(1)_{Y}:$ $\displaystyle\
\mathcal{A}^{(4)}$ (5)
All the remaining anomalies that involve $U(1)^{\prime}$s vanish identically
due to group theoretical arguments (see Chapter 22 of [17]). Consistency of
the model is achieved by the contribution of a Stückelberg field $S$ and its
appropriate couplings to the anomalous $U(1)^{\prime}$. The Stückelberg
lagrangian written in terms of superfields is [15]
$\mathcal{L}_{S}={\frac{1}{4}}\left.\left(S+S^{\dagger}+4b_{3}V^{(0)}\right)^{2}\right|_{\theta^{2}\bar{\theta}^{2}}\\!\\!-{\frac{1}{2}}\left\\{\left[\sum_{a=0}^{2}b^{(a)}_{2}S\textnormal{Tr}\left(W^{(a)}W^{(a)}\right)+b^{(4)}_{2}SW^{(1)}W^{(0)}\right]_{\theta^{2}}\\!\\!\\!\\!\\!\\!+h.c.\\!\right\\}$
(6)
where the index $a=0,\ldots,3$ runs over the
$U(1)^{\prime},\,U(1)_{Y},\,SU(2)$ and $SU(3)$ gauge groups respectively. The
Stückelberg multiplet is a chiral superfield
$S=s+i\sqrt{2}\theta\psi_{S}+\theta^{2}F_{S}-i\theta\sigma^{\mu}\bar{\theta}\partial_{\mu}s+{\frac{\sqrt{2}}{2}}\theta^{2}\bar{\theta}\bar{\sigma}^{\mu}\partial_{\mu}\psi_{S}-{\frac{1}{4}}\theta^{2}\bar{\theta}^{2}\Box
s$ (7)
The lowest component of $S$ is a complex scalar field $s=\alpha+i\phi$. In our
scenario the scalar $\phi$ is eaten up in the Stückelberg mechanism to give
mass to the gauge field. On the other end, the scalar $\alpha$ is the dilaton
of string theory and its value must be determined somehow. We will not
investigate the way in which this happens, but we will just retain the final
result: $\alpha$ drops out of our effective lagrangian. In the string
literature (see for instance [7]-[16]) the fields $\phi$, $\alpha$ and
$\psi_{S}$ are respectively known as axion, saxion and axino. In this paper,
to avoid confusion with the much better known QCD axion/axino system (see for
instance [18]-[22]), we will adopt the convention of [23] and, from now on, we
will refer to $s$ as the Stückelberg scalar and to $\psi_{S}$ as the
Stückelino.
The Stückelberg multiplet $S$ transforms under $U(1)^{\prime}$ as
$\displaystyle V^{(0)}$ $\displaystyle\to$ $\displaystyle
V^{(0)}+i\left(\Lambda-\Lambda^{\dagger}\right)$ $\displaystyle S$
$\displaystyle\to$ $\displaystyle S-4i~{}b_{3}~{}\Lambda$ (8)
where $b_{3}$ is a constant related to the $Z^{\prime}$ mass. In our model
there are two mechanisms that give mass to the gauge bosons: (i) the
Stückelberg mechanism and (ii) the Higgs mechanism. In this extension of the
MSSM, the mass terms for the gauge fields for
${Q_{H_{u}}}=-{Q_{H_{d}}}=0$666We impose this condition to simplify our
computations and to give analytical expressions of limited dimensions. There
are no obstructions to set ${Q_{H_{u}}}=-{Q_{H_{d}}}\neq 0$. are given by
$\mathcal{L}_{M}=\frac{1}{2}\left(V^{(0)}_{\mu}\ V^{(1)}_{\mu}\
V^{(2)}_{3\mu}\right)M^{2}\left(\begin{array}[]{c}V^{(0)\mu}\\\ V^{(1)\mu}\\\
V^{(2)\mu}_{3}\end{array}\right)$ (9)
with $M^{2}$ being the gauge boson mass matrix
$M^{2}=\left(\begin{array}[]{ccc}M_{V^{(0)}}&~{}~{}~{}0&~{}~{}~{}0\\\
...&g_{1}^{2}\frac{v^{2}}{4}&-g_{1}g_{2}\frac{v^{2}}{4}\\\
...&...&g_{2}^{2}\frac{v^{2}}{4}\\\ \end{array}\right)$ (10)
where $M_{V^{(0)}}=4b_{3}g_{0}$ is the mass parameter for the anomalous $U(1)$
and it is assumed to be in the TeV range. The lower dots denote the obvious
terms under symmetrization. After diagonalization, we obtain the eigenstates
$\displaystyle A_{\mu}$ $\displaystyle=$
$\displaystyle\frac{g_{2}V^{(1)}_{\mu}+g_{1}V^{(2)}_{3\mu}}{\sqrt{g_{1}^{2}+g_{2}^{2}}}$
(11) $\displaystyle Z_{0\mu}$ $\displaystyle=$
$\displaystyle\frac{g_{2}V^{(2)}_{3\mu}-g_{1}V^{(1)}_{\mu}}{\sqrt{g_{1}^{2}+g_{2}^{2}}}$
(12) $\displaystyle Z^{\prime}_{\mu}$ $\displaystyle=$ $\displaystyle
V^{(0)}_{\mu}$ (13)
and the corresponding masses
$\displaystyle M^{2}_{\gamma}$ $\displaystyle=$ $\displaystyle 0$ (14)
$\displaystyle M^{2}_{Z_{0}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(g_{1}^{2}+g_{2}^{2}\right)v^{2}$ (15)
$\displaystyle M^{2}_{Z^{\prime}}$ $\displaystyle=$ $\displaystyle
M_{V^{(0)}}^{2}$ (16)
Finally the rotation matrix from the hypercharge to the photon basis is
$\displaystyle\left(\begin{array}[]{c}Z^{\prime}_{\mu}\\\ Z_{0\mu}\\\
A_{\mu}\end{array}\right)$ $\displaystyle=$ $\displaystyle
O_{ij}\left(\begin{array}[]{c}V^{(0)}_{\mu}\\\ V^{(1)}_{\mu}\\\
V^{(2)}_{3\mu}\end{array}\right)=\left(\begin{array}[]{ccc}1&0{}{}&0\\\
0&-\sin\theta_{W}&\cos\theta_{W}\\\ 0&\cos\theta_{W}&\sin\theta_{W}\\\
\end{array}\right)\left(\begin{array}[]{c}V^{(0)}_{\mu}\\\ V^{(1)}_{\mu}\\\
V^{(2)}_{3\mu}\end{array}\right)$ (29)
where $i,j=0,1,2$.
We now give the expansion of the lagrangian piece $\mathcal{L}_{S}$ defined in
(6) in component fields only for the part that is needed in the following
sections. Using the Wess-Zumino gauge we get
$\displaystyle\mathcal{L}_{\text{St\"{u}ckelino}}$ $\displaystyle=$
$\displaystyle{\frac{i}{4}}\psi_{S}\sigma^{\mu}\partial_{\mu}{\bar{\psi}}_{S}-\sqrt{2}b_{3}\psi_{S}\lambda^{(0)}-{\frac{i}{2\sqrt{2}}}\sum_{a=0}^{2}b^{(a)}_{2}\textnormal{Tr}\left(\lambda^{(a)}\sigma^{\mu}\bar{\sigma}^{\nu}F_{\mu\nu}^{(a)}\right)\psi_{S}$
(30)
$\displaystyle-{\frac{i}{2\sqrt{2}}}b^{(4)}_{2}\left[{\frac{1}{2}}\lambda^{(1)}\sigma^{\mu}\bar{\sigma}^{\nu}F_{\mu\nu}^{(0)}\psi_{S}+(0\leftrightarrow
1)\right]+h.c.$
As it was pointed out in [8], the Stückelberg mechanism is not enough to
cancel all the anomalies. Mixed anomalies between anomalous and non-anomalous
factors require an additional mechanism to ensure consistency of the model:
non-gauge invariant GCS terms must be added. In our case, the GCS terms have
the form [10]
$\displaystyle\mathcal{L}_{GCS}$ $\displaystyle=$ $\displaystyle-
d_{4}\left[\left(V^{(1)}D^{\alpha}V^{(0)}-V^{(0)}D^{\alpha}V^{(1)}\right)W^{(0)}_{\alpha}+h.c.\right]_{\theta^{2}\bar{\theta}^{2}}+$
(31)
$\displaystyle+d_{5}\left[\left(V^{(1)}D^{\alpha}V^{(0)}-V^{(0)}D^{\alpha}V^{(1)}\right)W^{(1)}_{\alpha}+h.c.\right]_{\theta^{2}\bar{\theta}^{2}}+$
$\displaystyle+d_{6}\textnormal{Tr}\bigg{[}\left(V^{(2)}D^{\alpha}V^{(0)}-V^{(0)}D^{\alpha}V^{(2)}\right)W^{(2)}_{\alpha}+n.a.c+h.c.\bigg{]}_{\theta^{2}\bar{\theta}^{2}}$
where $n.a.c.$ refers to non-abelian completion terms. The $b$ constants in
(6) and the $d$ constants in (31) are fixed by the anomaly cancelation
procedure (for details see [7]).
For a symmetric distribution of the anomaly, we have
$\displaystyle b^{(0)}_{2}b_{3}=-\frac{\mathcal{A}^{(0)}}{384\pi^{2}}~{}\qquad
b^{(1)}_{2}b_{3}=-\frac{\mathcal{A}^{(1)}}{128\pi^{2}}\qquad
b^{(2)}_{2}b_{3}=-\frac{\mathcal{A}^{(2)}}{64\pi^{2}}~{}\qquad
b^{(4)}_{2}b_{3}=-\frac{\mathcal{A}^{(4)}}{128\pi^{2}}$
$\displaystyle~{}~{}~{}~{}d_{4}=-\frac{\mathcal{A}^{(4)}}{384\pi^{2}}~{}~{}~{}~{}~{}\qquad
d_{5}=\frac{\mathcal{A}^{(1)}}{192\pi^{2}}~{}~{}~{}~{}~{}~{}\qquad
d_{6}=\frac{\mathcal{A}^{(2)}}{96\pi^{2}}$ (32)
It is worth noting that the GCS coefficients $d_{4,5,6}$ are fully determined
in terms of the $\mathcal{A}$’s by the gauge invariance, while the
$b_{2}^{(a)}$’s depend only on the free parameter $b_{3}$, which is related to
the mass of the anomalous $U(1)$.
The soft breaking sector of the model is given by
$\mathcal{L}_{soft}=\mathcal{L}_{soft}^{MSSM}-{\frac{1}{2}}\left(M_{0}\lambda^{(0)}\lambda^{(0)}+h.c.\right)-{\frac{1}{2}}\left(\frac{M_{S}}{2}\psi_{S}\psi_{S}+h.c.\right)$
(33)
where $\mathcal{L}_{soft}^{MSSM}$ is the usual soft susy breaking lagrangian
while $\lambda^{(0)}$ is the gaugino of the added $U(1)^{\prime}$ and
$\psi_{S}$ is the Stückelino. The Stückelino soft mass term deserves some
comment: from [24] we know that a fermionic mass term for a chiral multiplet
is not allowed in presence of Yukawa interactions in which this chiral
multiplet is involved. But in the classical Lagrangian the Stückelberg
multiplet cannot contribute to superpotential terms given that the gauge
invariance given from our U(1)’ symmetry (8) requires non-holomorphicity in
the chiral fields. In fact in our model both the Stückelino and the scalar
$\phi$ couple only through GS interactions. It is worth noting that a mass
term for the scalar $\phi$ is instead not allowed since it transforms non
trivially under the anomalous $U(1)^{\prime}$ gauge transformation (8).
## 3 Neutralino Sector
Assuming the conservation of R-parity the LSP is a good weak interacting
massive particle (WIMP) dark matter candidate. As in the MSSM the LSP is given
by a linear combination of fields in the neutralino sector. The general form
of the neutralino mass matrix is given in [7]. Written in the interaction
eigenstate basis $(\psi^{0})^{T}=(\psi_{S},\ \lambda^{(0)},\ \lambda^{(1)},\
\lambda^{(2)}_{3},\ \tilde{h}_{d}^{0},\ \tilde{h}_{u}^{0})$ it is a six-by-six
matrix. From the point of view of the strength of the interactions the two
extra states are not on the same footing with respect to the standard ones.
The Stückelino and the extra gaugino $\lambda^{(0)}$ dubbed primeino are in
fact extremely weak interacting massive particle (XWIMP). Thus we are
interested in situations in which the extremely weak sector is decoupled from
the standard one and the LSP belongs to this sector. This can be achieved at
tree level with the choice
${Q_{H_{u}}}={Q_{H_{d}}}=0$ (34)
The neutralino mass matrix ${\bf M}_{\tilde{N}}$ becomes
${\bf
M}_{\tilde{N}}=\left(\begin{array}[]{cccccc}\frac{M_{S}}{2}&\frac{M_{V^{(0)}}}{\sqrt{2}}&0&0&0&0\\\
\dots&M_{0}&0&0&0&0\\\
\dots&\dots&M_{1}&0&-\frac{g_{1}v_{d}}{2}&\frac{g_{1}v_{u}}{2}\\\
\dots&\dots&\dots&M_{2}&\frac{g_{2}v_{d}}{2}&-\frac{g_{2}v_{u}}{2}\\\
\dots&\dots&\dots&\dots&0&-\mu\\\
\dots&\dots&\dots&\dots&\dots&0\end{array}\right)~{}~{}~{}~{}$ (35)
where $M_{S},~{}M_{0},~{}M_{1},~{}M_{2}$ are the soft masses coming from the
soft breaking terms (33) while $M_{V^{(0)}}$ is given in (10). It is worth
noting that the D terms and kinetic mixing terms can be neglected in the tree-
level computations of the eigenvalues and eigenstates.
Moreover, we make the assumption that
$\ M_{0}\gg M_{S},M_{V^{(0)}}$ (36)
so that the Stückelino is the LSP. This assumption is motivated by the
interaction strengths of the two extra states: the Stückelino interacts via
the vertex shown in Fig. 1b which can easily be seen (from (30)) to be
proportional to the coefficient $b^{(a)}_{2}$ of (32) which, in turn, is
inversely proportional to $b_{3}$ that is the $Z^{\prime}$ mass given that
$M_{Z^{\prime}}=M_{V^{(0)}}=4b_{3}g_{0}$. Given these considerations the
vertex shown in Fig. 1b is then of order $\sim g_{0}g_{a}^{2}/M_{Z^{\prime}}$.
The primeino interacts via the vertex in Fig. 1a which is of order $\sim
g_{a}$ that is the standard strength of weak interactions. Assuming the two
decoupling relations (34) and (36) we will see in the following sections that
the dominant contribution in the (co)annihilation processes is that of the
primeino, which is of the type of a standard gaugino interaction.
Figure 1: (a) Gaugino-fermion-sfermion interaction vertex. (b) Stückelino-
gaugino-vector interaction vertex.
## 4 Stückelino Interactions
The Stückelino interactions can be read off from the interaction lagrangian
(30). The relevant Stückelino-MSSM neutralino interaction term, written in
terms of four components Majorana spinors777The gamma matrices $\gamma^{\mu}$
are in the Weyl representation. We use capital letters for four components
spinors and lower case letters for two components spinors. , is given by:
$\mathcal{L}=i\sqrt{2}g_{1}^{2}b_{2}^{(1)}\bar{\Lambda}^{(1)}\gamma_{5}[\gamma^{\mu},\gamma^{\nu}](\partial_{\mu}V^{(1)}_{\nu})\Psi_{S}+i{\frac{\sqrt{2}}{2}}g_{2}^{2}b_{2}^{(2)}\bar{\Lambda}^{(2)}_{3}\gamma_{5}[\gamma^{\mu},\gamma^{\nu}](\partial_{\mu}V^{(2)}_{3\nu})\Psi_{S}$
(37)
where the $b_{2}^{(a)}$ coefficients are given in (32). The related
interaction vertex Feynman rule is
$C^{(a)}\gamma_{5}[\gamma^{\mu},\gamma^{\nu}]ik_{\mu}$ (38)
where $k_{\mu}$ is the momentum of the outgoing vector and the $C^{(a)}$’s are
$\displaystyle C^{(1)}=\sqrt{2}g_{1}^{2}b_{2}^{(1)}$ $\displaystyle
C^{(2)}={\frac{\sqrt{2}}{2}}g_{2}^{2}b_{2}^{(2)}$ (39)
The interaction Lagrangian (37) expressed in the mass eigenstates basis is
$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
i\sum_{j}\tilde{N_{j}}\left(\cos\theta_{W}C^{(1)}N_{(1)j}+\sin\theta_{W}C^{(2)}N_{(2)j}\right)\gamma_{5}[\gamma^{\mu},\gamma^{\nu}](\partial_{\mu}A_{\nu})\Psi_{S}+$
(40) $\displaystyle
i\sum_{j}\tilde{N_{j}}\left(-\sin\theta_{W}C^{(1)}N_{(1)j}+\cos\theta_{W}C^{(2)}N_{(2)j}\right)\gamma_{5}[\gamma^{\mu},\gamma^{\nu}](\partial_{\mu}{Z_{0}}_{\nu})\Psi_{S}$
where $\tilde{N_{j}}$ is a generic neutralino and $N_{ij}$ is the matrix that
diagonalizes (35). We remind that the Stückelino $\Psi_{S}$ is directly a mass
eigenstate because of the decoupling (36). We stress that these are only
interactions between the Stückelino and the MSSM neutralinos. All the
Stuckelino interactions in (30) include also analogous interactions involving
the charged wino or the primeino.
The factors in (39) are of naive dimension higher than four. They then contain
the parameters $b_{2}^{(a)}$ which are inversely proportional to the mass of
the $Z^{\prime}$ (see (32)). Given these interactions, our Stückelino will be
an extremely weak interacting particle, according to the definition we gave in
the introduction. The simplifying assumption (34) has now decoupled our
Stückelino and primeino from the standard MSSM sectors and, at tree level, the
relevant diagrams are given in Figs. 2 and 3. We can now give a rough estimate
of the effective interaction, comparing with the standard Fermi coupling,
$G_{F}$, of weak interactions: given a mass of the $Z^{\prime}$ boson of the
order of the TeV, the effective coupling in Fig. 2 is $G_{F}^{\prime}\approx
10^{-4}G_{F}$ and that in Fig. 3 is $G_{F}^{\prime}\approx 10^{-2}G_{F}$.
Figure 2: Annihilation of two Stückelinos into two gauge vectors via the
exchange of a gaugino.
Let us go back now to Fig. 2, where we denoted with $p_{1}$ and $p_{2}$ the
incoming momenta of the Stückelinos while $k_{1}$ and $k_{2}$ are the two
outcoming momenta of the gauge bosons in the final state. We will concentrate
on the case with two photons in the final state. In this case the result for
the differential cross section is given by
$\
\frac{d\sigma}{d\Omega}=\frac{4M_{S}^{2}\omega_{1}}{16\pi^{2}(\omega_{1}+\omega_{2})^{2}(\sqrt{M_{S}^{2}-E_{2}^{2}})}\sum_{i,j=1}^{2}\mathcal{M}_{i}\mathcal{M}^{*}_{j}$
(41)
where $\omega_{1}$ and $\omega_{2}$ are the energies of the two outcoming
photons. Each amplitude $\mathcal{M}_{i}$ is proportional to the relative
coefficient $(C^{(a)})^{2}$ whose generic form is given in (39). In our
scenario the Stückelino annihilations alone, being the cross section (41)
extremely weak, cannot give a relic density in the WMAP preferred range. Thus,
in the scenario of an XWIMP Stückelino, we are forced to consider
coannihilations between the Stückelino and the NLSP [16]. Several scenarios
can be considered for the NLSP. We can split them into two major classes: one
in which the NLSP is either a pure bino or a pure wino, and thus a
coannihilation with a third MSSM particle is needed in order to recover the
WMAP result, and one in which the NLSP is a generic MSSM neutralino with a
non-negligible bino and/or wino component. In both classes in order to have
effective coannihilations, the NLSP (and eventually the other MSSM particle
involved in the coannihilation process) must be almost degenerate in mass.
Furthermore, in the most common applications, the cross sections for the
annihilations of the the LSP and the NLSP and that of the coannihilations
between the LSP and NSLP are roughly of the same order of magnitude. In our
case this last condition will be stretched and the cross section we are going
to discuss will differ for some orders of magnitude. This situation is not
completely new in literature: already in [25] in which the NLSP is the stop
these differences are of order $10^{-2}$. In [16] differences of order $\leq
10^{-4}$ are considered, while in [26] $10^{-4}\div 10^{-5}$ differences are
found888This can be extracted from Fig.4 of the previous reference after an
appropriate rescaling.. But what assures us that the two species are still in
thermal equilibrium and do not decouple separately? The existence of
interactions of the type $LSP+MSSM1\to NLSP+MSSM2$ is then required to keep
the LSP in equilibrium [27]. $MSSM1,MSSM2$ are two MSSM particles. Furthermore
$MSSM1$ better be relativistic so that its abundance is much larger of any
cold particle to foster the above reaction. The above reaction will then keep
the LSP at equilibrium and the formalism of coannihilation can be safely
employed. As we will see in the next section, all of these requirements are
met in our scenario.
As a first example we then consider a pure bino as the NLSP. The allowed
coannihilation processes with the Stückelino are those which involve an
exchange of a photon or a $Z_{0}$ in the intermediate state and with a SM
fermion-antifermion pair, Higgses and $W$’s in the final state. The diagram
with the fermion-antifermion in the final state is sketched in Fig. 3. The
differential cross section in the center of mass frame has the following
general form
$\frac{d\sigma}{d\Omega}\propto\frac{1}{s}\frac{p_{f}}{p_{i}}|\mathcal{M}|^{2}$
(42)
where $s$ is the usual Mandelstam variable and $p_{f,i}$ is the spatial
momentum of the outgoing (incoming) particles. On dimensional ground
$|\mathcal{M}|^{2}$ has at least a linear dependence on $p_{f}$ and this
implies that the dominant contribution comes from the diagram with the SM
fermion-antifermion pair $f$ and $\bar{f}$ in the final state:
$\Psi_{S}\lambda^{(a)}\rightarrow f\bar{f}$ (43)
Figure 3: Coannihilation of a Stückelino and a bino into a $f\bar{f}$ pair via
the exchange of a photon or a $Z_{0}$.
The resulting differential cross section, computed in the center of mass
frame, is
$\frac{d\sigma}{d\Omega}=\sum_{f}c_{f}\frac{\sqrt{(E_{3}-m_{f})^{2}}}{64\pi^{2}(E_{1}+E_{2})^{2}\sqrt{(E_{1}^{2}-M_{S}^{2})}}(\mathcal{M}_{\gamma}^{2}+\mathcal{M}_{Z_{0}}^{2}+\mathcal{M}^{*}_{\gamma}\mathcal{M}_{Z_{0}}+\mathcal{M}_{\gamma}\mathcal{M}_{Z_{0}}^{*})$
(44)
where the sum is extended to all the SM fermions (with mass $m_{f}$) while
$c_{f}$ is a color factor. Details of the amplitude computation can be found
in Appendix A.
## 5 Stückelino Relic Density
In this section we compute the relic density of the Stückelino. The case of
the Stückelino as a cold dark matter candidate has been studied for the first
time in [22]. As we said in the previous section we study two scenarios: the
first in which the Stückelino coannihilates with only one NLSP degenerate in
mass (a generic MSSM neutralino), the second in which there is an additional
supersymmetric particle (either a chargino or a stau) involved in the
coannihilation process with the Stückelino and the NLSP. In the following we
will be largely following [16]. Since this is a first study and given also the
simplifying choice (34) we will defer a complete analysis to a future work and
will content ourselves with showing that our model can accommodate for WMAP
data. Then, following this philosophy we will not solve the Boltzmann equation
numerically but, in agreement with [16], we will argue that, if the ratio
between the thermally averaged cross sections of the coannihilation of Fig. 3
and that of a typical neutralino annihilation is much less than one, a relic
density satisfying the WMAP requirements can be found.
Just to fix the notation we briefly review the relic density computation for
$N$ interacting species [25, 26, 27]. The requirements discussed in the
previous section are met by our model given the lagrangian (30) and the
condition of (34). In this case all channels are open to interactions and the
Stückelino has an interaction with the photon and the bino of strength
$b_{2}^{(1)}$999There is another interesting possibility if we insist on
imposing (34): taking also $Q_{L}=0$ SUSY terms of the type $\int
d^{2}\theta\int
d^{2}\bar{\theta}\log(S+S^{\dagger}+V^{(0)})/M_{S}(LH_{u}/M_{S}^{2}+H_{u}^{\dagger}L^{\dagger}/M_{S}^{2})$
can be safely added to the lagrangian. This term induces a vertex between the
Stückelino, neutrino and Higgs field which can be a viable candidate to keep
the Stückelino in thermal equilibrium..
The Boltzmann equation for $N$ particle species is given by:
$\frac{dn}{dt}=-3Hn-\sum_{i,j=1}^{N}\langle\sigma_{ij}v_{ij}\rangle(n_{i}n_{j}-n^{eq}_{i}n^{eq}_{j})$
(45)
where $n_{i}$ denotes the number density per unit of comoving volume of the
species $i=1,\ldots,N$ ($i=1$ refers to the LSP, $i=2$ refers to the NLSP, and
so on), $n=\sum_{i}n_{i}$, $H$ is the Hubble constant, $\sigma_{ij}$ is the
annihilation cross section between a species $i$ and a species $j$, $v_{ij}$
is the modulus of the relative velocity while $n_{i}^{eq}$ is the equilibrium
number density of the species $i$ given by:
$\frac{n_{i}^{eq}}{n^{eq}}=\frac{g_{i}\left(1+\Delta_{i}\right)^{3/2}e^{-\Delta_{i}x_{f}}}{\sum_{i}g_{i}\left(1+\Delta_{i}\right)^{3/2}e^{-\Delta_{i}x_{f}}}$
(46)
where $g_{i}$ are the internal degrees of freedom,
$\Delta_{i}=(m_{i}-m_{1})/m_{1}$. $x_{f}=m_{1}/T$ is known once (45) is
solved. A preliminary estimate of $x_{f}$ can be obtained by solving the
equation
$x\simeq\ln\left(x^{1/2}M_{P}\,m_{LSP}\left<\sigma v\right>\right)$ (47)
which can be obtained from the decoupling condition. This can be done using
the estimate101010To check this crude estimate in the case of our Stückelino,
we have also solved numerically $\left<\sigma v\right>$ sweeping the
temperatures range $1\div 100$ GeV. We have then fit the results to recover a
function in good agreement with (48).
$\left<\sigma v\right>\simeq G^{2}\,m_{LSP}^{2}x^{-5/2}$ (48)
where the effective coupling $G$ can be the $G_{F},G_{F}^{\prime}$ introduced
in Section 4. In the mass range $m_{LSP}=10\div 1000$ GeV, by plugging in (48)
$G=G_{F}$ we would get $x_{f}=25\div 30$ (if we would take our Stückelino as a
separate species, that is we would use $G=G_{F}^{\prime}$, we would get those
values divided by half).
Eq. (45) can be rewritten in a useful way by defining the thermal average of
the effective cross section
$\langle\sigma_{eff}v\rangle\equiv\sum_{i,j=1}^{N}\langle\sigma_{ij}v_{ij}\rangle\frac{n_{i}^{eq}}{n^{eq}}\frac{n_{j}^{eq}}{n^{eq}}$
(49)
obtaining
$\frac{dn}{dt}=-3Hn-\langle\sigma_{eff}v\rangle(n^{2}-(n^{eq})^{2})$ (50)
where $n^{eq}=\sum_{i}n_{i}^{eq}$. It is sensible to use these approximations
when the LSP is kept in equilibrium by a relativistic particle in the thermal
background as discussed in the previous section. Given the typical values of
$x_{f}$ discussed above, the ratio between the number density per comoving
volume of this relativistic species and that of a cold relic is
$n^{eq}_{rel}/n^{eq}_{cold}=10^{4}\div 10^{6}$, which is sufficient to keep
the Stückelino coupled until the end of coannihilations.
As a rule of thumb [28] a first order estimate of the relic density is given
by
$\Omega_{\chi}h^{2}\simeq\frac{10^{-27}\,{\rm
cm^{3}\,s^{-1}}}{\langle\sigma_{eff}v\rangle}$ (51)
To give a rough idea of the role played by coannihilations we plotted in Fig.
4 the relic density estimate (51) induced by an electro-weak cross section
((48) with $G=G_{F}$), a Stückelino cross section ((48) with
$G=G^{\prime}_{F}$) and two coannihilations cross sections estimations. We can
see that the Stückelino annihilations cannot give a relic density in the WMAP
data range, while coannihilations can do it. Moreover we can see that the
effect of coannihilations on the MSSM is to decrease the efficiency of the
MSSM annihilations (while they increase the Stückelino one) and to increase
the LSP mass value (according to an increasing mass gap) in order to agree
with WMAP data. We stress that Fig. 4 does not take into account several MSSM
parameters such as the sfermion masses, neutralino composition etc., so it is
just a rough estimate that, however, clarifies the role played by the
coannihilations.
Figure 4: Relic density estimation for a pure Stückelino cross section
(black), a pure electro-weak cross section (red), a coannihilating cross
section with a Stückelino-neutralino mass gap $\Delta=1\%$ (blue) and a
coannihilating cross section with $\Delta=5\%$ (dashed blue). The WMAP data
range is the gray strip. The Figure on the right is a zoom of that on the left
around the region of interest.
In the following we will now give a better estimation of
$\langle\sigma_{eff}v\rangle$ in the two cases $N=2$ and $N=3$ using what we
have learnt in our scenario and the coannihilation cross section of Fig. 3
presented in detail in Appendix A.
* •
$N=2$ case. Assuming that the relative velocities are all equal $v_{ij}\equiv
v$ we get:
$\langle\sigma_{eff}^{(2)}v\rangle=\langle\sigma_{22}v\rangle\frac{\langle\sigma_{11}v\rangle/\langle\sigma_{22}v\rangle+2\langle\sigma_{12}v\rangle/\langle\sigma_{22}v\rangle
Q+Q^{2}}{(1+Q)^{2}}$ (52)
where $Q=n_{2}^{eq}/n_{1}^{eq}$. The first term in the numerator can be
neglected because the Stückelino annihilation cross section is suppressed by a
factor $(C^{(a)})^{4}$ with respect to the MSSM neutralino annihilations (see
the previous section) and thus
$\langle\sigma_{11}v\rangle\ll\langle\sigma_{22}v\rangle$. The second term
involves the coannihilation cross section. Let us consider the case in which
the NLSP is a generic MSSM neutralino (a linear combination of
$\lambda^{(1)},\ \lambda^{(2)}_{3},\ \tilde{h}_{d}^{0},\ \tilde{h}_{u}^{0}$)
with a non-vanishing bino or wino components. As we saw in the previous
section each amplitude is generically proportional to $C^{(a)}g_{i}$ with
$i=1,2$. Without loss of generality we consider the diagram which involves the
bino component $\Psi_{S}\lambda^{(1)}\rightarrow f\bar{f}$ and a photon
exchange in the intermediate channel, i.e the $\mathcal{M}_{\gamma}^{2}$
amplitude in (44). We get
$C^{2}_{\gamma}=(C^{(1)}\cos\theta_{W})^{2}=2(b_{2}^{(1)})^{2}g_{1}^{4}\cos^{2}\theta_{W}$
(53)
From the expression of the mixed $U(1)^{\prime}-U(1)_{Y}-U(1)_{Y}$ anomaly
(see [7]) and from the (32) we have the following relation
$b_{2}^{(1)}=\frac{3(3Q_{Q}+Q_{L})}{256\pi^{2}b_{3}}$ (54)
where $b_{3}=M_{Z^{\prime}}/4g_{0}$. With the assumption $M_{Z^{\prime}}=1$
TeV as in [7] we finally get
$\frac{C^{2}_{\gamma}}{e^{2}}\simeq 5.76\times
10^{-12}(3g_{0}Q_{Q}+g_{0}Q_{L})^{2}\,{\rm GeV}^{-2}$ (55)
where $e$ is the electric charge. We get similar expressions for the other
three terms in (44). This result has to be compared to the typical weak cross
section $\langle\sigma_{22}v\rangle\simeq 10^{-9}\,{\rm GeV}^{-2}$. As long as
the charges and the coupling constant of the extra $U(1)$ satisfy the
perturbative requirement
$g_{0}^{2}\cdot(3Q_{Q}+Q_{L})^{2}<16$ (56)
the following upper bound is satisfied:
$\frac{\langle\sigma_{12}v\rangle}{\langle\sigma_{22}v\rangle}\lesssim
10^{-6}$ (57)
in the case of a pure bino, while
$\frac{\langle\sigma_{12}v\rangle}{\langle\sigma_{22}v\rangle}\lesssim
10^{-5}$ (58)
in the case of a pure wino. Accordingly to eqs. (51), (52), (57) and (58) the
relic density gets rescaled as [16]
$\left(\Omega
h^{2}\right)^{(2)}\simeq\left[\frac{1+Q}{Q}\right]^{2}\left(\Omega
h^{2}\right)^{(1)}$ (59)
We performed a random sampling of MSSM models in which the NLSP is a pure bino
or a mixed bino-higgsino (the case of a pure wino falls back into the $N=3$
case due to the wino-chargino mass degeneracy) and we computed the relic
density in presence of coannihilations using the DarkSUSY package [29]. These
two situations are easily realized in some corners of the mSUGRA parameter
space. Thus in our scan we assumed this scenario in order to fix the pattern
of the supersymmetry breaking parameters at weak scale. We emphasize here that
this choice is completely arbitrary, and it is assumed only for simplicity,
since in our model [7] the supersymmetry breaking mechanism is not specified.
In the former case there is no model which satisfies the WMAP constraints
[30]:
$0.0913\leq\Omega h^{2}\leq 0.1285$ (60)
since the annihilation cross section of a pure bino is too low and the
rescaling (59) is not enough to get the right relic density. In the latter
case the higgsino component tends to increase the annihilation cross section
and thus we find models which satisfy the WMAP constraints. The results are
summarized in Fig. 5 for $\Delta_{2}=1\%$ and $\Delta_{2}=5\%$.
Figure 5: Stückelino relic density in the case in which the NLSP is a linear
combination bino-higgsino. Red (darker) points denote models which satisfy
WMAP data. Left panel: $\Delta_{2}=1\%$. Right panel: $\Delta_{2}=5\%$.
In order to fulfill the WMAP data (red (darker) points in the plot (5)) the
Stückelino mass must be in the range $50\;\text{GeV}\lesssim M_{S}\lesssim
700\;\text{GeV}$ in the limit $\Delta_{2}\to 0$, where the lowest bound is
given by the current experimental constraints [31].
* •
$N=3$ case. This is the case in which there is a third MSSM particle almost
degenerate in mass with the LSP and the NLSP. Typical situations of this kind
arise when the NLSP and the next to next to lightest supersymmetric particle
(NNLSP) are respectively the bino and the stau or the wino and the lightest
chargino. Expanding in an explicit way all the terms in the sum (49) we get:
$\displaystyle\langle\sigma_{\text{eff}}^{(3)}v\rangle$ $\displaystyle=$
$\displaystyle\langle\sigma_{11}v\rangle\gamma_{1}^{2}+\langle\sigma_{12}v\rangle\gamma_{1}\gamma_{2}+\langle\sigma_{13}v\rangle\gamma_{1}\gamma_{3}+$
(61)
$\displaystyle\langle\sigma_{21}v\rangle\gamma_{2}\gamma_{1}+\langle\sigma_{22}v\rangle\gamma_{2}^{2}+\langle\sigma_{23}v\rangle\gamma_{2}\gamma_{3}+$
$\displaystyle\langle\sigma_{31}v\rangle\gamma_{3}\gamma_{1}+\langle\sigma_{32}v\rangle\gamma_{3}\gamma_{2}+\langle\sigma_{33}v\rangle\gamma_{3}^{2}$
$\displaystyle=$
$\displaystyle\Big{[}\langle\sigma_{11}v\rangle(n^{\text{eq}}_{1})^{2}+\langle\sigma_{12}v\rangle
n^{\text{eq}}_{1}n^{\text{eq}}_{2}+\langle\sigma_{13}v\rangle
n^{\text{eq}}_{1}n^{\text{eq}}_{3}+$ $\displaystyle\
\langle\sigma_{21}v\rangle
n^{\text{eq}}_{2}n^{\text{eq}}_{1}+\langle\sigma_{22}v\rangle(n^{\text{eq}}_{2})^{2}+\langle\sigma_{23}v\rangle
n^{\text{eq}}_{2}n^{\text{eq}}_{3}+$
$\displaystyle\phantom{\Big{[}}\langle\sigma_{31}v\rangle
n^{\text{eq}}_{3}n^{\text{eq}}_{1}+\langle\sigma_{32}v\rangle
n^{\text{eq}}_{3}n^{\text{eq}}_{2}+\langle\sigma_{33}v\rangle(n^{\text{eq}}_{3})^{2}\Big{]}\frac{1}{(n^{\text{eq}})^{2}}$
$\displaystyle\simeq$
$\displaystyle\frac{\big{[}\langle\sigma_{22}v\rangle(n^{\text{eq}}_{2})^{2}+2\langle\sigma_{23}v\rangle
n^{\text{eq}}_{2}n^{\text{eq}}_{3}+\langle\sigma_{33}v\rangle(n^{\text{eq}}_{3})^{2}\big{]}}{(n^{\text{eq}})^{2}}$
where in the last line we have neglected the terms
$\langle\sigma_{11}v\rangle$, $\langle\sigma_{12}v\rangle$ and
$\langle\sigma_{13}v\rangle$ since these are the thermal averaged cross
sections which involve the Stückelino. By introducing a new set of variables
defined by
$Q_{i}=\frac{n^{\text{eq}}_{i}}{n^{\text{eq}}_{1}}=\frac{g_{i}}{g_{1}}(1+\Delta_{i})^{3/2}e^{-x_{f}\Delta_{i}}\qquad\text{for}\
i=2,3$ (62)
where $g_{i}$ are the internal degrees of freedom of the particle species,
$x_{f}=m_{1}/T$ and $\Delta_{i}=(m_{i}-m_{1})/m_{1}$, we obtain
$\displaystyle\langle\sigma_{\text{eff}}^{(3)}v\rangle$ $\displaystyle\simeq$
$\displaystyle\frac{\langle\sigma_{22}v\rangle
Q_{2}^{2}+2\langle\sigma_{23}v\rangle Q_{2}Q_{3}+\langle\sigma_{33}v\rangle
Q_{3}^{2}}{\left(1+Q_{2}+Q_{3}\right)^{2}}$ (63)
Under the assumption $(m_{3}-m_{2})/m_{1}\ll 1/x_{f}$, $Q_{3}/Q_{2}\simeq
g_{3}/g_{2}$ we finally get
$\langle\sigma_{\text{eff}}^{(3)}v\rangle\simeq\frac{Q_{2}^{2}}{\left[1+\left(1+\frac{g_{3}}{g_{2}}\right)Q_{2}\right]^{2}}\langle\sigma_{\text{MSSM}}v\rangle$
(64)
where
$\langle\sigma_{\text{MSSM}}v\rangle=\langle\sigma_{22}v\rangle+2\frac{g_{3}}{g_{2}}\langle\sigma_{23}v\rangle+\left(\frac{g_{3}}{g_{2}}\right)^{2}\langle\sigma_{33}v\rangle$
(65)
In order to compute the rescaling factor between the relic density of our
model and the MSSM relic density we have to express $\sigma_{\text{MSSM}}$ in
terms of a two coannihilating species effective cross section. This is given
by
$\displaystyle\langle\sigma_{\text{eff}}^{(2)}v\rangle$ $\displaystyle=$
$\displaystyle\frac{\langle\sigma_{22}v\rangle(n^{\text{eq}}_{2})^{2}+2\langle\sigma_{23}v\rangle
n^{\text{eq}}_{2}n^{\text{eq}}_{3}+\langle\sigma_{33}v\rangle(n^{\text{eq}}_{3})^{2}}{(n^{\text{eq}})^{2}}$
(66) $\displaystyle=$
$\displaystyle\frac{\langle\sigma_{22}v\rangle(n^{\text{eq}}_{2})^{2}+2\langle\sigma_{23}v\rangle
n^{\text{eq}}_{2}n^{\text{eq}}_{3}+\langle\sigma_{33}v\rangle(n^{\text{eq}}_{3})^{2}}{(n^{\text{eq}}_{2}+n^{\text{eq}}_{3})^{2}}$
$\displaystyle=$
$\displaystyle\frac{\langle\sigma_{22}v\rangle(n^{\text{eq}}_{2})^{2}+2\langle\sigma_{23}v\rangle
n^{\text{eq}}_{2}n^{\text{eq}}_{3}+\langle\sigma_{33}v\rangle(n^{\text{eq}}_{3})^{2}}{(n^{\text{eq}}_{2})^{2}(1+n^{\text{eq}}_{3}/n^{\text{eq}}_{2})^{2}}$
$\displaystyle=$
$\displaystyle\frac{\langle\sigma_{22}v\rangle+2\langle\sigma_{23}v\rangle
Q_{23}+\langle\sigma_{33}v\rangle Q_{23}^{2}}{(1+Q_{23})^{2}}$
where
$\displaystyle Q_{23}$ $\displaystyle=$ $\displaystyle
n^{\text{eq}}_{3}/n^{\text{eq}}_{2}=\frac{g_{3}}{g_{2}}\left(1+\frac{m_{3}-m_{2}}{m_{2}}\right)^{3/2}e^{-x_{f}\frac{m_{3}-m_{2}}{m_{2}}}$
(67) $\displaystyle\simeq$ $\displaystyle\frac{g_{3}}{g_{2}}$
since $(m_{3}-m_{2})/m_{1}\ll 1/x_{f}$ and $m_{2}>m_{1}$ then
$(m_{3}-m_{2})/m_{2}\ll 1/x_{f}$. We remind the reader that the values of
$n^{\text{eq}}_{2}$, $n^{\text{eq}}_{3}$ and $n^{\text{eq}}$ are different
with respect to those in the former case since now there are only two species
in the thermal bath. We then find
$\displaystyle\langle\sigma_{\text{eff}}^{(2)}v\rangle$ $\displaystyle\simeq$
$\displaystyle\frac{\langle\sigma_{22}v\rangle+2\frac{g_{3}}{g_{2}}\langle\sigma_{23}v\rangle+\left(\frac{g_{3}}{g_{2}}\right)^{2}\langle\sigma_{33}v\rangle}{\left(1+\frac{g_{3}}{g_{2}}\right)^{2}}$
(68) $\displaystyle\simeq$
$\displaystyle\frac{\langle\sigma_{\text{MSSM}}v\rangle}{\left(1+\frac{g_{3}}{g_{2}}\right)^{2}}$
and inserting back this relation into (64) we obtain
$\langle\sigma_{\text{eff}}^{(3)}v\rangle\simeq\left[\frac{\left(1+\frac{g_{3}}{g_{2}}\right)Q_{2}}{1+\left(1+\frac{g_{3}}{g_{2}}\right)Q_{2}}\right]^{2}\langle\sigma_{\text{eff}}^{(2)}v\rangle$
(69)
The rescaling factor between the three and two particle species relic density
is given by the following relation
$\left(\Omega
h^{2}\right)^{(3)}\simeq\left[\frac{1+\left(1+\frac{g_{3}}{g_{2}}\right)Q_{2}}{\left(1+\frac{g_{3}}{g_{2}}\right)Q_{2}}\right]^{2}\left(\Omega
h^{2}\right)^{(2)}$ (70)
We performed a random sampling of MSSM models with bino-stau and wino-chargino
coannihilations. The first situation is realized in some corners of the mSUGRA
parameter space111111Or in the so called Constrained MSSM (CMSSM). while the
second situation is naturally realized in anomaly mediated supersymmetry
breaking scenarios. For each model we computed the relic density $\left(\Omega
h^{2}\right)^{(2)}$ for the two coannihilating species with the DarkSUSY
package [29]. We finally computed $\left(\Omega h^{2}\right)^{(3)}$ using
(70). The bino-stau models which satisfy the WMAP constraints have a
Stückelino mass in the range $100\;\text{GeV}\lesssim M_{S}\lesssim
350\;\text{GeV}$ in the limit $\Delta_{2}\to 0$. As the mass gap increases the
number of allowed models drastically decreases and eventually vanishes for
$\Delta\simeq 5\%$. In the wino-chargino case, models which satisfy the WMAP
constraints are shown in Fig. 6 for four reference values of $\Delta_{2}$. The
space of parameters with $\Delta_{2}\lesssim 5\%$ and a Stückelino mass
$M_{S}\gtrsim 700$ GeV is favored while as the mass gap increases lower
Stückelino masses become favored, e.g. $100\;\text{GeV}\lesssim M_{S}<200$ GeV
($\Delta_{2}\simeq 20\%$).
Figure 6: Stückelino relic density in the case in which the NLSP is a wino
while the NNLSP is the lightest chargino. Red (darker) points denote models
which satisfy WMAP data. Upper left panel: $\Delta_{2}=20\%$. Upper right
panel: $\Delta_{2}=10\%$. Lower left panel: $\Delta_{2}=5\%$. Lower right
panel: $\Delta_{2}=1\%$.
## 6 Conclusions
We studied a possible dark matter candidate in the framework of our minimal
anomalous $U(1)^{\prime}$ extension of the MSSM [7]. In the decoupling limit
(34) and under the assumption $M_{0}\gg M_{S},M_{V^{(0)}}$ the Stückelino
turns out to be the LSP. Being an XWIMP the Stückelino annihilation cross
section is suppressed with respect to the typical weak interaction cross
sections. This implies that in order to satisfy the WMAP constraints on the
relic density we must have at least a NLSP almost degenerate in mass with the
Stückelino. We considered the case with two and three coannihilating particles
and we found some configuration which satisfies the WMAP constraints. The
results depend on the mass gap between the Stückelino and the NLSP. In the
exact degeneracy limit $\Delta_{2}\to 0$ the allowed models have a Stückelino
mass in the range $50\;\text{GeV}\lesssim M_{S}\lesssim 700\;\text{GeV}$ for
the bino-higgsino coannihilation case while $900\;\text{GeV}\lesssim
M_{S}\lesssim 2\;\text{TeV}$ for the wino-chargino coannihilation case. When
the mass gap is $\Delta_{2}\simeq 20\%$ the allowed models are those with
wino-chargino coannihilations and a Stückelino mass of
$100\;\text{GeV}\lesssim M_{S}<200$ GeV. Finally let us comment on the
differences between our scenario and that studied in the work [16]. In our
framework the $U(1)^{\prime}$ does not arise from a hidden sector and thus all
the MSSM fields can be charged under this extra abelian gauge group. This is
the most relevant feature which could also be detected experimentally (see for
example [6]). Moreover, in our scenario the Stückelino interactions are
suppressed with respect to the weak interactions due to the GS couplings while
in [16] the mechanism to suppress the couplings and give an XWIMP is provided
by the kinetic mixing between the $U(1)^{\prime}$ and $U(1)_{Y}$.
## Appendix A Amplitude for $\lambda_{1}+\psi_{S}\to f\bar{f}$
In this Appendix we give some details of the amplitude computation for the
process $\lambda_{1}+\psi_{S}\to f\bar{f}$,
$\mathcal{M}=-ik^{\mu}\bar{v}_{S}\gamma_{5}\left[\gamma_{\mu},\gamma_{\nu}\right]u_{1}\left[eq_{f}C_{\gamma}\frac{\eta^{\nu\rho}}{k^{2}}\bar{u}_{f}\gamma_{\rho}v_{f}+\frac{g_{Z_{0}}}{2}C_{Z_{0}}\frac{\eta^{\nu\rho}}{k^{2}-M_{Z_{0}}^{2}}\bar{u}_{f}\gamma_{\rho}(v_{f}^{Z_{0}}-a_{f}^{Z_{0}}\gamma_{5})v_{f}\right]$
(71)
where $q_{f}$ denote the electric charges, $v_{f}^{Z_{0}}$ and $a_{f}^{Z_{0}}$
are the vectorial and axial couplings with $Z_{0}$,
$C_{\gamma}=C^{(1)}\cos\theta_{W}$, $C_{Z_{0}}=-C^{(1)}\sin\theta_{W}$ while
$k^{2}=s$ is the momentum of the intermediate gauge boson. The corresponding
square modulus is
$\displaystyle|\mathcal{M}|^{2}$ $\displaystyle=$
$\displaystyle-64\left[T_{a}\left(\frac{a_{f}C_{Z_{0}}g_{Z_{0}}}{k^{2}-M_{Z_{0}}^{2}}\right)^{2}+T_{v}\left(\frac{2C_{\gamma}eq_{f}}{k^{2}}+\frac{C_{Z_{0}}g_{Z_{0}}v_{f}}{k^{2}-M_{Z_{0}}^{2}}\right)^{2}\right]$
(72)
with
$\displaystyle T_{v}$ $\displaystyle=$ $\displaystyle
m_{f}^{4}(p_{\lambda_{1}}p_{S})+M_{1}M_{S}\Big{[}2m_{f}^{4}+3(p_{f}p_{\bar{f}})m_{f}^{2}+(p_{f}p_{\bar{f}})^{2}\Big{]}+$
$\displaystyle-(p_{f}p_{\bar{f}})\Big{[}(p_{\lambda_{1}}p_{f})(p_{f}p_{S})+(p_{\lambda_{1}}p_{\bar{f}})(p_{\bar{f}}p_{S})\Big{]}+m_{f}^{2}\Big{[}(p_{\lambda_{1}}p_{S})(p_{f}p_{\bar{f}})+$
$\displaystyle-2(p_{\lambda_{1}}p_{f})(p_{f}p_{S})-(p_{\lambda_{1}}p_{\bar{f}})(p_{f}p_{S})-(p_{\lambda_{1}}p_{f})(p_{\bar{f}}p_{S})-2(p_{\lambda_{1}}p_{\bar{f}})(p_{\bar{f}}p_{S})\Big{]}$
$\displaystyle T_{a}$ $\displaystyle=$
$\displaystyle\Big{[}(p_{\lambda_{1}}p_{\bar{f}})(p_{f}p_{S})+(p_{\lambda_{1}}p_{f})(p_{\bar{f}}p_{S})\Big{]}m_{f}^{2}-M_{1}M_{S}\left[m_{f}^{4}-(p_{f}p_{\bar{f}})^{2}\right]+$
(73)
$\displaystyle-(p_{f}p_{\bar{f}})\Big{[}(p_{\lambda_{1}}p_{f})(p_{f}p_{S})+(p_{\lambda_{1}}p_{\bar{f}})(p_{\bar{f}}p_{S})\Big{]}$
where $p_{\lambda_{1}}$, $p_{S}$, $p_{f}$ and $p_{\bar{f}}$ are the bino,
Stückelino and SM fermions 4-momenta respectively. Writing all the momenta in
function of $s$ and integrating over the solid angle we get
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\sigma=c_{f}\left(g_{1}^{2}b_{2}^{(1)}\right)^{2}\sqrt{s-4m_{f}^{2}}\times$
(74)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\times\frac{\Big{[}-2M_{1}^{4}+\left(4M_{S}^{2}+s\right)M_{1}^{2}-6M_{S}sM_{1}-2M_{S}^{4}+s^{2}+M_{S}^{2}s\Big{]}}{12\pi\left(M_{Z_{0}}^{2}-s\right)^{2}s^{5/2}\sqrt{M_{1}^{4}-2\left(M_{S}^{2}+s\right)M_{1}^{2}+\left(M_{S}^{2}-s\right)^{2}}}\times$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\times\Bigg{[}\left(2m_{f}^{2}+s\right)\Big{(}2\cos\theta_{W}eq_{f}\left(M_{Z_{0}}^{2}-s\right)+\sin\theta_{W}g_{Z_{0}}v_{f}s\Big{)}^{2}+\left(\sin\theta_{W}g_{Z_{0}}a_{f}\right)^{2}s^{2}\left(s-4m_{f}^{2}\right)\Bigg{]}$
Acknowledgements
A. R. would like to thank Prof. Michael Green and DAMTP for hospitality and
the Marie Curie Research Training Network “Superstring Theory”, contract MRTN-
CT-2004-512194 and contract ESF JD164 for financial support during the
completion of this paper. F.Fucito wants to thank N.Fornengo and P.Ullio for
discussions.
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|
arxiv-papers
| 2008-11-12T17:38:03 |
2024-09-04T02:48:58.757413
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Francesco Fucito, Andrea Lionetto, Andrea Mammarella, Antonio Racioppi",
"submitter": "Antonio Racioppi",
"url": "https://arxiv.org/abs/0811.1953"
}
|
0811.2115
|
# Searching for white dwarfs candidates in Sloan Digital Sky Survey Data
Mirosław Należyty1 Agnieszka Majczyna1,2 Anna Ciechanowska1 and Jerzy Madej1
1University of Warsaw Astronomical Observatory, Al. Ujazdowskie 4, 00-478
Warsaw, Poland 2The Andrzej Sołtan Institute for Nuclear Studies, Hoża 69,
00-681 Warsaw, Poland nalezyty@astrouw.edu.pl
###### Abstract
Large amount of observational spectroscopic data are recently available from
different observational projects, like Sloan Digital Sky Survey. It’s become
more urgent to identify white dwarfs stars based on data itself i.e. without
modelling white dwarf atmospheres. In particular, existing methods of white
dwarfs identification presented in Kleinman et al. (2004) and in Eisenstein et
al. (2006) did not allow to find all the white dwarfs in examined data. We
intend to test various criteria of searching for white dwarf candidates, based
on photometric and spectral features.
## 1 Introduction
Because of their physical properties, white dwarfs stars are difficult to
observe and identification. The first version of the McCook & Sion catalogue
(1987) contains 1279 white dwarfs only. Recently, observational techniques was
considerable improve, so the amount of photometric and spectral data rapidly
grows. For this reason also the number of identified white dwarfs grows, but
it is very needed to elaborate efficient identification and classification
methods.
Until now, many of the white dwarf samples were selected, based on different
selection criteria. For example Fleming et al. (1996) published a white dwarfs
catalogue based on ROSAT data (Trumper 1983, 1992). A classification was made
by searching for position coincidence between ROSAT sources and hot white
dwarfs from McCook & Sion catalogue. Sources with no optical counterparts were
marked as white dwarf if was visible only in the softest X-ray energy range.
White dwarfs in EUV, selected also from the ROSAT all-sky survey by Marsh et
al. (1997a,b), 129 white dwarf stars from Bergeron, Saffer & Liebert (1992),
200 white dwarfs from the Palomar Green survey (Green, Schmidt & Liebert
(1986) analyzed by Liebert & Bergeron (1995) - it is only a few samples of
these objects.
The Sloan Digital Sky Survey project (York et al. 2000) provide us large
amount of very interesting data. Based on Data Release 1 and 4 (DR1 and DR4,
respectively) there were made two catalogues by Kleinman et al. (2004) and
Eisenstein et al. (2006), which contain 2551 and 9316 objects, respectively.
Used identification methods did not allow to find all white dwarfs with whole
range of effective temperatures and types in data sample. Kleinman et al.
(2004) chose white dwarf candidates based on dereddened colour indexes $u-g$
and $g-r$ and a magnitude in $u$ filter. Selected white dwarf candidates was
next verified by visual inspection and prescribed its type - DA, DB, DZ etc.
After this procedure, spectra of selected objects were fitted by theoretical
DA and DB spectra.
Eisenstein et al. (2006) disposed larger amount of data so they used more
automatic selection procedure. They select stars fulfilled applicable criteria
(see details in Eisenstein et al. 2006). In this manner they obtain sample of
blue stars, which observational spectra were fitted by theoretical spectra.
Stars with parameters characteristic for white dwarfs were next verified and
classified during different tests including visual inspection. This method has
some limitations, for example does not work for stars cooler than 8000K.
We intend to find a method of the white dwarf candidates selection - or, if it
is possible, white dwarfs itself - based only on photometric and spectral
features, chosen to obtain the best distinguishing between white dwarf stars
and other type objects. Before we start fitting theoretical model atmospheres
to observing spectra.
Not for the first time we have used data from the Sloan Digital Sky Survey
(York et al. 2000). We chose Data Release 5 (DR5) and we selected spectral
data for 10% randomly choosing objects. In this manner we obtained very
imposing amount of nearly 16000 different type objects, including galaxies,
quasars, stars, and of course white dwarfs. Additionally we also used white
dwarfs identified in SDSS DR4 by Eisenstein et al. (2006), treating these
stars on the one hand as a potential help to improve our white dwarfs
selection method, on the other hand as a control data.
## 2 Balmer lines and their parameters
The most characteristic feature of many DA white dwarf spectra is a presence
of wide, hydrogen absorption Balmer lines (except for the hottest objects). It
would appear that it should be a good selection criteria of degenerated stars
with high surface gravity, but not every white dwarf have wide Balmer lines.
Nevertheless we decided to check this
Figure 1: Colour index $u-g$ versus FWHM of the $H_{\alpha}$ Balmer line
objects with $H_{\alpha}$, $H_{\beta}$, and $H_{\gamma}$ lines present in
their spectra (dark gray dots). White dwarfs from SDSS DR4 belonging to our
sample with detected Balmer lines was marked with black dots.
possibility. Because of large amount (15998) of objects in our sample we had
to create proper software, which automatically finds hydrogen lines from
Balmer series, and then calculates its various parameters like line widths at
given depth, fluxes in lines etc. Fig. 1 shows one of the obtained results, a
dependency of FWHM of Balmer $H_{\alpha}$ line on colour index $u-g$ for
objects with $H_{\alpha}$, $H_{\beta}$, and $H_{\gamma}$ lines present in
their spectra. Clearly seeing bimodal structure is not a by-product of our
Balmer lines searching, but it is a result of a bimodal distribution of the
colour index $u-g$. A location of the white dwarfs identified in SDSS DR4
(Kleinman et al. 2004), and belonged to our sample with detected Balmer lines,
denoted by black dots, suggests, that all of the white dwarfs lies in the left
part of this diagram, with colour indexes $u-g<0.7$. In general, this is not a
truth. Visual inspection shows, that some amount of white dwarf stars also
lies in the right part of the diagram, and have larger values of colour
indexes $u-g$. However, our requirement for the presence of hydrogen Balmer
lines in spectra reduces our sample to practically star-like type objects
only. In fact, it is decreases a number of white dwarf candidates from 15998
to 4613. Unfortunately, reduced in this way sample does not contain the
hottest white dwarf stars, because there is no Balmer lines visible in their
spectra.
## 3 Fluxes in given ranges
Colour indexes based on $ugriz$ photometry are not very useful for us,
generally because the filters are too wide. So we decide to define own
artificial filters, i.e. own wavelength ranges, in which we calculated fluxes.
Choosing our filters we intended to not contain any Balmer
Figure 2: Logarithms of fluxes relation, calculated in two wavelength ranges:
$4550-4700$Å and $8000-8500$Å for all of 15998 objects in our sample (light
gray dots). Objects with detected $H_{\alpha}$, $H_{\beta}$, and $H_{\gamma}$
lines was marked by dark gray dots.
Figure 3: The same as in Fig. 2, but for 4613 objects with Balmer lines only
(dark gray dots). Black dots denote white dwarfs from SDSS DR4. Linear
function $y=0.94118x+0.33235$ is an approximate boundary between two regions
visible in this diagram. Preliminary results of the visual inspection shows,
that almost all of the 2202 objects located in the lower-right region looks
like white dwarf stars.
lines. In Fig. 2 we present a dependency of two flux logarithms, calculated in
two arbitrarily chosen wavelength ranges: $4550-4700$Å and $8000-8500$Å for
all of 15998 objects in our sample. Objects with detected $H_{\alpha}$,
$H_{\beta}$, and $H_{\gamma}$ lines (dark gray dots) occupy well defined area,
so we decided to watch them closer. Fig. 3 shows logarithms of fluxes in the
same wavelength ranges as in Fig. 2, but for 4613 objects with Balmer lines
only. It is not very difficult to see some structure in this diagram. At least
part of the objects divides to, not so bad, separate regions. This impression
is additionally magnified by black dots showing positions of the white dwarfs
from DR4, which occupy one of the mentioned above regions only. We decided to
approximate a boundary between these two areas by the linear function
$y=0.94118x+0.33235$, with empirically chosen constants. Preliminary results
of the visual inspection shows, that spectra of almost all of the 2202 objects
located in the lower-right region looks like white dwarf spectra. Because of
relatively poor separation of these two areas, between 2411 objects belonging
to the upper-left region we still could find relatively small amount of white
dwarf stars, located close to the borderline.
## 4 Conclusions
Our method of selecting white dwarf candidates is based on searching for
objects with hydrogen Balmer lines visible in their spectra, and on flux
calculations in well selected wavelength ranges. Preliminary results show that
our method allows to select quite complete sample of white dwarf stars, under
the above assumptions. Of course this method should be tested in detail, and
it needs some improvements (for example increasing separation between two
areas in Fig. 3, probably by choosing better wavelength ranges). It is quite
possible, that using this method we shall be able to find also the hottest
white dwarfs, which are too hot to show Balmer lines in their spectra,
although we will need the other criteria to select star-like type objects. We
plan to do this in future.
### 4.1 Acknowledgments
This work has been supported by the Polish Ministry of Science and Higher
Education grant No. N N203 4061 33. We also thank Institut d’Estudis Espacials
de Catalunya (IEEC) for financial support.
Funding for creation and distribution of the SDSS Archive has been provided by
the Alfred P. Sloan Foundation, the Participating Institutions, the National
Aeronautics and Space Administration, the National Science Foundation, the US
Department of Energy, the Japanese Monbukagakusho, and Max Planck Society. The
SDSS Web site is http://www.sdss.org/ .
The SDSS is managed by the Astrophysical Research Consortium (ARC) for the
Participanting Institutions. The Participanting Institutions are The
University of Chicago, Femilab, the Institute for Advanced Study, the Japan
Participation Group, the Johns Hopkins University, Los Alamos National
Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-
Institute for Astrophysics (MPA), New Mexico State University of Pittsburgh,
Princeton University, the United States Naval Observatory, and the University
of Washington.
## References
Bergeron P, Saffer R A and Liebert J 1992 ApJ 477 313
Eisenstein D J, Liebert J, Harris H C et al. 2006 ApJS 167 40
Fleming T A et al. 1996 A&A 316 147
Green R F, Schmidt M and Liebert J 1986 ApJS 61 305
Kleinman S J, Harris H C, Eisenstein D J et al. 2004 ApJ 607 426
Liebert J and Bergeron P 1995 White Dwarfs eds D Koester and K Werner (Berlin:
Springer) p12
Marsh M C, Barstow M A, Buckley D A et al. 1997a MNRAS 286 369
Marsh M C, Barstow M A, Buckley D A et al. 1997b, MNRAS 287 705
McCook G P and Sion E M 1987 ApJS 65 603
Trumper J 1983 Adv.Sp.Res. 2 241
Trumper J 1992 QJRAS 33 165
York D G, Adelman J, Anderson J E Jr et al. 2000 AJ 120 1579
|
arxiv-papers
| 2008-11-13T14:23:29 |
2024-09-04T02:48:58.768603
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Miros{\\l}aw Nale\\.zyty (1), Agnieszka Majczyna (1,2), Anna\n Ciechanowska (1) and Jerzy Madej (1) ((1) University of Warsaw Astronomical\n Observatory, Warsaw, Poland; (2) The Andrzej So{\\l}tan Institute for Nuclear\n Studies, Warsaw, Poland)",
"submitter": "Agnieszka Majczyna",
"url": "https://arxiv.org/abs/0811.2115"
}
|
0811.2142
|
# Submillimeter narrow emission lines from
the inner envelope of IRC$+$10216
Nimesh A. Patel11affiliation: Harvard-Smithsonian Center for Astrophysics, 60
Garden Street, Cambridge, MA; npatel@cfa.harvard.edu , Ken H.
Young11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden
Street, Cambridge, MA; npatel@cfa.harvard.edu , Sandra Brünken11affiliation:
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA;
npatel@cfa.harvard.edu , Robert W. Wilson11affiliation: Harvard-Smithsonian
Center for Astrophysics, 60 Garden Street, Cambridge, MA;
npatel@cfa.harvard.edu ,
Patrick Thaddeus11affiliation: Harvard-Smithsonian Center for Astrophysics, 60
Garden Street, Cambridge, MA; npatel@cfa.harvard.edu , Karl M.
Menten22affiliation: Max-Planck-Institut für Radio Astronomie, Auf dem Hügel
69, D-53121, Bonn, Germany , Mark Reid11affiliation: Harvard-Smithsonian
Center for Astrophysics, 60 Garden Street, Cambridge, MA;
npatel@cfa.harvard.edu , Michael C. McCarthy11affiliation: Harvard-Smithsonian
Center for Astrophysics, 60 Garden Street, Cambridge, MA;
npatel@cfa.harvard.edu ,
Dinh-V- Trung33affiliation: Academia Sinica Institute for Astronomy and
Astrophysics, Taipei, Taiwan 44affiliation: On leave from Institute of
Physics, Vietnamese Academy of Science and Technology, 10 Daotan, Badinh,
Hanoi, Vietnam , Carl A. Gottlieb11affiliation: Harvard-Smithsonian Center for
Astrophysics, 60 Garden Street, Cambridge, MA; npatel@cfa.harvard.edu ,
Abigail Hedden11affiliation: Harvard-Smithsonian Center for Astrophysics, 60
Garden Street, Cambridge, MA; npatel@cfa.harvard.edu
###### Abstract
A spectral-line survey of IRC+10216 in the 345 GHz band has been undertaken
with the Submillimeter Array. Although not yet completed, it has already
yielded a fairly large sample of narrow molecular emission lines with line-
widths indicating expansion velocities of $\sim$4 km s-1, less than 3 times
the well-known value of the terminal expansion velocity (14.5 km s-1) of the
outer envelope. Five of these narrow lines have now been identified as
rotational transitions in vibrationally excited states of previously detected
molecules: the v=1, J=17–16 and J=19–18 lines of Si34S and 29SiS and the v=2,
J=7–6 line of CS. Maps of these lines show that the emission is confined to a
region within $\sim$60 AU of the star, indicating that the narrow-line
emission is probing the region of dust-formation where the stellar wind is
still being accelerated.
###### Subject headings:
stars: individual (IRC$+$10216 (catalog )) — stars: late-type — circumstellar
matter — submillimeter — radio lines: stars
## 1\. Introduction
Massive circumstellar envelopes of Asymptotic Giant Branch (AGB) stars are
believed to be a major contributor of molecules and grains to the interstellar
medium. IRC+10216 (CW Leo), at a distance of 150 pc, is the archetypal AGB
carbon star with a high mass-loss rate ($>10^{-5}$ M⊙ yr-1) (Young et al.
1993; Crosas & Menten 1997). Owing to its proximity, this star is an ideal
target for detailed studies of physical and chemical processes in AGB
circumstellar envelopes (e.g. Olofsson et al. 1982). Nearly 60 molecular
species have been discovered in its circumstellar shell from single-dish
spectral-line surveys (Kawaguchi et al. 1995; Avery et al. 1992; Groesbeck et
al. 1994; Cernicharo et al. 2000; He et al. 2008).
Interferometric mapping allows us to check predictions of abundances of
various molecular species as a function of radius in the circumstellar
envelope. “Parent” molecules, such as CO, C2H2, CS, HCN, & SiS, are formed in
the stellar atmosphere in thermo-chemical equilibrium (Tsuji 1964, 1973). Once
they get levitated to a distance from the star at which the density is too low
for chemical reactions, their abundances “freeze out” at the values prevailing
in that region (McCabe et al. 1979). At around that distance ($\sim 20$ AU),
the temperature has dropped below the dust condensation value ($\sim 1200$ K)
and dust grains start forming (Monnier et al. 2000). Radiation pressure
accelerates the grains and, by friction, the molecules too, in an outflow
which, for IRC+10216, reaches a terminal velocity of 14.5 km s-1. In the outer
parts of the expanding envelope, a rich carbon-dominated photo chemistry
driven by the ambient ultraviolet field produces species such as CN and C4H
whose mostly optically thin emission can be observed as ring-like
distributions with radii, at a few times $10^{16}$–$10^{17}$ cm depending on
the chemical reactions at work (see the reviews of Glassgold 1996; Ziurys
2006).
At submillimeter wavelengths, we can observe lines requiring elevated
excitation conditions, allowing us to probe the physical conditions in the
inner circumstellar envelope (radius $\lesssim 10^{16}$ cm) where densities
and temperatures are relatively high (temperatures $\sim$100–1000 K, column
densities $\sim 10^{22}$–$10^{24}$ cm-2) (Keady & Ridgway 1993).
We have begun a spectral-line survey of IRC+10216 with the Submillimeter
Array111The Submillimeter Array is a joint project between the Smithsonian
Astrophysical Observatory and the Academia Sinica Institute of Astronomy and
Astrophysics, and is funded by the Smithsonian Institution and the Academia
Sinica. (SMA, Ho et al. 2004). Previous 345 GHz single-dish line surveys
include those done with the JCMT (Avery et al. 1992) and CSO (Groesbeck et al.
1994) in the frequency ranges of 339.6–364.6 GHz and 330.2–358.1 GHz,
respectively. Our SMA line survey will eventually cover the frequency range of
300–355 GHz with higher sensitivity and spatial resolution than previous
surveys; about 40% of the survey has so far been completed. The frequency
range 300–330 GHz will be observed for the first time. Here we present several
new results, including the discovery of many lines having narrow widths, and
detections of vibrationally excited rotational transitions in SiS, 29SiS,
Si34S and CS. A full account of the observed lines will be presented on
completion of the survey.
## 2\. Observations and Data reduction
We observed IRC$+$10216 with the SMA in 2007 February in the subcompact
configuration with baselines from 9.5 m to 69.1 m in the frequency range of
300 to 355 GHz. To follow up on some of the detected narrow lines with higher
angular and spectral resolution, we repeated the observations at 337.5 GHz
with the SMA in the extended configuration on 2008 February 19. The baseline
lengths in this configuration range from 44.2 m to 225.9 m. The synthesized
beam sizes were $3^{\prime\prime}\times 2^{\prime\prime}$ and
$0.^{\prime\prime}8\times 0.^{\prime\prime}6$ in the subcompact and extended
array observations, respectively. Table 1 summarizes the observational
parameters on the five tracks of observations which are relevant for the data
presented here. The duration of each track was from 7 to 9 hours. The phase
center was
$\alpha(2000)=09^{h}47^{m}57.38^{s},\delta(2000)=+13^{\circ}16^{\prime}43.^{\prime\prime}70$
for all observations. All the tracks in subcompact configuration were carried
out in mosaiced mode, with 5 pointings with offsets in RA and DEC:
$(0^{\prime\prime},0^{\prime\prime})$ and $(\pm 12^{\prime\prime},\pm
12^{\prime\prime})$. The extended configuration observations were carried out
with single pointing toward IRC+10216. Titan and the quasars 0851+202 and
1055+018 were observed every 20 minutes for gain calibration. The spectral
band-pass was calibrated using observations of Mars and Jupiter. Absolute flux
calibration was determined from observations of Titan and Ganymede.
The visibility data were calibrated using the Miriad package (Sault, Teuben &
Wright 1995). The mosaiced images are deconvolved using the Miriad task
mossdi; the resulting synthesized beams are summarized in Table 1. Maps of
continuum emission show the peak to have a position offset of
($\Delta\alpha,\Delta\delta)\approx(0.^{\prime\prime}7,0.^{\prime\prime}2)$
from the phase center position quoted above. The absolute position
measurements in the continuum emission are estimated to be accurate to $\sim
0.^{\prime\prime}1$. Taking into account the proper motion of IRC$+$10216 of
($\dot{\alpha},\dot{\delta})\approx(26,4)$ mas yr-1 determined by Menten et
al. (2006) our position is, within the mutual uncertainties consistent with
that determined by those authors. The continuum emission was unresolved at the
highest angular resolution of $\sim 0.^{\prime\prime}8$. The integrated
continuum flux density was 0.84 Jy at 301.1 GHz and 1.17 Jy at 337.5 GHz, with
an uncertainty of about 15% in the absolute flux calibration. All the spectra
shown below were produced by integrating the continuum-subtracted line
intensity in a $2^{\prime\prime}\times 2^{\prime\prime}$ rectangle centered on
continuum peak (by means of the Miriad task imspec).
## 3\. Expansion velocities
Within a 3′′ beam a total of 92 lines were detected in the first phase of the
SMA line survey of IRC+10216 at the central position. Truncated parabolic line
profiles were fitted using the CLASS package (using the shell model for the
line-profile) with one of the fitted parameters being $V_{exp}$, the expansion
velocity of the circumstellar shell. From this sample, 25 lines have
$V_{exp}\leq$ 7 km s-1and of these, 12 are as yet unidentified. Several are
tentatively identified to be lines of salts such as KCN, NaCl and NaCN. Some
of the lines may result from known molecules in vibrationally excited states,
such as the v3=1 $15_{7,9}-14_{7,8}$, $15_{7,8}-14_{7,7}$ doublet of SiCC at
345727.3 MHz. Figure 2 shows a sample spectrum toward IRC+10216 observed on
2008 February 9 over a 2 GHz wide band centered at 337.5 GHz. A comparison
with the line survey of Groesbeck et al. (1994) (see their Figure 1) shows
that only the C34S J=7–6 line at 337.396 GHz was detected in their
observations. All of the new narrow lines were missed in this previous survey
due to poorer sensitivity. Here we present results on the lines which are
securely identified as rotational transitions in vibrationally excited states
of molecules that are well known to be abundant in IRC+10216’s envelope.
In previous single-dish line surveys of IRC+10216 (Cernicharo et al. 2000;
Kawaguchi et al. 1995; Avery et al. 1992; Groesbeck et al. 1994) $V_{exp}$ is
not tabulated, but it can be seen from published spectra to be $\sim$14.5 km
s-1 for all lines. From the latest published line survey by He et al. (2008)
(see their Table 9.), we can plot a distribution of $V_{exp}$ which is shown
in Figure 1 as empty bars with bold outlines. This histogram peaks at 14 km
s-1. The distribution of $V_{exp}$ from our line survey is shown in Figure 1
as grey bars. Our line survey shows a peak in the same bin of 14 km s-1but
reveals a significant number of narrow lines with velocities around $\sim$4 km
s-1. Both histograms show a continuous distribution of expansion velocities
between these two peaks at 4 and 14 km s-1.
Figure 1.— Distribution of expansion velocities derived from lines detected in
the circumstellar envelope of IRC+10216. The uncertainty in $V_{exp}$ is
$\sim$0.2 km s-1($1\sigma$). White bars with bold outlines are from the recent
line survey of He et al. (2008) which is representative of all single-dish
line surveys toward IRC+10216 at cm to submm wavelengths. The bin at 14 km
s-1is shown truncated here; it actually consists of 170 lines. Grey bars
represent SMA observations, showing a new population of narrow lines which
peaks at $\sim$4 km s-1.
Lines with V${}_{exp}\leq 10$ km s-1 from IRC+10216 have been reported by
Highberger et al. (2000) and were assigned as vibrationally excited SiS and CS
lines. He et al. (2008) found four lines with $V_{exp}$=7–10.2 km s-1 (see
their Table 14) – all from vibrationally excited SiS. Narrow maser lines in
IRC+10216 from SiS and HCN were reported by Fonfría-Expósito et al. (2006) and
Schilke & Menten (2003), respectively. Savik-Ford et al. (2004) detected OH –
which has a narrow width (5.8 km s-1), but this line appears at $-$37 km s-1,
blue-shifted with respect to the systemic velocity of the star of $-26$ km
s-1. All the narrow lines we have observed are centered at the systemic
velocity of about -26 km s-1to within $\sim$2 km s-1. Infrared molecular line
profiles in the 10 $\mu$m band were observed and analyzed by Keady & Ridgway
(1993). They proposed (see their Figure 3) an expansion velocity as a function
of radius with: (1) V${}_{exp}=3\sim 4$ km s-1 from 1$\sim 8$ R∗, (2) rapid
acceleration from 4 to 11 km s-1 from 8 to 10 R∗ and (3) slower acceleration
from 11 km s-1to the terminal velocity of 14 km s-1 from 10 to 20 R∗. The
histogram of expansion velocities (Figure 1) peaks at $\sim$4 km s-1 and
$\sim$14 km s-1, consistent with this velocity structure.
Figure 2.— A sample spectrum toward IRC+10216 showing several examples of
narrow lines. Over this frequency range, the only line detected in a previous
line survey (Groesbeck et al. 1994) was the C34S J=7–6 line at 337.396 GHz.
This line has a Vexp =13.9 km s-1. All the other lines are new detections and
several of them are narrow (V${}_{exp}<7$ km s-1).
## 4\. Vibrationally excited SiS, 29SiS and Si34S
SiS is an abundant molecule in IRC+10216, one of the parent molecules found
close to the star (Glassgold 1996), and an important source of Si for the
formation of silicon carbides such as SiC, SiC2 and SiC4 (e.g., McCarthy et
al. 2003). Figure 3 (top) shows the spectrum of SiS v=1 J=19-18 emission at
343100.98 MHz (rest frequency), observed on 2007 February 12. This line was
previously detected (but not mapped) in the single-dish line-survey of
Groesbeck et al. (1994), where it is significantly weaker than here because of
poorer sensitivity (we have longer integration time as well as larger
collecting area with the SMA). Figure 3 (bottom) shows a map of the integrated
intensity emission, over the velocity interval of -40 to -10 km s-1. The total
integrated flux density is 69.1 Jy km s-1. The deconvolved source size is
1.1”x0.5” with P.A. of $-27.6^{\circ}$. Assuming a size of
$0.^{\prime\prime}5\times 0.^{\prime\prime}5$, we estimate a lower limit to
the brightness temperature of 200 K.
The emission appears to have an extended weak feature towards the southeast.
High angular resolution near-IR adaptive-optics images of IRC+10216 show
evidence of azimuthally asymmetric structures in the inner circumstellar
envelope, over angular scales of $\sim 2^{\prime\prime}$ (Menut et al. 2007).
The 3′′ angular resolution of our observations is insufficient to allow a
detailed comparison between the submillimeter line emission maps and near-IR
images.
The expansion velocity of the SiS v=1 J=19–18 line is 10.6 km s-1. This is an
example of a line of intermediate expansion velocity (see Figure 1), in the
rapidly accelerating zone of the envelope where presumably dust has already
formed.
Figure 4 shows spectra of Si34S and 29SiS v=1, J=17–16 and 19–18 lines,
detected toward IRC+10216 for the first time. This emission is unresolved and
the source-size upper limits are shown in Table 2. Assuming the size of the
emitting region is 0.′′2, we estimate a brightness temperature to be
140$\sim$200 K, which should be considered as lower limits. We estimate a
column density of 7$\times 10^{17}$ cm-2 and abundance of 7$\times 10^{-7}$
for 29SiS for an assumed excitation temperature of 550 K.
Figure 3.— Upper: Spectrum of SiS v=1 J=19-18 line obtained from a 2′′ square
centered on the star, with Vexp=10.6 km s-1. For comparison, the SiS, v=0
J=19–18 line is shown by the dashed line, (scaled down by a factor of 10). The
v=0 line has an expansion velocity of 13.5 km s-1. Note that the intensity
values for this line are scaled down by a factor of 10 in this figure. Lower:
Integrated intensity emission from the SiS v=1 J=19-18 line. The contours
levels are -5, 5, 10, 20, 40, 80, 120 $\times$ 0.45 Jy/beam km/s. The
coordinate offsets are with respect to
$\alpha(2000)=09^{h}47^{m}57.43^{s},\delta(2000)=+13^{\circ}16^{\prime}43.^{\prime\prime}98$.The
synthesized beam is shown in the lower right corner. See also Table 2.
From the observed integrated intensities of SiS, 29SiS and Si34S v=1, J=19–18
emission, we derive the isotopic abundance ratios $[^{28}\rm
Si/^{29}Si]=15.1\pm 0.7$ and $[^{32}\rm S/^{34}S]=19.6\pm 1.3$ (uncertainties
are $1\sigma$). For comparison, previously published values from single-dish
observations (Kahane et al. 1988; He et al. 2008) are $20.2\pm 2$ for
$[^{32}\rm S/^{34}S]$(in good agreement with our estimate here) and $18.7\pm
1$ for $[\rm Si/^{29}\rm Si]$ (marginally larger than our value). A possible
cause for disagreement might be non-negligible optical depths.
Figure 4.— Spectra of vibrationally excited rotational lines in isotopes of
SiS obtained from the imaged data cube. The flux density was calculated over
the area enclosed in a 2 ′′ square centered on the star. See also Table 2.
## 5\. Vibrationally excited CS
Vibrationally excited CS radio emission in IRC+10216 was first detected by
Turner (1987) in the v=1, J=2–1 and 5–4 transitions. These lines were re-
observed along with new detections of the J=3–2, 6–5 and 7–6 transitions by
Highberger et al. (2000). The J=7-6 emission reported here is the first
detection in the v=2 state (Figure 5). The triangular line-profile is
indicative of spatially unresolved emission from accelerating gas (Bujarrabal
et al. 1986). In subcompact configuration observations with angular resolution
of $\sim 3^{\prime\prime}$, the emission appears highly concentrated and
unresolved. There is no emission detected at the angular radius of $\sim
12^{\prime\prime}$ (where several other species have shown a peak in abundance
in previous interferometric maps). The observations of this line were repeated
in the extended configuration of the SMA at a beam size of 0.′′8, confirming
that the emission is unresolved. The deconvolved source size is
$<0.^{\prime\prime}2$. The lower limit for brightness temperature is 237 K.
Assuming an excitation temperature of 550 K, we estimate a column density of
$7\times 10^{17}$ cm-2 and a lower limit for CS abundance with respect to H2
of $9.3\times 10^{-6}$ within a radius of $\sim 4.5\times 10^{14}$ cm ($\sim
7R_{*}$).
Figure 5.— Spectrum of CS v=2 J=7-6 emission at 337912.19 GHz. See Figure 4
caption and also Table 2.
Previous interferometric observations of the v=1, J=5–4 line show the emission
to be within a radius of $0.^{\prime\prime}35$ ($\sim$10 R∗ or $\sim 7\times
10^{14}$ cm) (Lucas and Guelin 1999). Young et al. (2004) derived a lower
limit on the CS abundance to be $3.4\times 10^{-9}$ within $\sim 34R_{*}$.
From single-dish observations of CS v=1, J=3–2, 6–5 and 7–6 lines, Highberger
et al. (2000) estimate an abundance of 3–7$\times 10^{-5}$ relative to H2.
From comparison with the CS radial abundance predicted by chemical models, we
conclude, as did Young et al. (2004), that the model of Millar et al. (2001)
predicts too low a value ($\sim 10^{-11}$) at the radius of $\sim 10^{16}$ cm.
In this model, assuming CS to be a parent molecule (see Figure 1, right-panel,
in Millar et al. 2001), the initial abundance is still lower by about an order
of magnitude relative to that derived from submillimeter observations.
Moreover, the drop in CS abundance with radius (owing to the production of
other sulphur-bearing molecules), is too small, and not consistent with the
compact distribution of CS seen in the SMA observations. There is better
agreement in a more recent study of non-equilibrium chemistry of the inner
wind, which takes into account shocks induced by stellar pulsation (Cherchneff
2006), although these models seem more relevant for S stars (C/O$\approx 1$).
The CS v=2, J=7–6 transition requires extreme excitation conditions since it
corresponds to an energy Eu/k=3707 K. As noted by Highberger et al. (2001),
even for the v=1 line, collisional excitation with H2 would require very high
gas densities ($\sim 1-5\times 10^{14}$ cm-3. This line emission is most
plausibly excited by 8 $\mu$m stellar thermal radiation.
## 6\. Conclusions
Preliminary results from the SMA line survey of IRC+10216 have yielded a
population of narrow lines with expansion velocities of $\sim$ 4 km s-1. About
half of these can be assigned to vibrationally excited rotational transitions
of abundant species such as CS, SiS and their isotopomers. The emission is
found to occur in a very compact region smaller than $0.^{\prime\prime}2$
around the star. This is thought to be the region where dust is forming in the
envelope and in which the material has just begun accelerating and has yet to
attain the terminal velocity of $\sim$14 km s-1. The CS v=2, J=7-6 line is
most likely radiatively excited, because collisional excitation would require
an unrealistically high gas density and abundance of CS.
It is a pleasure to thank Ray Blundell for his help and support present on the
SMA IRC+10216 line-survey. We thank Mark Gurwell, Thushara Pillai and Jun-Hui
Zhao for helpful discussions on SMA data reduction. This research has
benefitted from the Cologne Molecular Spectroscopy Database (Müller et al.
2001; Müller et al. 2005) (http://www.astro.uni-koeln.de/site/vorhersagen/),
and the ALMA group’s spectral line catalog website:
http://www.splatalogue.net (Remijan et al. 2007) and the CASSIS (Centre
d’Analyse Scientifique de Spectres Infrarouges et Submillimétriques) software
(http://cassis.cesr.fr).
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Table 1Summary of observations Date | SMA | Tuning | Synthesized | $\tau_{225GHz}$ | Tsys
---|---|---|---|---|---
| configuration | | beam | | (SSB, K)
2007 February 7 | subcompact | 299.1 GHz (LSB) | $3.^{\prime\prime}2\times 2.^{\prime\prime}4$, P.A.= -4∘ | 0.08 | 180–270
2007 Feburary 8 | subcompact | 301.1 GHz (LSB) | $3.^{\prime\prime}3\times 2.^{\prime\prime}4$, P.A.=-6∘ | 0.09 | 160–230
2007 Feburary 9 | subcompact | 337.5 GHz (LSB) | $3.^{\prime\prime}0\times 2.^{\prime\prime}4$, P.A.=-8∘ | 0.08 | 180–350
2007 Feburary 12 | subcompact | 334.4 GHz (LSB) | $3.^{\prime\prime}0\times 2.^{\prime\prime}2$, P.A.=-3∘ | 0.05 | 130–260
2008 Feburary 19 | extended | 337.5 GHz (LSB) | $0.^{\prime\prime}8\times 0.^{\prime\prime}6$, P.A.=-88∘ | 0.03 | 100–250
Table 2Summary of narrow lines Species | Transition | Rest frequency11Assuming a systemic velocity of $-26.2$ km s-1. | Catalog freq22From the Cologne Database of Molecular Spectroscopy.. | Vexp | Peak | Integrated | Deconvolved | TB44Lower limit in brightness temperature assuming source size of $0.^{\prime\prime}2\times 0.^{\prime\prime}2$
---|---|---|---|---|---|---|---|---
| | | | | flux density33Typical uncertainty is 0.1 Jy. | flux density | size |
| | (MHz) | (MHz) | (km s-1) | (Jy) | (Jy km s-1) | | (K)
CS | v=2 J=7–6 | 337913.246 $\pm$ 0.18 | 337912.189 | 5.1$\pm$0.1 | 0.78 | 7.0 | $<0.^{\prime\prime}2$ | $>$237.0
Si34S | v=1 J=17–16 | 298630.441 $\pm$ 0.69 | 298629.989 | 4.5$\pm$0.4 | 0.36 | 2.6 | $<1.^{\prime\prime}9\times 0.^{\prime\prime}2$ | $>$138.7
Si34S | v=1 J=19–18 | 333733.581 $\pm$ 0.32 | 333731.998 | 5.7$\pm$0.1 | 0.49 | 4.5 | $<2.^{\prime\prime}3\times 0.^{\prime\prime}5$ | $>$183.3
29SiS | v=1 J=17–16 | 301390.406 $\pm$ 0.25 | 301388.939 | 4.8$\pm$0.2 | 0.52 | 3.8 | $<1.^{\prime\prime}9\times 0.^{\prime\prime}8$ | $>$198.3
29SiS | v=1 J=19–18 | 336819.842 $\pm$ 0.32 | 336814.954 | 7.5$\pm$0.1 | 0.50 | 4.6 | $<0.^{\prime\prime}9\times 0.^{\prime\prime}6$ | $>$152.9
|
arxiv-papers
| 2008-11-13T15:47:18 |
2024-09-04T02:48:58.773676
|
{
"license": "Public Domain",
"authors": "Nimesh A. Patel (1), Ken H. Young (1), Sandra Br\\\"unken (1), Robert W.\n Wilson (1), Patrick Thaddeus (1), Karl M. Menten (2), Mark Reid (1), Michael\n C. McCarthy (1), Dinh-V-Trung (3), Carl A. Gottlieb (1), Abigail Hedden (1)\n ((1) Harvard-Smithsonian Center for Astrophysics, (2) Max Planck Institut\n f\\\"ur Radio Astronomie, (3) Academia Sinica Institute for Astronomy and\n Astrophysics)",
"submitter": "Nimesh Patel",
"url": "https://arxiv.org/abs/0811.2142"
}
|
0811.2230
|
# Lorentz Invariance Violation and the Observed Spectrum of Ultrahigh Energy
Cosmic Rays
S.T. Scully Department of Physics and Astronomy
James Madison University, Harrisonburg, VA 22807 F.W. Stecker Astrophysics
Science Division
NASA Goddard Space Flight Center, Greenbelt, MD 20771
###### Abstract
There has been much interest in possible violations of Lorentz invariance,
particularly motivated by quantum gravity theories. It has been suggested that
a small amount of Lorentz invariance violation (LIV) could turn off photomeson
interactions of ultrahigh energy cosmic rays (UHECRs) with photons of the
cosmic background radiation and thereby eliminate the resulting sharp
steepening in the spectrum of the highest energy CRs predicted by Greisen
Zatsepin and Kuzmin (GZK). Recent measurements of the UHECR spectrum reported
by the HiRes and Auger collaborations, however, indicate the presence of the
GZK effect. We present the results of a detailed calculation of the
modification of the UHECR spectrum caused by LIV using the formalism of
Coleman and Glashow. We then compare these results with the experimental UHECR
data from Auger and HiRes. Based on these data, we find a best fit amount of
LIV of $4.5^{+1.5}_{-4.5}\times 10^{-23}$,consistent with an upper limit of
$6\times 10^{-23}$. This possible amount of LIV can lead to a recovery of the
cosmic ray spectrum at higher energies than presently observed. Such an LIV
recovery effect can be tested observationally using future detectors.
###### keywords:
cosmic rays; Lorentz invariance; quantum gravity
## 1 Introduction
Because of their extreme energy and isotropic distribution, it is believed
that UHECRs are extragalactic in origin. After the discovery of the cosmic
background radiation (CBR), Greisen [1] and Zatsepin and Kuzmin [2] pointed
out that photomeson interactions should deplete the flux of cosmic rays with
energies above $\sim$ 50 EeV. One of us [3], using data on the energy
dependence of the photomeson production cross section, then made a
quantitative calculation of this “GZK effect” deriving the mean photomeson
energy loss attenuation length for protons as a function of proton energy.
These results indicated that the attenuation length of a proton with an energy
greater than 100 EeV is less than 100 Mpc, which is much less than the visible
radius of the universe. Thus, what is sometimes referred to as the GZK
“cutoff” is not a true cutoff, but a suppression of the ultrahigh energy
cosmic ray flux arising from a limitation of the proton propagation length
through the cosmic background radiation owing to energy losses.
From time to time there have been reports in the literature of the detection
of giant air shower events from primaries with energies above the GZK
suppression energy (trans-GZK events) (e.g., Refs. [4] – [6]). Such events
have stimulated suggestions that a violation of Lorentz invariance or a
modification of the Lorentz transformation relations at ultrahigh energies
could result in a nullification of the GZK effect [7],[8]. Most significantly,
the AGASA group reported 11 events above the GZK suppression energy [6],
increasing the interest in the possibility of such new physics [9]. See
Ref.[10] for a recent review of this topic.
However, a reanalysis of the AGASA data (unpublished) has resulted in cutting
their originally reported number of trans-GZK events by half. More
importantly, the HiRes [11] and Auger groups [12], with larger exposures, have
very recently claimed to have found a GZK suppression effect. Motivated by
these new results, we have undertaken new detailed calculations of the effect
of a very small amount of Lorentz invariance violation (LIV) on the spectrum
of UHECRs at Earth. We present our results here and compare them with the
HiRes and Auger data separately.
## 2 Violating Lorentz Invariance
With the idea of spontaneous symmetry breaking in particle physics came the
suggestion that Lorentz invariance (LI) might be weakly broken at high
energies. Some of the more recent motivation for LIV has been in relating it
to possible Planck scale phenomena that could lead to astrophysically
observable consequences [13]. The idea that Planck scale physics may lead to a
natural abrogation of the GZK effect has been of particular interest, since
this would lead to a direct observational test. Significant fluxes of UHECRs
at trans-GZK energies, could be the result of a very small amount of LIV [14]
– [18]. Such a test would have important implications for some quantum gravity
and large extra dimension models, since those models may predict very small
amount of LIV.
Although no true quantum theory of gravity exists, it is natural to tie LIV to
various quantum gravity models. A few examples of such work can be found in
Refs. [15] – [20]. For more references, we refer to an excellent review by
Mattingly [21]. A data table of constraints on LIV and CPT violation
parameters within the framework of the “Standard Model Extension” model [22]
has recently been given by Kostelecky and Russell [23].
In this paper, we reinvestigate the observational implications of the possible
effect of a very small amount of LIV, viz., that cosmic rays could indeed
reach us after originating at distances greater than 100 Mpc without
undergoing large energy losses from photomeson interactions. We considered
this topic before in a more simplistic manner [24] when there was a clear
discrepancy between the AGASA group data [6] and the earlier HiRes data.
However, as discussed above, the observational situation has changed and now
requires a more detailed approach. We therefore undertook a detailed
calculation of the modification of the UHECR spectrum caused by LIV using the
formalism of Coleman and Glashow and the kinematical approach originally given
by Alfaro and Palma [18] in the context of the Loop Quantum Gravity model
[25],[26]. (See also Ref. [27].) Then, by comparing our results with the
observational UHECR data we can place a quantitative limit on the amount of
LIV. We also discuss how a small amount of LIV that is consistent with the
observational data can still lead to a recovery of the cosmic ray flux at
higher energies than presently observed.
## 3 LIV Framework
Coleman and Glashow have proposed a simple formulation for breaking LI by a
small first order perturbation in the free particle Lagrangian [14]. This
formalism has the advantages of (1) simplicity, (2) preserving the
$SU(3)\otimes SU(2)\otimes U(1)$ standard model of strong and electroweak
interactions, (3) having the perturbative term in the Lagrangian to consist of
operators of mass dimension $4$ that thus preserves power counting
renormalizability, and (4) being rotationally invariant in a preferred frame
that can be taken to be the rest frame of the 2.7 K cosmic background
radiation. This formalism has proven useful in exploring astrophysical data
for testing LIV [14],[19],[28].
To accomplish this, Coleman and Glashow start with the free particle
Lagrangian
${\cal L}=\partial_{\mu}\Psi^{*}{\bf Z}\partial^{\mu}\Psi-\Psi^{*}{\bf
M}^{2}\Psi$ (1)
where $\Psi$ is a column vector of $n$ fields with U(1) invariance and the
positive Hermitian matrices ${\bf Z}$ and M2 can be transformed so that ${\bf
Z}$ is the identity and M2 is diagonalized to produce the standard theory of
$n$ decoupled free fields.
They then add a leading order perturbative, Lorentz violating term constructed
from only spatial derivatives so that
${{\cal L}\rightarrow{\cal L}+\partial_{i}\Psi{\bf\epsilon}\partial^{i}\Psi},$
(2)
where $\epsilon$ is a dimensionless Hermitian matrix that commutes with M2 so
that the fields remain separable and the resulting single particle energy-
momentum eigenstates go from eigenstates of M2 at low energy to eigenstates of
$\epsilon$ at high energies.
The Lorentz violating perturbative term shifts the poles of the propagators,
resulting in free particle dispersion relations of the form
$E^{2}~{}=~{}\vec{p}\ ^{2}+m^{2}+\epsilon\vec{p}\ ^{2}.$ (3)
These can be put in the standard form for the dispersion relations
$E^{2}~{}=~{}\vec{p\ }{{}^{2}}c_{MAV}^{2}+m^{2}c_{MAV}^{4},$ (4)
by shifting the renormalized mass by the small amount $m\rightarrow
m/(1+\epsilon)$ and shifting the velocity from c (=1) by the amount
$c_{MAV}=\sqrt{(1+\epsilon)}\simeq 1+\epsilon/2$.
Since the group velocity is given by
${{\partial
E}\over{\partial|\vec{p}|}}={{|\vec{p}|}\over{\sqrt{|\vec{p}|^{2}+m^{2}c_{MAV}^{2}}}}c_{MAV}~{}~{}\rightarrow~{}~{}c_{MAV}~{}~{}{\rm
as}~{}~{}|\vec{p}|~{}~{}\rightarrow~{}\infty,$ (5)
Coleman and Glashow thus identify $c_{MAV}$ as the maximum attainable velocity
of the free particle.
Using this formalism, it becomes apparent that, in principle, different
particles can have different maximum attainable velocities (MAVs) resulting
from the individually distinguishable eigenstates of the $\epsilon$ matrix.
These various MAVs can all be different from $c$ as well as different from
each other. Hereafter, we denote the MAV of a particle of type $i$ by $c_{i}$
and the difference
$c_{i}-c_{j}~{}=~{}{{\epsilon_{i}-\epsilon_{j}}\over{2}}~{}\equiv~{}\delta_{ij}$
(6)
There are other popular formalisms that are inspired by quantum gravity models
or by speculations on the nature of space-time at the Planck scale,
$1/M_{Pl}\simeq 1.5\times 10^{-35}$ m, where $M_{Pl}=1/\sqrt{G}\simeq
1.2\times 10^{19}$ GeV. Such formalisms, in the context of effective field
theory, can be expressed by postulating Lagrangians containing operators of
dimension$\geq$ 5 with suppression factors as multiples of
$M_{Pl}~{}$[19],[29]. This leads to dispersion relations having a series of
smaller and smaller terms proportional to $p^{n+2}/M_{Pl}^{n}\simeq
E^{n+2}/M_{Pl}^{n}$, with $n\geq 1$. However, in relating LIV to the
observational data on UHECRs, we find it useful to use the simpler formalism
of Coleman and Glashow. Given the limited energy range of the UHECR data
relevant to the GZK effect, this formalism can later be related to possible
Planck scale phenomena and quantum gravity models of various sorts.
We now consider the photomeson production process near threshold where single
pion production dominates,
$p+\gamma\rightarrow N+\pi.$ (7)
Using the normal Lorentz invariant kinematics, the energy threshold for
photomeson interactions of UHECR protons of initial laboratory energy $E$ with
low energy photons of the CBR with laboratory energy $\omega$ is determined by
the relativistic invariance of the square of the total four-momentum of the
proton-photon system. This relation, together with the threshold inelasticity
relation $E_{\pi}=[m/(M+m)]E$ for single pion production, yields the threshold
conditions for head on collisions in the laboratory frame. In terms of the
pion energy for single pion production at threshold
$4\omega E_{\pi}~{}=~{}{{m^{2}(2M+m)}\over{M+m}},$ (8)
where M is the rest mass of the proton and m is the rest mass of the pion [3].
If LI is broken so that $c_{\pi}~{}>~{}c_{p}$, it follows from equations (3),
(6) and (8) that the threshold energy for photomeson production is altered
because the square of the four-momentum is shifted from its LI form so that
the threshold condition becomes
$4\omega E_{\pi}~{}=~{}{{m^{2}(2M+m)}\over{M+m}}+2\delta_{\pi p}E_{\pi}^{2}$
(9)
Equation (9) is a quadratic equation with real roots only under the condition
$\delta_{\pi
p}\leq{{2\omega^{2}(M+m)}\over{m^{2}(2M+m)}}\simeq\omega^{2}/m^{2}.$ (10)
Defining $\omega_{0}\equiv kT=2.35\times 10^{-4}$ eV, equation (10) can be
rewritten
$\delta_{\pi p}\leq 3.23\times 10^{-24}(\omega/\omega_{0})^{2}.$ (11)
If LIV occurs and $\delta_{\pi p}>0$, photomeson production can only take
place for interactions of CBR photons with energies large enough to satisfy
equation (11). Single photon photomeson production takes is dominated by the
$\Delta$ resonance and takes place close to the interaction threshold. This
fact, together with equation (9) implies that under some conditions photomeson
interactions leading to GZK suppression can occur for “lower energy” UHE
protons interacting with relatively higher energy CBR photons on the Wien tail
of the Planck spectrum, but such interactions for higher energy protons, which
would normally interact with photons having smaller values of $\omega$, will
be forbidden. Thus, the observed UHECR spectrum may exhibit the
characteristics of GZK suppression near the normal GZK threshold, but the
UHECR spectrum can “recover” at higher energies owing to the possibility that
photomeson interactions at higher proton energies may be forbidden.
## 4 Kinematics
We now consider a detailed quantitative treatment of this possibility, viz.,
GZK coexisting with LIV. We first give the kinematical relations needed to
perform our calculations in the presence of a small violation of Lorentz
invariance. We will denote quantities in the proton rest frame by a prime and
quantities in the cms system of the proton-photon collision by an asterisk.
Quantities in the laboratory frame are left unprimed. Following equations (3)
and (6), we denote
$E^{2}=p^{2}+2\delta_{a}p^{2}+{m_{a}}^{2}$ (12)
where $\delta_{a}$ is the difference between the MAV for the particle a and
the speed of light in the low momentum limit ($c=1$).
The cms energy of particle $a$ is then given by
$\sqrt{s_{a}}=\sqrt{E^{2}-p^{2}}=\sqrt{2\delta_{a}p^{2}+m_{a}^{2}}$ (13)
where, of course, we must have the condition $s_{a}\geq 0$. It is important to
note that, owing to LIV, in the cms where p = 0 the particle will not
generally be at rest because
$v={{\partial E}\over{\partial p}}\neq{{p}\over{E}}.$ (14)
We follow Ref. [18] in defining the square of the total rest energy in the cms
by
$s\equiv E_{tot}^{2}-p_{tot}^{2}$ (15)
Then denoting $s_{p}$ to be the square of the initial proton energy in the
system where the initial proton momentum is zero, it follows that
$s=2\sqrt{s_{p}}\epsilon+s_{p}$ (16)
where we now define $\epsilon$ as the energy of the photon in this system.
Now let us obtain the expression for the modified inelasticity, $K$, for the
photopion producing reaction $p+\gamma\rightarrow N+\pi$. Since the
inelasticity is defined by the fraction of the total energy carried away by
the pion, we can relate the energy of the emerging proton and pion to the
total energy in the laboratory system (essentially the initial energy of the
proton) by
$\displaystyle E_{\pi}$ $\displaystyle=$ $\displaystyle K_{\theta}E_{p}$
$\displaystyle E_{N}$ $\displaystyle=$ $\displaystyle(1-K_{\theta})E_{p}$ (17)
where $K_{\theta}$ is the inelasticity for a given $\theta$ which is the angle
between the momentum vectors of the photon and the proton in the laboratory
system. In order to solve for the inelasticity, we calculate the cms energy of
the nucleon in two different ways. On one hand, we can use the Lorentz
transformation of the laboratory nucleon energy to relate it to the cms
energy:
$\displaystyle E_{N}$ $\displaystyle=$
$\displaystyle\gamma^{*}({E_{N}}^{*}+\beta^{*}{p_{N}}^{*}\cos\theta)$ (18)
$\displaystyle=$
$\displaystyle\gamma^{*}({E_{N}}^{*}+\beta^{*}\sqrt{{E_{N}}^{*}-s_{N}(E_{N})}\cos\theta),$
where the Lorentz factor for the cms frame is the ratio of the total
laboratory energy $E_{p}+\omega\approx E_{p}$ to the total cms energy which is
given by equation (13) and where we now define $\omega$ to be the observed
energy of the CBR photon in the laboratory system..
On the other hand, we can derive the cms energy of the nucleon from the
threshold conditions by replacing the masses of the particles with their rest
energies as prescribed by equation (13). This yields the relationship
$2\sqrt{s}{E_{N}}^{*}=s+s_{N}-s_{\pi}$ (19)
where the quantities $s_{N}$ and $s_{\pi}$ can be determined from equation
(13) and are given by
$\displaystyle s_{\pi}$ $\displaystyle=$
$\displaystyle\delta_{\pi}(K_{\theta}E_{p_{i}})^{2}+{m_{\pi}}^{2}$
$\displaystyle s_{N}$ $\displaystyle=$
$\displaystyle\delta_{N}[(1-K_{\theta})E_{p_{i}}]^{2}+{m_{p}}^{2}.$ (20)
Here we have replaced $p$ with $E$ since we can exchange momentum for energy,
given the high Lorentz factor. We can now combine equations (18) and (19) to
yield a transcendental equation for $K_{\theta}$:
$\displaystyle\noindent(1-K_{\theta})\sqrt{s}$ $\displaystyle=$
$\displaystyle{{(s+s_{N}(K_{\theta})-s_{\pi}(K_{\theta}))}\over{2\sqrt{s}}}$
(21) $\displaystyle+$
$\displaystyle\beta\sqrt{{(s+s_{N}(K_{\theta})-s_{\pi}(K_{\theta}))^{2}\over{4s}}-s_{N}}\cos\theta.$
The total inelasticity, $K$, will be an average of $K_{\theta}$ with respect
to the angle between the proton and photon momenta, $\theta$:
$K=\frac{1}{\pi}\int\limits_{0}^{\pi}{K_{\theta}d\theta}.$ (22)
The primary effect of LIV on photopion production is a reduction of phase
space allowed for the interaction. This results from the limits on the allowed
range of interaction angles implied by equations (21) and (22). As the pion
rest energy grows, the cosine term in equation (21) becomes larger. For
collisions with $\theta<\pi/2$, kinematically allowed solutions become
severely restricted. The modified inelasticity that results is the key in
determining the effects of LIV on photopion production. The inelasticity
rapidly drops for higher incident proton energies.
As shown in Ref.[14], in order to modify the effect of photopion production on
the UHECR spectrum above the GZK energy we must have
$\delta_{\pi}>\delta_{p}$. It is shown in Figure 10 of Ref. [30] that for most
of the allowed parameter space near threshold $\delta_{\pi}$ can be as much as
an order of magnitude greater than $\delta_{p}$. Therefore, in this paper we
will assume that $\delta_{\pi}\gg\delta_{p}$ at or near threshold. This
assumption is also made in Ref. [18]. We will thus take $\delta_{\pi
p}\simeq\delta_{\pi}\equiv\delta$. We have numerically determined that the
dependence of our results on the $\delta_{\pi p}$ parameter dominates over
that on the $\delta_{p}$ parameter, as concluded in Ref. [14]. The effect of
taking a value of $\delta_{p}$ comparable to $\delta_{\pi}$ on the UHECR
spectrum will be presented in a future paper.
Figure 1 shows the calculated proton inelasticity modified by LIV for a value
of $\delta_{\pi p}=3\times 10^{-23}$ as a function of both CBR photon energy
and proton energy. Other choices for $\delta_{\pi p}$ yield similar plots. The
principal result of changing the value of $\delta_{\pi p}$ is to change the
energy at which LIV effects become significant. For a choice of $\delta_{\pi
p}=3\times 10^{-23}$, there is no observable effect from LIV for $E_{p}$ less
than $\sim 2\times 10^{20}$ eV. Above this energy, the inelasticity
precipitously drops as the LIV term in the pion rest energy approaches
$m_{\pi}$.
Figure 1: The calculated proton inelasticity modified by LIV for $\delta_{\pi
p}=3\times 10^{-23}$ as a function of CBR photon energy and proton energy.
With this modified inelasticity, the proton energy loss rate by photomeson
production is given by
${{1}\over{E}}{{dE}\over{dt}}=-{{\omega_{0}c}\over{2\pi^{2}\gamma^{2}}\hbar^{3}c^{3}}\int\limits_{\eta}^{\infty}d\epsilon~{}\epsilon~{}\sigma(\epsilon)K(\epsilon)\ln[1-e^{-\epsilon/2\gamma\omega_{0}}]$
(23)
where $\eta$ is the photon threshold energy for the interaction in the cms and
$\sigma(\epsilon)$ is the total $\gamma$-p cross section with contributions
from direct pion production, multipion production, and the $\Delta$ resonance.
The corresponding proton attenuation length is given by $cE/(dE/dt)$. This
attenuation length is plotted in Figure 2 for various values of $\delta_{\pi
p}$ along with the unmodified pair production attenuation for comparison. We
do not explore the effects of modifying pair production through LIV in this
paper.
Figure 2: The calculated proton attenuation lengths as a function proton
energy modified by LIV for various values of $\delta_{\pi p}$ (solid lines),
shown with the attenuation length for pair production unmodified by LIV
(dashed lines). From top to bottom, the curves are for $\delta_{\pi p}=1\times
10^{-22},3\times 10^{-23},2\times 10^{-23},1\times 10^{-23},3\times
10^{-24},0$ (no Lorentz violation).
## 5 UHECR Spectra with LIV and Comparison with Present Observations
We will start our calculation of LIV modified UHECR spectra by assuming power-
law source spectra for the UHECRs that are chosen to fit the UHECR data below
60 EeV. We then consider the propagation of high energy protons, including
energy losses resulting from cosmological redshifting, pair production and
pion production through interactions with CBR photons.
We shall assume for this calculation a flat $\Lambda$CDM universe with a
Hubble constant of H0 = 70 km s-1 Mpc-1, taking $\Omega_{\Lambda}$ = 0.7 and
$\Omega_{m}$ = 0.3. The energy loss owing to redshifting for a $\Lambda$CDM
universe is then given by
$\displaystyle-(\partial\log E/\partial
t)_{redshift}=H_{0}\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}.$ (24)
The attenuation length for protons against pair production is given by
$\displaystyle-(\partial\log E/\partial t)_{\gamma p}\equiv r_{\gamma
p}=r_{\pi}(E)+r_{e^{+}e^{-}}(E),$ (25)
The attenuation lengths, $\ell=cE/r(E)$, for protons against energy loss by
both pion production, with and without LIV, are shown together with that for
pair production in Figure 2. The CBR photon number density increases as
$(1+z)^{3}$ and the CBR photon energies increase linearly with $(1+z)$. The
corresponding energy loss for protons at any redshift $z$ is thus given by
$\displaystyle r_{\gamma p}(E,z)=(1+z)^{3}r[(1+z)E].$ (26)
We take the photomeson loss rate, $r_{\pi}(E)$, by updating [3] using the
latest cross sections listed in the Particle Data Group (http://pdg.lbl.gov)
and in Ref. [31]. We take the pair-production loss rate, $r_{e^{+}e^{-}}(E)$
from Ref. [32].
We calculate the initial energy, $E_{i}(z)$, at which a proton is created at a
redshift $z$ whose observed energy today is $E$ following the methods detailed
in Refs. [33] and [34]. We neglect the effect of possible small intergalactic
magnetic fields on the paths of these ultrahigh energy protons and assume that
they will propagate along straight lines from their source. The total flux of
emitted particles from a volume element $dV=R^{3}(z)r^{2}drd\Omega$ from
redshift $z$ and distance $r$ with measured energy $E$ is given by
$\displaystyle J(E)dE={{q(E_{i},z)dE_{i}dV}\over{(1+z)4\pi R_{0}^{2}r^{2}}}.$
(27)
We assume that the average UHECR volume emissivity is given by
$q(E_{i},z)=K(z)E_{i}^{-\Gamma}$.
We will assume a source evolution $q(E_{i},z)\propto(1+z)^{\zeta}$ with
$\zeta$ = 3.6, out to a maximum redshift of 2.5. This assumption corresponds
to a redshift evolution that is proportional to the star formation rate. Our
results on LIV are insensitive to the evolution model assumed because
evolution does not affect the shape of the UHECR spectrum near the GZK cutoff
energy [24, 33]. At higher energies where the attenuation length may again
become large owing to an LIV effect, we find the effect of evolution to be
less than 10% when compared to the no-evolution case ($\zeta=0$).
Since $R_{0}=(1+z)R(z)$ and $R(z)dr=cdt$, by integrating equation (27), one
obtains
$\displaystyle J(E)={{3cK(0)}\over{8\pi
H_{0}}}E^{-\Gamma}\int_{0}^{z_{max}}{{(1+z)^{(\zeta-1)}}\over{\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}}}\left({E_{i}\over{E}}\right)^{-\Gamma}{{dE_{i}}\over{dE}}dz.$
(28)
In this expression, K(0) is determined by fitting our final calculated
spectrum to the observational UHECR data [24] assuming $\Gamma=2.55$, which is
consistent with the Auger data below 100 EeV.
The results are shown in Figures 3 and 4.
Figure 3: Comparison of the HiRes II data with calculated spectra for various
values of $\delta_{\pi p}.$ From top to bottom, the curves give the predicted
spectra for $\delta_{\pi p}=1\times 10^{-22},3\times 10^{-23},2\times
10^{-23},1\times 10^{-23},3\times 10^{-24},0$ (no Lorentz violation).
Figure 4: Comparison of the Auger data with calculated spectra for various
values of $\delta_{\pi p}.$ From top to bottom, the curves give the predicted
spectra for $\delta_{\pi p}=1\times 10^{-22},6\times 10^{-23},4.5\times
10^{-23},3\times 10^{-23},2\times 10^{-23},1\times 10^{-23},3\times
10^{-24},0$ (no Lorentz violation).
## 6 Discussion of Results
It has been suggested that a small amount of Lorentz invariance violation
(LIV) could turn off photomeson interactions of ultrahigh energy cosmic rays
(UHECRs) with photons of the cosmic background radiation and thereby eliminate
the resulting sharp steepening in the spectrum of the highest energy CRs
predicted by Greisen Zatsepin and Kuzmin (GZK). Recent measurements of the
UHECR spectrum reported by the HiRes [11] and Auger [12] collaborations,
however, indicate the possible presence of a GZK effect.
In order to determine the implications for the search for Lorentz invariance
violation at ultrahigh energies from the analysis of the air shower events
observed by HiRes and AGASA, we undertook a detailed analysis of the spectral
features produced by modifications of the kinematical relationships caused by
LIV at ultrahigh energies. In our analysis, we calculate modified UHECR
spectra for various values of the Coleman-Glashow parameter, $\delta_{\pi p}$,
defined as the difference between the maximum attainable velocities of the
pion and the proton produced by LIV. We then compare our results with the
experimental UHECR data and thereby place limits on the amount of LIV as
defined by the $\delta_{\pi p}$ parameter.
Our results show that the amount of presently observed GZK suppression in the
UHECR data is consistent with the possible existence of a small amount of LIV.
In order to quantify this, we determined the value of $\delta_{\pi p}$ that
results in the smallest $\chi^{2}$ for the modeled UHECR spectral fit using
the observational data above the GZK energy. We find this value to be
$4.5\times 10^{-23}$. We then determined the range of acceptable values for
$\delta_{\pi p}$. This was done by computing the probablity of getting a
$\chi^{2}$ value at least as small as the $\chi^{2}$ value determined from the
fit. We rejected $\delta_{\pi p}$ values outside of the confidence level
associated with 1$\sigma$. We thus obtained a best-fit range of $\delta_{\pi
p}$ = $4.5^{+1.5}_{-4.5}\times 10^{-23}$, corresponding to an upper limit on
$\delta_{\pi p}$ of $6\times 10^{-23}$, as shown in Figure 4.
The HiRes spectral data (see Figure 3) do not go to high enough energy to
quantitatively constrain LIV. We also note that the Auger spectrum, being
consistent with no obvious pair-production feature, does not constrain LIV for
the pair-production interaction.
A small LIV effect can be distinguished from a higher energy component
produced by so-called top-down models because the latter predict relatively
large fluxes of UHE photons and neutrinos as well as a significant diffuse GeV
background flux that could be searched for by the Fermi $\gamma$-ray space
telescope. The Pierre Auger Observatory collaboration has provided
observational upper limits on the UHE photon flux that have already disfavored
top-down models [35]. The upper limits form Auger indicate that UHE photons at
best make up only a small percentage of the total UHE flux. This contradicts
predictions of top-down models that the flux of UHE photons should be larger
than that of UHE protons (See Ref. [9] for a review).
As opposed to the predictions of the top-down models, the LIV effect cuts off
UHE pion production at the higher energies and consequent UHE neutrino and
photon production from UHE pion decay. LIV would also not produce a GeV photon
flux.
It is also possible that the apparent modified GZK suppression in the data may
be related to an overdensity of nearby sources related to the local
supergalactic enhancement [3].111A correlation with nearby AGN has been hinted
at in the Auger data [37]. However, the HiRes group has found no significant
correlation [38]. More and better data will be required in order to resolve
this question. An LIV effect can be distinguished from a local source
enhancement by looking for UHECRs at energies above $\sim$200 EeV, as can be
seen from Figures 3 and 4. This is because the small amount of LIV that fits
the observational UHECR spectra can lead to a recovery of the cosmic ray flux
at higher energies than presently observed. Searching for such an effect will
require obtaining a data set containing a much higher number of UHECR air
shower events.
In the future, such an increased number of events may be obtained. The Auger
collaboration has proposed to build an “Auger North” array that would be seven
times larger than the present southern hemisphere Auger array
(http://www.augernorth.org). Further into the future, space-based telescopes
designed to look downward at large areas of the Earth’s atmosphere as a
sensitive detector system for giant air-showers caused by trans-GZK cosmic
rays [36]. We look forward to these developments that may have important
implications for fundamental high energy physics.
## Acknowledgment
STS gratefully acknowledges partial support from the Thomas F. & Kate Miller
Jeffress Memorial Trust grant no. J-805.
## References
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* [13] G. Amelino-Camilia et al., Nature 393 (1998) 763.
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* [29] R. C. Meyers and M. Pospelov, Phys. Rev. Letters 90 (2003) 211601.
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* [32] G.R. Blumenthal, Phys. Rev. D 1 (1970) 1596.
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|
arxiv-papers
| 2008-11-13T21:22:14 |
2024-09-04T02:48:58.781826
|
{
"license": "Public Domain",
"authors": "S. T. Scully (JMU) and F. W. Stecker (NASA/GSFC)",
"submitter": "Floyd Stecker",
"url": "https://arxiv.org/abs/0811.2230"
}
|
0811.2331
|
Observation of the magnetic domain structures… Observation of the magnetic
domain structures in Cu0,47Ni0,53 thin films at low temperatures I. S.
Veshchunov, V. A. Oboznov, A. N. Rossolenko, A. S. Prokofiev, L. Ya. Vinnikov,
Yu. Rusanov and D. V. Matveev Veshchunov, Oboznov, Rossolenko, Prokofiev,
Vinnikov, Rusanov and Matveev 22 October 2008*
# Observation of the magnetic domain structures in Cu0,47Ni0,53 thin films at
low temperatures
I. S. Veshchunov V. A. Oboznov A. N. Rossolenko A. S. Prokofiev L. Ya.
Vinnikov e-mail: vinnik@issp.ac.ru A. Yu. Rusanov and D. V. Matveev Institute
of Solid State Physics RAS, 142432 Chernogolovka, Moscow distr., Russia
###### Abstract
We report on the first experimental visualization of domain structure in films
of weakly ferromagnetic Cu0,47Ni0,53 alloy with different thickness at liquid
helium temperatures. Improved high-resolution Bitter decoration technique was
used to map the magnetic contrast on the top of the films well below the Curie
temperature TCurie ($\sim$ 60 K). In contrast to magnetic force microscopy,
this technique allowed visualization of the domain structure without its
disturbance while the larger areas of the sample were probed. Maze-like domain
patterns, typical for perpendicular magnetic anisotropy, were observed. The
average domain width was found to be about 100 nm.
75.70.-i, 75.60.Ch
The interplay between superconductivity and ferromagnetism leads to a number
of interesting phenomena [1, 2], which can be utilized in various
applications. The interest of investigating the domain structure of weakly
ferromagnetic Cu1-xNix (x$\sim$0.5) alloys, in particular, is caused by their
active use in thin film superconductor (S)/ ferromagnet (F) heterostructures.
The most promising microelectronic devices based on such heterostructures are
basic elements of digital rapid single flux quantum (RSFQ) and quantum logic
circuits. Apart from that, using Cu1-xNix films as weak ferromagnets in
fundamental S/F system properties research looks quite promising, which was
confirmed in numerous theoretical and experimental investigations [2, 3, 4, 5,
6, 7]. The physical properties of CuNi alloys are relatively well studied. The
average magnetic moment and the Curie temperature of uniform CuNi alloys
decrease linearly with Ni concentration and both approach zero at $\sim$45
at.% Ni content. The magnetism of CuNi films is weaker than that in bulk
material. Although Cu1-xNix films with Ni concentration close to the critical
value seem to be good candidates for using them in S/F proximity systems, at
the same time they have several disadvantages. One of them is that the film
structure is very sensitive to the fabrication conditions. In particular, the
homogeneity of the sputtered films is not ideal, (at least close to x=0.5)
since there is a tendency of Ni-rich clusters forming [8]. But the features of
CuNi films domain structure, which have huge influence on the transport and
magnetic properties of S/F systems [6] have never been revealed. In this paper
we present results of studying magnetic domains in thin films of Cu0,47Ni0,53
(hereafter called CuNi) alloy.
In the past decade magnetic force microscopy (MFM) has become a well-
established technique for the observation of the distribution of magnetic
domains with submicron resolution [9, 10, 11, 12, 13, 14]. It is widely used,
except for the cases when MFM magnetic probe might bring distortions into the
scanned image. That can happen, for instance, because of the sample local
magnetization change by the probe itself during the scan. It’s known that even
well below the Curie temperature, the coercive field for thin films of some
ferromagnets (CuNi in particular) might become almost zero, so using even
magnetically soft MFM probes can disturb the picture of local magnetization
when performing scans. Therefore, to study domain structure in such films we
used the improved low temperature Bitter decoration technique [15], which also
has such additional advantages as high spatial resolution and magnetic
sensitivity. The Bitter technique is based on the deposition of fine dispersed
magnetic nanoparticles, driven by the stray field gradients in the vicinity of
magnetic material, at places where the magnetic field is higher. Decoration
patterns can be examined by means of scanning electron microscopy (SEM). The
patterns provide no information about the magnitude of the magnetization, yet
in materials even with low stray fields, Bitter patterns can quickly yield
information about the size and shape of domains of various types that might be
present.
Figure 1: (a). Saturation magnetization $M$ as function of the film thickness
$d_{F}$ for Cu0.47Ni0.53. The dependence has a logarithmic like behavior. The
line between experimental points serves to guide an eye. (b). Anomalous Hall
voltage VHall dependence with temperature $T$ for Cu0.47Ni0.53 film with
$d_{F}$=22 nm.
For our experiments Cu0,47Ni0,53 thin films were grown by RF-sputtering in Ar
atmosphere of PAr= 4$\times$10-2mbar on silicon substrates at room
temperature. The deposition rate was 0.25 nm/s. The Cu and Ni contents in the
sputtered films were determined by Rutherford Backscattering (RBS) analysis.
It confirmed that the Ni concentration in sputtered films was of the same
value as in used CuNi targets.
First, the magnetic properties of Cu0.47Ni0.53 films structured in the shape
of narrow bridges with thickness $d_{F}$ ranging from 5 to 30 nm were studied
by measuring anomalous Hall voltage VHall, which is proportional to the film
magnetization [16]. Fig.1a shows the dependence of VHall corresponding to the
saturation magnetization of the sample with a particular thickness on the
sample thickness. In all cases the applied magnetic field was perpendicular to
the film surface. Typical temperature dependence of the Hall voltage for the
CuNi film with dF=22 nm is presented in Fig.1b. For all film thicknesses it
appeared to be non linear, with a weakly pronounced saturation at low
temperatures and tail-like behavior close to the TCurie. TCurie for different
samples was estimated by extrapolating the linear part of the V${}_{Hall}(T)$
dependence as presented in Fig.1b. In order to estimate the field range for
the most effective magnetic domain decoration the hysteresis magnetization
loops were measured as well. Results of magnetization reversal for
Cu0.47Ni0.53 film with dF=20 nm are presented in Fig.2. The measurements were
performed at 4.2K well below the TCurie ( 60 K) of the sample. Several field
sweeps were performed with different values of maximum field in the range of
150 - 700 Oe as shown in Fig.2. The coercive field $H_{Coer}$ for those sweeps
was found in the range between 50 \- 150 Oe.
Figure 2: Hall voltage VHall as function of applied magnetic field $H$ for
Cu0.47Ni0.53 film with $d_{F}$=20nm measured at 4.2 K. Different curves
correspond to different maximal sweep fields $H$ as indicated. For all curves
$H$ was perpendicular to the film surface. Arrows show the direction of
magnetic field sweep.
Special consideration should be given to the CuNi thin film decoration
procedure. The sequence of the entire experiment can be described as follows.
The sample was initially cooled in zero magnetic field down to 4,2 K. During
the decoration the temperature of the sample increased (of about 3-4 K) up to
the decoration temperature Td (the temperature measured by the resistive
thermometer in the end of the iron evaporation process). The first series of
decoration procedures was performed at magnetic fields $H_{dec}$=100, 250, 300
Oe, on the virgin curve of the hysteresis loop (see Fig.2). For each new field
value a new sample (geometrically identical to the previous one) was used.
Additional experiment was done to make sure that multiple domain situation
occurs after the magnetization switch. For that the magnetic field was
gradually increased from 0 to 300 Oe and then swept back to $H_{dec}$= -150
Oe, which corresponds to $-H_{Coer}$ for this value of maximum sweeping field,
as shown in Fig.2. In all cases the applied magnetic field was perpendicular
to the sample surface.
Fig.3a, b, c, d present the distribution of the iron particles mapping
magnetic contrast related to the domain structure on the surface of the sample
obtained with SEM. Fig.3a, b, c show the domain structure for decoration
fields $H_{dec}$=100, 250 and 300 Oe respectively. That, as it was mention
before, corresponds to the evolution of the domain state on the virgin curve
of the hysteresis loop. At the lowest applied field $H_{dec}$=100 Oe the
decorated domain structure implies practically demagnetized state on the
surface of the sample, which is believed to be perpendicular to the
spontaneous magnetization axis. Domains form a maze-like pattern with a
typical domain width of about 100 nm. Increasing the decorating field
$H_{dec}$ up to 250 Oe as indicated in Fig.3b results in widening of the
positive (magnetization is pointed up) domains. Degradation of the pattern
quality occurs because of the decrease of the local field gradients at the
film surface. The domain structure (i.e. decorated magnetic contrast) almost
disappears when approaching $H_{dec}$=300 Oe, see Fig.3c. A maze-like domain
structure shows up clearly again, as it can be seen from Fig.3d after
magnetization switching, at the $H_{dec}=H_{Coer}$= - 150 Oe.
Figure 3: Evolution of the domain structure with external magnetic field
applied perpendicular to the film plane: (a) H=150 Oe, (b) H=250 Oe, (c) H=300
Oe, (d) H= -150 Oe.
The results of investigations reveal several advantages of using CuNi alloys,
which justify effectiveness of their utilization in Josephson SFS junctions.
First, for this particular ferromagnetic material the exchange energy is
relatively small (Eex/kB $\sim$ 800 K and TCurie $\sim$ 60 K correspondingly).
That implies the superconducting order parameter decay length $\xi_{F1}$ and
the period of its spatial oscillation $\xi_{F2}$ can be of the order of
several nanometers instead of $\sim$ 1 nm for the weak link of Josephson SFS
junction made of non-diluted ferromagnet. Therefore, making Josephson SFS
junctions with comparatively thick F-layers using simple thin film
technologies becomes possible. Second, the domain structure of CuNi films has
a spatial period of about 0.1 $\mu$m, as the decoration experiments showed,
which provides a good averaging of the net magnetization in F-layer. This, in
its turn, allows to fabricate submicron ($\sim$0.2-0.3 $\mu$m) SFS sandwiches
without having the undesired macroscopic stray fields. The final remark should
be made about the possibility to manipulate the Josephson characteristics of
SFS junctions (critical current, phase difference) with external magnetic
field up to 20 Oe without disturbing the domain structure of F-layer, since it
has a strong perpendicular anisotropy.
Several notes should be made on the decoration technique. Clearly, in order to
obtain the highest possible resolution of the method a special care has to be
taken of the decorating particle size as well as of their magnetic properties
[17]. It’s important to remind that when the applied field is higher than the
local stray fields pointing up from the film surface, the magnetic particles
become polarized along the applied field before landing on the ferromagnet
surface [18].
Figure 4: Size distribution of Fe decoration particles. N-number of Fe
particles on the scanned area of $2\mu m^{2}$. He gas pressure was
$\sim$2x10-2 Torr.
Therefore, the particles aggregate to the positive domain areas in which the
magnetization is aligned in the direction of an external magnetic field while
negative domain areas are free from iron particles. Our additional experiments
demonstrated that actual particle size distribution strongly depends on the
buffer He pressure as was determined by SEM. Typical size distribution of the
particle size is presented in Fig.4. The highest resolution in the range of
10-100nm was reached when the average particle size was of about 10 nm, which
is reached at (2-3)$\times$10-2 Torr of buffer He. The serious limitation of
the decoration method lies in fact that magnetic properties of the decorating
particles strongly depend on their size: the saturation magnetization
decreases with particle size (about 25% of that in bulk iron for 10 nm size
particles [19]). Increasing the particle size implies larger magnetic moment
and thus higher magnetic sensitivity, but the spatial resolution of the
pattern gets reduced. Those particles with larger magnetic moment tend to form
irregular clusters on the sample surface due to large interparticle
interaction.
Recently, the perpendicular magnetic anisotropy in dc-magnetron sputtered
Ni60Cu40/Cu multilayers was detected by hysteresis loop measurements for CuNi
layer thickness between 4,2 nm and 34 nm [20]. However, in that experiments
the features of the domain structure of CuNi films were not revealed.
In summary, the improved Bitter technique allowed visualizing the domain
structure of weakly ferromagnetic Cu0,47Ni0,53 on large area (tens of square
millimeters) at low temperatures. The image of magnetic contrast on the top of
Cu0,47Ni0,53 films was seen for the first time. It was experimentally shown
that thin CuNi films tend to have small scale domain structure. The films with
thickness in the range of 10-30 nm have perpendicular magnetic anisotropy
which results in maze-like domain patterns and nearly rectangular hysteresis
loop. The characteristic domain structure scale is found to be about 100 nm.
We are grateful to V. V. Ryazanov and L. S. Uspenskaya for helpful discussions
and L. G. Isaeva for help in preparation of evaporators. This work is
supported by RFBR (07-02-00174), Joint Russian-Israeli project MOST-RFBR
(06-02-72025) and InQubit, Ltd.
## References
* [1] A. I. Buzdin, Rev. Mod. Phys.77, 935 (2005).
* [2] V. V. Ryazanov, V. A. Oboznov, A. Yu. Rusanov, A. V. Veretennikov, A. A. Golubov, and J. Aarts, Phys. Rev. Lett. 86, 2427 (2001).
* [3] V. V. Ryazanov, V. A. Oboznov, A. Yu. Rusanov, A. V. Veretennikov, Phys. Rev. B 65, 020501 (2001).
* [4] V. A. Oboznov, V. V. Bol ginov, A. K. Feofanov, V. V. Ryazanov, and A. I. Buzdin, Phys. Rev. Lett. 96, 197003 (2006).
* [5] S. M. Frolov, M. J. A. Stoutimore, T. A. Crane, V. V. Ryazanov, V. A. Oboznov, D. J. Van Harlingen, Nature Physics 4, 32-36 (2008).
* [6] V. V. Ryazanov, V. A. Oboznov, A. S. Prokofiev, and S. V. Dubonos, Pis’ma Zh. Eksp. Teor. Fiz. 77, 43, (2003); JETP Lett. 77, 39 (2003).
* [7] A. Rusanov, M. Hesselberth, S. Habraken, J. Aarts, Physica C 404, 322 (2004)
* [8] K. Levin, and D. L. Mills. Phys. Rev. B 9, 2354 (1974).
* [9] M. Lange, M. J. Van Bael, V. V. Moshchalkov, and Y. Bruynseraede, Appl. Phys. Lett. 81, 322 (2002).
* [10] V. Vlasko-Vlasov, U. Welp, G. Karapetrov, V. Novosad, D. Rosenmann, M. Iavarone, A. Belkin, and W.-K. Kwok, Phys. Rev. B 77, 134518 (2008).
* [11] A. Garcia-Santiago, F. Sa’nchez, M. Varela, and J. Tejada, Appl. Phys. Lett. 77, 2900 (2000).
* [12] D. B. Jan, J. Y. Coulter, M. E. Hawley, L. N. Bulaevskii, M. P. Maley, Q. X. Jia, B. B. Maranville, F. Hellman, and X. Q. Pan, Appl. Phys. Lett. 82, 778 (2003).
* [13] P. Mazalski, I. Sveklo, M. Tekielak, A. Kolendo, A. Mazievski, P. Kuswik, B. Szymanski, and F. Stobiecki, Materials Science-Poland 25, 4 (2007).
* [14] L. Y. Zhu, T. Y. Chen, and C. L. Chien, Phys. Rev. Lett. 101, 017004 (2008).
* [15] L. Ya. Vinnikov, I. V. Grigor’eva, and L. A. Gurevich, in The Real Structure of High-Tc Superconductors, edited by V. Sh. Shekhtman, Springer Series in Materials Science Vol. 23 Springer-Verlag, Berlin, (1993) p. 89.
* [16] C. M. Hurd, The Hall Effect in Metals and alloys (Plenum, New York, 1972).
* [17] In our experiments done at low temperature magnetic particles were formed by evaporation of the host material (in our case it was iron) from the surface of a tungsten wire.
* [18] T. Sakurai and Y. Shimada, Jpn. J. Appl. Phys., vol. 31 pp. 1905-1908, Part 1, No.6A, June (1992).
* [19] M. V. Marchevsky, Ph.D. thesis, Leiden University (1997).
* [20] A. Ruotolo, C. Bell, C. W. Leung, and M. G. Blamire, J. Appl. Phys., 96 pp. 512 (2004).
|
arxiv-papers
| 2008-11-14T12:44:57 |
2024-09-04T02:48:58.790246
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "I.S. Veshchunov, V.A. Oboznov, A.N. Rossolenko, A.S. Prokofiev, L.Ya.\n Vinnikov, A.Yu. Rusanov and D.V. Matveev",
"submitter": "Andrey S. Prokofiev",
"url": "https://arxiv.org/abs/0811.2331"
}
|
0811.2451
|
# GLOBAL SLIM ACCRETION DISK SOLUTIONS REVISITED
Cheng-Liang Jiao, Li Xue, Wei-Min Gu, and Ju-Fu Lu Department of Physics and
Institute of Theoretical Physics and Astrophysics,
Xiamen University, Xiamen, Fujian 361005, China; lujf@xmu.edu.cn
###### Abstract
We show that there exists a maximal possible accretion rate, beyond which
global slim disk solutions cannot be constructed because in the vertical
direction the gravitational force would be unable to balance the pressure
force to gather the accreted matter. The principle for this restriction is the
same as that for the Eddington luminosity and the corresponding critical
accretion rate, which were derived for spherical accretion by considering the
same force balance in the radial direction. If the assumption of hydrostatic
equilibrium is waived and vertical motion is included, this restriction may
become even more serious as the value of the maximal possible accretion rate
becomes smaller. Previous understanding in the literature that global slim
disk solutions could stand for any large accretion rates is due to the
overestimation of the vertical gravitational force by using an approximate
potential. For accretion flows with large accretion rates at large radii,
outflows seem unavoidable in order for the accretion flow to reduce the
accretion rate and follow a global solution till the central black hole.
accretion, accretion disks - black hole physics - hydrodynamics
## 1 INTRODUCTION
Although great progress has been made in recent years in increasingly
sophisticated numerical accretion disk simulations, simple analytic disk
models still are the only accessible way of making direct link between the
theory and observations, as only these models can be used to estimate, e.g.,
the spectra of accretion-powered astrophysical systems. This fact justifies
the continuous effort to improve the understanding of accretion processes
using a simple viscosity parameterization.
The slim disk model is one of such simple analytic models for black hole
accretion (Abramowicz et al. 1988; Kato et al. 1998). This model was developed
upon the standard Shakura-Sunyaev disk (SSD) model (Shakura & Sunyaev 1973) by
considering two processes in black hole accretion flows, namely the transonic
motion and the advective heat transport, which were neglected in the SSD
model. In the limit of low mass accretion rates, i.e., $\dot{M}$ is
substantially lower than its critical value $\dot{M}_{\mathrm{Edd}}$
corresponding to the Eddington luminosity $L_{\mathrm{Edd}}$, say,
$\dot{M}\lesssim 0.1\dot{M}_{\mathrm{Edd}}$, the advective heat transport is
unimportant, and the structure of slim disks is similar to that of SSDs, with
a difference that slim disk flows are transonic in their inner regions (e.g.,
Chen & Taam 1993). In this sense, the theoretical basis of black hole disk
models for low and even moderate accretion rates is well established. But the
slim disk model is supposed to have the advantage over the SSD model that it
can be extended to the case of high accretion rates, i.e., with $\dot{M}$
approaching or surpassing $\dot{M}_{\mathrm{Edd}}$. In this case, the very
basic assumption of the SSD model that is valid only for low accretion rates,
i.e., the geometrical thinness, $H\ll r$, where $H$ is the half-thickness of
the disk and $r$ is the cylindrical radius, would break down (Frank et al.
2002, p. 98). It was then suggested in the slim disk model that the disk
becomes geometrically slim, i.e., with $H\lesssim r$; and accordingly, the
process of advective heat transport becomes important or even dominant over
the radiative cooling because, as given by Abramowicz et al. (1986), there is
a relation $f_{\mathrm{adv}}\varpropto(H/r)^{2}$, where
$f_{\mathrm{adv}}\equiv Q_{\mathrm{adv}}^{-}/Q_{\mathrm{vis}}^{+}$ is the
advective factor, with $Q_{\mathrm{adv}}^{-}$ and $Q_{\mathrm{vis}}^{+}$ being
the advective cooling and viscous heating rates per unit area of the disk,
respectively.
However, the self-consistency of the slim disk model and its applicability to
the high accretion rate case are not so obvious. In particular, uncertainties
seem to concentrate on the treatment of the vertical structure of the disk.
First, vertical hydrostatic equilibrium is a reasonable assumption for SSDs
because, for geometrically thin disks, vertical motion of the disk matter must
be negligible compared with radial motion; but it is questionable to adopt
this assumption for slim disks that are not thin. Second, as noticed recently
by Gu & Lu (2007, hereafter GL07), even the vertical hydrostatic equilibrium
is assumed, in the slim disk model there is a serious inconsistency that the
vertical gravitational force was greatly magnified by using the approximate
form of potential due to Hōshi (1977), which is valid only for thin disks
again; and accordingly, the previous understanding that slim disks could exist
for any large accretion rates seems doubtful. Third, the above mentioned
relation $f_{\mathrm{adv}}\varpropto(H/r)^{2}$ was derived with the Hōshi
approximation of potential, so its applicability to not thin disks with large
accretion rates is not justified; or in other words, it is not clear whether
an accretion disk can ensure advection dominance (i.e.,
$f_{\mathrm{adv}}>0.5$) while remaining to be geometrically slim. According to
the analyses of Narayan & Yi (1995) and Gu et al. (2008), in order for
advection to be dominant, the disk needs to be geometrically thick, i.e., with
$H>r$, rather than slim. This is probably the reason why $H>r$ was obtained in
many numerical calculations of slim disks and other accretion disks (see
references in GL07).
What about another popular black hole accretion disk model, the advection-
dominated accretion flow (ADAF) model (Narayan & Yi 1994; Abramowicz et al.
1995)? ADAFs are also supposed to be geometrically slim and advection-
dominated, and their vertical structure was treated in a way similar to that
for slim disks. The above mentioned uncertainties should also apply to the
ADAF model, because they are based on purely hydrostatic considerations and
are related only to the geometrical thickness of the disk. Many two- and
three-dimensional numerical simulations of viscous radiatively inefficient
accretion flows revealed the existence of convection-dominated accretion flows
rather than ADAFs (see references in GL07). This fact is probably an
indication that the ADAF model might have hidden inconsistencies, and one of
which might be related to the treatment of vertical structure. What is
different from slim disks is that ADAFs are expected to correspond to very low
accretion rates and are known to have a maximal possible accretion rate at
each radius (e.g., Abramowicz et al. 1995), so the problem addressed in GL07
regarding the allowed accretion rate may have no impact on ADAFs. In addition,
ADAFs are optically thin and ion pressure-supported, the radiation processes
in them are more complicated, making the vertical structure more difficult to
deal with, than for slim disks that are optically thick and radiation
pressure-supported.
Our present work is devoted to discuss the slim disk model and is a
straightforward continuation of GL07. All the results of GL07 were based on a
local analysis, i.e., only for a certain radius. Although a similar local
analysis was often used in the literature (e.g., Abramowicz et al. 1995; Chen
et al. 1995; Kato et al. 1998), one should be cautious of the fact that
conclusions made in the local sense do not necessarily hold in the global
sense. For example, it has been shown that, even though a disk is locally
unstable at a given radius according to a local stability analysis, the disk
can be globally stable in global numerical simulations (Janiuk et al. 2002;
Gierliński & Done 2004). Therefore, it is worthwhile to check and extend the
results of GL07 by investigating global solutions of original differential
equations for black hole accretion flows, similar to what was done by, e.g.,
Chen & Wang (2004), Watarai et al. (2005), Artemova et al. (2006), and Watarai
(2006), but with a revised vertical gravitational force.
## 2 SOLUTIONS WITH VERTICAL HYDROSTATIC EQUILIBRIUM
### 2.1 Equations
The basic equations to be solved for slim disks can be written in cylindrical
coordinates as (cf. Kato et al. 1998, p. 236; GL07):
$\dot{M}=-2\pi r\Sigma v_{r}={\mathrm{constant}},$ (1)
$v_{r}\frac{dv_{r}}{dr}+\frac{1}{\rho_{0}}\frac{dp_{0}}{dr}+(\Omega_{\mathrm{K}}^{2}-\Omega^{2})r=0,$
(2)
$\dot{M}(\Omega r^{2}-j)=2\pi\alpha r^{2}\Pi,$ (3)
$Q_{\mathrm{vis}}^{+}=Q_{\mathrm{adv}}^{-}+Q_{\mathrm{rad}}^{-},$ (4)
$p_{0}=\frac{k_{\mathrm{B}}\rho_{0}T_{0}}{\mu m_{p}}+\frac{1}{3}aT^{4}_{0},$
(5)
where $\Sigma=2\int_{0}^{H}\rho dz$ is the surface density, $\rho$ is the
density, $p$ is the pressure, $v_{r}$ is the radial velocity, $\Omega$ is the
angular velocity and $\Omega_{\mathrm{K}}$ is its Keplerian value, $j$ is an
integration constant representing the specific angular momentum accreted by
the black hole, $\alpha$ is the Shakura-Sunyaev viscosity parameter,
$\Pi=2\int_{0}^{H}pdz$ is the vertically integrated pressure,
$Q_{\mathrm{rad}}^{-}$ is the radiative cooling rate per unit area, $T$ is the
temperature, $\mu$ is the mean molecular weight and is taken to be 0.62, and
the subscript “0” represents quantities on the equatorial plane.
There are some differences between equations (1 - 5) and the basic equations
in GL07 (their eqs. [6 - 10]). In GL07 all the five equations were written in
the vertically integrated form; here the continuity equations (1), angular
momentum equation (3), and energy equation (4) are in the same form, because
it is obviously convenient to write equation (1) in this form and equations
(3) and (4) also contain $\dot{M}$; but the radial momentum equation (2) and
state equation (5) are given for equatorial quantities. The state equation of
GL07 (their eq. [10]) is a trivial vertical integration of our equation (5)
and makes no difference. But in GL07 (and also in Kato et al. 1998), in
obtaining the vertically integrated radial momentum equation (eq. [7] of GL07
or eq. [8.25] of Kato et al. 1998), some simplifications were made. First,
both $v_{r}$ and $\Omega$ were regarded to be independent of the coordinate
$z$, and this is also adopted in our equations here. Second, a step
$\int_{-H}^{H}\frac{dp}{dr}dz=\frac{d\Pi}{dr}$ was taken, but in the strict
sense this equality holds only when $H$ does not vary with $r$, in studies of
global solutions here we should better not to take it. This is the reason why
we prefer to write equation (2) for equatorial quantities. The differential
equation (4) was reduced to be an algebraic one in GL07; and here instead, it
keeps its original form with the explicit expressions for
$Q_{\mathrm{vis}}^{+}$, $Q_{\mathrm{adv}}^{-}$, and $Q_{\mathrm{rad}}^{-}$
given as
$Q_{\mathrm{vis}}^{+}=-\frac{\dot{M}\Omega(\Omega r^{2}-j)}{2\pi
r^{2}}\frac{d\ln{\Omega}}{d\ln{r}},$ (6)
$Q_{\mathrm{adv}}^{-}=-\frac{3\dot{M}}{2\pi
r^{2}}\frac{\Pi}{\Sigma}\left(\frac{d\ln p_{0}}{d\ln
r}-\frac{4}{3}\frac{d\ln\rho_{0}}{d\ln r}\right),$ (7)
$Q_{\mathrm{rad}}^{-}=\frac{32\sigma T_{0}^{4}}{3\overline{\kappa}\rho_{0}H}$
(8)
(Kato et al. 1998), where
$\overline{\kappa}=\kappa_{{\mathrm{es}}}+\kappa_{{\mathrm{ff}}}=0.34+6.4\times{10^{22}}\overline{\rho}\overline{T}^{\
-7/2}\mathrm{cm^{2}\ g^{-1}}$, $\overline{\rho}(=\Sigma/2H)$ and
$\overline{T}(=\int_{0}^{H}Tdz/H)$ are the vertically averaged density and
temperature, respectively.
The key difference between the existing slim disk model and our work here is
in the treatment of the vertical hydrostatic equilibrium equation,
$\frac{\partial p}{\partial z}+\rho\frac{\partial\psi}{\partial z}=0,$ (9)
where $\psi$ is the potential. In the slim disk model (e.g., Kato et al. 1998,
p. 241), the potential of Paczyński & Wiita (1980),
$\psi(r,z)=-\frac{GM}{\sqrt{r^{2}+z^{2}}-r_{g}},$ (10)
where $r_{g}\equiv 2GM/c^{2}$ is the gravitational radius, was approximated in
the form of Hōshi (1977), i.e.,
$\psi(r,z)\approx\psi(r,0)+\frac{\Omega_{\mathrm{K}}^{2}z^{2}}{2}.$ (11)
Using equation (11) and assuming a polytropic relation in the vertical
direction, $p=K\rho^{1+1/N}$, where $K$ and $N$ are constants, the vertical
integration of equation (9) gave (for $N=3$)
$\left(\frac{\rho}{\rho_{0}}\right)^{1/3}=\left(\frac{p}{p_{0}}\right)^{1/4}=\frac{T}{T_{0}}=1-\frac{z^{2}}{H^{2}},$
(12)
$8\frac{p_{0}}{\rho_{0}}=\Omega_{\mathrm{K}}^{2}H^{2},$ (13)
and $\Sigma$ and $\Pi$ were
$\Sigma=2\times\frac{16}{35}\rho_{0}H,$ (14)
$\Pi=2\times\frac{128}{315}p_{0}H.$ (15)
The simple relation ${c_{s}}/\Omega_{\mathrm{K}}H=\mathrm{constant}$ was
obtained, with the sound speed $c_{s}$ being defined either as
$c_{s}^{2}=p_{0}/\rho_{0}$ or as $c_{s}^{2}=\Pi/\Sigma$.
However, as the main point made in GL07, the approximation of equation (11) is
invalid for slim disks. By using the explicit Paczyński & Wiita potential,
equation (10), to integrate equation (9), one obtains instead of equations
(12) and (13):
$g=\left(\frac{\rho}{\rho_{0}}\right)^{1/3}=\left(\frac{p}{p_{0}}\right)^{1/4}=\frac{T}{T_{0}}=\frac{\frac{1}{\sqrt{r^{2}+z^{2}}-r_{g}}-\frac{1}{\sqrt{r^{2}+H^{2}}-r_{g}}}{\frac{1}{r-r_{g}}-\frac{1}{\sqrt{r^{2}+H^{2}}-r_{g}}},$
(16)
$4\frac{p_{0}}{\rho_{0}}=GM\left(\frac{1}{r-r_{g}}-\frac{1}{\sqrt{r^{2}+H^{2}}-r_{g}}\right).$
(17)
Accordingly, $\Sigma$ and $\Pi$ become
$\Sigma=2\rho_{0}\int_{0}^{H}g^{3}dz,$ (18)
$\Pi=2p_{0}\int_{0}^{H}g^{4}dz,$ (19)
which are much more complicated than equations (14) and (15). It is seen that
the relation ${c_{s}}/\Omega_{\mathrm{K}}H=\mathrm{constant}$ does not hold.
To summarize, in the existing slim disk model, the set of eight equations,
namely equations (1 - 5) and (13 - 15), could be solved for the eight unknown
quantities $\rho_{0}$, $p_{0}$, $\Sigma$, $\Pi$, $T_{0}$, $v_{r}$, $\Omega$,
and $H$ as functions of $r$, with given constant parameters $M$, $\dot{M}$,
$\alpha$, and $j$; while in our work here, equations (17 - 19), instead of
equations (13 - 15), still along with equations (1 - 5) form a new set to be
solved for the same eight unknowns. There are two differential equations,
i.e., equations (2) and (4); and other equations in each set are algebraic,
including equations (18) and (19) once the vertical integration is
analytically made.
### 2.2 Solutions
We numerically solve the set of equations (1 - 5) and (17 - 19), i.e., with
the correctly calculated vertical gravitational force using the explicit
Paczyński & Wiita potential. For comparisons, we also solve the set of
equations (1 - 5) and (13 - 15), as done already in the existing slim disk
model, i.e., with the magnified vertical gravitational force using the Hōshi
approximation of the Paczyński & Wiita potential. After some algebra, the
equations in each set can be combined into only two differential equations. A
physically acceptable global solution should be able to extend from a large
radius to the vicinity of the central black hole, passing the sonic point
regularly. We start the integration at $r=10^{6}r_{g}$, where the outer
boundary conditions are set to be corresponding to an SSD, i.e., being
geometrically thin and Keplerian rotating. If a solution is found to be
transonic, we extend it to $r=2r_{g}$ (in the following figures only a smaller
radial range is shown in order to see more clearly the solution behavior in
the inner region). We fix $M=10M_{\sun}$ and $\alpha=0.1$, and vary the values
of $\dot{M}$ for different solutions. For a transonic solution, the other
constant parameter $j$ is an eigenvalue of the problem and has to be adjusted
correctly and accurately (e.g., Chen & Taam 1993; Chen & Wang 2004).
Rather than showing the radial variation of every physical quantity, our main
purpose here is to see how the correction of the vertical gravitational force
changes the previous understanding of global slim disk solutions, that is,
there was no upper limit of $\dot{M}$ for slim disks, any large value of
$\dot{M}$ could correspond to a global solution (e.g., Kato et al. 1998; Chen
& Wang 2004; Watarai et al. 2005; Watarai 2006). Figures 1 and 2 are
sufficient for this purpose, which are for the relative thickness $H/r$ and
the advective factor $f_{\mathrm{adv}}$, respectively. In the figures, solid
lines are solutions of equations (1 - 5) and (17 - 19), and dashed lines are
solutions of equations (1 - 5) and (13 - 15). Solid line $a$ and dashed line
$a$’ are for the same accretion rate $\dot{m}=1$, where
$\dot{m}\equiv\dot{M}/\dot{M}_{\mathrm{Edd}}$, with
$\dot{M}_{\mathrm{Edd}}=64\pi GM/c\kappa_{\mathrm{es}}$ being the Eddington
accretion rate; and similarly, solid line $b$ and dashed line $b$’, solid line
$c$ and dashed line $c$’, solid line $d$ and dashed line $d$’, and solid line
$e$ and dashed line $e$’ are for $\dot{m}=10,31.5,31.7$, and 100,
respectively. It is seen that for a moderate accretion rate $\dot{m}=1$ (lines
$a$ and $a$’), global solutions obtained from the two sets of equations are
almost the same, i.e., both the solutions have $H/r\lesssim 1$ (Fig. 1), and
are radiation-dominated in the outer region and with important advection in
the inner region (Fig. 2). This proves that using the Hōshi approximation of
potential for moderate accretion rates is acceptable. However, deviations
appear and become serious as $\dot{m}$ increases. For $\dot{m}=10$, our new
solution (lines $b$ in the two figures) has $H$ and $f_{\mathrm{adv}}$
significantly larger than that in the existing slim disk solution (lines
$b$’). Such a situation continues till a critical value $\dot{m}=31.5$, for
which our new solution can still be constructed (lines $c$). For a slightly
larger $\dot{m}=31.7$ and a still larger $\dot{m}=100$, global solutions can
be no longer obtained from our new set of equations (1 - 5) and (17 - 19). It
is seen that $H$ tends to infinity (lines $d$ and $e$ in Fig. 1) and
$f_{\mathrm{adv}}$ tends to exceed 1 (its maximal possible value, lines $d$
and $e$ in Fig. 2), so the inward integration cannot go on. These solutions
(lines $d$ and $e$) are not global solutions at all, since they cannot extend
to a sonic point. On the other hand, no matter how large $\dot{M}$ is, global
solutions can always be found from the set of existing slim disk equations (1
- 5) and (13 - 15), as drawn by lines $b$’, $c$’, $d$’, and $e$’ in the two
figures (lines $c$’ and $d$’ coincide with each other).
Unfortunately, the existing global slim disk solutions, though being formally
constructed for any high accretion rates, have hidden inconsistencies. As seen
clearly in Figure 1 of GL07, the vertical gravitational force,
$\partial\psi/\partial z$ in equation (9), was greatly overestimated, and
accordingly, the geometrical thickness $H$ was greatly underestimated, by
using the approximation of equation (11). This is the reason why in the slim
disk model, the gravitational force seemed to be always able to balance the
pressure force and ensure vertical hydrostatic equilibrium. Even so, slim disk
solutions may still have $H/r>1$ (lines $c$’, $d$’, and $e$’ in Fig. 1), the
assumption of slimness is violated.
As the main result of our work, it is found that, when the vertical
gravitational force is correctly calculated from equation (10), there exists a
maximal possible accretion rate, $\dot{M}_{\mathrm{max}}\approx
31.5\dot{M}_{\mathrm{Edd}}$, beyond which there are no global solutions at all
(lines $d$ and $e$ in Figs. 1 and 2). The physical reason for this is the
following. The amount of accreted matter that can be gathered by the black
hole’s gravitational force must be limited. If
$\dot{M}>\dot{M}_{\mathrm{max}}$, the pressure force of the matter and
radiation in the vertical direction would be too large to be balanced by the
gravitational force, the disk would be huffed by the pressure force to tend to
an infinite thickness (lines $d$ and $e$ in Fig. 1), and the accretion
processes would never be maintained. This reason is based on the same
principle as that for the Eddington luminosity $L_{\mathrm{Edd}}$ and the
corresponding critical accretion rate $\dot{M}_{\mathrm{Edd}}$ to be defined.
The only difference is that $L_{\mathrm{Edd}}$ and $\dot{M}_{\mathrm{Edd}}$
were derived for spherical accretion, i.e., by considering the balance between
the gravitational force and the pressure force in the radial direction; and
here the maximal possible accretion rate is found by considering the balance
of the same two forces in the vertical direction of accretion disks.
In GL07, by a local analysis, i.e., considering the vertical balance of
gravitational and pressure forces at a certain radius, a similar maximal
possible accretion rate, $\dot{M}_{\mathrm{max}}(r)$, was found for each
radius. Their result is confirmed here. But a global solution is with a
constant accretion rate, so the allowed accretion rate has a unique value,
rather than being radius-dependent.
As a check of our global solutions, the total optical depth
$\tau=\overline{\kappa}\Sigma/2$ is calculated and is shown in Figure 3, where
lines $a$, $b$, $c$, $a$’, $b$’, and $c$’ are the correspondents of lines $a$,
$b$, $c$, $a$’, $b$’, and $c$’ in Figures 1 and 2, respectively. It is seen
that, similar to the existing slim disk solutions (lines $a$’, $b$’, and
$c$’), our global solutions are also optically thick everywhere (lines $a$,
$b$, and $c$). But $\tau$ in our solutions is somewhat smaller than that in
the existing slim disk solutions, especially in the inner regions. This is
because in our solutions, the correctly calculated vertical gravitational
force is smaller, then $H$ is larger, $v_{r}$ is larger, and $\Sigma$ is
smaller for the same $\dot{M}$; and $\overline{\kappa}$ is almost unchanged,
especially in the inner regions where the electron scattering opacity is
dominant and is a constant.
## 3 DISCUSSION
### 3.1 Waiving hydrostatic equilibrium
Though global solutions with vertical gravitational force correctly calculated
can be obtained for $\dot{M}<\dot{M}_{\mathrm{max}}$ as shown in Figures 1, 2,
and 3, there seems to be also some inconsistency hidden in these solutions.
For high accretion rates, the disk’s relative thickness $H/r$ becomes
substantially larger than 1 (lines $b$ and $c$ in Fig. 1); and for the
critical value $\dot{M}=31.5\dot{M}_{\mathrm{Edd}}$, $H/r$ reaches up to $\sim
20$ in the middle region of the solution. As argued by Abramowicz et al.
(1997), if it is assumed that there is no velocity component in the direction
orthogonal to the surface of the disk (i.e., no outflows leaving the disk),
then there is a relation $v_{H}/v_{r}=dH/dr$, where $v_{H}$ is the vertical
velocity on the surface, and vertical motion should not be neglected if $H$
does not very slowly vary with $r$. In particular, at the maximum of $H/r$
(see lines $b$ and $c$ in Fig. 1), it follows from $d(H/r)/dr=0$ that
$dH/dr=H/r$. This means that in the middle region of the solution the vertical
velocity could greatly exceed the radial velocity, and the assumption of
hydrostatic equilibrium would not be valid. In this case, instead of
hydrostatic equilibrium equation (9), one should use the more general form of
vertical momentum equation,
$\frac{1}{\rho}\frac{\partial p}{\partial z}+\frac{\partial\psi}{\partial
z}+v_{r}\frac{\partial v_{z}}{\partial r}+v_{z}\frac{\partial v_{z}}{\partial
z}=0$ (20)
(Abramowicz et al. 1997).
Solving this partial differential equation is beyond the capacity of the slim
disk model that is one-dimensional. In an illustrative sense, here we wish to
try to consider the non-negligible vertical velocity $v_{z}$ with a very
simple treatment, which is similar to what was done in Abramowicz et al.
(1997). We assume
$v_{z}(r,z)=\frac{z}{H}v_{H}=\frac{z}{r}u,$ (21)
where
$u(r)=v_{r}\frac{d\mathrm{ln}H}{d\mathrm{ln}r},$ (22)
then equation (20) is reduced to be
$\frac{1}{\rho}\frac{\partial p}{\partial z}+\frac{\partial\psi}{\partial
z}+zY=0,$ (23)
where
$Y(r)=\frac{1}{r^{2}}\left(rv_{r}\frac{du}{dr}-v_{r}u+u^{2}\right),$ (24)
which looks similar to equation (9) of Abramowicz et al. (1997).
Instead of equations (16) and (17), the vertical integration of equation (23)
gives (of course, still with the explicit potential eq. [10])
$g=\left(\frac{\rho}{\rho_{0}}\right)^{1/3}=\left(\frac{p}{p_{0}}\right)^{1/4}=\frac{T}{T_{0}}=\frac{GM(\frac{1}{\sqrt{r^{2}+z^{2}}-r_{g}}-\frac{1}{\sqrt{r^{2}+H^{2}}-r_{g}})+\frac{1}{2}Y(H^{2}-z^{2})}{GM(\frac{1}{r-r_{g}}-\frac{1}{\sqrt{r^{2}+H^{2}}-r_{g}})+\frac{1}{2}YH^{2}},$
(25)
$4\frac{p_{0}}{\rho_{0}}=GM\left(\frac{1}{r-r_{g}}-\frac{1}{\sqrt{r^{2}+H^{2}}-r_{g}}\right)+\frac{1}{2}YH^{2}.$
(26)
Equations (18) and (19) are formally unchanged, but the quantity g in these
two equations is given by equation (25) now.
The nine equations, i.e., equations (1 - 5), (18), (19), (22), and (26) can be
solved for nine unknowns, $\rho_{0}$, $p_{0}$, $\Sigma$, $\Pi$, $T_{0}$,
$v_{r}$, $\Omega$, $H$ and $u$ (eq. [1] keeps unchanged even there is a non-
zero $v_{z}$, because of the no outflow assumption; and eqs. [2 - 5] still
hold as well). The situation of solutions is seen in Figure 4 that shows $H/r$
as a function of $r$. Global solutions exist till a new maximal possible
accretion rate $\dot{M}_{\mathrm{max}}\approx 8.5\dot{M}_{\mathrm{Edd}}$
(lines $a$ and $b$ that are for $\dot{m}=1$ and 8.5, respectively), above
which there are no global solutions (lines $c$ and $d$ that are for
$\dot{m}=9$ and 30, respectively). But this time, for
$\dot{M}>\dot{M}_{\mathrm{max}}$, it is not the case of Figure 1 that the
inward integration stops at a radius where the thickness $H$ tends to
infinity; rather, it is the case that the inward integration stops at a radius
where the quantity $g$ of equation (25) tends to become unphysically negative
at some height $z$ between the equatorial plane $z=0$ (where $g=1$) and the
disk surface $z=H$ (where $g=0$), even though $H$ is finite at that radius. It
is not surprising that the value of $\dot{M}_{\mathrm{max}}$ here is even
smaller than that in the above section where vertical hydrostatic equilibrium
is assumed. The reason for this is the following. As seen from Figure 5, in
global solutions $a$ and $b$, $v_{H}$ is always negative, i.e., the vertical
motion is always inward towards the equatorial plane; and the absolute value
of $v_{H}$ increases with decreasing $r$. This means clearly that there is a
vertical acceleration towards the equatorial plane. That is, in the vertical
direction, the gravitational force has to overcome the pressure force to
accelerate the accreted matter, rather than only balancing the pressure force
to ensure hydrostatic equilibrium. The upper limit of the amount of the
accreted matter, to which the gravitational force is able to do this harder
job, must be smaller than in the hydrostatic equilibrium case. For
$\dot{M}>\dot{M}_{\mathrm{max}}$, the gravitational force would be unable to
do the job. To keep equation (23) formally holding, $g$ (and $\rho$ and $T$)
of equation (25) would have to become mathematically negative, which are
physically unacceptable.
Though the treatment of vertical motion made here is crude, it is sayable that
inclusion of this motion should not change the main conclusion reached in the
above section, i.e., there is a maximal possible accretion rate
$\dot{M}_{\mathrm{max}}$ for global slim disk solutions to exist; and the
value of $\dot{M}_{\mathrm{max}}$ becomes even smaller than in the case that
this motion was omitted.
### 3.2 Outflows
If the mass accretion rate given at large radii exceeds its maximal possible
value, $\dot{M}>\dot{M}_{\mathrm{max}}$, it seems that the only possible way
for a global slim disk solution to be realized is that the accretion flow
loses its matter at large radii in the form of outflows, such that at smaller
radii $\dot{M}$ is reduced to be below $\dot{M}_{\mathrm{max}}$.
Outflows have been observed in many high energy astrophysical systems that are
believed to be powered by black hole accretion, but the mechanism of outflow
formation remains unclear from the theoretical point of view. At the same time
when the ADAF model was proposed, it was suggested that ADAFs are likely to be
able to produce outflows because they have a positive Bernoulli constant in
their self-similar solutions (Narayan & Yi 1994). Later, Abramowicz et al.
(2000) showed that in global solutions of ADAFs the Bernoulli function, rather
than the Bernoulli constant, can have either positive or negative values, and
that even a positive Bernoulli function is only a necessary, not a sufficient,
condition for outflow formation.
Historically, the Bernoulli constant or function was defined as the sum of the
specific enthalpy, kinetic energy, and gravitational potential energy. For
ADAFs Narayan & Yi (1994) and Abramowicz et al. (2000) discussed, the
Bernoulli function has a simple expression since the enthalpy is totally due
to the disk gas; it has also a clear physical meaning that its positivity may
imply a possibility of outflows. For slim disks, however, the contribution
from radiation to the enthalpy becomes important or even dominant over that
from gas. To our knowledge, in this case it is unclear whether and how can the
Bernoulli function be defined, or whether this function would have the same
meaning as for ADAFs if its historical definition is copied. Therefore, we use
instead the specific total energy of the matter in slim disks to consider the
possibility of outflows, whose equatorial-plane value is
$E=\left(\frac{3}{2}\frac{k_{\mathrm{B}}T_{0}}{\mu
m_{p}}+\frac{aT_{0}^{4}}{\rho_{0}}\right)+\frac{1}{2}\left(v_{r}^{2}+\Omega^{2}r^{2}\right)-\frac{GM}{r-r_{g}}$
(27)
(cf. eq. [11.33] of Kato et al. 1998).
Figure 6 shows $E$ corresponding to the solutions with the correct
gravitational force in Figures 1 and 2. Lines $a$, $b$, $c$, $d$, and $e$ in
Figure 6 are for the same physical parameters as lines $a$, $b$, $c$, $d$, and
$e$ in Figures 1 and 2, respectively. For accretion rates that allow global
solutions to exist, $\dot{M}=1,10$ and $31.5\dot{M}_{\mathrm{Edd}}$ (lines
$a$, $b$, and $c$), $E$ is negative everywhere in the disk, and outflows are
unlikely to originate. However, for accretion rates exceeding
$\dot{M}_{\mathrm{max}}(r)$, $\dot{M}=31.7$ and $100\dot{M}_{\mathrm{Edd}}$
(lines $d$ and $e$), it is seen that as $r$ decreases, $E$ tends to become
positive first, and then the solution stops extending. Further, it is
noticeable that, when a flow changes from a state represented by line $e$ to a
state represented by line $d$, its $E$ increases, and at the same time its
$\dot{M}$ decreases. The increase of $E$ favors outflow formation, and the
decrease of $\dot{M}$ is just the result of outflows. Such a process of the
change of the flow’s state continues until $\dot{M}$ is sufficiently reduced
and the flow reaches a state represented by line $c$, then $E$ becomes
negative and outflows cease.
To summarize, in this subsection we make two arguments for outflows produced
in an accretion flow with large $\dot{M}$ at large radii. First, outflows are
necessary because the accretion flow must lose its matter in this way in order
to follow a global solution till the central black hole. Second, outflows are
possible because the accretion flow can have a positive specific total energy;
what is additionally needed for outflows to be realized is that the matter in
the accretion flow comes under an outward perturbation in the vertical
direction.
We thank Lin-Hong Chen and Jian-Min Wang for providing us their numerical code
of global slim disk solutions and the referee for very helpful comments. This
work was supported by the National Basic Research Program of China under Grant
No. 2009CB824800 and the National Natural Science Foundation of China under
Grants No. 10503003, 10673009 and 10833002.
## References
* (1) Abramowicz, M. A., Chen, X., Kato, S., Lasota, J.-P., & Regev, O. 1995, ApJ, 438, L37
* (2) Abramowicz, M. A., Czerny, B., Lasota, J.-P., & Szuszkiewicz, E. 1988, ApJ, 332, 646
* (3) Abramowicz, M. A., Lanza, A., & Percival, M. J. 1997, ApJ, 479, 179
* (4) Abramowicz, M. A., Lasota, J.-P., & Igumenshchev, I. V. 2000, MNRAS, 314, 775
* (5) Abramowicz, M. A., Lasota, J.-P., & Xu, C. 1986, in IAU Symp. 119, Quasars, ed. G. Swarup & V. K. Kapahi (Dordrecht: Reidel), 371
* (6) Artemova, Y. V., Bisnovatyi-Kpgan, G. S., Igumenshchev, I. V., & Novikov, I. D. 2006, ApJ, 637, 968
* (7) Chen, L.-H., & Wang, J.-M. 2004, ApJ, 614, 101
* (8) Chen, X., Abramowicz, M. A., Lasota, J.-P., Narayan, R., & Yi, I. 1995, ApJ, 443, L61
* (9) Chen, X., & Taam, R. E. 1993, ApJ, 412, 254
* (10) Frank, J., King, A., & Raine, D. 2002, Accretion Power in Astrophysics (Cambridge: Cambridge Univ. Press)
* (11) Gierliński, M., & Done, C. 2004, MNRAS, 347, 885
* (12) Gu, W.-M., & Lu, J.-F. 2007, ApJ, 660, 541 (GL07)
* (13) Gu, W.-M., Xue, L., Liu, T., & Lu, J.-F. 2008, in preparation
* (14) Hōshi, R. 1977, Prog. Theor. Phys., 58, 1191
* (15) Janiuk, A., Czerny, B., & Siemiginowska, A. 2002, ApJ, 576, 908
* (16) Kato, S., Fukue, J., & Mineshige, S. 1998, Black-Hole Accretion Disks (Kyoto: Kyoto Univ. Press)
* (17) Narayan, R., & Yi, I. 1994, ApJ, 428, L13
* (18) Narayan, R., & Yi, I. 1995, ApJ, 444, 231
* (19) Paczyński, B., & Wiita, P. J. 1980, A&A, 88, 23
* (20) Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337
* (21) Watarai, K. 2006, ApJ, 648, 523
* (22) Watarai, K., Ohsuga, K., Takahashi, R., & Fukue, J. 2005, PASJ, 57, 513
Figure 1: Disk’s relative thickness $H/r$ as a function of $r$. Solid lines
represent our results with the vertical gravitational force revised, and
dashed lines the results in the existing slim disk model. Lines $a$ and $a$’,
$b$ and $b$’, $c$ and $c$’, $d$ and $d$’, and $e$ and $e$’ are for accretion
rates $\dot{M}/\dot{M}_{\mathrm{Edd}}=$ 1, 10, 31.5, 31.7, and 100,
respectively. There exists a maximal possible accretion rate
$\dot{M}_{\mathrm{max}}\approx 31.5\dot{M}_{\mathrm{Edd}}$, above which global
solutions cannot be constructed as $H$ tends to infinity (lines $d$ and $e$).
Here and in Figs. 2, 3, and 6 the vertical hydrostatic equilibrium is assumed.
Figure 2: Advective factor $f_{\mathrm{adv}}$ as a function of $r$. All the
lines are the correspondents of those in Fig. 1, respectively. Lines $d$ and
$e$ are not global solutions as $f_{\mathrm{adv}}$ tends to exceed its maximal
possible value 1. Figure 3: Total optical depth $\tau$ as a function of $r$.
Lines $a$, $b$, $c$, $a$’, $b$’, and $c$’ are the correspondents of those in
Fig. 1, respectively. Figure 4: Quantity $H/r$ as a function of $r$ in the
case that vertical motion is included. Lines $a$, $b$, $c$, and $d$ are for
$\dot{M}/\dot{M}_{\mathrm{Edd}}=$ 1, 8.5, 9, and 30, respectively. There is a
new $\dot{M}_{\mathrm{max}}\approx 8.5\dot{M}_{\mathrm{Edd}}$, above which
there are no global solutions (lines $c$ and $d$). Figure 5: Radial velocity
$v_{r}$ and vertical velocity on the disk surface $v_{H}$ in global solutions.
Lines $a$ and $b$ are the correspondents of those in Fig. 4, respectively.
Figure 6: Specific total energy $E$ as a function of $r$. Lines $a$, $b$, $c$,
$d$ and $e$ are the correspondents of those in Fig.1, respectively.
|
arxiv-papers
| 2008-11-15T01:08:04 |
2024-09-04T02:48:58.797877
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Cheng-Liang Jiao, Li Xue, Wei-Min Gu, and Ju-Fu Lu",
"submitter": "Li Xue",
"url": "https://arxiv.org/abs/0811.2451"
}
|
0811.2508
|
XXXXXX
# Brown Dwarfs as Galactic Chronometers
Adam J. Burgasser1 1Massachusetts Institute of Technology, Kavli Institute for
Astrophysics and Space Research,
Building 37, Room 664B, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
email: ajb@mit.edu
(2008)
###### Abstract
Brown dwarfs are natural clocks, cooling and dimming over time due to
insufficient core fusion. They are also numerous and present in nearly all
Galactic environments, making them potentially useful chronometers for a
variety of Galactic studies. For this potential to be realized, however,
precise and accurate ages for individual sources are required, a prospect made
difficult by the complex atmospheres and spectra of low-temperature brown
dwarfs; degeneracy between mass, age and luminosity; and the lack of useful
age trends in magnetic activity and rotation. In this contribution, I review
five ways in which ages for brown dwarfs are uniquely determined, discuss
their applicability and limitations, and give current empirical precisions.
###### keywords:
stars: binaries, stars: fundamental parameters, stars: kinematics, stars:
late-type, stars: low-mass, brown dwarfs.
††volume: 258††journal: Ages of Stars††editors: E. Mamajek, D. Soderblom & J.
Valenti, eds.
## 1 Introduction
Brown dwarfs are very low-mass stars whose masses (M $\lesssim$ 0.075 M⊙) are
insufficient to sustain the core hydrogen fusion reactions that balance
radiative energy losses (Kumar, 1963; Hayashi & Nakano, 1963). Supported from
further gravitational contraction by electron degeneracy pressure, evolved
brown dwarfs continually cool and dim over time as they radiate away their
initial contraction energy, ultimately achieving photospheric conditions that
can be similar to those of giant planets. The first examples of brown dwarfs
were identified as recently as 1995 (Nakajima et al., 1995; Rebolo et al.,
1995). Today, there are hundreds known in nearly all Galactic environments,
identified largely in wide-field, red and near-infrared imaging surveys such
as 2MASS, DENIS, SDSS and UKIDSS. The known population of brown dwarfs
encompasses the late-type M (Teff $\approx$ 2500–3500 K), L (Teff $\approx$
1400–2500 K) and T spectral classes (Teff $\approx$ 600–1400 K; e.g., Vrba et
al. 2004), while efforts are currently underway to find even colder members of
the putative Y dwarf class (see review by Kirkpatrick 2005).
Because brown dwarfs cool over time, their spectral properties are inherently
time-dependent, making them potentially useful chronometers for Galactic
studies (much like white dwarfs; see contributions by M. Salaris, J. Kalirai,
S. Catalán and H. Richer). However, the primary observables of a brown
dwarf—temperature, luminosity and spectral type—depend on both mass and age
(and weakly on metallicity). This degeneracy complicates characterizations of
individual sources and mixed populations. Unfortunately, traditional stellar
age-dating methods do not appear to be applicable for brown dwarfs. Magnetic
activity metrics, such as the frequency and strength of optical or X-ray non-
thermal emission, appear to be largely age-invariant (e.g., Stelzer et al.
2006) and quiescent emission drops off precipitously in the early L dwarfs as
cool photospheres are decoupled from field lines 111Interestingly, radio
emission does not drop off for cooler brown dwarfs, and may even increase
relative to bolometric flux, although there are currently few detections
(e.g., Berger 2006). (e.g., Mohanty et al. 2002; see contribution by A. West).
Long-term angular momentum loss in brown dwarfs is far more muted than in
stars, and there is no clear rotation-activity relation for L dwarfs (e.g.,
Reiners & Basri 2008). Exploitation of the cooling properties of brown dwarfs
is therefore a favorable approach for determining their ages.
In this contribution, I review five methods currently employed to age-date
brown dwarfs and summarize their applicability, inherent limitations and
current (typical) precisions. Table 1 provides a summary of the methods
discussed in detail below.
Table 1: Age-dating Methods for Brown Dwarfs.
Technique | Applicable for | Pros | Cons | Precision | Examples
---|---|---|---|---|---
| | | | | in the
| | | | | Literature
Cluster | nearby clusters; | precise ages based on | generally restricted to | $\sim$10% clusters; | 1,2,3,4
members & | companions to | stellar/cluster work; | young clusters; wide | $\sim$50–100% for |
companions | age-dated stars | calibration for other | (resolvable) & close | companions |
| | techniques and | (RV) companions rare; | |
| | evolutionary models | must verify coevality/ | |
| | | association; atmospheres | |
| | | variable for $t\lesssim$ 10 Myr | |
6708 Å Li I | $t$ $\sim$ 10–200 Myr; | consistent predictions | requires high sensitivity, | $\sim$10% for | 5,6,7
absorption | resolved binaries; | from different models; | high resolution spectra; | young clusters; |
| individual field | straightforward test; | low brightness region; | upper/lower |
| sources with | largely insensitive to | not useful for T dwarfs | limits for all |
| Teff $>$ 1500 K | atmospheric properties | or for $t\lesssim$ 10 Myr; | others |
| and $t\lesssim$ 2 Gyr | | relies on accurate | |
| (limits only) | | evolutionary models | |
Mass | astrometric/RV | precise masses | suitable systems are | $\sim$10–20% | 8,9,10,11
standards | binaries | yield precise ages; | rare; long-term | |
| | weakly sensitive | follow-up required; | |
| | to atmospheric | relies on accurate | |
| | properties | evolutionary models | |
Surface | any source with | applicable to | low precision; other | $\sim$50–100% | 12,13,14
gravities | a well-measured | individual sources; | factors (e.g., metallicity) | |
| spectrum | particularly useful | complicate analysis; | |
| | for T dwarfs | relies on accurate | |
| | | evolutionary and | |
| | | atmospheric models | |
Kinematics | well-defined | useful statistical test | very low precision; | $\sim$100–300% | 15,16,17
| groups or | for various subclasses; | large groups required | |
| populations | insensitive to | to beat statistical noise; | |
| | evolutionary or | susceptible to | |
| | atmospheric models | selection biases | |
References: (1) Bouvier et al. (1998); (2) Luhman et al. (2003); (3) Geballe
et al. (2001); (4) Burgasser (2007); (5) Stauffer et al. (1998); (6) Jeffries
& Oliveira (2005); (7) Liu & Leggett (2005); (8) Zapatero Osorio et al.
(2004); (9) Stassun et al. (2006); (10) Liu et al. (2008); (11) Dupuy et al.
(2008); (12) Mohanty et al. (2004); (13) Burgasser et al. (2006a); (14) Saumon
et al. (2007); (15) Schmidt et al. (2007); (16) Zapatero Osorio et al. (2007);
(17) Faherty et al. (2008).
## 2 Cluster Members and Companions
The most straightforward way to age-date an individual brown dwarf is to
borrow from its environment, a tactic that is suitable for members of coeval
clusters/associations and companions to age-dated stars. Brown dwarfs are
well-sampled down to and below the deuterium fusing mass limit (M $\lesssim$
0.013 M⊙) in the youngest nearby clusters ($t$ $\lesssim$ 5 Myr), as their
luminosities are greater at early ages. Brown dwarfs have also been identified
in somewhat older (10–50 Myr)“loose associations” in the vicinity of the Sun
($d$ $\lesssim$ 50–100 pc; e.g., Kirkpatrick et al. 2008). For older and more
distant clusters ($d$ $\gtrsim$ 1 kpc), decreasing surface temperatures and
compact radii exacerbate the sensitivity issues that plague low-mass stellar
studies (see contribution by G. Piotto). There are as yet no known brown
dwarfs in globular clusters, despite detection of the end of the stellar main
sequence in systems such as NGC 6397 (Richer et al., 2008).
For brown dwarfs in young clusters, numerous studies have examined age-related
trends in colors (e.g., Jameson et al. 2008), spectral characteristics (e.g.,
Allers et al. 2007), accretion timescales (e.g., Mohanty et al. 2005) and
circum(sub)stellar disk evolution (e.g., Scholz et al. 2007). These have been
coarsely quantified, and appear to be most useful at very young ages
($t\lesssim$ 10 Myr). Surface properties and luminosities are highly variable
at these ages due to sensitivity to formation conditions (e.g., Baraffe et al.
2002), ongoing accretion, complex magnetic effects (e.g., Reiners et al. 2007)
and age spreads within a cluster (see contribution by R. Jeffries). Hence,
while brown dwarfs in clusters with ages spanning $\sim$1–650 Myr are now
well-documented, with age uncertainties as good as 10% (for the LDB technique;
see $\S$ 3), useful predictive trends are probably limited to ages of
$\gtrsim$10 Myr.
Known brown dwarf companions to main sequence stars now number a few dozen,
spanning a wide range of separation, age and composition. Many of these
systems are widely-separated so that their brown dwarf companions can be
directly studied. Age uncertainties for companion brown dwarfs depend on
stellar dating methods which are generally more uncertain (50–100%; e.g., Liu
et al. 2008) than cluster ages. Searches for substellar companions to more
precisely age-dated white dwarfs (e.g. Day-Jones et al. 2008; Farihi et al.
2008) and subgiants (Pinfield et al., 2006) have so far met with limited
success. Nevertheless, brown dwarf companions to age-dated stars serve as
important benchmarks for calibrating other age-dating methods at late ages
($\gtrsim$500 Myr) and are fundamental for testing evolutionary models (see
contribution by T. Dupuy).
## 3 6708 Å Li I Absorption
Lithium is fused at a lower temperature than hydrogen (2.5$\times$106 K),
resulting in a somewhat lower fusing mass limit (0.065 M⊙; Bildsten et al.
1997). Because the interiors of low mass stars and brown dwarfs are fully
convective at early ages, an object with a mass above this limit will fully
deplete its initial reservoir of lithium within $\sim$200 Myr. Hence, any
system older than this which exhibits Li I absorption has a mass less than
0.065 M⊙ and is therefore a brown dwarf (e.g., Rebolo et al. 1992). With a
mass limit, one can use evolutionary models to determine an age limit.
In the age range $\sim$10–200 Myr, the degree of lithium depletion in low-mass
stars and brown dwarfs is itself mass-dependent, occurring earlier in more
massive stars which first achieve the necessary core temperatures. Hence, over
this range the age of an individual source can be precisely constrained if its
mass is known. A more practical approach, however, is to ascertain the age of
a group of coeval low-mass objects based on which sources do or do not exhibit
Li I absorption; this is the lithium depletion boundary (LDB) technique.
Different evolutionary models yield remarkably similar predictions for the
location of the LDB over a broad range of ages (Burke et al., 2004), and the
boundary itself is readily identifiable in color-magnitude diagrams. As such,
this technique has been used to age-date several nearby young clusters and
associations (e.g., Stauffer et al. 1998; Barrado y Navascués et al. 1999;
Jeffries & Oliveira 2005; Mentuch et al. 2008). LDB studies have also provided
independent confirmation of other cluster-dating methods such as isochrone
fitting (Jeffries & Oliveira, 2005). A variant of the LDB technique for coeval
binary systems has been proposed by Liu & Leggett (2005), in which a system
that exhibits Li I absorption in the secondary but not in the primary can have
both lower and upper age limits assigned to it (note that the presence/absence
of Li I in both components simply sets a single upper/lower age limit). This
technique requires resolved optical spectroscopy of both components and can be
pursued only for a few (rare) wide low-mass pairs (e.g., Burgasser et al., in
prep.) or using high spatial-resolution spectroscopy (e.g. Martín et al.
2006). No single brown dwarf pair straddling the LDB has yet been identified.
Despite its utility, the detection of LI I absorption in brown dwarf spectra
has limitations. The 6708 Å line lies in an relatively faint spectral region
for cool L-type dwarfs, so sensitive spectral observations on large telescopes
are typically required to detect (or convincingly rule out) this feature. For
optically-brighter M-type brown dwarfs, high-resolution observations are
generally required to distinguish Li I absorption from overlapping molecular
absorption features. Young brown dwarfs ($t\lesssim$ 50 Myr) with low surface
gravities show weakened alkali line absorption (see $\S$ 5), including Li I,
making it again necessary to obtain sensitive, high-resolution observations
(Kirkpatrick et al., 2008). For brown dwarfs cooler than $\sim$1500 K (i.e.,
the T dwarfs), lithium is chemically depleted in the photosphere through its
conversion to LiCl, LiF or LiOH (Lodders, 1999). As such, practical age
constraints using Li I can only be made for systems younger than $\sim$2 Gyr.
## 4 Mass standards
One way of breaking the mass/age/luminosity degeneracy for an individual brown
dwarf is to explicitly measure its mass. This is feasible for sufficiently
tight brown dwarf binaries for which radial velocity (RV) and/or astrometric
orbits can be measured. Of the $\sim$100 very low mass (M1,M2 $\leq$ 0.1 M⊙)
binary systems now known, only a handful have sufficiently short periods that
large portions of their RV orbits (e.g., Joergens & Müller 2007; Blake et al.
2008), astrometric orbits (e.g., Lane et al. 2001; Bouy et al. 2004; Liu et
al. 2008; Dupuy et al. 2008), or both (Zapatero Osorio et al., 2004; Stassun
et al., 2006) have been measured. With measurable total system masses or mass
functions, individual masses can be estimated from relative photometry or
directly determined from recoil motion in both components (e.g., Stassun et
al. 2006). The individual masses and component luminosities can then be
compared to evolutionary models to determine ages.
Liu et al. (2008) have suggested that such “mass standards” provide more
precise constraints on the physical properties (including ages) of brown
dwarfs as compared to “age standards”, namely companions to main sequence
stars. Orbital masses can currently be constrained to roughly 5–10% precision,
translating into 10–20% uncertainties in ages based on evolutionary tracks
(versus 50–100% for main-sequence stars). More importantly, brown dwarf
binaries with mass measurements and independent age determinations—i.e.,
companions to age-dated stars and cluster members—can provide specific tests
of the evolutionary models themselves. Further details are provided in the
contribution by T. Dupuy.
## 5 Surface gravity
Only 10–20% of brown dwarfs are found to be multiple (e.g., Burgasser et al.
2006b) and few of these are suitable for orbital mass measurements. A proxy
for mass is surface gravity, which can be determined directly from a brown
dwarf’s spectrum. For Teff $\lesssim$ 2500 K and ages greater than $\sim$50
Myr, evolutionary models predict that brown dwarf surface gravities ($g\propto
M/R^{2}$) are roughly proportional to mass due to near-constant radii (roughly
equal to Jupiter’s radius). Surface gravity is also proportional to
photospheric pressure ($P_{ph}\propto g/{\kappa}_{R}$, where $\kappa_{R}$ is
the Rosseland mean opacity), which in turn influences the chemistry, line
broadening and (in some cases) opacities of absorbing species in the
photosphere. Hence, “gravity-sensitive” features in a brown dwarf’s spectrum
can be used to infer its mass and, through evolutionary models, its age.
Examples of gravity-sensitive features include the optical and near-infrared
VO bands and alkali lines in late-type M and L dwarfs, all of which evolve
considerably between field dwarfs ($\log{g}$ $\approx$ 5 cgs), young cluster
dwarfs ($\log{g}$ $\approx$ 3–4) and giant stars ($\log{g}$ $\approx$ 0; e.g.,
Luhman 1999). Enhanced VO absorption and weakened alkali line absorption is a
characteristic trait of young brown dwarf spectra (e.g., Gorlova et al. 2003;
Allers et al. 2007). Alkali features in particular are useful for cooler brown
dwarfs as VO condenses out of the photosphere. Quantitative analyses of these
features have been used to distinguish “young” ($\gtrsim$10 Myr) from “very
young” sources thus far ($\lesssim$5 Myr; e.g., Cruz et al. 2007). More robust
metrics await larger and more fully-characterized samples.
Another important surface gravity diagnostic is collision-induced H2
absorption, a smooth opacity source spanning a broad swath of the infrared
(e.g., Linsky 1969; Borysow et al. 1997). H2 absorption is weakened in the
low-pressure atmospheres of young cluster brown dwarfs, resulting in reddened
near-infrared spectral energy distributions and colors; in particular, a
characteristic, triangular-shaped H-band (1.7 $\mu$m) flux peak (e.g., Lucas
et al. 2001; Kirkpatrick et al. 2006). The proximity of many young and
reddened brown dwarfs ($<$ 100 pc) rules out ISM absorption as a primary
source for this reddening. Jameson et al. (2008) have exploited this trend by
using a proper-motion selected sample of nearby young cluster candidate
members to infer an age/color/luminosity relation for brown dwarfs younger
than 0.7 Gyr, with a stated accuracy of $\pm$0.2 dex in log(age), or about 60%
fractional uncertainty. Kinematically older low-mass stars and brown dwarfs in
the Galactic disk (e.g., Faherty et al. 2008) and halo populations (e.g.,
Burgasser et al. 2003) exhibit unusually blue near-infrared colors due to
enhanced H2 absorption. However, differences in metallicities and condensate
cloud properties can muddle surface gravity determinations in these sources by
modulating the photospheric pressure through opacity effects (changing
$\kappa_{R}$; e.g., Leggett et al. 2000; Looper et al. 2008).
The use of H2 absorption as a surface gravity indicator is particularly useful
for T dwarfs, as H2 dominates the $K$-band (2.1 $\mu$m) opacity and
significantly influences near-infrared colors (e.g., Burgasser et al. 2002;
Knapp et al. 2004). Several groups now employ this feature to estimate the
atmospheric properties of individual T dwarfs (e.g., Burgasser et al. 2006a;
Warren et al. 2007; Burningham et al. 2008), typically through the use of
spectral indices that separately sample surface gravity (e.g., the $K$-band)
and temperature variations (e.g., H2O or CH4 bands). These indices are
compared to atmospheric models calibrated by one or more benchmarks (e.g., a
companion to a precisely age-dated star), and evolutionary models are used to
determine individual masses and ages. Typical uncertainties of $\log{g}$
$\sim$0.3 dex translate into 50-100% uncertainties in age, comparable to
uncertainties for main sequence stars. Again, variations in metallicity can
mimic variations in surface gravity, although a third diagnostic such as
luminosity can break this degeneracy (e.g., Burgasser 2007). As atmosphere
models improve in fidelity, parameters are increasingly inferred from direct
fits to spectral data, with comparable uncertainties (e.g., Saumon et al.
2007; Cushing et al. 2008).
## 6 Kinematics
When a sufficiently large enough sample of brown dwarfs is assembled, one can
apply standard kinematic analyses, building from the assumption that
gravitational perturbations lead to increased velocity dispersions with time
(e.g., Spitzer & Schwarzschild 1953; see contribution by B. Nordström).
Velocity dispersions are typically characterized by a time-dependent power-law
form, i.e., $\sigma\propto(1+t/\tau)^{\alpha}$ (e.g., Wielen 1977; Hänninen &
Flynn 2002). Other statistics, such as Galactic scale height, can also be tied
to age through kinematic simulations (e.g., West et al. 2008) to calibrate
secondary age diagnostics such as magnetic activity (see contribution by A.
West).
Samples of very low mass stars and brown dwarfs have only recently become
large enough that kinematic studies are feasible. The largest samples (over
800 sources) have been based on proper motion measurements (e.g., Schmidt et
al. 2007; Casewell et al. 2008; Faherty et al. 2008). For field dwarfs,
dispersion in tangential velocities for both magnitude- and volume-limited
samples indicate a mean age in the range 2–8 Gyr, largely invariant with
spectral type. This is consistent with the ages of more massive field stars
and population synthesis models (e.g., Burgasser 2004). However, when field
samples are broken down by color (Faherty et al., 2008) or presence of
magnetic activity (Schmidt et al., 2007), distinct age groupings are inferred,
indicating that both very old (i.e., thick disk or halo) and very young (i.e.,
thin disk or young association) brown dwarf populations coexist in the
immediate vicinity of the Sun. Indeed, one of the major results from these
studies is the identification of widely-dispersed brown dwarf members of
nearby, young moving groups such as the Hyades (e.g., Bannister & Jameson
2007).
With only two dimensions of motion measured, proper motion samples may produce
biased dispersion measurements depending on the area of sky covered by a
sample. Full 3D velocities require RV measurements which are more expensive
and have thus far been obtained only for a small fraction of the known brown
dwarf population (e.g., Basri & Reiners 2006; Blake et al. 2007). A seminal
study by Zapatero Osorio et al. (2007) of 21 nearby, late-type dwarfs with
parallax, proper motion and RV measurements found considerably smaller 3D
velocity dispersions for L and T dwarfs than GKM stars, suggesting that local
brown dwarfs are young ($t\sim$ 0.5–4 Gyr). The discrepancy between this
result and the proper motion studies may be attributable to small number
statistics and/or contamination by young moving groups; $\sim$40% of the brown
dwarfs in the Zapatero Osorio et al. (2007) sample appear to be associated
with the Hyades. Resolving this discrepancy requires larger RV samples, which
has the side benefit of potentially uncovering RV variables that can be used
as mass standards (e.g., Basri & Reiners 2006).
## 7 Improvements and Future Work
With several methods for age-dating brown dwarfs over a broad range of phase
space now available, opportunities to use these objects as chronometers for
various Galactic studies look to be increasingly promising; e.g., age-dating
planetary systems, examining cluster age spreads, testing Galactic disk
dynamical heating models, and direct measures of the substellar mass function
and birthrate in the field and other populations. However, there are areas
where improvements in uncertainties are needed and basic assumptions tested.
Surface gravity determinations in particular require better constraints, since
these enable age-dating of individual sources. In the short term, improvements
in spectral models can help in this endeavor; however, a sufficiently sampled
grid of benchmark sources may obviate the need for models entirely. Benchmarks
should increasingly arise from mass standards, for which age constraints are
more precise; these additionally provide necessary tests of evolutionary
models upon which most of the age-dating techniques hinge. Improved angular
resolution and sensitivity with $JWST$ and the next generation of large
($>25$m) telescopes will increase resolved binary sample sizes by expanding
the volume in which they can be found and monitored. These facilities will
also aid searches for substellar cluster members in old open field and
globular clusters and, perhaps more importantly, mass standards in these
clusters to facilitate more stringent tests of evolutionary models (T. Dupuy,
priv. comm.). Finally, a larger, more complete sample of brown dwarfs with
precise RV measurements will both improve our statistical constraints on the
age of the local brown dwarf population (and subpopulations) while
additionally aiding in the search for mass standards.
The author thanks T. Dupuy, S. Leggett, M. Liu, & A. West for helpful
comments.
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|
arxiv-papers
| 2008-11-15T17:18:24 |
2024-09-04T02:48:58.806178
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Adam J. Burgasser",
"submitter": "Adam J. Burgasser",
"url": "https://arxiv.org/abs/0811.2508"
}
|
0811.2640
|
# Double charmonium production at B-factories within light cone formalism.
V.V. Braguta braguta@mail.ru Institute for High Energy Physics, Protvino,
Russia
###### Abstract
This paper is devoted to the study of the processes $e^{+}e^{-}\to
J/\Psi\eta_{c},J/\Psi\eta_{c}^{\prime},\psi^{\prime}\eta_{c},\psi^{\prime}\eta_{c}^{\prime}$
within light cone formalism. It is shown that if one disregards the
contribution of higher fock states, the twist-3 distribution amplitudes needed
in the calculation can be unambiguously determined from the twist-2
distribution amplitudes and equations of motion. Using models of the twist-2
distribution amplitudes the cross sections of the processes under study have
been calculated. The results of the calculation are in agreement with Belle
and BaBar experiments. It is also shown that relativistic and radiative
corrections to the cross sections play crucial role in the achievement of the
agreement between the theory and experiments. The comparison of the results of
this paper with the results obtained in other papers has been carried out. In
particular, it is shown that the results of papers where relativistic and
radiative corrections were calculated within NRQCD are overestimated by a
factor of $\sim 1.5$.
###### pacs:
12.38.-t, 12.38.Bx, 13.66.Bc,
## I Introduction
Double charmonium production at B-factories is very interesting problem from
theoretical point of view. At the beginning, the interest to this problem was
caused by large discrepancy between theoretical predictions for the cross
section of the process $e^{+}e^{-}\to J/\Psi\eta_{c}$ Braaten:2002fi ; Liu:1 ;
Liu:2 and it’s first experimental measurement by Belle collaboration
Abe:2002rb . Lately, large discrepancy between theory and experiment was found
by Belle Abe:2004ww and BaBar Aubert:2005tj collaborations for the processes
$e^{+}e^{-}\to
J/\Psi\eta_{c}^{\prime},\psi^{\prime}\eta_{c},\psi^{\prime}\eta_{c}^{\prime},J/\Psi\chi_{c0},\psi^{\prime}\chi_{c0}$.
A number of attempts were made to explain this discrepancy. For instance, the
authors of papers Bodwin:2002fk ; Bodwin:2002kk ; Luchinsky:2003yh studied
the possibility of admixture of $e^{+}e^{-}\to J/\psi J/\psi$ events in the
$e^{+}e^{-}\to J/\Psi\eta_{c}$. Another attempt was to attribute the
discrepancy to large radiative corrections Zhang:2005ch ; Gong:2007db ;
Zhang:2008gp . One more very popular approach to the problem Ma:2004qf ;
Bondar:2004sv ; Braguta:2005gw ; Braguta:2005kr ; Braguta:2006nf ;
Ebert:2006xq ; Choi:2007ze ; Berezhnoy:2007sp ; Ebert:2008kj consisted in
taking into account internal motion of the quark-antiquark pair in charmonium.
Some other interesting points of view on this problem were proposed in papers
Lee:2003db ; Zhang:2008ab ; Berezhnoy:2006mz .
Among many approaches to the resolution of this challenging problem one should
mention nonrelativistic QCD (NRQCD) Bodwin:1994jh . Contrary to many other
models of quarkonium production this approach allows one to improve
systematically the accuracy of the calculation. Thus at the leading order
approximation Braaten:2002fi ; Liu:1 ; Liu:2 there is very large discrepancy
between NRQCD predictions for the cross section of the process $e^{+}e^{-}\to
J/\Psi\eta_{c}$ and the experiments. However, after the inclusion of the one
loop radiative corrections Zhang:2005ch ; Gong:2007db this discrepancy
becomes smaller. At the next step, after taking into account radiative and
relativistic corrections simultaneously He:2007te ; Bodwin:2007ga , the
problem can be resolved. From this example one can draw a conclusion, that
relativistic and radiative corrections play very important role in the
processes with charmonia production. Probably, in a similar way one can
resolve many other puzzles with quarkonia production Bodwin .
It should be noted that within NRQCD the discrepancy between the theory and
experiments was resolved only for the process $e^{+}e^{-}\to J/\Psi\eta_{c}$.
At the same time the discrepancies were observed also for the processes
$e^{+}e^{-}\to
J/\Psi\eta_{c}^{\prime},\psi^{\prime}\eta_{c},\psi^{\prime}\eta_{c}^{\prime},J/\Psi\chi_{c0},\psi^{\prime}\chi_{c0}$.
It is not clear is it possible to apply the same approach for these processes,
since excited charmonia states are rather relativistic and the application of
NRQCD to the production of such states is questionable.
Another systematic approach to the study of hard exclusive processes is light
cone formalism (LC) Lepage:1980fj ; Chernyak:1983ej . Within this approach the
amplitude of hard exclusive process can be separated into two parts. The first
part is partons production at very small distances, which can be treated
within perturbative QCD. The second part is the hadronization of the partons
at larger distances. This part contains information about nonperturbative
dynamic of the strong interactions. For hard exclusive processes it can be
parameterized by process independent distribution amplitudes (DA), which can
be considered as hadrons’ wave functions at light like separation between the
partons in the hadron. It should be noted that within LC one does not assume
that the mesons are nonrelativistic. This approach can equally well be applied
to the production of light and heavy mesons, if the DAs of the produced meson
are known. For this reason, one can hope that within this approach one can
study the production of excited charmonia states.
The first attempts to describe the experimental results obtained at Belle and
BaBar collaborations within LC were done in papers Ma:2004qf ; Bondar:2004sv ;
Braguta:2005kr ; Braguta:2006nf . There are two very important problems common
for all these papers. The first one is that in these papers the
renormalization group evolution of the DAs was disregarded. What is very
important since the evolution of the DAs takes into account very important
part of radiative corrections – the leading logarithmic radiative corrections.
The second problem is poor knowledge of charmonia DAs, which are the key
ingredient of any calculation done within LC.
Recently, the leading twist DAs of the $S$-wave charmonia mesons have become
the objects of intensive study Choi:2007ze ; Bodwin:2006dm ; Ma:2006hc ;
Braguta:2006wr ; Braguta:2007fh ; Braguta:2007tq ; Feldmann:2007id ;
Bell:2008er ; Hwang:2008qi . Knowledge about these DAs allowed one to build
some models for the $S$-wave charmonia DAs, that can be used in practical
calculations. In this paper LC will be applied to the study the processes
$e^{+}e^{-}\to
J/\Psi\eta_{c},J/\Psi\eta_{c}^{\prime},\psi^{\prime}\eta_{c},\psi^{\prime}\eta_{c}^{\prime}$.
It will be shown that with the models of DAs proposed in papers Braguta:2006wr
; Braguta:2007fh ; Braguta:2007tq LC predictions are in agreement with the
results obtained at Belle and BaBar experiments.
This paper is organized as follows. In the next sections a brief description
of LC is given. In section III the formula for the amplitude of the process
$e^{+}e^{-}\to VP$, where $V$ and $P$ are vector and pseudoscalar mesons, is
derived. In section IV this formula is used to calculate the cross sections of
the processes $e^{+}e^{-}\to
J/\Psi\eta_{c},J/\Psi\eta_{c}^{\prime},\psi^{\prime}\eta_{c},\psi^{\prime}\eta_{c}^{\prime}$.
The results of the calculation are discussed in section V. Finally, in the
last section the results of this paper are summarized.
## II Brief description of light cone formalism.
In this section a brief description of light cone formalism (LC) will be
given. As an example, let us consider hard exclusive process with single meson
production. And let us assume that this meson is a pseudoscalar meson $P$. The
presence of high energy scale $E_{h}$, which is of order of the characteristic
energy of the hard exclusive process, allows one to apply factorization
theorem for the amplitude of the process $T$
$\displaystyle T=\sum_{n}C_{n}\times\langle P|O_{n}|0\rangle,$ (1)
where the coefficient $C_{n}$ describes partons production at small distances,
the matrix element $\langle P|O_{n}|0\rangle$ describes hadronization of the
partons which takes place at large distances. The sum is taken over all
possible operators $O_{n}$. For instance, the operators
$\bar{Q}\gamma_{\mu}\gamma_{5}Q,\bar{Q}\gamma_{\mu}\gamma_{5}{\overset{\leftrightarrow}{D}_{\mu_{1}}}{\overset{\leftrightarrow}{D}}_{\mu_{2}}Q,\bar{Q}\sigma_{\mu\nu}\gamma_{5}G_{\alpha\beta}Q$
are few examples of the operator $O_{n}$. Actually, there are infinite number
of the operators $O_{n}$ that contribute to the pseudoscalar meson production.
The cross section of hard exclusive process can be expanded in inverse powers
of the high energy scale $E_{h}$
$\displaystyle\sigma=\frac{\sigma_{0}}{E_{h}^{n}}+\frac{\sigma_{1}}{E_{h}^{n+1}}+...$
(2)
To determine if some operator contributes to a given term in $1/E_{h}$
expansion one uses the concept of the twist of this operator Braun:2003rp .
Thus only the leading twist – the twist-2 operators
$\bar{Q}\hat{z}\gamma_{5}Q,\bar{Q}\hat{z}\gamma_{5}({z\overset{\leftrightarrow}{D}})Q,\bar{Q}\hat{z}\gamma_{5}({z\overset{\leftrightarrow}{D}})^{2}Q,...$111$z$
here is lightlike fourvector $z^{2}=0$. contribute to the leading term in
expansion (2). From this one sees that already at the leading order
approximation infinite number of operators contribute to the cross section.
Nevertheless, it is possible to cope with infinite number of contributions if
one parameterizes all the twist-2 operators by the moments of some function
$\phi(x)$
$\displaystyle\langle
P(q)|\bar{Q}\hat{z}\gamma_{5}({-i\overset{\leftrightarrow}{D}}_{\mu_{1}})...({-i\overset{\leftrightarrow}{D}}_{\mu_{n}})Q|0\rangle\times
z^{\mu_{1}}...z^{\mu_{n}}=if_{P}(qz)^{n+1}\int_{0}^{1}dx\phi(x)(2x-1)^{n},$
(3)
where $q$ is the momentum of the pseudoscalar meson $P$, $f_{P}$ is the
constant which is defined as $\langle
P(q)|\bar{Q}\gamma_{\mu}\gamma_{5}Q|0\rangle=if_{P}q_{\mu}$, $x$ is the
fraction of momentum of the whole meson $P$ carried by quark. The function
$\phi(x)$ is called the leading twist distribution amplitude (DA). One can
think of it as about the Fourier transform of the wave function of the meson
$P$ with lightlike distance between quarks. Using the definition of this
function, factorization theorem (1) can be rewritten as
$\displaystyle T=\int_{0}^{1}dxH(x)\phi(x),$ (4)
where $H(x)$ is the hard part of the amplitude, which describes small distance
effects. This part of the amplitude can be calculated within perturbative QCD.
As it was noted, the leading twist DA parameterizes infinite set of the
twist-2 operators. This part of the amplitude describes hadronization of
quark-antiquark pair at large distances and parameterizes nonperturbative
effects in the amplitude. It should be noted that formula (4) resums the
contributions of all the twist-2 operators. If the meson $P$ is a
nonrelativistic meson, formula (4) resums relativistic corrections to the
amplitude $T$.
Now let us consider radiative corrections to formula (4). The presence of two
different energy scales, which are strongly separated $E_{h}\gg M_{P}$, gives
rise to the appearance of large logarithm $\log{E_{h}^{2}/M_{P}^{2}}$. This
logarithm enhances the role of radiative corrections. The main contribution to
amplitude (4) comes from the leading logarithmic radiative corrections
$\sim(\alpha_{s}\log{E_{h}^{2}/M_{P}^{2}})^{n}$. It turns out that these
corrections can be taken into the account in formula (4) as follows
Chernyak:1983ej ; Lepage:1980fj
$\displaystyle T=\int_{0}^{1}dxH(x,\mu)\phi(x,\mu).$ (5)
To resum the leading logarithmic corrections coming from all loops the scale
$\mu$ should be taken of order of $\sim E_{h}$. The hard part of the amplitude
$H(x,\mu)$ should be calculated at the tree level approximation. At this level
$H(x,\mu)$ depends on the renormalization scale $\mu$ only through the running
of the strong coupling constant $\alpha_{s}(\mu)$. The rest of the leading
logarithms are resummed in the DA $\phi(x,\mu)$ using renormalization group
method (see Appendix B). It should be stressed that formula (5) exactly resums
the leading logarithmic radiative corrections.
Commonly, to study the production of nonrelativistic mesons one uses effective
theory NRQCD Bodwin:1994jh . NRQCD deals with three energy scales $m_{Q}\gg
m_{Q}v\gg m_{Q}v^{2}$, where $m_{Q}$ is the mass of the heavy quark $Q$, $v\ll
1$ is relative velocity of quark antiquark pair. In the process of hard
nonrelativistic meson production there appears one additional energy scale
$E_{h}$ which is much greater than all scales $m_{Q},m_{Q}v,m_{Q}v^{2}$.
Evidently, it is not possible to apply NRQCD at this scale. From the effective
theory perspective, first, this large energy scale must be integrated out. And
this is done through the taking into account renormalization group evolution
of the DA $\phi(x,\mu)$.
This paper is devoted to the study of the process $e^{+}e^{-}\to VP$, where
$V$ and $P$ are vector and pseudoscalar mesons. Essential feature of this
process is that for it the leading order contribution in $1/E_{h}$ expansion
is zero Bondar:2004sv . So, we deal with the next-to-leading twist process.
For this process formula (4) remains valid. The only difference is that now we
have contributions coming from different twist-2 and twist-3 DAs (see next
section). As in the case of the leading twist process, formula (4) resums
relativistic corrections to the amplitude. However, if we consider the leading
logarithmic radiative corrections to the amplitude, formula (5) is incorrect.
One can expect that some leading logarithms are lost. It is only possible to
state that formula (5) resums the leading logarithms which appear in the
amplitude due to the running of the $\alpha_{s}$ and DAs. Below this
approximation will be used.
There is one common feature of all next-to-leading twist process. It is
connected to the following fact: along with two particles twist-3 operators
(for instance $\bar{Q}\gamma_{5}Q$) there appears operators of the type
$\bar{Q}\gamma_{\mu}\gamma_{5}G_{\alpha\beta}Q$. Evidently, these operators
describe higher fock state $|\bar{Q}Qg>$ contribution to the amplitude of the
process. NRQCD predicts that for nonrelativistic mesons such states are
suppressed by higher powers of relative velocity of quark-aniquark pair inside
the meson Bodwin:1994jh . For this reason, in this paper the contribution of
such states will be disregarded.
## III The amplitude of the process: $e^{+}e^{-}\to VP$.
In this section the amplitude of the process $e^{+}e^{-}\to VP$, where
$V=J/\Psi,\psi^{\prime}$ and $P=\eta_{c},\eta_{c}^{\prime}$ will be
considered. Two diagrams that give contribution to the amplitude of this
process are shown in Fig 1. The other two can be obtained from the depicted
ones by the charge conjugation. The amplitude of the process involved can be
written in the following form:
$\displaystyle
M=-4\pi\alpha\frac{\bar{u}(k_{1})\gamma_{\mu}u(k_{2})}{s}\langle
V(p_{1},\lambda)P(p_{2})|J_{\mu}^{em}|0\rangle,$ (6)
where $\alpha$ is the electromagnetic coupling constant,
$\bar{u}(k_{1}),u(k_{2})$ are the electron and positron bispinors, $\sqrt{s}$
is the invariant mass of $e^{+}e^{-}$ system, $J_{\mu}^{em}$ is the
electromagnetic current. The matrix element $\langle
V(p_{1},\lambda)P(p_{2})|J_{\mu}^{em}|0\rangle$ can be parameterized by the
only formfactor $F(s)$:
$\displaystyle\langle
V(p_{1},\lambda)P(p_{2})|J_{\mu}^{em}|0\rangle=iq_{c}~{}F(s)~{}e_{\mu\nu\rho\sigma}\epsilon_{\lambda}^{\nu}p_{1}^{\rho}p_{2}^{\sigma},$
(7)
where $q_{c}$ is the charge of $c$ quark, $\epsilon_{\lambda}^{\nu}$ is the
polarization vector of the meson $V(p_{1},\lambda)$. The cross section of the
process under consideration can be written as follows
$\displaystyle\sigma(e^{+}e^{-}\to\ VP)$ $\displaystyle=$
$\displaystyle\frac{\pi\alpha^{2}q_{c}^{2}}{6}\biggl{(}\frac{2|\mathbf{p}|}{\sqrt{s}}\biggr{)}^{3}|F(s)|^{2}.$
(8)
In the last formula $\mathbf{p}$ is the momentum of the meson $V$ in the
center mass frame of the final mesons.
To calculate the formfactor $F(s)$ LC will be applied. As it was noted in the
previous section, within LC the formfactor is a series in inverse powers of
the characteristic energy of the process $\sqrt{s}$. The leading order
contribution to the formfactor $F(s)$ in the $1/s$ expansion was obtained in
paper Bondar:2004sv . In derivation of this expression the authors of paper
Bondar:2004sv disregarded the mass difference of the final mesons. For the
mesons with different masses the expression for the formfactor $F(s)$, which
was derived in paper Braguta:2005kr , can be written as follows
$\displaystyle|F(s)|$ $\displaystyle=$
$\displaystyle\frac{32\pi}{9}\left|\frac{f_{V}f_{P}M_{P}M_{V}}{q_{0}^{4}}\right|\,I_{0}\,,$
(9) $\displaystyle I_{0}$ $\displaystyle=$
$\displaystyle\int^{1}_{0}dx\int^{1}_{0}dy~{}\alpha_{s}(\mu)\left\\{\frac{M_{P}}{M_{V}^{2}}\frac{Z_{t}Z_{p}V_{T}(x,\mu)P_{P}(y,\mu)}{d(x,y)\,s(x)}-\frac{1}{M_{P}}\frac{\overline{M}_{c}^{2}}{{M_{V}}^{2}}\,\frac{Z_{m}Z_{t}V_{T}(x,\mu)P_{A}(y,\mu)}{d(x,y)\,s(x)}+\right.$
$\displaystyle+$
$\displaystyle\frac{1}{2M_{P}}\frac{V_{L}(x,\mu)\,P_{A}(y,\mu)}{d(x,y)}+\frac{1}{2M_{P}}\frac{(1-2y)}{s(y)}\frac{V_{\perp}(x,\mu)\,P_{A}(y,\mu)}{d(x,y)}+$
$\displaystyle+$
$\displaystyle\left.\frac{1}{8}\biggl{(}1-Z_{t}Z_{m}\frac{4{\overline{M}}_{c}^{2}}{{M_{V}}^{2}}\biggr{)}\frac{1}{M}_{P}\,\frac{(1+y)V_{A}(x,\mu)P_{A}(y,\mu)}{d^{2}(x,y)}\right\\}.$
(11)
Where $q_{0}^{2}\simeq(s-M_{V}^{2}-M_{P}^{2})$, $x$ and $y$ are the fractions
of momenta carried by quark in the meson V and by quark in the meson P
correspondingly, $P_{A},P_{P},V_{T},V_{L},V_{\perp},V_{A}$ are the DAs defined
in Appendix B, $M_{V},M_{P}$ are the masses of the vector and pseudoscalar
mesons correspondingly,
$\overline{M}_{c}=M^{\overline{MS}}_{c}(\mu=M^{\overline{MS}}_{c})$, $Z_{t}$
and $Z_{p}$ are the renormalization factors of the local tensor and
pseudoscalar currents, the dimensionless propagators $d(x,y),s(x),s(y)$ are
defined as follows:
$\displaystyle d(x,y)$ $\displaystyle=$
$\displaystyle\frac{k^{2}}{q_{0}^{2}}=\left(x+\frac{\delta}{y}\right)\left(y+\frac{\delta}{x}\right),~{}s(x)=\frac{k_{x}^{2}}{q_{0}^{2}}-\delta=\left(x+\frac{\delta}{y(1-y)}\right),~{}s(y)=\frac{k_{y}^{2}}{q_{0}^{2}}-\delta=\left(y+\frac{\delta}{x(1-x)}\right),$
(12) $\displaystyle Z_{p}$ $\displaystyle=$
$\displaystyle\left[\frac{\alpha_{s}(\mu)}{\alpha_{s}({\overline{M}}_{c}^{2})}\right]^{\frac{-3c_{F}}{b_{o}}},\quad
Z_{t}=\left[\frac{\alpha_{s}(\mu)}{\alpha_{s}({\overline{M}}_{c}^{2})}\right]^{\frac{c_{F}}{b_{o}}},\quad
Z_{m}=\left[\frac{\alpha_{s}(\mu)}{\alpha_{s}({\overline{M}}_{c}^{2})}\right]^{\frac{3c_{F}}{b_{o}}},\quad\delta=\Biggl{(}Z_{m}\frac{{\overline{M}}_{c}}{q_{0}}\Biggr{)}^{2}\,,$
where $c_{F}=4/3,\,b_{o}=25/3$, the definitions of the fourvectors $k,k_{x}$
and $k_{y}$ are shown in Fig. 1.
Figure 1: The diagrams that contribute to the process $e^{+}e^{-}\to
V(p_{1},\lambda)P(p_{2})$ at the leading order approximation in the strong
coupling constant.
It should be noted here that if one ignores renormalization group running,
formula (9) is the analog of formula (4). So, in this case, it takes into
account infinite series of the relativistic corrections to double charmonium
production. If one takes into account the leading logarithmic corrections due
to the DAs and the running of the strong coupling constant, the scale $\mu$
should be taken of order of the characteristic energy of the process
$\sim\sqrt{s}$.
Before we proceed with the calculation of double charmonium production, the
expression for the formfactor $F(s)$ will be modified as follows:
1\. Formula (9) was obtained at the first nonvanishing approximation in the
$1/s$ expansion. This implies that all parameters in this formula must be
taken at the same level of accuracy. For instance, in the expression
$q_{0}^{2}\simeq(s-M_{V}^{2}-M_{P}^{2})$ last two terms are beyond the
accuracy of calculation and these terms must be omitted. Below the following
approximation will be used $q_{0}^{2}=s$.
2\. Now let us consider expressions (12) for the propagators
$d(x,y),s(x),s(y)$. It is seen that all these expressions contain the terms
proportional to $\sim\delta$. These terms are of NLO approximation, so,
according to the previous item, they must be omitted. Thus the propagators can
be written as
$\displaystyle d(x,y)=xy,\quad s(x)=x,\quad s(y)=y.$ (13)
However, if one substitutes these expressions to (III) and takes into account
the renormalization group evolution of the DAs, the divergence at the end
point region ($x,y\sim 0$) appears. It should be noted that the motion of
quark-antiquark pair in the end point regions $x,y\sim 0,1$ is relativistic.
So, if we consider the production of a nonrelativistic meson without the
evolution of the DAs, the end point regions in the DA are strongly suppressed.
For this reason the expression for the formfactor $F(s)$ is free from the
divergence. However, if the evolution of the DAs is taken into account, the
divergence in the $F(s)$ appears. In this paper this problem will be solved as
follows. According to the definition of $x$
$\displaystyle x=\frac{E^{c}+p^{c}_{z}}{E^{M}+p^{M}_{z}},$ (14)
where $E^{c},p^{c}_{z}$ are the energy and $z$ component of the momenum of
$c$-quark, $E^{M},p^{M}_{z}$ are the energy and $z$ component of the momenum
of meson $M$. If the energy and momenum of $c$-quark is much greater than it’s
mass, one can use the propagators in form (13), since corrections to this
approximation are suppressed. Let us now consider the kinematic region where
$c$-quark is approximately at rest and $x$ can be estimated as
$\displaystyle
x\sim\frac{M^{*}_{c}}{E^{M}+p^{M}_{z}}\simeq\frac{M_{c}^{*}}{\sqrt{s}}=x_{min}.$
(15)
$M_{c}^{*}$ here is the pole mass of $c$ quark. In this region the accuracy of
propagators (13) is not sufficient and one must take into account the
corrections. It is clear that in any approach the corrections to the
propagators regularize the whole expression for the formfactor $F(s)$. This
fact can be seen as follows: for any kinematical region of double quark and
double antiquark production shown in Fig. 1 the squares of momenta of the
gluon and quark propagators cannot be smaller than $(2M_{c}^{*})^{2}$ and
$(M_{c}^{*}+M_{V,P})^{2}$ correspondingly. So, there is no divergence in the
exact expressions for the propagators.
In this paper this effect will be taken into account as follows: the
propagators will be taken in form (13), but the integration in the expression
for the formfactor $F(s)$ will be done in the region $x,y\in(x_{min},~{}1)$.
In other words to get rid of the singularity the cut off parameters $x_{min}$
is introduced.
In principle, the calculation with propagators (12) is also possible. As it
was noted already, in this case the terms proportional to the $\delta$ play
role of the regulator of expression (III). However, one can expect that the
calculation done in this manner is less accurate and the result must be
smaller than it is. To understand this one can apply the idea of duality of
NRQCD and LC descriptions of the hard exclusive nonrelativistic mesons
production: if LC expression for the formfactor $F(s)$ is expanded in relative
velocities of quark antiquark pairs of the mesons $V$ and $P$, one will get
NRQCD result for the formfactor. 222 It should be noted that these is no
strict proof of this statement. However, one can expect that this statement is
indeed true since the amplitude in NRQCD and LC can be expended in series of
equivalent operators. Assuming that both theories can describe experiment one
can expect that these expansions in both theories coincide.
At the leading order approximation of NRQCD this idea can be reproduced if one
takes infinitely narrow approximation for the DAs
($P_{i}(x)=V_{j}(x)\sim\delta(x-1/2),~{}i=A,P,~{}~{}j=T,L,{\perp},A$ ) and
uses the following values of the parameters in formula (9)
$M_{V}=M_{P}=2M_{c}^{*},{\overline{M}}_{c}=M_{c}^{*},\mu=M_{c}^{*},f_{V,P,T}^{2}=\langle
O_{1}\rangle_{S}/M_{c}^{*}$. If this procedure is applied to (9) with
propagators (13), we will exactly reproduce the expression for the formfactor
$F(s)$ obtained within the leading order of NRQCD Braaten:2002fi
$\displaystyle F_{NRQCD}=\frac{2^{9}\pi\alpha_{s}}{9s^{2}}\langle
O_{1}\rangle_{S},$ (16)
where $\langle O_{1}\rangle_{S}$ is the NRQCD matrix element Braaten:2002fi .
Note that at the leading order approximation of NRQCD the formfactor $F(s)$
scales exactly as $1/s^{2}$. So, within NRQCD $1/s^{3}$ terms appear due to
the relativistic corrections, which are suppressed as $v^{2}$ ($v^{2}$ is the
characteristic velocity in charmonium). In LC $1/s^{3}$ terms can appear only
due to the power corrections of the leading order result. So, applying the
idea of duality of NRQCD and LC one can state that the power $1/s$ corrections
to formula (9) with propagators (13) are of order of $\sim
v^{2}(M_{c}^{*})^{2}/s$. If we further apply the same procedure but with
propagators (12), the formfactor $F(s)$ will be different from the leading
order NRQCD prediction
$\displaystyle
F(s)=F_{NRQCD}(1-14\delta+O(\delta^{2}))=F_{NRQCD}\biggl{(}1-14\frac{(M_{c}^{*})^{2}}{s}+O\biggl{(}\frac{1}{s^{2}}\biggr{)}\biggr{)}.$
(17)
To get the agreement with NRQCD prediction for the formfactor, one must expect
rather large power correction to this result ($14\delta\times F_{NRQCD}$) in
LC so that to cancel the second term in the last equation. This correction
appears at next-to-leading order approximation in $1/s$ expansion. So, the
leading order approximation of LC with propagators (12) underestimates real
result. It is not difficult to estimate the size of this effect using formula
(17). For $\sqrt{s}=10.6$ GeV and $m_{c}^{*}=1.4$ GeV the cross section
calculated with propagators (12) is smaller than that with propagators (13) by
$\sim 50\%$. Numerical calculation confirms this estimation.
3\. To calculate the cross section of the processes $e^{+}e^{-}\to
J/\Psi\eta_{c},J/\Psi\eta_{c}^{\prime},\psi^{\prime}\eta_{c},\psi^{\prime}\eta_{c}^{\prime}$
one needs to know the following constants:
$\displaystyle\langle J/\Psi(p,\epsilon)|\bar{C}\gamma_{\alpha}C|0\rangle$
$\displaystyle=$ $\displaystyle
f_{V1}M_{J/\Psi}\epsilon_{\alpha}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\langle\psi^{\prime}(p,\epsilon)|\bar{C}\gamma_{\alpha}C|0\rangle=f_{V2}M_{\psi^{\prime}}\epsilon_{\alpha},$
(18) $\displaystyle\langle
J/\Psi(p,\epsilon)|\bar{C}\sigma_{\alpha\beta}C|0\rangle_{\mu}$
$\displaystyle=$ $\displaystyle
if_{T1}(\mu)(p_{\alpha}\epsilon_{\beta}-p_{\beta}\epsilon_{\alpha})~{}~{}~{}~{}\langle\psi^{\prime}(p,\epsilon)|\bar{C}\sigma_{\alpha\beta}C|0\rangle_{\mu}=if_{T2}(\mu)(p_{\alpha}\epsilon_{\beta}-p_{\beta}\epsilon_{\alpha}),$
$\displaystyle\langle\eta_{c}(p,\epsilon)|\bar{C}\gamma_{\alpha}\gamma_{5}C|0\rangle$
$\displaystyle=$ $\displaystyle
if_{P1}p_{\alpha}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\langle\eta_{c}^{\prime}(p,\epsilon)|\bar{C}\gamma_{\alpha}\gamma_{5}C|0\rangle=if_{P2}p_{\alpha}.$
It should be noted that the operator ${\bar{C}}\sigma_{\alpha\beta}C$ is not
renormalization group invariant. For this reason the constants $f_{T1}$ and
$f_{T2}$ depend on scale as
$\displaystyle
f_{T}(\mu)=\biggl{(}\frac{\alpha_{s}(\mu)}{\alpha_{s}(\mu_{0})}\biggr{)}^{\frac{c_{F}}{b_{0}}}f_{T}(\mu_{0}),~{}~{}~{}i=1,2.$
(19)
In the derivation of formula (9) it was assumed that the tensor and vector
constants $f_{Vi},f_{Ti}(\overline{M}_{c})$ are connected to each other as
$f_{Ti}(\overline{M}_{c})/f_{Vi}=2\,{M}_{c}/M_{Vi},~{}i=1,2$ Bondar:2004sv ;
Braguta:2005kr . At the leading order approximation in relative velocity and
strong coupling constant these relations are correct. However, they are
violated due to radiative and relativistic corrections especially for the
excited mesons. For this reason, below $f_{V},f_{T}$ will be treated as
independent constants.
Introducing the modifications described above, the expression for the
formfactor $F(s)$ can be rewritten as follows
$\displaystyle|F(s)|$ $\displaystyle=$
$\displaystyle\frac{32\pi}{9}\left|\frac{f_{V}f_{P}M_{P}M_{V}}{s^{2}}\right|\,I_{0}\,,$
(20) $\displaystyle
I_{0}=\int^{1}_{x_{min}}dx\int^{1}_{y_{min}}dy~{}\alpha_{s}(\mu)\left\\{\frac{\tilde{f}_{t}}{M_{V}}\frac{M_{P}}{2{\overline{M}}_{c}}\frac{Z_{p}V_{T}(x,\mu)P_{P}(y,\mu)}{x^{2}\,y}-\frac{\tilde{f}_{t}}{M_{P}}\frac{\overline{M}_{c}}{2M_{V}}\,\frac{Z_{m}V_{T}(x,\mu)P_{A}(y,\mu)}{x^{2}\,y}+\right.$
$\displaystyle+$
$\displaystyle\frac{1}{2M_{P}}\frac{V_{L}(x,\mu)\,P_{A}(y,\mu)}{x\,y}+\frac{(1-2y)}{2M_{P}}\frac{V_{\perp}(x,\mu)\,P_{A}(y,\mu)}{x\,y^{2}}+\left.\frac{1}{8}\biggl{(}1-{\tilde{f}_{t}}Z_{m}\frac{2{\overline{M}}_{c}}{{M_{V}}}\biggr{)}\frac{1}{M}_{P}\,\frac{(1+y)V_{A}(x,\mu)P_{A}(y,\mu)}{x^{2}\,y^{2}}\right\\},$
where $\tilde{f}_{t}=f_{T}(\mu)/f_{V}$. Now we are ready to proceed with the
calculation.
## IV Numerical results.
To calculate the cross sections of the processes $e^{+}e^{-}\to
J/\Psi\eta_{c},J/\Psi\eta_{c}^{\prime},\psi^{\prime}\eta_{c},\psi^{\prime}\eta_{c}^{\prime}$
the following values of input parameters will be used:
1. The strong coupling constant $\alpha_{s}(\mu)$ will be taken at the one loop approximation
$\displaystyle\alpha_{s}(\mu)=\frac{4\pi}{\beta_{0}\log(\mu^{2}/\Lambda^{2})},$
(22)
with $\Lambda=0.2$ GeV, $\beta_{0}=25/3$.
2. For the $\overline{MS}$ mass and the pole mass of $c$-quark the values ${\overline{M}}_{c}=1.2$ GeV and $M_{c}^{*}=1.4$ GeV will be used.
3. As it was noted in the previous section, to calculate the cross sections one needs constants (18). The constants $f_{V1}$ and $f_{V2}$ can be determined directly from the experiment. The constants $f_{T1}$ and $f_{T2}$ were calculated within NRQCD in paper Braguta:2007ge . NRQCD formalism can also be used to determine the values of the last two constants $f_{P1}$ and $f_{P2}$ (see Appendix A). So, the calculation will be done with the following values of these constants
$\displaystyle f_{V1}^{2}$ $\displaystyle=$ $\displaystyle 0.173\pm
0.004~{}\mbox{GeV}^{2},\qquad~{}~{}~{}~{}~{}~{}f_{V2}^{2}=0.092\pm
0.002~{}\mbox{GeV}^{2},$ (23) $\displaystyle f_{T1}^{2}(M_{J/\Psi})$
$\displaystyle=$ $\displaystyle 0.144\pm 0.016~{}\mbox{GeV}^{2},\quad
f_{T2}^{2}(M_{J/\Psi})=0.068\pm 0.022~{}\mbox{GeV}^{2},$ $\displaystyle
f_{P1}^{2}$ $\displaystyle=$ $\displaystyle 0.139\pm
0.048~{}\mbox{GeV}^{2},\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}f_{P2}^{2}=0.068\pm
0.040~{}\mbox{GeV}^{2}.$
4. The last inputs to formula (20) are the DAs. The models of the leading twist DAs will be taken from papers Braguta:2006wr ; Braguta:2007fh ; Braguta:2007tq . To build the models for the twist-3 DAs one can apply equations of motion. This procedure is described in detail in Appendix B.
$\begin{array}[]{|c|c|c|c|c|c|c|c|c|}\hline\cr
H_{1}H_{2}&\lx@intercol\hfil\sigma_{Exp}\times Br_{H_{2}\to
charged>2}(\mbox{fb})\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\sigma_{LO~{}NRQCD}(\mbox{fb})\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\sigma_{NRQCD}(\mbox{fb})\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&\sigma_{poten~{}model}(\mbox{fb})&\sigma_{LC}(\mbox{fb})\\\
\hline\cr&\mbox{Belle}\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Abe:2004ww}{\@@citephrase{(}}{\@@citephrase{)}}}}&\mbox{BaBar}\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Aubert:2005tj}{\@@citephrase{(}}{\@@citephrase{)}}}}&\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Braaten:2002fi}{\@@citephrase{(}}{\@@citephrase{)}}}}&~{}~{}\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Liu:1}{\@@citephrase{(}}{\@@citephrase{)}}}}{}{}&\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{He:2007te}{\@@citephrase{(}}{\@@citephrase{)}}}}&\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Bodwin:2007ga}{\@@citephrase{(}}{\@@citephrase{)}}}}&\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Ebert:2008kj}{\@@citephrase{(}}{\@@citephrase{)}}}}&\\\
\hline\cr\psi(1S)\eta_{c}(1S)&25.6\pm 2.8\pm 3.4&17.6\pm
2.8^{+1.5}_{-2.1}&3.78\pm 1.26&5.5&20.4&17.6^{+10.7}_{-8.3}&22.2\pm
1.1&14.4^{+11.2}_{-9.8}\\\ \hline\cr\psi(2S)\eta_{c}(1S)&16.3\pm 4.6\pm
3.9&-&1.57\pm 0.52&3.7&-&-&15.3\pm 0.8&10.4^{+9.2}_{-7.8}\\\
\hline\cr\psi(1S)\eta_{c}(2S)&16.5\pm 3.0\pm 2.4&16.4\pm
3.7^{+2.4}_{-3.0}&1.57\pm 0.52&3.7&-&-&16.4\pm 0.8&13.0^{+12.2}_{-11.0}\\\
\hline\cr\psi(2S)\eta_{c}(2S)&16.0\pm 5.1\pm 3.8&-&0.65\pm 0.22&2.5&-&-&9.6\pm
0.5&9.0^{+9.7}_{-8.5}\\\ \hline\cr\end{array}$ Table 1: The second and third
columns contain experimental results measured at Belle and Babar experiments.
The $Br_{H_{2}\to charged>2}$ means the branching ratio of the decay of the
hadron $H_{2}$ into two charged particles. In the fourth and fifth columns the
results of the leading order NRQCD obtained in papers Liu:1 ; Braaten:2002fi
are shown. The NRQCD results obtained with inclusion of radiative and
relativistic corrections He:2007te ; Bodwin:2007ga are shown in columns six
and seven. Potential model predictions Ebert:2008kj for the cross sections
are presented in column eight. Last column contains the values of the cross
sections obtained in this paper.
There are different sources of uncertainty to the results obtained in this
paper. The most important uncertainties can be divided into the following
groups:
1. The uncertainty in the models of the distribution amplitudes $\phi_{i}(x,\mu)$, which can be modeled by the variation of the parameters of these models (39). The calculation shows that for the processes $e^{+}e^{-}\to J/\Psi\eta_{c},J/\Psi\eta_{c}^{\prime},\psi^{\prime}\eta_{c},\psi^{\prime}\eta_{c}^{\prime}$. these uncertainties are not greater than $\sim 5\%,~{}12\%,~{}28\%,~{}40\%$ correspondingly. It is seen that these uncertainties are not very large. This fact results from the property found in papers Braguta:2006wr : the evolution improves any model of DA.
2. The uncertainty due to the radiative corrections. In the approach applied in this paper the leading logarithmic radiative corrections due to the evolution of the DAs and strong coupling constant were resummed. As it was noted above for the leading twist processes the resummation of the leading logarithms in the DAs and strong coupling constant is equivalent to the resummation of the leading logarithms in the whole amplitude. Unfortunately, this is not valid for the next-to-leading twist processes, for which some of the leading logarithms are lost. For this reason the radiative corrections should be estimated as $\sim\alpha_{s}(\sqrt{s}/2)\log(s/4/(M_{c}^{*})^{2})\sim 50\%$.
3. The uncertainty due to the power corrections. This uncertainty is determined by the next-to-leading order contribution in the $1/s$ expansion. As it was noted above due to the application of the propagators in form (13) one can hope that these corrections are suppressed by the square of relative velocity of quark antiquark pair in the meson. In the calculation these corrections will be estimated as $\sim 4v^{2}_{\psi^{\prime}}M_{\psi^{\prime}}^{2}/s\sim 25\%$.
4. The uncertainty due to the regularization procedure. As it was noted above, to get rid of the divergence in the formfactor $F(s)$ the cut of parameter was introduced. Evidently, our results depend on the value of the cut of parameter $x_{min}\cdot\sqrt{s}$, which is of order of the mass of $c$-quark (see formula (15) ). To estimate this source of uncertainty the cut off parameter is varied in the region $x_{min}\cdot\sqrt{s}=1.0-1.6$ GeV. The calculation shows that at $x_{min}\cdot\sqrt{s}=1.0$ GeV the cross sections are increased by $\sim 50\%$ and at $x_{min}\cdot\sqrt{s}=1.6$ GeV the cross sections are decreased by $\sim 20\%$. It should be noted that this source of uncertainty is closely connected with the uncertainty due to the radiative corrections. However, to understand this in detail one needs the theory which takes into account all the leading logarithmic corrections.
5. The uncertainty in the values of constants (23). It should be noted that this source of uncertainty is very important especially for the production of the excited states $\psi^{\prime},\eta_{c}^{\prime}$. Thus, for the processes $e^{+}e^{-}\to J/\Psi\eta_{c},\psi^{\prime}\eta_{c},J/\Psi\eta_{c}^{\prime},\psi^{\prime}\eta_{c}^{\prime}$ the errors due to the uncertainties in the values of constants (23) are $\sim 35\%,40\%,60\%,65\%$ correspondingly.
Adding all the uncertainties in quadrature one gets the total errors of the
calculations.
The results of the calculation are presented in Table I. The second and third
columns contain experimental results measured at Belle and Babar experiments.
In the fourth and fifth columns the results of the leading order of NRQCD
approach obtained in papers Liu:1 ; Braaten:2002fi are shown. The NRQCD
results obtained with inclusion of radiative and relativistic corrections
He:2007te ; Bodwin:2007ga are shown in columns six and seven. Potential model
predictions Ebert:2008kj for the cross sections are presented in column
eight. Last column contains the values of the cross sections obtained in this
paper.
## V Discussion.
It is seen from Tab. I that within the accuracy of the calculation the results
of this paper are in agreement with Belle and BaBar experiments. It is also
seen that the uncertainty of the calculation is rather large. There are two
very important sources of uncertainty. The first one is the theoretical
problem with taking into account of all leading logarithmic radiative
corrections to the amplitude of the next-to-leading twist processes. It can be
estimated as $\sim 50-70\%$ of the cross sections. This source of uncertainty
can be reduced if the theory, which takes into account all leading logarithmic
corrections, is created. The second very important source of uncertainty is
poor knowledge of constants (18). For some processes this uncertainty can
reach $60\%$. The values of these constants used in this paper can be
considered as the first estimation of their real values. So, theoretical and
experimental study of these constants can greatly improve the accuracy of any
predictions done within LC.
$\begin{array}[]{|c|c|c|c|c|}\hline\cr
H_{1}H_{2}&\sigma_{LO~{}NRQCD}(\mbox{fb})&\sigma_{rel~{}corr}(\mbox{fb})&\sigma_{tot}(\mbox{fb})\\\
\hline\cr\psi(1S)\eta_{c}(1S)&1.9&6.3&14.4\\\
\hline\cr\psi(2S)\eta_{c}(1S)&1.0&6.2&13.0\\\
\hline\cr\psi(1S)\eta_{c}(2S)&1.0&7.8&10.4\\\
\hline\cr\psi(2S)\eta_{c}(2S)&0.53&7.2&9.0\\\ \hline\cr\end{array}$ Table 2:
The second column contains the values of the cross sections obtained at the
leading NRQCD approximation. The third column contains the cross sections
calculated in the following approximation: all relativistic corrections are
resummed, but the leading logarithmic radiative corrections are not taken into
account. The last column represents the results obtained if the relativistic
and leading logarithmic radiative corrections to the amplitude are taken into
account simultaneously.
Next, let us discuss the results obtained in other papers and compare them
with the results of this paper. First, let us consider the results of the
leading order NRQCD predictions Braaten:2002fi ; Liu:1 shown in columns four
and five of Tab. I. These results are approximately by an order of magnitude
smaller than the cross sections measured at the experiments. At the same time
LC predictions are in reasonable agreement with the experiments. This facts
lead to the question: why LC predictions are much greater than the leading
order NRQCD predictions? The answer to this question can be given within LC.
As it was noted in section III, LC can reproduce the leading order NRQCD
results. To do this the renormalization group evolution of the constants and
DAs will be disregarded and all parameters will be taken at the leading NRQCD
approximation: the constants $f_{Ti},f_{Pi}$ are equal to the constant
$f_{Vi},i=1,2$, which can be determined from the leptonic decay width, the
mass $\overline{M}_{c}=M_{c}^{*}=1.4$ GeV, $M_{V}=M_{P}=2M_{c}^{*}$, all DAs
are taken at the infinitely narrow approximation $\sim\delta(x-1/2)$. Thus one
gets the results shown in the second column of Tab. II. At the second step all
parameters used in the calculation are taken at their central values, but
without renormalization group evolution. At this step infinite series of the
relativistic corrections are resummed, but the leading logarithmic radiative
corrections are not taken into account (see section II). The results are shown
in the third column of Tab. II. At the last step, renormalization group
evolution is taken into account and the results are presented in the last
column of Tab. II. Within LC this means that the relativistic and leading
logarithmic radiative corrections are taken into account simultaneously.
From Tab. II one sees that the relativistic and leading logarithmic radiative
corrections taken into account simultaneously dramatically enhance the leading
NRQCD predictions and bring the agreement with Belle and BaBar experiments.
Very important conclusion which can be drawn from this result is that in hard
exclusive processes relativistic and leading logarithmic radiative corrections
play very important role and the consideration of such processes at the
leading NRQCD approximation is unreliable.
Looking to the results of Tab. II one can also draw a conclusion that
relativistic corrections alone cannot describe the experimental results. This
conclusion is in agreement with the results of papers Braaten:2002fi ;
Braguta:2005gw ; Ebert:2006xq ; Berezhnoy:2007sp . In these papers the cross
section of the process $e^{+}e^{-}\to J/\Psi\eta_{c}$ is enhanced due to the
relativistic corrections by a factor of $\sim 2-3$ what is in agreement with
the results obtained in this paper. At the same time this conclusion is in
disagreement with the results of paper Ebert:2008kj (see Tab.I column eight),
where the authors tried to attribute the disagreement between theory and
experiment only to the relativistic corrections, which were calculated within
potential model. From the results shown in Tab.II one sees that this
approximation is realistic only for the production of excited states
$e^{+}e^{-}\to\psi^{\prime}\eta_{c}^{\prime}$. In this case the relativistic
corrections are much more important than the leading logarithmic radiative
corrections.
The authors of papers Bondar:2004sv ; Braguta:2005kr took into account only
the part of the leading logarithmic radiative corrections which appears due to
the evolution of the constant $f_{T}$ and the running mass of $c$-quark. The
evolution of the DAs was disregarded. The calculation done in this paper shows
that this approximation can be applied only to the DAs of the $2S$ charmonia
mesons, for which the evolution is not very important ( see paper
Braguta:2007tq ). As it was shown in papers Braguta:2006wr ; Braguta:2007fh
the evolution of the $1S$ state charmonia DAs is very important. To compensate
the effect of the evolution of the DAs the authors of paper Bondar:2004sv
proposed rather wide model of the DAs with so called relativistic tail333It
should be noted that relativistic tail of the DAs used in this paper is absent
at the scale $\mu\sim M_{c}^{*}$ and appears due to the evolution of the DAs
at larger scales. (see paper Bodwin:2006dm ). For instance, the characteristic
velocity of the $1S$ charmonia (see formula (32)) with this DA can be
estimated as
$\displaystyle\langle v^{2}\rangle_{1S}\sim 3\langle\xi^{2}\rangle_{1S}=0.39,$
(24)
which is much larger than $\langle v^{2}\rangle_{1S}=0.2-0.3$ calculated in
papers Bodwin:2006dn ; Braguta:2006wr ; Braguta:2007fh ; Choi:2007ze ;
Bodwin:2007fz .
At the end of this section let us consider how the disagreement between theory
and experiment can be resolved within NRQCD. The authors of papers He:2007te ;
Bodwin:2007ga took into account the relativistic and one loop radiative
corrections and got the following values of the cross section of the process
$e^{+}e^{-}\to J/\Psi\eta_{c}$
$\displaystyle\sigma(e^{+}e^{-}\to J/\Psi\eta_{c})$ $\displaystyle=$
$\displaystyle
20.4~{}\mbox{fb}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{He:2007te}{\@@citephrase{(}}{\@@citephrase{)}}}},$
$\displaystyle\sigma(e^{+}e^{-}\to J/\Psi\eta_{c})$ $\displaystyle=$
$\displaystyle
17.6^{+10.7}_{-8.3}~{}\mbox{fb}~{}~{}~{}~{}\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Bodwin:2007ga}{\@@citephrase{(}}{\@@citephrase{)}}}}.$
(25)
So, similarly to this paper, the authors of papers He:2007te ; Bodwin:2007ga
resolved the disagreement through the taking into account of the relativistic
and radiative corrections.
Central values (25) are larger than the central values of the cross section
obtained in this paper. Below it will be shown that central values (25) are
overestimated. A simple way to understand why the cross section was
overestimated is to consider paper Gong:2007db which has similar problem. In
this paper explicit expression for the one loop radiative corrections to the
cross section of the process $e^{+}e^{-}\to J/\Psi\eta_{c}$ were calculated.
The result of this paper can be written in the form
$\displaystyle\sigma=\sigma^{0}\times\biggl{(}1+\frac{\alpha_{s}(\mu)}{\pi}K\biggr{)},$
(26)
where $\sigma^{0}$ is the cross section at the leading order approximation,
the expression for the factor $K$ can be found in Gong:2007db . The cross
section $\sigma^{0}$ is proportional to
$|R_{J/\Psi}(0)|^{2}\times|R_{\eta_{c}}(0)|^{2}$, where
$R_{J/\Psi}(r),R_{\eta_{c}}(r)$ are the radial wave functions of the $J/\Psi$
and $\eta_{c}$ mesons. It should be noted here that the cross section $\sigma$
is very sensitive to the values of the wave functions at the origin
$R_{J/\Psi}(0),R_{\eta_{c}}(0)$, so it is very important how these parameters
were calculated. In the calculation the authors of Gong:2007db took
$|R_{J/\Psi}(0)|=|R_{\eta_{c}}(0)|$ and the value of the $|R_{J/\Psi}(0)|$ was
taken from the leptonic decay width $\Gamma_{ee}$
$\displaystyle|R_{J/\Psi}(0)|^{2}=\frac{1}{1-\frac{16}{3}\frac{\alpha_{s}}{\pi}}\frac{M_{J/\Psi}^{2}\Gamma_{ee}}{4\alpha^{2}q_{c}^{2}}.$
(27)
Now few comments are in order. First, this expression is taken at the next-to-
leading order approximation in the strong coupling constant, what means that
some part of the one loop radiative corrections is put to the $\sigma^{0}$.
Second, if formula (27) is expanded in the $\alpha_{s}$, one will get infinite
series in the strong coupling constant. So, the application of formula (27) in
the calculation of the wave function at the origin is equivalent to the
statement that one knows all infinite series of the radiative corrections to
the wave function. Evidently, this is not correct. To be consistent with the
one loop approximation applied in paper Gong:2007db , one should expand
expression (27) in the strong coupling constant and than leave only the first
term in this expansion
$\displaystyle|R_{J/\Psi}(0)|^{2}=\biggl{(}1+\frac{16}{3}\frac{\alpha_{s}}{\pi}\biggr{)}\frac{M_{J/\Psi}^{2}\Gamma_{ee}}{4\alpha^{2}q_{c}^{2}}.$
(28)
It is not difficult to see that the $|R_{J/\Psi}(0)|^{2}$ calculated from
formula (27) is greater than that calculated form formula (28) by a factor
$1/(1-(16\alpha_{s}/3\pi)^{2})$. The cross section calculated using formula
(27) is greater than the cross section calculated using formula (28) by a
factor $1/(1-(16\alpha_{s}/3\pi)^{2})^{2}\sim 1.5$ for $\alpha_{s}=0.25$. So,
the values of the wave functions at the origin calculated in paper Gong:2007db
were overestimated, what led to the overestimation of the cross section by a
factor of $\sim 1.5$.
Similar overestimation of the wave functions at the origin and, as the result,
overestimation of the cross section takes place in papers He:2007te ;
Bodwin:2007ga . For instance, in paper Bodwin:2007ga the authors used the
value of the NRQCD matrix element $\langle O_{1}\rangle_{J/\Psi}$, which is
analog of the wave function at the origin, calculated in paper Bodwin:2007fz
using the formula
$\displaystyle\langle O_{1}\rangle_{J/\Psi}=\frac{1}{(1-f(\langle
v^{2}\rangle_{J/\Psi})-\frac{8}{3}\frac{\alpha_{s}}{\pi})^{2}}\frac{3M_{J/\Psi}^{2}\Gamma_{ee}}{8\pi\alpha^{2}q_{c}^{2}},$
(29)
where $\langle v^{2}\rangle_{J/\Psi}$ is defined in (32),
$f(x)=x/(3(1+x+\sqrt{1+x}))$. Similar expression can be written for the
$\langle O_{1}\rangle_{\eta_{c}}$. It is clear that the application of formula
(29) is equivalent to the statement that one knows not only full $\alpha_{s}$
corrections in front of the leading NRQCD operator $\langle
O_{1}\rangle_{J/\Psi}$ but also full $\alpha_{s}$ corrections in front of all
operators that control relativistic corrections $\langle
v^{n}\rangle_{J/\Psi}$ to the amplitude of the leptonic decay. Evidently, this
is not correct. To make formula (29) more consistent with the approach applied
in paper Bodwin:2007ga , it should be modified as follows
$\displaystyle\langle
O_{1}\rangle_{J/\Psi}=\frac{3M_{J/\Psi}^{2}\Gamma_{ee}}{8\pi\alpha^{2}q_{c}^{2}}\biggl{(}\frac{1}{(1-f(\langle
v^{2}\rangle_{J/\Psi}))^{2}}+\frac{16}{3}\frac{\alpha_{s}}{\pi}\biggr{)}.$
(30)
In last formula all problems mentioned above are resolved. Similarly, one can
improve the expression for the operator $\langle O_{1}\rangle_{\eta_{c}}$.
With numerical values of paper Bodwin:2007fz the factor of the overestimation
is $\sim 1.5$. Taking into account this factor, the cross section $17.6$ fb
obtained in paper Bodwin:2007ga should be changed to $11.7$ fb.
Similar analysis can be done for the result of paper He:2007te . The
calculation shows that the cross section is overestimated by a factor of $\sim
1.5$. Taking into account this factor the central value of the cross sections
now is $\sim 13.6$ fb.
## VI Conclusion.
This paper is devoted to the study of the processes $e^{+}e^{-}\to
J/\Psi\eta_{c},J/\Psi\eta_{c}^{\prime},\psi^{\prime}\eta_{c},\psi^{\prime}\eta_{c}^{\prime}$
within Light Cone Formalism (LC). The amplitude for these processes was first
derived in paper Bondar:2004sv . In the present paper the amplitude was
modified so that to archive better accuracy of the calculation and to resolve
some questions raised in Bodwin:2006dm .
To calculate the cross sections of the processes under study one needs the
twist-2 and twist-3 distribution amplitudes (DA). The models of the twist-2
DAs of the $S$-wave charmonia were proposed in papers Braguta:2006wr ;
Braguta:2007fh ; Braguta:2007tq . To get the twist-3 DAs, equations of motion
were applied. It turns out that if one ignores the contribution arising from
higher fock states, the twist-3 DAs can be unambiguously determined.
Using the models of the DAs, the cross sections of the processes
$e^{+}e^{-}\to
J/\Psi\eta_{c},J/\Psi\eta_{c}^{\prime},\psi^{\prime}\eta_{c},\psi^{\prime}\eta_{c}^{\prime}$
were calculated. Within the error of the calculation the results of this study
are in agreement with Belle and BaBar experiments. In addition, the question -
why LC predictions are much greater than the leading order NRQCD predictions -
was studied. Numerical results of the calculation shows that large
disagreement between LC and the leading NRQCD predictions can be attributed to
large contribution of relativistic and radiative corrections. From this
results one can draw a conclusion that in hard exclusive processes
relativistic and radiative corrections play very important role and the
consideration of such processes at the leading NRQCD approximation is
unreliable.
The results of this paper are in agreement with recent NRQCD study of the
process $e^{+}e^{-}\to J/\Psi\eta_{c}$ He:2007te ; Bodwin:2007ga where the
authors took into account relativistic and one loop radiative corrections.
However, in the present paper it was shown that the results of these papers
are overestimated by a factor 1.5.
###### Acknowledgements.
The author thanks A.K. Likhoded, A.V. Luchinsky for useful discussion. This
work was partially supported by Russian Foundation of Basic Research under
grant 07-02-00417, CRDF grant Y3-P-11-05 and president grant MK-2996.2007.2.
## Appendix A Calculation of the constants $f_{P1},f_{P2}$.
To calculate the values of the constants $f_{Pi},~{}i=1,2$ one can apply NRQCD
formalism. At the NLO approximation of NRQCD the constants $f_{Pi}$ can be
written as follows Bodwin:1994jh ; Braaten:1998au
$\displaystyle f_{Pi}^{2}M_{P_{i}}^{2}$ $\displaystyle=$ $\displaystyle\langle
P_{i}|\varphi^{+}\chi|0\rangle\langle
0|\chi^{+}\varphi|P_{i}\rangle\times\biggl{(}1-4\frac{\alpha_{s}}{\pi}-{\langle
v^{2}\rangle_{i}}\biggr{)},$ (31)
where
$\displaystyle\langle v^{2}\rangle_{i}=-\frac{1}{m_{c}^{2}}\frac{\langle
0|\chi^{+}({\overset{\leftrightarrow}{\bf
D}})^{2}\varphi|P_{i}\rangle}{\langle 0|\chi^{+}\varphi|P_{i}\rangle},$ (32)
$P_{i}$ here is the $\eta_{c}$ meson if $i=1$ and the $\eta_{c}^{\prime}$
meson if $i=2$. To continue the calculation one also needs NRQCD expression
for the decay width $\Gamma\left[\eta_{c}\to\gamma\gamma\right]$:
$\displaystyle\Gamma\left[\eta_{c}\to\gamma\gamma\right]=\frac{4\pi
q_{c}^{2}\alpha^{2}}{M_{\eta_{c}}^{3}}\langle\eta_{c}|\varphi^{+}\chi|0\rangle\langle
0|\chi^{+}\varphi|\eta_{c}\rangle\times\biggl{(}1+\frac{\pi^{2}-20}{3}\frac{\alpha_{s}}{\pi}-\frac{\langle
v^{2}\rangle_{\eta_{c}}}{3}\biggr{)}$ (33)
To determine the constant $f_{P1}$ let us express this constant through the
width $\Gamma\left[\eta_{c}\to\gamma\gamma\right]$444In the derivation of (34)
the expression $M_{\eta_{c}}=2M_{c}^{*}+M_{c}^{*}\langle
v^{2}\rangle_{\eta_{c}}$ was used.
$\displaystyle f_{P1}^{2}=\frac{M_{c}^{*}}{2\pi
q_{c}^{2}\alpha^{2}}\Gamma\left[\eta_{c}\to\gamma\gamma\right]\biggl{(}1-\frac{\pi^{2}-8}{3}\frac{\alpha_{s}}{\pi}-\frac{1}{6}\langle
v^{2}\rangle_{\eta_{c}}\biggr{)},$ (34)
where $M_{c}^{*}$ is the mole mass of $c$-quark. The value of the constant
$f_{P1}$ will be calculated with the following set of parameters:
$\Gamma\left[\eta_{c}\to\gamma\gamma\right]=7.2\pm 0.7\pm 2.0$ Yao:2006px ,
$\alpha_{s}=0.25$, $\langle v^{2}\rangle_{\eta_{c}}=0.25$ Bodwin:2006dn ,
$\langle v^{2}\rangle_{\psi^{\prime}}=0.54$ Braguta:2007tq , $M_{c}^{*}=1.4\pm
0.2$ GeV. To estimate the error of the calculation one should take into
account that within NRQCD the constant is double series in the relativistic
and radiative corrections. At the NNLO approximation one has the relativistic
corrections $\sim\langle v^{2}\rangle^{2}$, the radiative corrections to the
short distance coefficient of the operator $\langle
0|\chi^{+}(\vec{\sigma}\vec{\epsilon})\varphi|V_{i}(\epsilon)\rangle$
$\sim\alpha_{s}^{2}$ and the radiative corrections to the short distance
coefficient of the operator $\langle
0|\chi^{+}(\vec{\sigma}\vec{\epsilon})({\overset{\leftrightarrow}{\bf
D}})^{2}\varphi|V_{i}(\epsilon)\rangle$ that can be estimated as
$\sim\alpha_{s}\langle v^{2}\rangle$. In addition, there is an experimental
uncertainty in the measurement of $\Gamma\left[\eta_{c}\to\gamma\gamma\right]$
and the uncertainty in the $m_{c}$. Adding all these uncertainties in
quadrature one can estimate the error of the calculation. Thus one gets
$\displaystyle f_{P1}^{2}=0.139\pm 0.048~{}~{}\mbox{GeV}^{2}.$ (35)
Unfortunately, one cannot apply formula (34) to get the value of the $f_{P2}$.
Since today only the product
$\Gamma\left[\eta_{c}^{\prime}\to\gamma\gamma\right]\times
Br(\eta_{c}^{\prime}\to K^{0}_{S}K^{\pm}\pi^{0})$ has been measured Yao:2006px
and there is no model independent way to determine
$\Gamma\left[\eta_{c}^{\prime}\to\gamma\gamma\right]$.
To estimate the value the $f_{P2}$ one could use the value of the constants
$f_{V2}$ of $f_{T2}(M_{J/\Psi})$, which are equal up to the relativistic
correction and radiative corrections. In this paper the constant
$f_{T2}(M_{J/\Psi})$ will be taken to get the value of $f_{P2}$
$\displaystyle f^{2}_{P2}=0.068\pm 0.040~{}~{}\mbox{GeV}^{2}.$ (36)
## Appendix B Models for the distribution amplitudes.
In this section models and renormalization group evolution of the DAs needed
in the calculation will be considered. The DAs of the vector meson $V$ are
defined as follows Bondar:2004sv
$\displaystyle\langle
V(p,\epsilon)|\bar{Q}_{\alpha}(z)Q_{\beta}(-z)|0\rangle_{\mu}=\frac{f_{V}M_{V}}{4}\int_{0}^{1}dxe^{i(pz)(2x-1)}\biggl{\\{}\biggl{(}\hat{\epsilon}-\hat{p}\frac{({\epsilon}z)}{(pz)}\biggr{)}V_{\perp}(x,\mu)+\hat{p}\frac{({\epsilon}z)}{(pz)}V_{L}(x,\mu)+$
$\displaystyle\frac{f_{T}(\mu)}{f_{V}}\frac{1}{M_{V}}~{}\sigma_{\mu\nu}{\epsilon}^{\mu}p^{\nu}~{}V_{T}(x,\mu)+\frac{1}{8}\biggl{(}1-\frac{f_{T}(\mu)}{f_{V}}\frac{2M_{c}(\mu)}{{M_{V}}}\biggr{)}~{}e_{\mu\nu\sigma\rho}\gamma^{\mu}\gamma_{5}{\epsilon}^{\nu}p^{\sigma}z^{\rho}~{}V_{A}(x,\mu)\biggr{\\}}_{\beta\alpha}.$
(37)
The DAs of the pseudoscalar meson are defined as
$\displaystyle\langle
P(p)|\bar{Q}_{\alpha}(z)Q_{\beta}(-z)|0\rangle_{\mu}=i\frac{f_{P}M_{V}}{4}\int_{0}^{1}dxe^{i(pz)(2x-1)}\biggl{\\{}\frac{\hat{p}\gamma_{5}}{M_{P}}P_{A}(x,\mu)-\frac{M_{P}}{2M_{c}(\mu)}\gamma_{5}P_{P}(x,\mu)\biggr{\\}}_{\beta\alpha}.$
(38)
where $x$ is the fraction of momentum of the meson $V$ carried by quark, the
constants $f_{T}(\mu),f_{V}$ are defined in equations (18), $M_{c}(\mu)$ is
the running mass of $c$-quark. Expressions (37, 38) are defined at the scale
$\mu$.
The models of the leading twist DAs $V_{L}(x,\mu),V_{T}(x,\mu),P_{A}(x,\mu)$
were proposed in papers Braguta:2006wr ; Braguta:2007fh ; Braguta:2007tq .
According to these models the functions
$P_{A}(x,\mu\sim\overline{M}_{c}),V_{T}(x,\mu\sim\overline{M}_{c}),V_{L}(x,\mu\sim\overline{M}_{c})$
are equal to the function $\phi_{1S}(x,\mu\sim\overline{M}_{c})$ for the $1S$
state mesons and to the function $\phi_{2S}(x,\mu\sim\overline{M}_{c})$ for
the $2S$ state mesons, which have the form Chernyak:1983ej
$\displaystyle\phi_{1S}(x,\mu\sim\overline{M}_{c})$ $\displaystyle\sim$
$\displaystyle\varphi_{as}(x)\mbox{ Exp}\biggl{[}-\frac{b}{4x(1-x)}\biggr{]},$
$\displaystyle\phi_{2S}(x,\mu\sim\overline{M}_{c})$ $\displaystyle\sim$
$\displaystyle\varphi_{as}(x)(a+(2x-1)^{2})\mbox{
Exp}\biggl{[}-\frac{b}{4x(1-x)}\biggr{]},$ (39)
where $\varphi_{as}(x)=6x(1-x)$ is the asymptotic function. For the $1S$
charmonium states the constant $b$ can vary within the interval $3.8\pm 0.7$.
For the $2S$ charmonium states the constants $a$ and $b$ can vary within the
intervals $0.03^{+0.32}_{-0.03}$ and $2.5^{+3.2}_{-0.8}$ correspondingly. The
renormalization group evolution of the leading twist DAs is well known
Chernyak:1983ej and it can be written in the form
$\displaystyle\phi(x,\mu)=6x(1-x)\biggl{[}1+\sum_{n=2,4..}\biggl{(}\frac{\alpha_{s}(\mu)}{\alpha_{s}(\mu_{0})}\biggr{)}^{\epsilon_{n}/{b_{0}}}a_{n}(\mu_{0})C_{n}^{3/2}(2x-1)\biggr{]},$
(40)
where $a_{n}(\mu_{0})$ is the coefficient of the expansion in Gegenbauer
polynomials $C_{n}^{3/2}(z)$ at scale $\mu_{0}$, the constant $b_{0}=25/3$,
the anomalous dimensions $\epsilon_{n}$ for the functions $V_{L},P_{A}$ are
defined as
$\displaystyle\epsilon_{n}$ $\displaystyle=$
$\displaystyle\frac{4}{3}\biggl{(}1-\frac{2}{(n+1)(n+2)}+4\sum_{j=2}^{n+1}\frac{1}{j}\biggr{)},$
(41)
the anomalous dimension $\epsilon_{n}$ for the function $V_{T}$
$\displaystyle\epsilon_{n}$ $\displaystyle=$
$\displaystyle\frac{4}{3}\biggl{(}4\sum_{j=2}^{n+1}\frac{1}{j}\biggr{)}.$ (42)
Further let us consider the twist-3 DA $P_{P}(x,\mu)$. It turns out the if one
ignores higher fock states it is possible to connect the twist-3 DA
$P_{P}(x,\mu)$ to the twist-2 DA $P_{A}(x,\mu)$ using equations of motion
Ball:1998je
$\displaystyle\langle\xi^{n}\rangle_{P}=\delta_{n0}+\frac{n-1}{n+1}\biggl{(}\langle\xi^{n-2}\rangle_{P}-r(\mu)\langle\xi^{n-2}\rangle_{A}\biggr{)},$
(43)
where $\xi=2x-1$, $\langle\xi^{n}\rangle_{P,A}$ are the moments of the DAs
$P_{P}$ and $P_{A}$ correspondingly, $r(\mu)=4M_{c}(\mu)^{2}/M_{\eta}^{2}$. To
solve these equations one can expand the $P_{P}$ in a series of Gegenbauer
polynomials $G^{1/2}_{n}(z)$ Braun:1989iv
$\displaystyle
P_{P}(x,\mu)=\biggl{[}1+\sum_{n=2,4..}b_{n}(\mu)C_{n}^{1/2}(2x-1)\biggr{]}.$
(44)
Substituting expressions for the DAs $P_{P}$ and $P_{A}$ (40) and (44) to
equations of motion (43), one can solve these equations recursively and relate
the coefficients $b_{n}(\mu)$ to the coefficients $a_{n}(\mu)$. For instance,
for the first tree coefficients one can get the following formulas
$\displaystyle
b_{2}(\mu)=-\frac{5}{2}r(\mu),~{}~{}~{}b_{4}(\mu)=-\frac{27}{20}r(\mu)-\frac{81}{10}r~{}a_{2}(\mu),~{}~{}~{}b_{6}(\mu)=-\frac{13}{14}r(\mu)-\frac{39}{7}r(\mu)a_{2}(\mu)-\frac{195}{14}r(\mu)a_{4}(\mu).$
(45)
It is clear that relativistic motion in the nonrelativistic system must be
strongly suppressed. The behavior of DA in the end point region $x\sim 0,1$ is
determined by the relativistic motion. So, the DA of nonrelativistic system
must be strongly suppressed in the end point region. As it was shown in paper
Braguta:2006wr for the leading twist DAs such suppression can be achieved if
there is fine tuning of the coefficients $a_{n}(\mu)$ at the scale
$\mu\sim\overline{M}_{c}$. Similarly, to get the suppression in the end point
region for the function $P_{P}$ one requires the fine tuning in the
coefficients $b_{n}(\overline{M}_{c})$, which can be achieved through the fine
tuning of the constants $r(\overline{M}_{c})$ and $a_{n}(\overline{M}_{c})$.
If we put $r(\overline{M}_{c})=4\overline{M}_{c}^{2}/M_{\eta}^{2}$ the fine
tuning for the higher moments will be broken what leads to large relativistic
motion in the end point region. To avoid this problem it will be assumed that
$r(\overline{M}_{c})$ is a free parameter, which will be adjusted through the
requirement that the moments of the $P_{P}(x,\mu\sim\overline{M}_{c})$ must be
equal to the moments of the $P_{A}(x,\mu\sim\overline{M}_{c})$ to the leading
order in relative velocity expansion. Using the last statement and (43) one
can obtain the expression for the $r(\overline{M}_{c})$
$\displaystyle
r(\overline{M}_{c})=1-3\langle\xi^{2}\rangle_{A}+9(\langle\xi^{2}\rangle_{A})^{2}-5\langle\xi^{4}\rangle_{A}-27(\langle\xi^{2}\rangle_{A})^{3}+30\langle\xi^{2}\rangle_{A}\langle\xi^{4}\rangle_{A}-7\langle\xi^{6}\rangle_{A}+O(v^{8}).$
(46)
All moments in this expression are taken at scale $\mu\sim\overline{M}_{c}$.
The exact expression for the $r(\overline{M}_{c})$ can be written in the
following form
$\displaystyle\frac{2}{r(\overline{M}_{c})}=\int_{-1}^{1}d\xi\frac{1+\xi^{2}}{(1-\xi^{2})^{2}}P_{A}\biggl{(}\frac{1+\xi}{2},\mu\sim\overline{M}_{c}\biggr{)},$
(47)
where $\xi=2x-1$.
The same approach can be applied to the twist-3 DAs $V_{A},V_{\perp}$. These
functions can be expanded in a series of Gegenbauer polynomials Ball:1998sk :
$\displaystyle
V_{\perp}(x,\mu)=\biggl{[}1+\sum_{n=2,4..}c_{n}(\mu)C_{n}^{1/2}(2x-1)\biggr{]},$
$\displaystyle
V_{A}(x,\mu)=6x(1-x)\biggl{[}1+\sum_{n=2,4..}d_{n}(\mu)C_{n}^{3/2}(2x-1)\biggr{]}.$
(48)
The coefficients $c_{n}(\mu)$ and $d_{n}(\mu)$ can be related to the
coefficients $a_{n}(\mu)$ of the functions $V_{L},V_{T}$ through the equations
of motion Ball:1998sk
$\displaystyle(n+1)\langle\xi^{n}\rangle_{\perp}=\langle\xi^{n}\rangle_{L}+\frac{n(n-1)}{2}(1-\delta(\mu))\langle\xi^{n-2}\rangle_{A},$
(49)
$\displaystyle\frac{1}{2}(n+2)(1-\delta(\mu))\langle\xi^{n}\rangle_{A}=\langle\xi^{n}\rangle_{\perp}-\delta(\mu)\langle\xi^{n}\rangle_{T},$
where $\langle\xi^{n}\rangle_{L,T,\perp,A}$ are the moments of the DAs
$V_{L},V_{T},V_{\perp},V_{A}$,
$\delta(\mu)=2f_{T}(\mu)/f_{V}M_{c}(\mu)/M_{V}$. Similarly, to the constant
$r(\overline{M}_{c})$ the value of the constant $\delta(\overline{M}_{c})$
must be adjusted. We adjust it through the requirement that the moments of the
DA $V_{\perp}$ must be equal to the moments of the $V_{L}$ to the leading
order approximation in relative velocity expansion. Thus one can get
$\displaystyle\delta(\overline{M}_{c})=1-\biggl{(}2\langle\xi^{2}\rangle_{L}-6(\langle\xi^{2}\rangle_{L})^{2}+4\langle\xi^{4}\rangle_{L}-22\langle\xi^{2}\rangle_{L}\langle\xi^{4}\rangle_{L}+18(\langle\xi^{2}\rangle_{L})^{3}+6\langle\xi^{6}\rangle_{L}\biggr{)}+O(v^{8}).$
(50)
It should be noted that with this value of the $\delta(\overline{M}_{c})$ the
moments of the function $V_{A}$ are equal to the moments of the $V_{L}$ to the
leading order in relative velocity. The exact expression for the
$\delta(\overline{M}_{c})$ is
$\displaystyle\delta(\overline{M}_{c})=\frac{\int_{-1}^{1}\frac{d\xi}{1-\xi^{2}}V_{L}\biggl{(}\frac{1+\xi}{2},\mu\sim\overline{M}_{c}\biggr{)}}{\int_{-1}^{1}{d\xi}\frac{1+\xi^{2}}{(1-\xi^{2})^{2}}V_{T}\biggl{(}\frac{1+\xi}{2},\mu\sim\overline{M}_{c}\biggr{)}}$
(51)
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|
arxiv-papers
| 2008-11-17T08:42:14 |
2024-09-04T02:48:58.815508
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V.V. Braguta",
"submitter": "V Braguta",
"url": "https://arxiv.org/abs/0811.2640"
}
|
0811.2706
|
# Double-Lepton Polarization Asymmetries in
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ Decay in the Fourth-Generation Standard
Model
S. M. Zebarjad111zebarjad@physics.susc.ac.ir, F. Falahati, H. Mehranfar
Physics Department, Shiraz University, Shiraz 71454, Iran
###### Abstract
In this paper, we investigate the effects of the fourth generation of quarks
on the double-lepton polarization asymmetries in the
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ decay. It is shown that these
asymmetries in $B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ decay compared with
those of $B\rightarrow K\ell^{+}\ell^{-}$ decay are more sensitive to the
fourth-generation parameters. We conclude that an efficient way to establish
the existence of the fourth generation of quarks could be the study of these
asymmetries in the $B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ decay.
PACS numbers: 12.60.-i, 13.30.-a, 14.20.Mr
## I Introduction
Although Standard Model (SM) is a successful theory, there is no clear
theoretical argument within this model to restrict the number of generations
to three, and therefore the possibility of a new generation should not be
ruled out. Based on this possibility, a number of theoretical and experimental
investigations have been performed. The measurement of the $Z$ decay widths
restricts the number of light neutrino for $m_{\nu}<m_{Z}/2$ to threeewwg .
However, if a heavy neutrino exits, the possibility of extra generations of
heavy quarks is not excluded from the experiment. Moreover the electro weak
data okun supports an extra generation of heavy quarks, if the mass
difference between the new up and down-type quarks is not too large.
Many authors who support the existence of fourth-generation studied those
effects in various areas, for instance Higgs and neutrino physics, cosmology
and dark matter Polonsky –Ginzburg . For example, in Ginzburg it is argued
that the fourth generation of quarks and leptons can be generated in the Higgs
boson production at the Tevatron and the LHC, before being actually detected.
By the detailed study of this process at the Tevatron and LHC, the number of
generations in the SM can be determined. Moreover, the flavor democracy
(Democratic Mass Matrix approach) FD1 favors the existence of the nearly
degenerate fourth SM family, while the fifth SM family is disfavored both by
the mass phenomenology and precision tests of the ${\rm SM}$ FD2 . The main
restrictions on the new SM families come from the experimental data on the
$\rho$ and $S$ parameters FD2 . However, the common mass of the fourth quark
($m_{t^{\prime}}$) lies between 320 $GeV$ and 730 $GeV$ considering the
experimental value of $\rho=1.0002^{+0.0007}_{-0.0004}$ PDG . The last value
is close to upper limit on heavy quark masses, $m_{q}\leq 700$ $GeV$ $\approx
4m_{t}$, which follows from partial-wave unitarity at high energies chanowitz
. It should be noted that with preferable value $a\approx g_{w}$ Flavor
Democracy predicts $m_{t^{\prime}}\approx 8m_{w}\approx 640$ $GeV$.
One of the promising areas in the experimental search for the fourth-
generation, via its indirect loop effects, is the rare B meson decays. Based
on this idea, serious attempts to probe the effects of the fourth-generation
on the rare B meson were made by many researchers. The fourth-generation can
affect physical observables, i.e. branching ratio, CP asymmetry, polarization
asymmetries and forward–backward asymmetries. The study of these physical
observables is a good tool to look for the fourth generation of up type quarks
Hou:2006jy –Turan:2005pf .
Recently, the sensitivity of the double-lepton polarization asymmetries to the
fourth-generation in the transition of $B$ to a pseudo scalar meson
($B\rightarrow K\ell^{+}\ell^{-}$) has been investigated and it is found out
that this observable is sensitive to the fourth-generation parameters
($m_{t^{\prime}}$, $V_{t^{\prime}b}V^{*}_{t^{\prime}s}$)Bashiry:2007tf . In
this work, we investigate the effects of the fourth generation of quarks
$(b^{\prime},t^{\prime})$ on the double-lepton polarizations in the transition
of $B$ to a vector meson ( $B_{s}\rightarrow\phi\ell^{+}\ell^{-}$) and compare
our results with those of $B\rightarrow K\ell^{+}\ell^{-}$ decay presented in
Ref.Bashiry:2007tf . It should be mentioned that both decays occur through
$b\rightarrow s$ transition in which the sequential fourth generation of up
quarks $(t^{\prime})$, like $u,c,t$ quarks, contributes at the loop level.
Hence, this new generation will change only the values of the Wilson
coefficients via virtual exchange of the fourth-generation up quark
$t^{\prime}$ and the full operator set is exactly the same as in SM.
The paper is organized as follows. In Section II, the expressions for the
matrix element and double-lepton polarizations of
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ in the SM have been presented. The
effect of the fourth generation of quarks on the effective Hamiltonian and the
double-lepton polarization asymmetries have been discussed in Section III. The
sensitivity of these polarizations to the fourth-generation parameters
$(m_{t^{\prime}},r_{sb},\phi_{sb})$ have been numerically analyzed in the
final Section.
## II The Matrix Element and Double-Lepton Polarizations of
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ in the SM
In the SM, the relevant effective Hamiltonian for
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ decay which is described by
$b\rightarrow s\ell^{+}\ell^{-}$ transition at quark level can be written as
${\cal H}_{\rm
eff}=-\frac{G_{F}}{\sqrt{2}}V_{tb}V^{*}_{ts}\sum_{i=1}^{10}C_{i}(\mu){\cal
O}_{i}(\mu)\,,$ (1)
where the complete set of the operators ${\cal O}_{i}(\mu)$ and the
corresponding expressions for the Wilson coefficients $C_{i}(\mu)$ are given
in R23 . Using the above effective Hamiltonian, the one-loop matrix elements
of $b\rightarrow s\ell^{+}\ell^{-}$ can be written in terms of the tree-level
matrix elements of the effective operators as:
$\displaystyle{\cal M}(b\rightarrow s\ell^{+}\ell^{-})$ $\displaystyle=$
$\displaystyle<s\ell^{+}\ell^{-}|{\cal H}_{\rm eff}|b>$ (2) $\displaystyle=$
$\displaystyle-\frac{G_{F}}{\sqrt{2}}V_{tb}V^{*}_{ts}\sum_{i}C^{\rm
eff}_{i}(\mu)<s\ell^{+}\ell^{-}|{\cal O}_{i}|b>^{tree}.$ $\displaystyle=$
$\displaystyle-\frac{G_{F}\alpha}{2\pi\sqrt{2}}V_{tb}V^{*}_{ts}\Bigg{[}\tilde{C}_{9}^{\rm
eff}\bar{s}\gamma_{\mu}(1-\gamma_{5})b~{}\bar{\ell}\gamma_{\mu}\ell+\tilde{C}_{10}^{\rm
eff}\bar{s}\gamma_{\mu}(1-\gamma_{5})b~{}\bar{\ell}\gamma_{\mu}\gamma_{5}\ell$
$\displaystyle\,\,\,\,\,\,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2C_{7}^{\rm
eff}\frac{m_{b}}{q^{2}}\bar{s}\sigma_{\mu\nu}q^{\nu}(1+\gamma_{5})b\,\bar{\ell}\gamma_{\mu}\ell\Bigg{]},$
where $q^{2}=(p_{1}+p_{2})^{2}$ and $p_{1}$ and $p_{2}$ are the final leptons
four–momenta and the effective Wilson coefficients at $\mu$ scale, are given
as R23 ; R24 :
$\displaystyle C_{7}^{\rm eff}$ $\displaystyle=$ $\displaystyle
C_{7}-\frac{1}{3}C_{5}-C_{6}$ $\displaystyle C_{10}^{\rm eff}$
$\displaystyle=$ $\displaystyle\frac{\alpha}{2\pi}\tilde{C}^{\rm
eff}_{10}=C_{10}$ $\displaystyle C_{9}^{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2\pi}\tilde{C}^{\rm
eff}_{9}=C_{9}+\frac{\alpha}{2\pi}Y(s).$ (3)
In Eq.(II), $s=q^{2}/m_{b}^{2}$ and the function $Y(s)$ contains the short-
distance contributions due to the one-loop matrix element of the four quark
operators, $Y_{per}(s)$, as well as the long-distance contributions coming
from the real $c\bar{c}$ intermediate states, i.e., $J/\psi$, $\psi^{\prime}$,
$\cdots$.The latter contributions are taken into account by introducing
Breit–Wigner form of the resonance propagator which leads to the second term
in the following formula (see Eq.II) R26 –R28 . As a result the function
$Y(s)$ can be written as:
$\displaystyle Y(s)$ $\displaystyle=$ $\displaystyle
Y_{per}(s)+\frac{3\pi}{\alpha^{2}}(3C_{1}+C_{2}+3C_{3}+C_{4}+3C_{5}+C_{6})$
$\displaystyle\times$
$\displaystyle\sum_{V_{i}=\psi_{i}}\kappa_{i}\frac{m_{V_{i}}\Gamma(V_{i}\rightarrow\ell^{+}\ell^{-})}{m_{V_{i}}^{2}-sm_{b}^{2}-im_{V_{i}}\Gamma_{V_{i}}},$
where
$\displaystyle Y_{per}(s)$ $\displaystyle=$ $\displaystyle
g(\frac{m_{c}}{m_{b}},s)(3C_{1}+C_{2}+3C_{3}+C_{4}+3C_{5}+C_{6})$ (5)
$\displaystyle-$ $\displaystyle\frac{1}{2}g(1,s)(4C_{3}+4C_{4}+3C_{5}+C_{6})$
$\displaystyle-$
$\displaystyle\frac{1}{2}g(0,s)(C_{3}+3C_{4})+\frac{2}{9}(3C_{3}+C_{4}+3C_{5}+C_{6}).$
The explicit expressions for the $g$ functions can be found in R23 and the
phenomenological parameters $\kappa_{i}$ in Eq.(II) can be determined from
${\cal B}(B\rightarrow K^{\ast}V_{i}\rightarrow
K^{\ast}\ell^{+}\ell^{-})={\cal B}(B\rightarrow K^{\ast}V_{i})\,{\cal
B}(V_{i}\rightarrow\ell^{+}\ell^{-}),$ (6)
where the data for the right hand side is given in R29 . For the lowest
resonances, $J/\psi$ and $\psi^{\prime}$ one can use $\kappa=1.65$ and
$\kappa=2.36$, respectively (see R30 ). In this study, we neglect the long-
distance contributions for simplicity and like Ref.R23 , to have a scheme
independent matrix element, we use the leading order as well as the next-to-
leading order QCD corrections to $C_{9}$ and the leading order QCD corrections
to the other Wilson coefficients.
In order to compute the decay width and other physical observables of
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ decay, we need to sandwich the matrix
elements in Eq.(2) between the final and initial meson states. Therefore, the
hadronic matrix elements for the $B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ can be
parameterized in terms of form factors. For the vector meson $\phi$ with
polarization vector $\varepsilon_{\mu}$ the semileptonic form factors of the
V–A current is defined as:
$\displaystyle<\phi(p_{\phi},\epsilon)\mid\bar{s}\gamma_{\mu}(1-\gamma_{5})b\mid
B(p_{B_{s}})>=-\frac{2V(q^{2})}{m_{B_{s}}+m_{\phi}}\epsilon_{\mu\nu\rho\sigma}p_{\phi}^{\rho}q^{\sigma}\epsilon^{*\nu}$
$\displaystyle-i\left[\epsilon_{\mu}^{*}(m_{B_{s}}+m_{\phi})A_{1}(q^{2})-(\epsilon^{*}q)(p_{B_{s}}+p_{\phi})_{\mu}\frac{A_{2}(q^{2})}{m_{B_{s}}+m_{\phi}}\right.$
$\displaystyle-\left.q_{\mu}(\epsilon^{*}q)\frac{2m_{\phi}}{q^{2}}(A_{3}(q^{2})-A_{0}(q^{2}))\right],$
(7)
where $q=p_{B_{s}}-p_{\phi}$, and $A_{3}(q^{2}=0)=A_{0}(q^{2}=0)$ (this
condition ensures that there is no kinematical singularity in the matrix
element at $q^{2}=0$). Also, the form factor $A_{3}(q^{2})$ can be written as
a linear combination of the form factors $A_{1}$ and $A_{2}$ :
$\displaystyle
A_{3}(q^{2})=\frac{1}{2m_{\phi}}\left[(m_{B_{s}}+m_{\phi})A_{1}(q^{2})-(m_{B_{s}}-m_{\phi})A_{2}(q^{2})\right].$
(8)
The other semileptonic form factors coming from the dipole operator
$\sigma_{\mu\nu}q^{\nu}(1+\gamma_{5})b$ can be defined as:
$\displaystyle\left<\phi(p_{\phi},\varepsilon)\left|\bar{s}i\sigma_{\mu\nu}q^{\nu}(1+\gamma_{5})b\right|B(p_{B_{s}})\right>=$
(9) $\displaystyle
4\epsilon_{\mu\nu\rho\sigma}\varepsilon^{\ast\nu}p^{\rho}q^{\sigma}T_{1}(q^{2})+2i\left[\varepsilon_{\mu}^{\ast}(m_{B_{s}}^{2}-m_{\phi}^{2})-(p_{B_{s}}+p_{\phi})_{\mu}(\varepsilon^{\ast}q)\right]T_{2}(q^{2})$
$\displaystyle+2i(\varepsilon^{\ast}q)\left[q_{\mu}-(p_{B_{s}}+p_{\phi})_{\mu}\frac{q^{2}}{m_{B_{s}}^{2}-m_{\phi}^{2}}\right]T_{3}(q^{2})~{}.$
As seen From Eqs. (II-9), we have to compute the form factors to obtain the
physical observables at hadronic level.The form factors are related to the
non-perturbative sector of QCD and can be evaluated only by using non-
perturbative methods. In the present work, we use light cone QCD sum rule
predictions for the form factors in which one-loop radiative corrections to
twist-2 and twist-3 contributions are taken into account. The form factors
$\displaystyle
F(q^{2})\in\\{V(q^{2}),A_{0}(q^{2}),A_{1}(q^{2}),A_{2}(q^{2}),A_{3}(q^{2}),T_{1}(q^{2}),T_{2}(q^{2}),T_{3}(q^{2})\\}~{},$
are fitted to the the following functions R31 ; R32 :
$F(q^{2})=\frac{F(0)}{1-a_{F}\frac{q^{2}}{m_{B_{s}}^{2}}+b_{F}(\frac{q^{2}}{m_{B_{s}}^{2}})^{2}},$
(10)
where the parameters $F(0)$, $a_{F}$ and $b_{F}$ are listed in the Table1.
$\begin{array}[]{|l|ccc|}\hline\cr&F(0)&a_{F}&b_{F}\\\ \hline\cr
A_{0}^{B_{s}\rightarrow\phi}&\phantom{-}0.382&1.77&\phantom{-}0.856\\\
A_{1}^{B_{s}\rightarrow\phi}&\phantom{-}0.296&0.87&-0.061\\\
A_{2}^{B_{s}\rightarrow\phi}&\phantom{-}0.255&1.55&\phantom{-}0.513\\\
V^{B_{s}\rightarrow\phi}&\phantom{-}0.433&1.75&\phantom{-}0.736\\\
T_{1}^{B_{s}\rightarrow\phi}&\phantom{-}0.174&1.82&\phantom{-}0.825\\\
T_{2}^{B_{s}\rightarrow\phi}&\phantom{-}0.174&0.70&-0.315\\\
T_{3}^{B_{s}\rightarrow\phi}&\phantom{-}0.125&1.52&\phantom{-}0.377\\\
\hline\cr\end{array}$ Table 1: The form factors for
$B_{s}\rightarrow\phi\,\ell^{+}\ell^{-}$ in a three–parameter fit R31 .
Using Eqs.(II-9), the matrix element of the
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ decay can be written as follows:
$\displaystyle{\cal
M}(B_{s}\rightarrow\phi\ell^{+}\ell^{-})=\frac{G\alpha}{4\sqrt{2}\pi}V_{tb}V_{ts}^{\ast}$
$\displaystyle\times\Bigg{\\{}\bar{\ell}\gamma^{\mu}(1-\gamma_{5})\ell\,\Big{[}-2B_{0}\epsilon_{\mu\nu\lambda\sigma}\varepsilon^{\ast\nu}p_{\phi}^{\lambda}q^{\sigma}-iB_{1}\varepsilon_{\mu}^{\ast}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+iB_{2}(\varepsilon^{\ast}q)(p_{B_{s}}+p_{\phi})_{\mu}+iB_{3}(\varepsilon^{\ast}q)q_{\mu}\Big{]}$
$\displaystyle~{}~{}~{}~{}+\bar{\ell}\gamma^{\mu}(1+\gamma_{5})\ell\,\Big{[}-2C_{1}\epsilon_{\mu\nu\lambda\sigma}\varepsilon^{\ast\nu}p_{\phi}^{\lambda}q^{\sigma}-iD_{1}\varepsilon_{\mu}^{\ast}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+iD_{2}(\varepsilon^{\ast}q)(p_{B_{s}}+p_{\phi})_{\mu}+iD_{3}(\varepsilon^{\ast}q)q_{\mu}\Big{]}\Bigg{\\}}~{},$
where
$\displaystyle B_{0}$ $\displaystyle=$ $\displaystyle(\tilde{C}^{\rm
eff}_{9}-\tilde{C}^{\rm
eff}_{10})\frac{V}{m_{B_{s}}+m_{\phi}}+4(m_{B_{s}}+m_{s}){C}^{\rm
eff}_{7}\frac{T_{1}}{q^{2}}~{},$ $\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle(\tilde{C}^{\rm eff}_{9}-\tilde{C}^{\rm
eff}_{10})(m_{B_{s}}+m_{\phi})A_{1}+4(m_{B_{s}}-m_{s}){C}^{\rm
eff}_{7}(m_{B_{s}}^{2}-m_{\phi}^{2})\frac{T_{2}}{q^{2}}~{},$ $\displaystyle
B_{2}$ $\displaystyle=$ $\displaystyle\frac{\tilde{C}^{\rm
eff}_{9}-\tilde{C}^{\rm
eff}_{10}}{m_{B_{s}}+m_{\phi}}A_{2}+4(m_{B_{s}}-m_{s}){C}^{\rm
eff}_{7}\frac{1}{q^{2}}\left[T_{2}+\frac{q^{2}}{m_{B_{s}}^{2}-m_{\phi}^{2}}T_{3}\right]~{},$
$\displaystyle B_{3}$ $\displaystyle=$ $\displaystyle 2(\tilde{C}^{\rm
eff}_{9}-\tilde{C}^{\rm
eff}_{10})m_{\phi}\frac{A_{3}-A_{0}}{q^{2}}-4(m_{B_{s}}-m_{s}){C}^{\rm
eff}_{7}\frac{T_{3}}{q^{2}}~{},$ $\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle B_{0}(\tilde{C}^{\rm eff}_{10}\rightarrow-\tilde{C}^{\rm
eff}_{10})~{},$ $\displaystyle D_{i}$ $\displaystyle=$ $\displaystyle
B_{i}(\tilde{C}^{\rm eff}_{10}\rightarrow-\tilde{C}^{\rm
eff}_{10})~{},~{}~{}~{}~{}(i=1,~{}2,~{}3).$
From the above equations for the differential decay width, we get the
following result:
$\displaystyle\frac{d\Gamma}{d\hat{s}}(B_{s}\rightarrow\phi\ell^{+}\ell^{-})=\frac{G^{2}\alpha^{2}m_{B_{s}}}{2^{14}\pi^{5}}\left|V_{tb}V_{ts}^{\ast}\right|^{2}\lambda^{1/2}(1,\hat{r},\hat{s})v\Delta(\hat{s})~{},$
(12)
with
$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\frac{2}{3\hat{r}_{\phi}\hat{s}}m_{B_{s}}^{2}Re[-12m_{B_{s}}^{2}\hat{m_{l}}^{2}\lambda\hat{s}\\{(B_{3}-D_{2}-D_{3})B_{1}^{*}-(B_{3}+B_{2}-D_{3})D_{1}^{*}\\}$
$\displaystyle+12m_{B_{s}}^{4}\hat{m_{l}}^{2}\lambda\hat{s}(1-\hat{r}_{\phi})(B_{2}-D_{2})(B_{3}^{*}-D_{3}^{*})$
$\displaystyle+48\hat{m_{l}}^{2}\hat{r}_{\phi}\hat{s}(3B_{1}D_{1}^{*}+2m_{B_{s}}^{4}\lambda
B_{0}C_{1}^{*})$
$\displaystyle-16m_{B_{s}}^{4}\hat{r}_{\phi}\hat{s}\lambda(\hat{m_{l}}^{2}-\hat{s})\\{|B_{0}|^{2}+|C_{1}|^{2}\\}$
$\displaystyle-6m_{B_{s}}^{4}\hat{m_{l}}^{2}\lambda\hat{s}\\{2(2+2\hat{r}_{\phi}-\hat{s})B_{2}D_{2}^{*}-\hat{s}|(B_{3}-D_{3})|^{2}\\}$
$\displaystyle-4m_{B_{s}}^{2}\lambda\\{\hat{m_{l}}^{2}(2-2\hat{r}_{\phi}+\hat{s})+\hat{s}(1-\hat{r}_{\phi}-\hat{s})\\}(B_{1}B_{2}^{*}+D_{1}D_{2}^{*})$
$\displaystyle+\hat{s}\\{6\hat{r}_{\phi}\hat{s}(3+v^{2})+\lambda(3-v^{2})\\}\\{|B_{1}|^{2}+|D_{1}|^{2}\\}$
$\displaystyle-2m_{B_{s}}^{4}\lambda\\{\hat{m_{l}}^{2}[\lambda-3(1-\hat{r}_{\phi})^{2}]-\lambda\hat{s}\\}\\{|B_{2}|^{2}+|D_{2}|^{2}\\}],$
where $\hat{s}=q^{2}/m_{B_{s}}^{2}$,
$\hat{r}_{\phi}=m_{\phi}^{2}/m_{B_{s}}^{2}$ and
$\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2ac-2bc$,
$\hat{m}_{\ell}=m_{\ell}/m_{B_{s}}$, $v=\sqrt{1-4\hat{m}_{\ell}^{2}/\hat{s}}$
is the final lepton velocity.
Having obtained the matrix element for the
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$, we can now calculate the
double–polarization asymmetries. For this purpose, we define the orthogonal
unit vectors $s^{\pm\mu}_{i}$ in the rest frame of leptons, where i=L,N or T
refer to the longitudinal, normal and transversal polarization directions,
respectively:
$\displaystyle s^{-\mu}_{L}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\left(0,\vec{e}_{L}^{\,-}\right)=\left(0,\frac{\vec{p}_{-}}{\left|\vec{p}_{-}\right|}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}s^{+\mu}_{L}=\left(0,\vec{e}_{L}^{\,+}\right)=\left(0,\frac{\vec{p}_{+}}{\left|\vec{p}_{+}\right|}\right),$
$\displaystyle s^{-\mu}_{N}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\left(0,\vec{e}_{N}^{\,-}\right)=\left(0,\frac{\vec{p}_{\phi}\times\vec{p}_{-}}{\left|\vec{p}_{\phi}\times\vec{p}_{-}\right|}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}s^{+\mu}_{N}=\left(0,\vec{e}_{N}^{\,+}\right)=\left(0,\frac{\vec{p}_{\phi}\times\vec{p}_{+}}{\left|\vec{p}_{\phi}\times\vec{p}_{+}\right|}\right),$
$\displaystyle s^{-\mu}_{T}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\left(0,\vec{e}_{T}^{\,-}\right)=\left(0,\vec{e}_{N}^{\,-}\times\vec{e}_{L}^{\,-}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}s^{+\mu}_{T}=\left(0,\vec{e}_{T}^{\,+}\right)=\left(0,\vec{e}_{N}^{\,+}\times\vec{e}_{L}^{\,+}\right).$
In the above equations $\vec{p}_{\mp}$ and $\vec{p}_{\phi}$ are the
three–momenta of the leptons $\ell^{\mp}$ and $\phi$ meson, respectively. Then
by Lorentz transformation these unit vectors are boosted from the rest frame
of leptons to the center of mass (CM) frame of leptons. Under this
transformation only the longitudinal unit vectors $s^{\pm\mu}_{L}$ change, but
the other two vectors remain unchanged. $s^{\pm\mu}_{L}$ in the CM frame of
leptons are obtained as:
$\displaystyle\left(s^{-\mu}_{L}\right)_{CM}=\left(\frac{\left|\vec{p}_{-}\right|}{m_{\ell}},\frac{E\vec{p}_{-}}{m_{\ell}\left|\vec{p}_{-}\right|}\right),~{}~{}~{}~{}~{}~{}~{}~{}\left(s^{+\mu}_{L}\right)_{CM}=\left(\frac{\left|\vec{p}_{-}\right|}{m_{\ell}},-\frac{E\vec{p}_{-}}{m_{\ell}\left|\vec{p}_{-}\right|}\right).$
(14)
The polarization asymmetries can now be calculated using the spin projector
${1\over 2}(1+\gamma_{5}\\!\\!\not\\!\\!s_{i}^{-})$ for $\ell^{-}$ and the
spin projector ${1\over 2}(1+\gamma_{5}\\!\not\\!\\!s_{i}^{+})$ for
$\ell^{+}$.
Considering the above explanations, we can define the double–lepton
polarization asymmetries as in Fukae :
$\displaystyle
P_{ij}(\hat{s})=\frac{\displaystyle{\Bigg{(}\frac{d\Gamma}{d\hat{s}}(\vec{s}_{i}^{-},\vec{s}_{j}^{+})}-\displaystyle{\frac{d\Gamma}{d\hat{s}}(-\vec{s}_{i}^{-},\vec{s}_{j}^{+})\Bigg{)}}-\displaystyle{\Bigg{(}\frac{d\Gamma}{d\hat{s}}(\vec{s}_{i}^{-},-\vec{s}_{j}^{+})}-\displaystyle{\frac{d\Gamma}{d\hat{s}}(-\vec{s}_{i}^{-},-\vec{s}_{j}^{+})\Bigg{)}}}{\displaystyle{\Bigg{(}\frac{d\Gamma}{d\hat{s}}(\vec{s}_{i}^{-},\vec{s}_{j}^{+})}+\displaystyle{\frac{d\Gamma}{d\hat{s}}(-\vec{s}_{i}^{-},\vec{s}_{j}^{+})\Bigg{)}}+\displaystyle{\Bigg{(}\frac{d\Gamma}{d\hat{s}}(\vec{s}_{i}^{-},-\vec{s}_{j}^{+})}+\displaystyle{\frac{d\Gamma}{d\hat{s}}(-\vec{s}_{i}^{-},-\vec{s}_{j}^{+})\Bigg{)}}}~{},$
(15)
where $i,j=L,~{}N,~{}T$, and the first index $i$ corresponds to lepton while
the second index $j$ corresponds to antilepton, respectively. After doing the
straightforward calculation we obtain the following expressions for
$P_{ij}(\hat{s})$:
$\displaystyle P_{LL}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{m_{B_{s}}^{2}}{3\hat{r}_{\phi}\hat{s}\Delta}\,\mbox{\rm
Re}\bigg{\\{}-24m_{B_{s}}^{2}\hat{m}_{\ell}^{2}\hat{s}\lambda\Big{[}(B_{1}-D_{1})(B_{3}^{\ast}-D_{3}^{\ast})\Big{]}$
(16) $\displaystyle+$
$\displaystyle\\!\\!\\!12m_{B_{s}}^{3}\hat{m}_{\ell}\hat{s}\lambda(1-\hat{r}_{\phi})\Big{[}2m_{B_{s}}\hat{m}_{\ell}(B_{2}-D_{2})(B_{3}^{\ast}-D_{3}^{\ast})\Big{]}$
$\displaystyle-$
$\displaystyle\\!\\!\\!8m_{B_{s}}^{4}\hat{r}_{\phi}\hat{s}^{2}\lambda(1+3v^{2})(\left|B_{0}\right|^{2}+\left|C_{1}\right|^{2})+12m_{B_{s}}^{4}\hat{m}_{\ell}^{2}\hat{s}^{2}\lambda\left|B_{3}-D_{3}\right|^{2}$
$\displaystyle+$
$\displaystyle\\!\\!\\!8m_{B_{s}}^{2}\hat{m}_{\ell}^{2}\lambda(4-4\hat{r}_{\phi}-\hat{s})(B_{1}D_{2}^{\ast}+B_{2}D_{1}^{\ast})-32\hat{m}_{\ell}^{2}(\lambda+3\hat{r}_{\phi}\hat{s})B_{1}D_{1}^{\ast}$
$\displaystyle-$
$\displaystyle\\!\\!\\!8m_{B_{s}}^{4}\hat{m}_{\ell}^{2}\lambda[\lambda+3(1-\hat{r}_{\phi})^{2}]B_{2}D_{2}^{\ast}-64m_{B_{s}}^{4}\hat{m}_{\ell}^{2}\hat{r}_{\phi}\hat{s}\lambda
B_{0}C_{1}^{\ast}$ $\displaystyle+$
$\displaystyle\\!\\!\\!8m_{B_{s}}^{2}\lambda[\hat{s}-\hat{s}(\hat{r}_{\phi}+\hat{s})-3\hat{m}_{\ell}^{2}(2-2\hat{r}_{\phi}-\hat{s})](B_{1}B_{2}^{\ast}+D_{1}D_{2}^{\ast})$
$\displaystyle-$
$\displaystyle\\!\\!\\!m_{B_{s}}^{4}\hat{s}\lambda[\lambda(1+3v^{2})-3(1-\hat{r}_{\phi})^{2}(1-v^{2})](\left|B_{2}\right|^{2}+\left|D_{2}\right|^{2})$
$\displaystyle+$
$\displaystyle\\!\\!\\!4[6\hat{m}_{\ell}^{2}(\lambda+6\hat{r}_{\phi}\hat{s})-\hat{s}(\lambda+12\hat{r}_{\phi}\hat{s})](\left|B_{1}\right|^{2}+\left|D_{1}\right|^{2})\bigg{\\}}~{},$
$\displaystyle P_{LN}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{\pi
m_{B_{s}}^{2}}{2\hat{r}_{\phi}\Delta}\sqrt{\frac{\lambda}{\hat{s}}}\,\mbox{\rm
Im}\bigg{\\{}-4m_{B_{s}}^{4}\hat{m}_{\ell}\lambda(1-\hat{r}_{\phi})B_{2}D_{2}^{\ast}$
(17) $\displaystyle+$
$\displaystyle\\!\\!\\!2m_{B_{s}}^{4}\hat{m}_{\ell}\hat{s}\lambda
B_{2}B_{3}^{\ast}-2m_{B_{s}}^{4}\hat{m}_{\ell}\hat{s}\lambda\Big{[}B_{3}D_{2}^{\ast}+(B_{2}+D_{2})D_{3}^{\ast}\Big{]}$
$\displaystyle-$
$\displaystyle\\!\\!\\!2m_{B_{s}}^{2}\hat{m}_{\ell}\hat{s}(1+3\hat{r}_{\phi}-\hat{s})\Big{(}B_{1}B_{2}^{\ast}-D_{1}D_{2}^{\ast}\Big{)}-4\hat{m}_{\ell}(1-\hat{r}_{\phi}-\hat{s})B_{1}D_{1}^{\ast}$
$\displaystyle-$
$\displaystyle\\!\\!\\!2m_{B_{s}}^{2}\hat{m}_{\ell}\hat{s}(1-\hat{r}_{\phi}-\hat{s})(B_{1}+D_{1})(B_{3}^{\ast}-D_{3}^{\ast})$
$\displaystyle+$
$\displaystyle\\!\\!\\!2m_{B_{s}}^{2}\hat{m}_{\ell}[\lambda+(1-\hat{r}_{\phi})(1-\hat{r}_{\phi}-\hat{s})]\Big{(}B_{2}D_{1}^{\ast}+B_{1}D_{2}^{\ast}\Big{)}\bigg{\\}}~{},$
$\displaystyle P_{NL}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!-P_{LN}~{},$ (18) $\displaystyle P_{LT}\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\frac{\pi
m_{B_{s}}^{2}v}{\hat{r}_{\phi}\Delta}\sqrt{\frac{\lambda}{\hat{s}}}\,\mbox{\rm
Re}\bigg{\\{}m_{B_{s}}^{4}\hat{m}_{\ell}\lambda(1-\hat{r}_{\phi})\left|B_{2}-D_{2}\right|^{2}$
(19) $\displaystyle-$
$\displaystyle\\!\\!\\!8m_{B_{s}}^{2}\hat{m}_{\ell}\hat{r}_{\phi}\hat{s}\Big{(}B_{0}B_{1}^{\ast}-C_{1}D_{1}^{\ast}\Big{)}+m_{B_{s}}^{4}\hat{s}\lambda\hat{m}_{\ell}B_{2}B_{3}^{\ast}$
$\displaystyle-$
$\displaystyle\\!\\!\\!m_{B_{s}}^{4}\hat{m}_{\ell}\hat{s}\lambda\Big{(}B_{2}D_{3}^{\ast}+B_{3}D_{2}^{\ast}-D_{2}D_{3}^{\ast}\Big{)}+\hat{m}_{\ell}(1-\hat{r}_{\phi}-\hat{s})\left|B_{1}-D_{1}\right|^{2}$
$\displaystyle+$
$\displaystyle\\!\\!\\!m_{B_{s}}\hat{s}(1-\hat{r}_{\phi}-\hat{s})\Big{[}-m_{B_{s}}\hat{m}_{\ell}(B_{1}-D_{1})(B_{3}^{\ast}-D_{3}^{\ast})\Big{]}$
$\displaystyle-$
$\displaystyle\\!\\!\\!m_{B_{s}}^{2}\hat{m}_{\ell}[\lambda+(1-\hat{r}_{\phi})(1-\hat{r}_{\phi}-\hat{s})](B_{1}-D_{1})(B_{2}^{\ast}-D_{2}^{\ast})\bigg{\\}}~{},$
$\displaystyle P_{TL}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{\pi
m_{B_{s}}^{2}v}{\hat{r}_{\phi}\Delta}\sqrt{\frac{\lambda}{\hat{s}}}\,\mbox{\rm
Re}\bigg{\\{}m_{B_{s}}^{4}\hat{m}_{\ell}\lambda(1-\hat{r}_{\phi})\left|B_{2}-D_{2}\right|^{2}$
(20) $\displaystyle+$
$\displaystyle\\!\\!\\!8m_{B_{s}}^{2}\hat{m}_{\ell}\hat{r}_{\phi}\hat{s}\Big{(}B_{0}B_{1}^{\ast}-C_{1}D_{1}^{\ast}\Big{)}+m_{B_{s}}^{4}\hat{s}\lambda\hat{m}_{\ell}B_{2}B_{3}^{\ast}$
$\displaystyle-$
$\displaystyle\\!\\!\\!m_{B_{s}}^{4}\hat{m}_{\ell}\hat{s}\lambda\Big{(}B_{2}D_{3}^{\ast}+B_{3}D_{2}^{\ast}-D_{2}D_{3}^{\ast}\Big{)}+\hat{m}_{\ell}(1-\hat{r}_{\phi}-\hat{s})\left|B_{1}-D_{1}\right|^{2}$
$\displaystyle-$
$\displaystyle\\!\\!\\!m_{B_{s}}\hat{s}(1-\hat{r}_{\phi}-\hat{s})\Big{[}m_{B_{s}}\hat{m}_{\ell}(B_{1}-D_{1})(B_{3}^{\ast}-D_{3}^{\ast})\Big{]}$
$\displaystyle-$
$\displaystyle\\!\\!\\!m_{B_{s}}^{2}\hat{m}_{\ell}[\lambda+(1-\hat{r}_{\phi})(1-\hat{r}_{\phi}-\hat{s})](B_{1}-D_{1})(B_{2}^{\ast}-D_{2}^{\ast})\bigg{\\}}~{},$
$\displaystyle P_{NT}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{2m_{B_{s}}^{2}v}{3\hat{r}_{\phi}\Delta}\,\mbox{\rm
Im}\bigg{\\{}4\lambda\Big{(}B_{1}D_{1}^{\ast}+m_{B_{s}}^{4}\lambda
B_{2}D_{2}^{\ast}\Big{)}-16m_{B_{s}}^{4}\hat{s}\lambda\hat{r}_{\phi}B_{0}C_{1}^{\ast}$
(21) $\displaystyle-$
$\displaystyle\\!\\!\\!4m_{B_{s}}^{2}\lambda(1-\hat{r}_{\phi}-\hat{s})\Big{(}B_{1}D_{2}^{\ast}+B_{2}D_{1}^{\ast}\Big{)}\bigg{\\}}~{},$
$\displaystyle P_{TN}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!-P_{NT}~{},$ (22) $\displaystyle P_{NN}\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\frac{2m_{B_{s}}^{2}}{3\hat{r}_{\phi}\Delta}\,\mbox{\rm
Re}\bigg{\\{}-24\hat{m}_{\ell}^{2}\hat{r}_{\phi}(\left|B_{1}\right|^{2}+\left|D_{1}\right|^{2})+16m_{B_{s}}^{4}\hat{s}\lambda\hat{r}_{\phi}v^{2}B_{0}C_{1}^{\ast}$
(23) $\displaystyle+$
$\displaystyle\\!\\!\\!6m_{B_{s}}^{2}\hat{m}_{\ell}^{2}\lambda\Big{[}-2B_{1}(B_{2}^{\ast}+B_{3}^{\ast}-D_{3}^{\ast})+2D_{1}(B_{3}^{\ast}-D_{2}^{\ast}-D_{3}^{\ast})\Big{]}$
$\displaystyle+$
$\displaystyle\\!\\!\\!6m_{B_{s}}^{3}\hat{m}_{\ell}\lambda(1-\hat{r}_{\phi})\Big{[}2m_{B_{s}}\hat{m}_{\ell}(B_{2}-D_{2})(B_{3}^{\ast}-D_{3}^{\ast})\Big{]}$
$\displaystyle+$
$\displaystyle\\!\\!\\!6m_{B_{s}}^{4}\hat{m}_{\ell}^{2}\lambda(2+2\hat{r}_{\phi}-\hat{s})(\left|B_{2}\right|^{2}+\left|D_{2}\right|^{2})+6m_{B_{s}}^{4}\hat{m}_{\ell}^{2}\hat{s}\lambda\left|B_{3}-D_{3}\right|^{2}$
$\displaystyle+$
$\displaystyle\\!\\!\\!m_{B_{s}}^{2}\lambda[3(2-2\hat{r}_{\phi}-\hat{s})-v^{2}(2-2\hat{r}_{\phi}+\hat{s})](B_{1}D_{2}^{\ast}+B_{2}D_{1}^{\ast})$
$\displaystyle-$
$\displaystyle\\!\\!\\!m_{B_{s}}^{4}\lambda\Big{[}(3+v^{2})\lambda+3(1-v^{2})(1-\hat{r}_{\phi})^{2}\Big{]}B_{2}D_{2}^{\ast}$
$\displaystyle-$
$\displaystyle\\!\\!\\!2[6\hat{r}_{\phi}\hat{s}(1-v^{2})+\lambda(3-v^{2})]B_{1}D_{1}^{\ast}\bigg{\\}}~{},$
$\displaystyle P_{TT}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{2m_{B_{s}}^{2}}{3\hat{r}_{\phi}\hat{s}\Delta}\,\mbox{\rm
Re}\bigg{\\{}8m_{B_{s}}^{4}\hat{r}_{\phi}\hat{s}\lambda\Big{[}4\hat{m}_{\ell}^{2}(\left|B_{0}\right|^{2}+\left|C_{1}\right|^{2})+2\hat{s}B_{0}C_{1}^{\ast}\Big{]}$
(24) $\displaystyle-$
$\displaystyle\\!\\!\\!6m_{B_{s}}^{2}\hat{m}_{\ell}^{2}\hat{s}\lambda\Big{[}-2(B_{1}-D_{1})(B_{3}^{\ast}-D_{3}^{\ast})\Big{]}$
$\displaystyle-$
$\displaystyle\\!\\!\\!6m_{B_{s}}^{3}\hat{m}_{\ell}\hat{s}\lambda(1-\hat{r}_{\phi})\Big{[}2m_{B_{s}}\hat{m}_{\ell}(B_{2}-D_{2})(B_{3}^{\ast}-D_{3}^{\ast})\Big{]}$
$\displaystyle-$
$\displaystyle\\!\\!\\!6m_{B_{s}}^{4}\hat{m}_{\ell}^{2}\hat{s}^{2}\lambda\left|B_{3}-D_{3}\right|^{2}+4m_{B_{s}}^{2}\hat{m}_{\ell}^{2}\lambda(4-4\hat{r}_{\phi}-\hat{s})(B_{1}B_{2}^{\ast}+D_{1}D_{2}^{\ast})$
$\displaystyle+$
$\displaystyle\\!\\!\\!2\hat{s}[6\hat{r}_{\phi}\hat{s}(1-v^{2})+\lambda(1-3v^{2})]B_{1}D_{1}^{\ast}$
$\displaystyle-$
$\displaystyle\\!\\!\\!2m_{B_{s}}^{4}\hat{m}_{\ell}^{2}\lambda[\lambda+3(1-\hat{r}_{\phi})^{2}](\left|B_{2}\right|^{2}+\left|D_{2}\right|^{2})$
$\displaystyle-$
$\displaystyle\\!\\!\\!m_{B_{s}}^{2}\hat{s}\lambda[2-2\hat{r}_{\phi}+\hat{s}-3v^{2}(2-2\hat{r}_{\phi}-\hat{s})](B_{1}D_{2}^{\ast}+B_{2}D_{1}^{\ast})$
$\displaystyle-$
$\displaystyle\\!\\!\\!8\hat{m}_{\ell}^{2}(\lambda-3\hat{r}_{\phi}\hat{s})(\left|B_{1}\right|^{2}+\left|D_{1}\right|^{2})$
$\displaystyle-$
$\displaystyle\\!\\!\\!m_{B_{s}}^{4}\hat{s}\lambda\Big{[}(1+3v^{2})\lambda-3(1-v^{2})(1-\hat{r}_{\phi})^{2}\Big{]}B_{2}D_{2}^{\ast}\bigg{\\}}~{}.$
The analytical dependence of the double–lepton polarizations on the fourth
quark mass($m_{t^{\prime}}$) and the product of quark mixing matrix elements
($V_{t^{\prime}b}^{\ast}V_{t^{\prime}s}=r_{sb}e^{i\phi_{sb}}$) are studied in
the next section.
## III Effects of the fourth-generation
As we mentioned in the introduction, the inclusion of the fourth-generation in
the Standard Model (SM4) does not lead to new operators in the ${\cal H}_{\rm
eff}$ and all Wilson coefficients receive additional terms as
$\frac{\lambda_{t^{\prime}}}{\lambda_{t}}C^{\rm SM4}_{i}$ either via virtual
exchange of the fourth-generation up-type quark $t^{\prime}$
$(C_{3},...,C_{10})$ or via using the unitarity of CKM matrix $(C_{1},C_{2})$
. Consequently, one can write the new effective Hamiltonian as:
${\cal H}_{\rm
eff}=-\frac{G_{F}}{\sqrt{2}}V_{tb}V^{*}_{ts}\sum_{i=1}^{10}C^{\rm
new}_{i}(\mu){\cal O}_{i}(\mu),$ (25)
where $C^{new}_{i}$ are:
$\displaystyle C^{\rm new}_{i}(\mu)$ $\displaystyle=$ $\displaystyle
C_{i}(\mu)+\frac{\lambda_{t^{\prime}}}{\lambda_{t}}C^{\rm
SM4}_{i}(\mu),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}i=1\ldots 10.$ (26)
In the above equation, $\lambda_{f}=V_{fb}^{\ast}V_{fs}$ and
$\lambda_{t^{\prime}}$ can be parameterized as:
$\lambda_{t^{\prime}}=V_{t^{\prime}b}V_{t^{\prime}s}^{*}=r_{sb}e^{i\phi_{sb}}.$
(27)
Now by using the above effective Hamiltonian, we can reobtain the one-loop
matrix elements of $b\rightarrow s\ell^{+}\ell^{-}$ by replacing $C^{\rm
eff}_{i}(\tilde{C}^{\rm eff}_{i})$ with ${C^{\rm
eff\,new}_{i}}({{\tilde{C}}^{\rm eff\,\rm new}_{i}})$ in Eq.(2), where
${C^{\rm eff\,new}_{i}}$ and ${{\tilde{C}}^{\rm eff\,\rm new}_{i}}$are given
as:
$\displaystyle C^{\rm eff\,\rm new}_{i}(\mu)$ $\displaystyle=$ $\displaystyle
C^{\rm eff}_{i}(\mu)+\frac{\lambda_{t^{\prime}}}{\lambda_{t}}C^{\rm eff\,\,\rm
SM4}_{i}(\mu),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}i=7,$
$\displaystyle\tilde{C}^{\rm eff\,\rm new}_{i}(\mu)$ $\displaystyle=$
$\displaystyle\tilde{C}^{\rm
eff}_{i}(\mu)+\frac{\lambda_{t^{\prime}}}{\lambda_{t}}\tilde{C}^{\rm
eff\,\,\rm SM4}_{i}(\mu),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}i=9,10.$ (28)
Here the effective Wilson coefficients ${C}^{\rm eff\,\,\rm SM4}_{i}$ and
$\tilde{C}^{\rm eff\,\,\rm SM4}_{i}$ are defined in the same way as Eqs.(II)
by substituting $C_{i}$ with $C_{i}^{SM4}$. It is worth nothing that the
explicit forms of ${C}^{\rm eff\,\,\rm SM4}_{i}$ and $\tilde{C}^{\rm
eff\,\,\rm SM4}_{i}$ can also be found from the corresponding Wilson
coefficients in SM by replacing $m_{t}\rightarrow m_{t^{\prime}}$ R23 . Based
on the preceding explanations, in order to obtain the matrix element and the
double-lepton polarization asymmetries for
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ decay in the presence of the fourth-
generation, one should replace $C^{\rm eff}_{i}(\tilde{C}^{\rm eff}_{i})$ with
${C^{\rm eff\,new}_{i}}({{\tilde{C}}^{\rm eff\,\rm new}_{i}})$ in all
equations of the previous section.
The unitary quark mixing matrix is now $4\times 4$ which can be written in
terms of $6$ mixing angles and $3$ CP violating phases. The relevant elements
of this matrix for $b\rightarrow s$ transition satisfy the relation:
$\lambda_{u}+\lambda_{c}+\lambda_{t}+\lambda_{t^{\prime}}=0.$ (29)
Consequently, as required by GIM mechanism, the factor $\lambda_{t}C_{i}^{\rm
new}$ should be modified to $\lambda_{t}C_{i}$ when $m_{t^{\prime}}\rightarrow
m_{t}$ or $\lambda_{t^{\prime}}\rightarrow 0$. We can easily check the
validity of this condition by using Eq.(29):
$\displaystyle\lambda_{t}C_{i}^{\rm
new}=\lambda_{t}C_{i}+\lambda_{t^{\prime}}C_{i}^{\rm SM4}$ $\displaystyle=$
$\displaystyle-(\lambda_{u}+\lambda_{c})C_{i}+\lambda_{t^{\prime}}(C_{i}^{\rm
SM4}-C_{i})$ (30) $\displaystyle=$
$\displaystyle-(\lambda_{u}+\lambda_{c})C_{i}$ $\displaystyle=$
$\displaystyle\lambda_{t}C_{i}.$
The numerical analysis of the dependence of the double–lepton polarizations on
the fourth quark mass ($m_{t^{\prime}}$) and the product of quark mixing
matrix elements ($V_{t^{\prime}b}^{\ast}V_{t^{\prime}s}=r_{sb}e^{i\phi_{sb}}$)
are presented in the next section.
## IV Results and Discussions
The main input parameters in the calculations are the form factors for which
we have chosen the predictions of light cone QCD sum rule method R31 ; R32 ,
as pointed out in section II. Besides the form factors, we use the other input
parameters as follow:
$\displaystyle
m_{B_{s}}=5.37\,\mbox{GeV}\,,\,m_{b}=4.8\,\mbox{GeV}\,,\,m_{c}=1.5\,\mbox{GeV}\,,\,m_{\tau}=1.77\,\mbox{GeV}\,,\,$
$\displaystyle
m_{\mu}=0.105\,\mbox{GeV},\,m_{\phi}=1.020\,\mbox{GeV}\,\,,\,|V_{tb}V_{ts}^{*}|=0.0385\,,\,\alpha^{-1}=129\,,\,$
$\displaystyle G_{f}=1.166\times
10^{-5}\,{\mbox{GeV}}^{-2}\,,\,\tau_{B_{s}}=1.46\times 10^{-12}\,s\,.$ (31)
In order to present a quantitative analysis of the double-lepton polarization
asymmetries, the values of fourth-generation parameters are needed.
Considering the experimental values of $B\longrightarrow X_{s}\gamma$ and
$B\longrightarrow X_{s}\ell^{+}\ell^{-}$ decays the value of the $r_{sb}$
parameter is restricted to the range $\\{.01-.03\\}$ for
$\phi_{sb}\sim\\{0^{\circ}-360^{\circ}\\}$ and
$m_{t^{\prime}}\sim\\{200-600\\}$ GeVArhrib:2002md ; Zolfagharpour:2007ez2 .
Using the $B_{s}$ mixing parameter $\Delta m_{B_{s}}$, a sharp restriction on
$\phi_{sb}$ has been obtained ($\phi_{sb}\sim 90^{\circ}$)Hou:2006jy .
Therefore in our following numerical analysis, the corresponding values of
above ranges are:
$r_{sb}=\\{.01,~{}.02,~{}.03\\},\phi_{sb}=\\{60^{\circ},~{}90^{\circ},~{}120^{\circ}\\},m_{t^{\prime}}=175\leq
m_{t^{\prime}}\leq 600$.
It is clear from the expressions of all nine double–lepton polarization
asymmetries that they depend on the momentum transfer $q^{2}$ and the new
parameters $(m_{t^{\prime}}$, $r_{sb}$, $\phi_{sb})$. Consequently, it may be
experimentally difficult to investigate these dependencies at the same time.
One way to deal with this problem is to integrate over $q^{2}$ and study the
averaged double-lepton polarization asymmetries. The average of $P_{ij}$ over
$q^{2}$ is defined as:
$\displaystyle\langle
P_{ij}\rangle=\frac{\displaystyle\int_{4\hat{m}_{\ell}^{2}}^{(1-\sqrt{\hat{r}_{\phi}})^{2}}P_{ij}\frac{d{\cal
B}}{d\hat{s}}d\hat{s}}{\displaystyle\int_{4\hat{m}_{\ell}^{2}}^{(1-\sqrt{\hat{r}_{\phi}})^{2}}\frac{d{\cal
B}}{d\hat{s}}d\hat{s}}~{}.$ (32)
We have used the above formula and depicted the dependency of $\langle
P_{ij}\rangle$ on the fourth-generation parameters in Fig.[1-7]. In the
following, we compare our results for $B_{s}\rightarrow\phi\ell^{+}\ell^{-}$
decay with the results of Ref.Bashiry:2007tf for $B\rightarrow
K\ell^{+}\ell^{-}$ decay. Since the overall behavior of $\langle
P_{ij}\rangle$ versus $m_{t^{\prime}},r_{sb}$ and $\phi_{sb}$ are almost the
same as that of $B\rightarrow K\ell^{+}\ell^{-}$ decay, we discuss the
differences of these two decays and some aspects which have not been discussed
in Ref.Bashiry:2007tf :
* •
Figrue(1): Similar to the $B\rightarrow K\mu^{+}\mu^{-}$ decay, $\langle
P_{LL}\rangle$ is not sensitive to the fourth-generation quark parameters,
therefore the $\langle P_{LL}\rangle$ plots for $\mu$ channel have been
omitted. However, for the $\tau$ channel, the maximum deviation from SM is
about $50\%$ which can be seen at $m_{t^{\prime}}\sim 600GeV$. In comparison
with the results of Ref.Bashiry:2007tf , it is understood that the deviation
from SM for $B_{s}\rightarrow\phi\tau^{+}\tau^{-}$ is twice that of
$B\rightarrow K\tau^{+}\tau^{-}$ decay. Therefore, the magnitude of $\langle
P_{LL}\rangle$ in $B_{s}\rightarrow\phi\tau^{+}\tau^{-}$ compared with that in
$B\rightarrow K\tau^{+}\tau^{-}$decay has more chance to show the existence of
the fourth-generation.
* •
Figrue(2): The value of $\langle P_{LN}\rangle_{max}$ for $\mu$ channel is
about 0.04 which is four times greater than that for $B\rightarrow K$ decay.
However, for $\tau$ channel such value is at most around $0.3$ which is
approximately equal to the maximum value of $\langle P_{LN}\rangle$ for
$B\rightarrow K$ decay. Furthermore, in $\mu$ and $\tau$ channels by
increasing $r_{sb}$ and keeping the values of $\phi_{sb}$ fixed, the maximum
deviation from SM occurs at smaller values of $m_{t^{\prime}}$. This result
can be interesting since the maximum deviation from SM happens for
$r_{sb}\sim\\{0.02-0.03\\}$ and $m_{t^{\prime}}\sim\\{300-400\\}$GeV.
Therefore, the new generation has a chance to be observed around
$m_{t^{\prime}}\sim\\{300-400\\}$GeV. Our analysis shows that to measure the
effect of the fourth-generation in $\langle P_{LN}\rangle$, the $\tau$ channel
of $B_{s}\rightarrow\phi$ and $B\rightarrow K$ are more important than $\mu$
channel of these decays, knowing that in the $\mu$ channel the
$B_{s}\rightarrow\phi$ decay is more significant than the $B\rightarrow K$
decay.
* •
Figrue(3): For $\mu$ channel, the magnitude of $\langle P_{LT}\rangle$ in
$B_{s}\rightarrow\phi$ decay changes at most about $80\%$ compared with the SM
prediction, while the maximum change in $B\rightarrow K$ decay reaches up to
$60\%$. For $\tau$ case, unlike $B\rightarrow K$ decay, the magnitude of
$\langle P_{LT}\rangle$ in $B_{s}\rightarrow\phi$ transition exhibits the
strong dependence on the fourth quark mass $(m_{t^{\prime}})$ and the product
of quark mixing matrix elements
$(|V_{t^{\prime}b}V_{t^{\prime}s}^{*}|=r_{sb})$. As seen from Fig.(3) the
maximum deviation from SM in $\tau$ channel is much more than that in $\mu$
channel. Therefore for establishing the fourth generation of quarks the
measurement of $\langle P_{LT}\rangle$ for
$B_{s}\rightarrow\phi\tau^{+}\tau^{-}$ decay is more suitable than such
measurement for $B_{s}\rightarrow\phi\mu^{+}\mu^{-}$ and $B\rightarrow
K\mu^{+}\mu^{-}$ decays .
* •
Figrue(4): It is seen from Eqs.(19) and (20) that contrary to $B\rightarrow K$
decay, $P_{TL}$ is neither symmetric nor anti-symmetric under the exchange of
subscripts L and T which leads to different values for $P_{TL}$ and $P_{LT}$.
For $\mu$ channel, the magnitude of $\langle P_{TL}\rangle$ in
$B_{s}\rightarrow\phi$ decay changes at most about $40\%$ compared with the SM
prediction, while the maximum change in the case of $B\rightarrow K$ decay
reaches up to $60\%$. For $\tau$ case, unlike $B\rightarrow K$ decay, the
magnitude of $\langle P_{TL}\rangle$ in $B_{s}\rightarrow\phi$ transition
changes at most about $60\%$ compared with the SM prediction. Therefore, in
the measurement of $\langle P_{TL}\rangle$, the decays
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$($\ell=\mu,\tau$) and $B\rightarrow
K\mu^{+}\mu^{-}$ have the same significance for finding the new generation of
quarks.
* •
Figrue(5): By comparing this figure with Fig.(2), one can find out that the
overall behavior of $\langle P_{TN}\rangle$ and $\langle P_{LN}\rangle$ are
the same. Furthermore, the magnitude of $\langle P_{TN}\rangle_{max}$ for
$\mu$ channel is about 0.22 which is four times smaller than that for
$B\rightarrow K$ decay and for $\tau$ channel such value is at most around
$0.0075$ which is approximately ten times smaller than $\langle
P_{TN}\rangle_{max}$ for $B\rightarrow K$ decay. Although the measurement of
$\langle P_{TN}\rangle$ in $B\rightarrow K\tau^{+}\tau^{-}$ decay for finding
the new generation is useful, such measurement in the decays
$B_{s}\rightarrow\phi\mu^{+}\mu^{-}$ and $B\rightarrow K\mu^{+}\mu^{-}$ are
more significant.
* •
Figrue(6): For both $\mu$ and $\tau$ channels in $B_{s}\rightarrow\phi$ decay,
the values of $\langle P_{NN}\rangle$ show stronger dependence on the fourth-
generation parameters $(m_{t^{\prime}},r_{sb},\phi_{sb})$ in comparison with
those in $B\rightarrow K$ decay. Furthermore, the situation for $\tau$ channel
is even more interesting than $\mu$ channel, since for fixed values of
$\phi_{sb}$ and $r_{sb}$, an increase in $m_{t^{\prime}}$ changes the sign of
$\langle P_{NN}\rangle$. So, for $B_{s}\rightarrow\phi$ decay, the study of
the magnitude and the sign of $\langle P_{NN}\rangle$ for $\tau$ channel and
the magnitude of this asymmetry in $\mu$ channel can serve as good tests for
discovering the new physics beyond the SM. It should also be mentioned that
for both $\mu$ and $\tau$ channels of $B\rightarrow K$ decay in general, and
specially for the $\mu$ channel, the deviation of $\langle P_{NN}\rangle$ from
SM can be a measurable quantity, even though it is less sensitive to the
fourth generation of quarks compered with that of $B\rightarrow\phi$ decay(see
Ref.Bashiry:2007tf ).
* •
Figrue(7): A comparison between this figure and an analogous figure for
$B\rightarrow K\ell^{+}\ell^{-}$ shows that the values of $\langle
P_{TT}\rangle$ for both $\mu$ and $\tau$ channels in $B_{s}\rightarrow\phi$
decay have considerable dependency on the fourth-generation parameters
$(m_{t^{\prime}},r_{sb},\phi_{sb})$. Therefore, compared with $B\rightarrow
K\ell^{+}\ell^{-}$ decay in Ref.Bashiry:2007tf , the study of the magnitude of
$\langle P_{TT}\rangle$ in $B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ provides a
better opportunity to see the effect of the new physics beyond the SM.
Finally, let us briefly discuss whether it is possible to measure the lepton
polarization asymmetries in experiments or not. Experimentally, to measure an
asymmetry $\left<P_{ij}\right>$ of the decay with branching ratio $\cal{B}$ at
$n\sigma$ level, the required number of events (i.e., the number of
$B\bar{B}$) is given by the formula
$\displaystyle N=\frac{n^{2}}{{\cal B}s_{1}s_{2}\langle
P_{ij}\rangle^{2}}~{},$
where $s_{1}$ and $s_{2}$ are the efficiencies of the leptons. Typical values
of the efficiencies of the $\tau$–leptons vary from $50\%$ to $90\%$ for their
different decay modesR6016 and the error in $\tau$–lepton polarization is
estimated to be about $(10-15)\%$ R6017 . So, the error in measurement of the
$\tau$–lepton asymmetries is approximately $(20-30)\%$, and the error in
obtaining the number of events is about $50\%$.
Looking at the expression of $N$, it can be understood that in order to detect
the lepton polarization asymmetries in the $\mu$ and $\tau$ channels at
$3\sigma$ level, the minimum number of required events are (for the efficiency
of $\tau$–lepton we take $0.5$):
* •
for $B_{s}\rightarrow\phi\mu^{+}\mu^{-}$ decay
$\displaystyle N\sim\left\\{\begin{array}[]{ll}10^{6}&(\mbox{\rm
for}\left<P_{LL}\right>)~{},\\\ 10^{7}&(\mbox{\rm
for}\left<P_{NT}\right>,\left<P_{TN}\right>)~{},\\\ 10^{8}&(\mbox{\rm
for}\left<P_{LT}\right>,\left<P_{TL}\right>,\left<P_{NN}\right>,\left<P_{TT}\right>)~{},\\\
10^{9}&(\mbox{\rm for}\left<P_{LN}\right>,\left<P_{NL}\right>)~{},\\\
\end{array}\right.$ (37)
* •
for $B_{s}\rightarrow\phi\tau^{+}\tau^{-}$ decay
$\displaystyle N\sim\left\\{\begin{array}[]{ll}10^{8}&(\mbox{\rm
for}\left<P_{LT}\right>,\left<P_{TL}\right>,\left<P_{NN}\right>,\left<P_{TT}\right>)~{},\\\
10^{9}&(\mbox{\rm
for}\left<P_{LL}\right>,\left<P_{LN}\right>,\left<P_{NL}\right>)~{},\\\
10^{12}&(\mbox{\rm for}\left<P_{NT}\right>,\left<P_{TN}\right>)~{}.\\\
\end{array}\right.$ (41)
Considering the above values for N and the number of $B\bar{B}$ pairs which
will be produced at LHC($\sim 10^{12}$), one can conclude that except
$\left<P_{NT}\right>$ and $\left<P_{TN}\right>$ for $\tau$ channel, all
double-lepton polarizations can be detected at the LHC.
In summary, in this paper we have presented the analyses of the double-lepton
polarization asymmetries in $B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ decay using
the SM with the fourth generation of quarks. We found out that these
asymmetries have strong dependency on the fourth-generation parameters which
can be detected at the LHC. We compared
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$decay with $B\rightarrow
K\ell^{+}\ell^{-}$ decay, and showed that the double-lepton polarizations of
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ are more sensitive to the fourth-
generation parameters and therefore by looking at
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ decay, one has more chance to
investigate the correctness of the fourth generation of quarks hypothesis in
the near future.
## V Acknowledgment
The authors would like to thank V. Bashiry for his useful discussions. Support
of Research Council of Shiraz University is gratefully acknowledged.
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Figure 1: The dependence of the $\left<P_{LL}\right>$ on the fourth-generation
quark mass $m_{t^{\prime}}$ for three different values of
$\phi_{sb}=\\{60^{\circ},~{}90^{\circ},~{}120^{\circ}\\}$ and
$r_{sb}=\\{0.01,~{}0.02,~{}0.03\\}$ for the $\tau$ channel.
Figure 2: The dependence of the $\left<P_{LN}\right>$ on the fourth-generation
quark mass $m_{t^{\prime}}$ for three different values of
$\phi_{sb}=\\{60^{\circ},~{}90^{\circ},~{}120^{\circ}\\}$ and
$r_{sb}=\\{0.01,~{}0.02,~{}0.03\\}$ for the $\mu$ and $\tau$ channels.
Figure 3: The dependence of the $\left<P_{LT}\right>$ on the fourth-generation
quark mass $m_{t^{\prime}}$ for three different values of
$\phi_{sb}=\\{60^{\circ},~{}90^{\circ},~{}120^{\circ}\\}$ and
$r_{sb}=\\{0.01,~{}0.02,~{}0.03\\}$ for the $\mu$ and $\tau$ channels.
Figure 4: The dependence of the $\left<P_{TL}\right>$ on the fourth-generation
quark mass $m_{t^{\prime}}$ for three different values of
$\phi_{sb}=\\{60^{\circ},~{}90^{\circ},~{}120^{\circ}\\}$ and
$r_{sb}=\\{0.01,~{}0.02,~{}0.03\\}$ for the $\mu$ and $\tau$ channels.
Figure 5: The dependence of the $\left<P_{TN}\right>$ on the fourth-generation
quark mass $m_{t^{\prime}}$ for three different values of
$\phi_{sb}=\\{60^{\circ},~{}90^{\circ},~{}120^{\circ}\\}$ and
$r_{sb}=\\{0.01,~{}0.02,~{}0.03\\}$ for the $\mu$ and $\tau$ channels.
Figure 6: The dependence of the $\left<P_{NN}\right>$ on the fourth-generation
quark mass $m_{t^{\prime}}$ for three different values of
$\phi_{sb}=\\{60^{\circ},~{}90^{\circ},~{}120^{\circ}\\}$ and
$r_{sb}=\\{0.01,~{}0.02,~{}0.03\\}$ for the $\mu$ and $\tau$ channels.
Figure 7: The dependence of the $\left<P_{TT}\right>$ on the fourth-generation
quark mass $m_{t^{\prime}}$ for three different values of
$\phi_{sb}=\\{60^{\circ},~{}90^{\circ},~{}120^{\circ}\\}$ and
$r_{sb}=\\{0.01,~{}0.02,~{}0.03\\}$ for the $\mu$ and $\tau$ channels.
|
arxiv-papers
| 2008-11-17T12:28:26 |
2024-09-04T02:48:58.826329
|
{
"license": "Public Domain",
"authors": "S. M. Zebarjad, F. Falahati and H. Mehranfar",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/0811.2706"
}
|
0811.2739
|
# Indistinguishability of independent single photons
F. W. Sun111fs2293@columbia.edu C. W. Wong Optical Nanostructures Laboratory,
Columbia University, New York, NY 10027.
###### Abstract
The indistinguishability of independent single photons is presented by
decomposing the single photon pulse into the mixed state of different
transform limited pulses. The entanglement between single photons and outer
environment or other photons induces the distribution of the center
frequencies of those transform limited pulses and makes photons
distinguishable. Only the single photons with the same transform limited form
are indistinguishable. In details, the indistinguishability of single photons
from the solid-state quantum emitter and spontaneous parametric down
conversion is examined with two-photon Hong-Ou-Mandel interferometer.
Moreover, experimental methods to enhance the indistinguishability are
discussed, where the usage of spectral filter is highlighted.
###### pacs:
42.50.Ar, 42.25.Hz, 03.65.Yz
## I Introduction
Linear optical quantum computation Kok07RMP is based on the interference
between different photons HOM , in which the indistinguishability of photons
is a fundamental and critical requirement. Any distinguishability will reduce
the visibility of interference and the fidelity of quantum computation
protocol Rohde05PRA . It will also directly affect the other applications with
photon interference, such as quantum key distribution Gisin02RMP and high
precision quantum phase measurement Sun08EPL . Moreover, photon
indistinguishability is fundamental to stimulated emission Lamas-Linares01NAT
; Sun07PRL and has been applied in quantum cloning Lamas-Linares02SCI ;
Fasel02PRL and entanglement measure Sun07PRA2 . Based on the spontaneous
parametric down conversion (SPDC), the indistinguishability in the multiphoton
interference has been intensely examined recently in experiment Ou99PRL ;
Tsujino04PRL ; Eisenberg06PRL ; Xiang06PRL ; Liu07EPL and theory Ou99PRA ;
OuPRAs ; Sun07PRA ; Sun08PRA . However, the kernel is the indistinguishability
of independent single photons. In SPDC, independent single photons are
heralded by detecting the twinning photons, with several experiments focusing
on their indistinguishability and interference Riedmatten03PRA ;
Kaltenbaek06PRL ; Mosley08PRL . In the solid-state quantum emitters, single
photons have been remarkably examined Martini96PRL ; Kim99NAT ; Michler00SCI ;
Santori01PRL ; Kuhn02PRL ; Santori02NAT , where, in addition to photon
statistics and quantum efficiencies, indistinguishability is another important
character of the single photon source Santori02NAT ; Mosley08PRL .
Generally, the distinguishability of the single photons comes from the
entanglement with extrinsic system, such as photons, phonons or outer
environment. Theoretically, the single photons can be described as the mixed
state by tracing out the entangling parts. In SPDC, the property of
entanglement can be achieved through the analysis of the phase matching
condition U'Ren07PRA ; Mosley08PRL . However, it is more complicated in the
solid-state quantum emitters. Many kinds of physical processes introduce the
entanglement between the environment and the emitted single photons. For
example, in the single quantum dot, the interaction with phonon results in
short dephasing time and gives rises to a very broad spectrum of the photons
Carmichael . This spectrum broadening will make photons distinguishable. Since
it is hard to measure all the physical information of each photon source, the
direct analysis of the photon state with interference is highly desired.
In this paper, we will give a brief description of single photons to show the
indistinguishability. In the frequency degree of freedom (DOF), the whole
photon state is a mixed state of transform limited pulses with different
center frequencies. For two independent single photons, there is no
entanglement between them. The indistinguishability describes the nature of
identicality of the transform limited pulses. To aid in the analysis, we
regarded the bandwidth of distribution of these center frequencies as the
extrinsic width, which comes form the entanglement between with extrinsic
system. The transformed limited pulse are pure state and its width is the
intrinsic width. For the same single photon source, single photons have the
same extrinsic width and the same intrinsic width. The total spectrum
bandwidth is the combination of the intrinsic width and the extrinsic width.
Generally, when the extrinsic width is much larger than the intrinsic width,
the single photons are totally distinguishable. Only when the extrinsic width
is zero, the single photon pulse is transform limited and indistinguishable.
In either Lorentzian or Gaussian distributions of the spectrum, the photon
indistinguishability is the ratio of intrinsic width to total bandwidth. In
experiment, the distinguishability can be measured with Hong-Ou-Mandel (HOM)
interferometer HOM , where the visibility shows the indistinguishability. In
the main section, we will examine the indistinguishability of single photons
from solid quantum emitters and SPDC after a general description of the single
photon state is given. In the discussion section, experimental methods to
enhance the indistinguishability are presented, where the effect of spectral
filter is highlighted.
## II Description and indistinguishability of single photons
We begin the description of single photons from the transform limited pulse,
which is a pure quantum state,
$\left|\omega\right\rangle=\int_{-\infty}^{+\infty}d\upsilon
g_{\omega}(\upsilon)a^{{\dagger}}(\upsilon)\left|\text{vac}\right\rangle\text{,}$
(1)
where $a^{{\dagger}}$ ($a$) is the single photon creation (annihilation)
operator. $|g_{\omega}(\upsilon)|^{2}$ is the spectrum of the transform
limited pulse with center frequency $\omega$ and width $\Delta_{g}$ (intrinsic
width). We will discuss the independent single photons from the same source
and assume the same $\Delta_{g}$, since the interactions between the single
photons and outer environment or other photons are highly similar during the
generation. Correspondingly, the transform limited pulse has the duration of
$T_{TL}=1/\Delta_{g}$. Also, $g_{\omega}(\upsilon)$ satisfies the
normalization condition
$\int_{-\infty}^{+\infty}d\upsilon|g_{\omega}(\upsilon)|^{2}=1$. The
indistinguishability of two independent transform limited photon pulse is
$K_{ij}^{TL}=\left|\left\langle\omega_{i}|\omega_{j}\right\rangle\right|^{2}$.
Roughly, the two photons are totally distinguishable when
$\left|\omega_{i}-\omega_{j}\right|\gg\Delta_{g}$ and indistinguishable for
$\omega_{i}=\omega_{j}$.
Figure 1: (color online) Illustration of total single photon pulse (red dashed
curve, width $\Delta_{s}$) composed of transform limited pulses (grey bold
curves, width $\Delta_{g}$) with different center frequencies. When
$\Delta_{S}=\Delta_{g}$, the single photon pulse is transform limited and
indistinguishable.
Since the single photons may be entangled with extrinsic system, the center
frequencies have the distribution $f(\omega)$
[$\int_{-\infty}^{+\infty}d\omega f(\omega)=1$] with width $\Delta_{f}$
(extrinsic width). Then, the whole state is written as
$\rho=\int_{-\infty}^{+\infty}d\omega
f(\omega)\left|\omega\right\rangle\left\langle\omega\right|\text{.}$ (2)
The total spectrum $S(\upsilon)=\int_{-\infty}^{+\infty}d\omega
f(\omega)\left|g_{\omega}(\upsilon)\right|^{2}$ is broadened to
$\Delta_{S}\geq$ $\Delta_{g}$ because of the distribution $f(\omega)$.
However, the lifetime of the single photon pulse is same with those transform
limited pulses, that is $T_{\rho}=T_{TL}$. Fig. 1 illustrates that the total
single photon pulse is composed of different transform limited pulses. Only
when $\Delta_{S}=\Delta_{g}=1/T_{\rho}$ is satisfied, the single photon pulse
becomes transform limited.
Formally, the indistinguishability of two independent single photons can be
described as
$K=tr(\rho\otimes\rho)=\iint\nolimits_{-\infty}^{+\infty}d\omega_{i}d\omega_{j}f(\omega_{i})f(\omega_{j})\left|\left\langle\omega_{i}|\omega_{j}\right\rangle\right|^{2}\text{.}$
(3)
If and only if $\rho$ is the pure state, $K=1$. That is with $\Delta_{f}=0$
and $\Delta_{S}=$ $\Delta_{g}$, the single photon states are
indistinguishable. On the other hand, when $\Delta_{S}\gg$ $\Delta_{g}$,
$K\rightarrow 0$, the single photon states are distinguishable. From this
view, two photons may be distinguishable even when they have the same
description. The indistinguishability describes the nature of identicality of
the pure state. Eq. (3) also describes the purity of the state $\rho$.
Generally, a mixed state comes from an entangled system. In principle,
distinguishable information of the state may be obtained by measuring the
entangling part. Therefore, the mixed state always has some distinguishability
and the purity of the state is a good scale to evaluate the
indistinguishability. Moreover, this definition of indistinguishability is
highly supported by the experiment. In experiment, the indistinguishable
photons will present photon bunching effect and the value of the
indistinguishability has the simple relationship with the interference
visibility. Based on the single photons from solid-state quantum emitters and
SPDC, we will now give detailed discussions on their indistinguishability.
### II.1 Indistinguishability of single photons from single solid quantum
emitter
Here we focus on the single photons from single quantum dot. The single photon
from two-level quantum dot spontaneous emission has the Lorentzian
distribution,
$g_{\omega}(\upsilon)=\frac{1}{\sqrt{\pi}}\frac{\sqrt{\Gamma/2}}{(\upsilon-\omega)+i\Gamma/2}\text{,}$
(4)
where $\Gamma/2$ is the intrinsic width and describes the rate of spontaneous
emission Scully . Correspondingly, the lifetime is $T_{1}=1/\Gamma$. In
addition to the intrinsic linewidth, the spectrum broadening mainly comes from
the dephasing process. Also, the spectral diffusion of single quantum dot
gives much more broader spectrum Neuhauser00PRL . All these spectrum
broadening can be included in the distribution of $f(\omega)$. For simplicity,
we only consider the spectral broadening from pure dephasing which can also be
described as the Lorentzian function,
$f(\omega)=\frac{1}{\pi}\frac{\Gamma^{\prime}}{(\omega-\omega_{c})^{2}+\Gamma^{\prime
2}}\text{,}$ (5)
where $\omega_{c}$ is the center frequency of the distribution $f(\omega)$.
The extrinsic width is $\Gamma^{\prime}=1/T_{2}^{\prime}$, where
$T_{2}^{\prime}$ is the pure dephasing time. The total state can be described
with Eq. (2). The whole spectrum is
$S(\upsilon)=\frac{1}{\pi}\frac{\Gamma_{2}}{(\upsilon-\omega_{c})^{2}+\Gamma_{2}^{2}}\text{,}$
(6)
where $\Delta_{S}^{L}=\Gamma_{2}=1/T_{2}=\Gamma^{\prime}+\Gamma/2$ is the
total spectral width and the superscript $L$ in $\Delta_{S}^{L}$ denotes
Lorentzian distribution. In the time domain, we get
$1/T_{2}=1/2T_{1}+1/T_{2}^{\prime}$. When $\Gamma^{\prime}=0$,
$\Gamma_{2}=\Gamma/2=1/2T_{1}$, the single photons are transform limited.
The indistinguishability of the two transform limited pulses centered at
$\omega_{i}$ and $\omega_{j}$ is
$K_{ij}^{TL}=\frac{\Gamma^{2}}{(\omega_{i}-\omega_{j})^{2}+\Gamma^{2}}\text{,}$
(7)
while the indistinguishability of the two single photons with Eq. (3) is
$K_{L}=\frac{\Gamma}{2\Gamma_{2}}\text{.}$ (8)
When $\Gamma^{\prime}=0$, $K_{L}=1$ and the single photon are the transform
limited and indistinguishable.
Experimentally, the HOM interferometer is usually used to measure the
indistinguishability of two single photons, as shown in Fig.2(a). Two single
photons are injected into the two input ports of a 50/50 beamsplitter
separately. The two-photon coalescence probability $C_{AB}$ of output ports$\
A$ and $B$ is null when two photons are indistinguishable and arrive at the
beamsplitter simultaneously. Any distinguishability will induce nonzero two
photon coalescence probability and reduce the interference visibility. In
order to obtain the coalescence probability $C_{AB}(\tau)$ with the interval
$\tau$ between the arrival times of two photons, we first calculate
probability of two photons exiting in the same output port $C_{AA}(\tau)$,
which shows photon bunching when the two photons are indistinguishable Ou99PRA
; HBT . Therefore,
$\displaystyle C_{AA}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{8}\left\langle
E^{(-)}(t)E^{(-)}(t+\tau)E^{(+)}(t+\tau)E^{(+)}(t)\right\rangle$ (9)
$\displaystyle=$
$\displaystyle\frac{1}{8}\iint\nolimits_{-\infty}^{+\infty}d\omega_{i}d\omega_{j}f(\omega_{i})f(\omega_{j})\int_{-\infty}^{+\infty}dt\left\langle\omega_{i}\right|\left\langle\omega_{j}\right|E^{(-)}(t)E^{(-)}(t+\tau)E^{(+)}(t+\tau)E^{(+)}(t)\left|\omega_{j}\right\rangle\left|\omega_{i}\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{4}\iint\nolimits_{-\infty}^{+\infty}d\omega_{i}d\omega_{j}f(\omega_{i})f(\omega_{j})\int_{-\infty}^{+\infty}dt[\left\langle\omega_{j}\right|E^{(-)}(t)E^{(+)}(t)\left|\omega_{j}\right\rangle\left\langle\omega_{i}\right|E^{(-)}(t+\tau)E^{(+)}(t+\tau)\left|\omega_{i}\right\rangle$
$\displaystyle+\left\langle\omega_{j}\right|E^{(-)}(t)E^{(+)}(t+\tau)\left|\omega_{i}\right\rangle\left\langle\omega_{i}\right|E^{(-)}(t+\tau)E^{(+)}(t)\left|\omega_{j}\right\rangle]$
$\displaystyle=$ $\displaystyle\frac{1}{4}[1+K(\tau)]$
where $E^{(+)}(t)=\int_{-\infty}^{+\infty}d\omega a(\omega)e^{-i\omega
t}/\sqrt{2\pi}$ is the detection operator. The coefficient $1/8$ comes from
photon loss of the beamsplitter ($1/4$) for two photons and the normalization
coefficient of two permutations of two photons detecting by two detectors
($1/2$). In the practical experiment, the detection duration is much larger
than the photon pulse lifetime and the integral time is extended to
$(-\infty,+\infty)$. In the above equation, $K(\tau)$ is the
indistinguishability of two photons with time interval $\tau$
$\displaystyle K(\tau)$ $\displaystyle=$
$\displaystyle\iint\nolimits_{-\infty}^{+\infty}d\omega_{i}d\omega_{j}f(\omega_{i})f(\omega_{j})\int_{-\infty}^{+\infty}dt$
(10)
$\displaystyle\times\left|\left\langle\omega_{i}\right|E^{(-)}(t)E^{(+)}(t+\tau)\left|\omega_{j}\right\rangle\right|^{2}\text{.}$
For the Lorentzian distribution, the indistinguishability is
$K_{L}(\tau)=\frac{\Gamma}{2\Gamma_{2}}e^{-\Gamma|\tau|}\text{,}$ (11)
and the two-photon probability is
$C_{AA}(\tau)=\frac{1}{4}(1+\frac{\Gamma}{2\Gamma_{2}}e^{-\Gamma|\tau|})\text{.}$
(12)
The excess probability of $K(\tau)$ is the signature of the photon
indistinguishability. It is the result of photon bunching from the permutation
symmetry of bosonic particles Sun07PRA ; Sun08PRA .
Because of the symmetry of the beamsplitter, the probability of two photons
together in the output port $B$ is same with $C_{AA}(\tau)$. Therefore, the
two-photon coincidence probability $C_{AB}(\tau)$ based on the energy
conservation law is
$\displaystyle C_{AB}(\tau)$ $\displaystyle=$ $\displaystyle 1-2C_{AA}(\tau)$
(13) $\displaystyle=$ $\displaystyle[1-K(\tau)]/2$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1-\frac{\Gamma}{2\Gamma_{2}}e^{-\Gamma|\tau|})\text{.}$
(14)
$C_{AB}(\tau)$ shows the typical HOM dip with the $1/e$ width of pulse
lifetime, $1/\Gamma=T_{1}$, as shown in the experimental report Santori02NAT .
The visibility shows the indistinguishability of $K_{L}$, which is illustrated
in Fig. 2(b) with different ratios of extrinsic width to intrinsic width,
$\eta=\Delta_{f}/\Delta_{g}$. For Lorentzian distribution,
$\eta=\eta_{L}=2\Gamma^{\prime}/\Gamma$. When extrinsic width is much larger
than the intrinsic width, $\eta_{L}\gg 1$, the indistinguishability is
approaching to $1/\eta_{L}$.
Figure 2: (color online)(a) Illustration of two-photon Hong-Ou-Mandel
interference. $\tau$ is the interval between arrival time of the two input
photons. (b) Two-photon Hong-Ou-Mandel interference visibility with different
ratios ($\eta$) of external width to intrinsical width.
$\eta=\eta_{L}=2\Gamma^{\prime}/\Gamma$ for Lorentzian distribution and
$\eta=\eta_{G}=\sigma_{f}/\sigma_{g}$ for Gaussian distribution. The
visibility approaches $1/\eta$ when $\eta$ is much larger than $1$.
### II.2 Indistinguishability of single photons from SPDC
In the SPDC, the distinguishability of single photons is induced by the
entanglement between the twinning photon. In order to obtain the information
of the heralded single photon, the entangling parts need to be traced out in
theory. From SPDC, the two-photon state can be written as Ou97QSO ; U'Ren07PRA
$\left|S,I\right\rangle=\iint\nolimits_{-\infty}^{+\infty}d\omega_{S}d\omega_{I}\Phi(\omega_{S},\omega_{I})a_{S}^{{\dagger}}(\omega_{S})a_{I}^{{\dagger}}(\omega_{I})\left|\text{vac}\right\rangle\text{,}$
(15)
where
$\Phi(\omega_{S},\omega_{I})=P(\omega_{S}+\omega_{I})H(\omega_{S},\omega_{I})$
is the two-photon wave function, which contains the information of the pump
beam spectrum $P(\omega_{S}+\omega_{I})$ and the phase matching condition
$H(\omega_{S},\omega_{I})$ in the nonlinear crystal. We assume the pump beam
is transform limited and the spectrum $|P(\omega_{S}+\omega_{I})|^{2}$ is
Gaussian distribution with width $\sigma_{g}$. By making the detection of the
idle photon ($I$) with a single frequency of $\Omega_{I}$, the signal photon
($S$) has the transform limited single photon state from Eq.(15),
$\left|S\right\rangle_{\Omega_{2}}=\int_{-\infty}^{+\infty}d\omega_{S}P(\omega_{S}+\Omega_{I})H(\omega_{S},\Omega_{I})a_{S}^{{\dagger}}(\omega_{S})\left|\text{vac}\right\rangle\text{.}$
(16)
Under the normal phase matching condition for thin nonlinear crystal, the
bandwidth of $H(\omega_{S},\Omega_{I})$ is much larger than the pump width
Grice97PRA ; Kim05OL . Therefore, the $H(\omega_{S},\Omega_{I})$ is slowly
varying function and can be taken outside of the integral. In this case, the
transform limited single photon pulse has the same shape and width of the pump
beam, which can be described with $g_{\omega}(\upsilon)=$
$e^{-(\upsilon-\omega)^{2}/4\sigma_{g}^{2}}/\sqrt[4]{2\pi\sigma_{g}^{2}}$.
Since the actual detection of the idle photon is the sum of the above
detections of different frequency $\Omega_{2}$, the center frequency of the
transform limited single photon pulse has the distribution of $f(\omega)$.
Without loss of generality, we assume that
$f(\omega)=e^{-(\omega-\omega_{c})^{2}/2\sigma_{f}^{2}}/\sqrt{2\pi\sigma_{f}^{2}}$
Grice97PRA ; Kim05OL . Therefore, the heralded single photon can be formally
described in Eq. (2) with intrinsic width $\sigma_{g}$ and extrinsic width
$\sigma_{f}$. The total spectrum is also Gaussian distribution with the width
$\sigma=\sqrt{\sigma_{g}^{2}+\sigma_{f}^{2}}\text{.}$ (17)
Moreover, the indistinguishability of two photons with interval $\tau$ is
calculated,
$K_{G}(\tau)=\frac{\sigma_{g}}{\sigma}e^{-\tau^{2}\sigma_{g}^{2}}\text{.}$
(18)
The two-photon coalescence probability $C_{AB}(\tau)$ for the HOM interference
is
$C_{AB}(\tau)=\frac{1}{2}(1-\frac{\sigma_{g}}{\sigma}e^{-\tau^{2}\sigma_{g}^{2}})$
(19)
with the visibility of $K_{G}(0)=\sigma_{g}/\sigma$, which is also shown in
Fig. 2(b) with different ratios of $\eta_{G}=\sigma_{f}/\sigma_{g}$. Moreover,
the indistinguishability approaches to $1/\eta_{G}$ with large extrinsic
width. If the two-photon wave function can be factorized, i.e.
$\Phi(\omega_{S},\omega_{I})=\Phi(\omega_{S})\Phi(\omega_{I})$, the single
photons is transform limited Ou97QSO ; U'Ren07PRA ; Mosley08PRL . In this
case, $\sigma_{f}=0$ and the single photons are indistinguishable.
From the results of HOM interference, Eq. (14) and Eq. (19), the width of
indistinguishability, or the two-photon fourth-order coherence, only depends
on the intrinsic width, or the lifetime of the transform limited pulse.
However, the total single photon spectral width determines the width of the
single photon second-order coherence time.
## III Discussion
### III.1 The definition and the experimental enhancement of
indistinguishability
From the single photon state, the indistinguishability is described in Eq.
(3), which is the purity of the state if the single photons are generated in
the same source. For the single photons from the different source, the
indistinguishability has the description of
$K_{ij}=tr(\rho_{i}\otimes\rho_{j})$. At the same time, from the multi-mode
theory, the indistinguishability is described as $\mathcal{E}/\mathcal{A}$,
where $\mathcal{E}$($\mathcal{A}$) is the excess (accidental) two-photon
probability Ou99PRA . In Refs. Sun07PRA ; Sun08PRA , the indistinguishability
is derived from the coefficients of Schmidt decomposition. All of these
definitions are equivalent.
It needs to be emphasized that the extrinsic spectral width comes from the
entanglement with extrinsic system. Only this extrinsic spectral width will
bring the distinguishability. In the above discussion, we assumed that all
other DOFs of the single photon have the same states and no entanglement with
the frequency DOF. Actually, the entanglement between the frequency DOF and
other inner DOF of the same photon may induce the mixed spectrum description.
However, for the same entanglement, the mixed spectrum will not induce the
distinguishability when all the DOFs are included, since the entangled state
can be described as a linear superposition form for the single photon in a
higher dimensional space.
Practically, in order to enhance the indistinguishability, different methods
are needed to narrow the extrinsic spectral width or broaden the intrinsic
spectral width. For the quantum emitters, low temperature is needed to reduce
the interaction with phonons. In this case, the dephasing time is extended
Borri01PRL and the extrinsic spectral broadening is controlled. Moreover, the
interaction with optical cavity mode will decrease the lifetime of the
spontaneous emission through Purcell effect Purcell46PR . Therefore, the
intrinsic width is broadened and the indistinguishability is enhanced
Santori02NAT . In SPDC, particular design on the phase matching condition
helps to generate indistinguishable single photons U'Ren07PRA ; Mosley08PRL .
However, the usage of spectral filter is the most feasible method to enhance
the indistinguishability, especially in the experiment on SPDC.
### III.2 The effect of spectral filter
In experiment, the narrow spectral filter is widely used to enhance the
indistinguishability and interference visibility. Theoretically, the detection
operator after the spectral filter can be described as
$E^{(+)}(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}d\omega
F(\omega)a(\omega)e^{-i\omega t}\text{,}$ (20)
where $|F(\omega)|^{2}$ is the spectral transmissivity of the filter. Here we
assume the Gaussian distribution of
$|F(\omega)|^{2}=e^{-(\omega-\omega_{c})^{2}/2\sigma_{F}^{2}}$, centered same
at $\omega_{c}$ with width $\sigma_{F}$.
With the spectral filter, the spectrum of the single photons is also Gaussian
distribution and its width narrows to
$\sigma^{\prime}=\frac{\sigma_{F}\sqrt{\sigma_{g}^{2}+\sigma_{f}^{2}}}{\sqrt{\sigma_{g}^{2}+\sigma_{f}^{2}+\sigma_{F}^{2}}}\text{.}$
(21)
At the same time, the filter narrows the intrinsic width,
$\sigma_{g}^{\prime}=\frac{\sigma_{F}\sigma_{g}}{\sqrt{\sigma_{g}^{2}+\sigma_{F}^{2}}}\text{.}$
(22)
Using Eq.(18), the indistinguishability is
$K_{G}^{\prime}=\frac{\sigma_{g}\sqrt{\sigma_{g}^{2}+\sigma_{f}^{2}+\sigma_{F}^{2}}}{\sqrt{\sigma_{g}^{2}+\sigma_{F}^{2}}\sqrt{\sigma_{g}^{2}+\sigma_{f}^{2}}}\text{,}$
(23)
for $\tau=0$.
More rigorously, the effect of the spectral filter extended from Eq.(10) is
described as:
$\displaystyle K_{G}^{\prime}$ $\displaystyle=$
$\displaystyle\iint\nolimits_{-\infty}^{+\infty}d\omega_{i}d\omega_{j}f(\omega_{i})f(\omega_{j})\int_{-\infty}^{+\infty}dt$
$\displaystyle\times\left|\left\langle\omega_{j}\right|E^{(-)}(t)E^{(+)}(t)\left|\omega_{i}\right\rangle\right|^{2}/C^{2}$
$\displaystyle=$
$\displaystyle\iint\nolimits_{-\infty}^{+\infty}d\omega_{i}d\omega_{j}f(\omega_{i})f(\omega_{j})$
$\displaystyle\times\left|\int d\upsilon
F(\upsilon)g_{\omega_{i}}(\upsilon)F^{\ast}(\upsilon)g_{\omega_{j}}^{\ast}(\upsilon)\right|^{2}/C^{2}$
$\displaystyle=$
$\displaystyle\frac{\sigma_{g}\sqrt{\sigma_{g}^{2}+\sigma_{f}^{2}+\sigma_{F}^{2}}}{\sqrt{\sigma_{g}^{2}+\sigma_{F}^{2}}\sqrt{\sigma_{g}^{2}+\sigma_{f}^{2}}}\text{,}$
(25)
where $C$ is the probability to detect the single-photon after the filter
$\displaystyle C$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}d\omega
f(\omega)\int_{-\infty}^{+\infty}dt\left\langle\omega\right|E^{(-)}(t)E^{(+)}(t)\left|\omega\right\rangle$
(26) $\displaystyle=$
$\displaystyle\frac{\sigma_{F}}{\sqrt{\sigma_{g}^{2}+\sigma_{f}^{2}+\sigma_{F}^{2}}}\text{.}$
(27)
Certainly there is photon loss for $C<1$ when using the filter to enhance the
indistinguishability.
Figure 3: (color online) Indistinguishability with different widths of
filters. (a) Two photons coalescence probability $C_{AB}(\tau)$ of the HOM
interference without filter (red dashed curve) and with filters (green solid
curve for $R=0.5$ and blue dotted curve for $R=0.1$). Here we set $\eta_{G}=3$
for all three cases. The two curves with filters are normalized to the maximal
probability of $1/2$ for total distinguishable cases
($\tau\gg\sigma_{g}^{\prime}$). (b) The red solid and the green dashed curves
show the indistinguishability for $\eta_{G}=3$ and $\eta_{G}=10$ with Gaussian
filters, respectively. The red solid ($\eta_{L}=3$) and green open
($\eta_{L}=10$) circles are for the corresponding results of Lorentzian filter
on Lorentzian spectrum distribution. In this case, $R=2\Gamma_{F}/\Gamma$,
where $\Gamma_{F}$ is the Lorenzian filter width.
Fig. 3 shows the effect on the indistinguishability with different ratios of
spectral filter width to intrinsic width, $R=\sigma_{F}/\sigma_{g}$. In Fig.
3(a), the width of HOM dip is broadened to $1/\sigma_{g}^{\prime}$, since the
intrinsic width is narrowed by the spectral filter in Eq.(22). Fig. 3(b) shows
the indistinguishability with $\eta_{G}=3$ (red solid) and $\eta_{G}=10$
(green dashed). In comparison, the results of the Lorentzian filter on
Lorentzian spectrum distribution are also shown in Fig. 3(b) with red solid
($\eta_{L}=3$) and green open ($\eta_{L}=10$) circles. These results are
numerically calculated with Eq.(III.2) and Eq.(26). Clearly, the
indistinguishability is approaching to $1$ when the filter width is closing to
$0$. For $\eta\gg 1$, the value of the indistinguishability shows the same
result as in Fig. 2(b), where the extrinsic width is replaced by the filter
width.
In SPDC, for the pump pulse duration of $110$fs (full width at half maximum),
the indistinguishability with a full width at half maximum $3$nm filter is
about $0.94$ for $\eta\gtrsim 3$ Kim05OL . It is little higher than the
experimental results in Xiang06PRL ; Sun08EPL because there may be
entanglement in other degrees of freedom between the twin photons besides the
frequency entanglement Sun07PRA . Here, we used the condition that the single
photon intrinsic width $\sigma_{g}$ is same with the pump beam width for thin
nonlinear crystal.
### III.3 Independent photons from many quantum emitters
In some cases, there is more than one independent photon from many quantum
emitters. The total state is
$\rho_{N}=\prod_{i=1}^{N}\rho_{k}\text{.}$ (28)
where
$\rho_{k}=(C\left|\text{vac}\right\rangle\left\langle\text{vac}\right|+\int_{-\infty}^{+\infty}d\omega
f_{k}(\omega)\left|\omega\right\rangle\left\langle\omega\right|)$ is for the
independent single photon with $C=1-\int_{-\infty}^{+\infty}d\omega
f_{k}(\omega)$. Considering the photon loss in the practical experiment and
quantum efficiency of the quantum emitters, $\int_{-\infty}^{+\infty}d\omega
f_{k}(\omega)<1$. Moreover, $f_{k}(\omega)$ may have different center
wavelengthes. For example, there is size distribution of quantum dots. In this
case, the total spectrum will include the broadening from size distribution.
Therefore, the spectrum is very broad and the photons will be distinguishable
even at the low temperature.
## IV Conclusion
The description of the single photons state in the spectrum domain is
presented to discuss the indistinguishability. The ratio of extrinsic spectrum
width to intrinsic width governs the indistinguishability. Single photons are
indistinguishable only when they have the same transform limited forms, while
they are highly distinguishable when the extrinsic spectrum width is much
larger than the intrinsic width. Fundamentally, the indistinguishability of
independent photons shows the sameness of part which can be described with
pure state and only the indistinguishable parts can interfere each other. In
experiment, the indistinguishability shows excess probability of two-photon
coincident detection in Hanbury-Brown-Twiss interferometer HBT or less
probability in HOM interferometer. Moreover, the indistinguishability can be
experimentally enhanced with the narrow spectral filter or by controlling the
generation condition.
###### Acknowledgements.
F.W.S. thanks Z. Y. Ou for helpful discussion. This work is funded by DARPA,
NSF Contract No. ECCS 0747787, and the New York State Office of Science,
Technology and Academic Research.
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|
arxiv-papers
| 2008-11-17T16:03:43 |
2024-09-04T02:48:58.834643
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. W. Sun and C. W. Wong",
"submitter": "Fangwen Sun",
"url": "https://arxiv.org/abs/0811.2739"
}
|
0811.2756
|
# Quantum Thermodynamic Cycles and Quantum Heat Engines (II)
H. T. Quan Theoretical Division, MS B213, Los Alamos National Laboratory, Los
Alamos, NM, 87545, U.S.A.
###### Abstract
We study the quantum mechanical generalization of force or pressure, and then
we extend the classical thermodynamic isobaric process to quantum mechanical
systems. Based on these efforts, we are able to study the quantum version of
thermodynamic cycles that consist of quantum isobaric process, such as quantum
Brayton cycle and quantum Diesel cycle. We also consider the implementation of
quantum Brayton cycle and quantum Diesel cycle with some model systems, such
as single particle in 1D box and single-mode radiation field in a cavity.
These studies lay the microscopic (quantum mechanical) foundation for Szilard-
Zurek single molecule engine.
###### pacs:
05.90.+m, 05.70.-a, 03.65.-w, 51.30.+i
## I INTRODUCTION
Quantum thermodynamics is the study of heat and work dynamics in quantum
mechanical systems quantum thermodynamics . In the extreme limit of small
systems with only a few degrees of freedom, both the finite-size effect and
quantum effects influence the thermodynamic properties of the system
dramatically phystoday05 ; scully03 ; quan06 . The traditional thermodynamic
theory based on classical systems of macroscopic size does not apply any more,
and the quantum mechanical generalization of thermodynamics becomes necessary.
The interplay between thermodynamics and quantum physics has been an
interesting research topic since 1950s earlydays . In recent years, with the
developments of nanotechnology and quantum information processing, the study
of the interface between quantum physics and thermodynamics begins to attract
more and more attention moreattention . Studies of quantum thermodynamics not
only promise important potential applications in nanotechnology and quantum
information processing, but also bring new insights to some fundamental
problems of thermodynamics, such as Maxwell’s demon and the universality of
the second law maxwell . Among all the studies about quantum thermodynamics, a
central concern is to make quantum mechanical extension of classical
thermodynamic processes and cycles quan07 .
It is well know that in classical thermodynamics there are four basic
thermodynamic processes: adiabatic process, isothermal processes, isochoric
process, and isobaric process fourprocesses . These four processes correspond
to constant entropy, constant temperature, constant volume, and constant
pressure, respectively. From these four basic thermodynamic processes, we can
construct all kinds of thermodynamic cycles, such as Carnot cycle, Otto cycle,
Brayton cycle, et al atoz . Among all the four kinds of basic thermodynamic
processes, adiabatic process has been extended to quantum domain, and has been
extensively studied ever since the born of quantum mechanics. Nevertheless no
attention was paid to the quantum mechanical generalization of the remaining
three basic thermodynamic processes until most recently. In a recent paper
quan07 , along with our collaborators, we systematically study the quantum
mechanical generalization of the isothermal and the isochoric process. Base on
these studies, the properties of quantum Carnot cycle and quantum Otto cycle
are clarified. Meanwhile in recent years, numerous studies on other quantum
thermodynamic cycles are also reported numerousstudy . However, as to our best
knowledge, the quantum mechanical generalization of isobaric process (constant
pressure) has not been studied systematically so far. Possibly the lack of the
consideration of quantum isobaric process is due to the fact that “pressure”
(force) pressure is not a well defined observable in a quantum mechanical
system. Because of the short of a well defined “pressure” (force) and thus the
quantum isobaric process, the properties of quantum thermodynamic cycles that
consist of quantum isobaric process, such as quantum Brayton cycle and quantum
Diesel cycle fourprocesses ; atoz cannot be clarified. We notice that some
discussions about quantum Brayton cycle have been reported brayton .
Nevertheless, their definitions of quantum isobaric process and quantum
Brayton cycle are ambiguous, and even in contradiction sometimes. As a result,
in their studies they cannot bridge the quantum and classical thermodynamic
cycles.
In this paper, along with our previous effort quan07 , we will focus on the
study of the quantum isobaric process atoz and its related quantum
thermodynamic cycles. We begin with the definition of “pressure” for an
arbitrary quantum system, and then generalize the isobaric process to quantum
mechanical systems. Based on this and our previous quan07 generalizations of
thermodynamic processes, we are able to study an arbitrary quantum
thermodynamic cycle constructed by any of these four quantum thermodynamic
processes. As an example, we will discuss the quantum Brayton cycle and the
quantum Diesel cycle and compare their properties with their classical
counterpart. Comparisons between these quantum thermodynamic cycles and their
classical counterparts enable us to extend our understanding about the
thermodynamics at the interface of classical and quantum physics. This paper
is organized as follows: In Sec. II, we define microscopically “pressure” for
an arbitrary quantum mechanical system and study the quantum mechanical
generalization of isobaric process; in Sec. III, we study quantum Brayton
cycle and study how the efficiency of Brayton cycle bridges quantum and
classical thermodynamic cycles; in Sec. IV we study quantum Diesel cycle in
comparison with their classical counter part; Sec. V is the remarks and
conclusion.
## II QUANTUM ISOBARIC PROCESS
### II.1 Pressure in quantum-mechanical system
In order to study the quantum isobaric process, we must first study pressure
in an arbitrary quantum mechanical system. Let us recall that in some previous
work quan07 ; workandheat , heat and work have been extended to quantum
mechanical systems and expressed as functions of the eigenenergies $E_{n}$ and
probability distributions $P_{n}$. The first law of thermodynamics has also
been generalized to quantum mechanical systems:
$\displaystyle\begin{split}{\mathchar
22\relax\mkern-11.0mud}Q&=\sum_{n}E_{n}dP_{n},\\\ {\mathchar
22\relax\mkern-11.0mud}W&=\sum_{n}P_{n}dE_{n},\\\ dU&={\mathchar
22\relax\mkern-11.0mud}Q+{\mathchar
22\relax\mkern-11.0mud}W=\sum_{n}(E_{n}dP_{n}+P_{n}dE_{n}),\end{split}$ (1)
where $E_{n}$ is the $n$th eigenenergy of the quantum mechanical system with
the Hamiltonian $H=\sum_{n}E_{n}\left|n\right\rangle\left\langle n\right|$
under consideration; $P_{n}$ is the occupation probability in the $n$th
eigenstate; $\left|n\right\rangle$ is the $n$th eigenstate of the Hamiltonian.
The density matrix of the system can be written as
$\rho=\sum_{n}P_{n}\left|n\right\rangle\left\langle n\right|$. ${\mathchar
22\relax\mkern-11.0mud}Q$ and ${\mathchar 22\relax\mkern-11.0mud}W$ depict the
heat exchange and work done respectively during a thermodynamic process. From
classical thermodynamics we know that the first law can be expressed as
$dU={\mathchar 22\relax\mkern-11.0mud}Q+{\mathchar
22\relax\mkern-11.0mud}W=TdS+\sum_{n}Y_{n}dy_{n}$. Here, $T$ and $S$ refer to
temperature and the thermodynamic entropy; $Y_{n}$ is the generalized force,
and $y_{n}$ is generalized coordinate corresponding to $Y_{n}$ ($dy_{n}$ is
the generalized displacement) generalizedforce . Inversely, the generalized
force conjugate to the generalized coordinate $y_{n}$ can be expressed as
onsager
$Y_{n}=-\frac{{\mathchar 22\relax\mkern-11.0mud}W}{dy_{n}}.$ (2)
For example, when the generalized coordinate is chosen to be the volume $V$,
we have its corresponding generalized force – pressure $P=-{\mathchar
22\relax\mkern-11.0mud}W/dV$. Motivated by the definition of the generalized
force for a classical system, we define analogously the force (for 1D system,
force is the same as pressure) for a quantum mechanical system
$F=-\frac{{\mathchar
22\relax\mkern-11.0mud}W}{dL}=-\sum_{n}P_{n}\frac{dE_{n}}{dL},$ (3)
where $L$ is the generalized coordinate corresponding to the force $F$. In
obtaining Eq. (3), we have used the expression of work for a quantum system
${\mathchar 22\relax\mkern-11.0mud}W=\sum_{n}P_{n}dE_{n}$. For a single
particle in a 1D box (1DB) szilard (see Fig. 1), the generalized coordinate
is the width of the potential, and the eigenenergies for such a system depend
on the generalized coordinate $E_{n}(L)=(\pi\hbar n)^{2}/(2mL^{2})$. Here
$\hbar$ is the Plank’s constant; $n$ is the quantum number; $m$ is the mass of
the particle. We obtain the derivative of $E_{n}(L)$ over $L$
straightforwardly $\frac{dE_{n}}{dL}=-2\frac{E_{n}(L)}{L}$. When the system is
in thermal equilibrium with a heat bath at a inverse temperature
$\beta=\frac{1}{kT}$, the force exerting on either wall of the potential can
be calculated by substituting $\frac{dE_{n}}{dL}$ and the Gibbs distribution
$P_{n}=\frac{1}{Z}e^{-\beta E_{n}}$ into Eq. (3). Here $k$ is the Boltzmann’s
constant, and $Z=\sum_{n}e^{-\beta E_{n}}$ is the partition function.
Alternatively, the expression of force (3) in a quantum mechanical system can
be obtained in a statistical-mechanical way pathria .
$\begin{split}F&=-\left(\frac{\partial\mathbb{F}}{\partial
L}\right)_{T}=kT\left(\frac{\partial\ln{Z}}{\partial
L}\right)_{T}=kT\frac{1}{Z}\frac{\partial}{\partial L}\sum_{n}e^{-\beta
E_{n}}\\\ &=-\sum_{n}\left(\frac{e^{-\beta E_{n}}}{Z}\right)\frac{\partial
E_{n}}{\partial L}=-\sum_{n}P_{n}\frac{dE_{n}}{dL},\end{split}$ (4)
where $\mathbb{F}=-kT\ln{Z}$ is the free energy of the quantum system. It
should be pointed out that Eq. (3) is more general than Eq. (4) because Eq.
(3) stands no matter the system is in equilibrium or not. When $P_{n}$ in Eq.
(3) satisfies Gibbs distribution $P_{n}=\frac{1}{Z}e^{-\beta E_{n}}$, or the
system is in thermal equilibrium, the expectation value of force $F$ of Eq.
(3) reproduce the usual force in classical thermodynamics.
Another model example is the single particle in a 1D harmonic oscillator
potential (1DH) (see Fig. 2). Its Hamiltonian is the same as the Hamiltonian
of a single-mode radiation field in a cavity scully03 . For such 1DH, we will
see later that the definition of force (3) for 1DH agrees with the radiation
pressure (4) of a single mode radiation field. We would like to mention that
the definition of force in (3) is a further step in quantum thermodynamics
after the definitions of heat and work (1). We will see all these definitions
of work, heat, entropy and pressure for a quantum mechanical system are self-
consistent, and consistent with classical thermodynamics.
Figure 1: Schematic diagram of pressure in quantum mechanical system (single
particle in 1D box). One wall (A) of the square well is fixed, while the other
one (B) is movable. The force acting on the wall B by the quantum system
constrained in the potential well can be calculated from Eq. (3)
### II.2 Quantum isobaric process
Figure 2: Radiation pressure of a single-mode radiation field. Here the width
of the potential is inversely proportional to the mode frequency of the cavity
$L\propto\frac{1}{\omega}$. The $n$th eigenenergy of the single mode radiation
field with the potential width $L_{\alpha}$ is given by
$E_{n}(L_{\alpha})=(n+\frac{1}{2})\frac{\hbar s\pi c}{L_{\alpha}}$,
$\alpha=A,B$. This pressure is equivalent to the pressure in a quantum
mechanical system – single particle in 1D harmonic oscillator. Hence quantum
isobaric process based on single-mode radiation field is equivalent to that
based on a 1D quantum harmonic oscillator.
Having clarified force for a quantum mechanical system, in the following we
study how to extend classical isobaric process to a quantum mechanical system.
Classical isobaric process is a quasi-static thermodynamic process, in which
the pressure of the system remains a constant fourprocesses ; atoz . The time
scale of relaxation of the system with the heat bath is much shorter than the
time scale of controlling the volume of the system relaxation . In a classical
isobaric process, in order to achieve a constant pressure, we must carefully
control the temperature of the system, i.e., carefully control the temperature
of the heat bath, when we change the volume of the classical system atoz . For
example, for the classical idea gas with the equation of state $PV=NkT$, the
temperature of the system in the isobaric process is required to be
proportional to the volume$(T\propto V)$ of the gas, so that the pressure can
remains a constant. For a quantum mechanical system, however, the change of
the temperature of the heat bath with the generalized coordinate may not be so
obvious as the classical ideal gas. Because we usually do not know the
equation of state of a quantum mechanical system. Let us consider the quantum
isobaric process based on 1DB (see Fig. 1). For such a quantum mechanical
system, the pressure on the wall can be obtained from Eq. (3)
$\begin{split}F&=-\sum_{n}P_{n}(L)\frac{dE_{n}(L)}{dL}\\\
&=-\sum_{n}\frac{\exp[-\beta(L)E_{n}(L)]}{Z(L)}\frac{dE_{n}(L)}{dL}\\\
&=-\sum_{n}\frac{\exp[-\beta(L)\frac{\pi^{2}\hbar^{2}n^{2}}{2mL^{2}}]}{\frac{1}{2}\sqrt{\frac{2mL^{2}}{\pi\hbar^{2}\beta(L)}}}\times\frac{(-2)}{L}\times\frac{\pi^{2}\hbar^{2}n^{2}}{2mL^{2}}\\\
&=\frac{4}{L}\sqrt{\frac{\pi\hbar^{2}\beta(L)}{2mL^{2}}}\left[-\frac{\partial}{\partial\beta(L)}\sum_{n}\exp[-\beta(L)\frac{\pi^{2}\hbar^{2}n^{2}}{2mL^{2}}]\right]\\\
&=\frac{4}{L}\sqrt{\frac{\pi\hbar^{2}\beta(L)}{2mL^{2}}}\left[-\frac{\partial}{\partial\beta(L)}\frac{1}{2}\sqrt{\frac{2mL^{2}}{\pi\hbar^{2}\beta(L)}}\right]\\\
&=\frac{1}{L\beta(L)}.\end{split}$ (5)
Eq. (5) can be regarded as the equation of state $FL=kT$ for 1DB obtained from
Eq. (3), and it means that if we want to keep the pressure $F$ as a constant,
we must control the temperature of the system to be proportional to the width
of the potential well $\beta(L)=1/(FL)$ when the system inside the box pushes
one of the walls to perform work. This property of 1DB is the same as the
classical ideal gas. We will see more analogues between them later. It should
be mentioned that the temperature function $\beta(L)$ of the “volume” in a
quantum isobaric process is system-dependent. I.e., for different quantum
systems, the function of the temperature over the “volume” in the quantum
isobaric process differs from one to another. In the following we consider the
quantum isobaric process based on a single mode radiation field in a cavity,
which was first proposed as the working substance for a quantum heat engine in
Ref. scully03 . We assume that the cavity of length $L$ and cross-section $A$
can support only a single mode of the field $\omega=\frac{s\pi c}{L}$, where
$s$ is an integer, and $c$ is the speed of light. The Hamiltonian reads
$H=\sum_{n}(n+\frac{1}{2})\hbar\omega\left|n\right\rangle\left\langle
n\right|,$ (6)
where $\left|n\right\rangle$ is the Fock state of the radiation field. From
Eq. (3) we obtain the radiation force $F$ as a function of the temperature
$\beta$ and the length of the cavity $L$ casimir
$\begin{split}F&=-\sum_{n}\frac{e^{-\beta(L)E_{n}(L)}}{Z(L)}\frac{dE_{n}(L)}{dL}\\\
&=-\frac{1}{1-e^{-\beta(L)\hbar\omega}}\sum_{n}e^{-\beta(L)n\hbar\omega}\left[(n+\frac{1}{2})\hbar\omega\right]\frac{1}{L},\\\
&=\left[\frac{\hbar\frac{s\pi c}{L}}{e^{\beta(L)\hbar\frac{s\pi
c}{L}}-1}+\frac{1}{2}\hbar\frac{s\pi c}{L}\right]\frac{1}{L}.\end{split}$ (7)
From Eq. (7) it can be inferred that in order to achieve a constant force, we
must carefully control the temperature of the heat bath in the following
subtle way
$\beta(L)=\frac{L}{\hbar s\pi c}\ln{\frac{2FL^{2}+\hbar s\pi c}{2FL^{2}-\hbar
s\pi c}}.$ (8)
It can be seen that in a quantum isobaric process, the temperature function
(8) for the single-mode radiation field is much more complicated than that
$(\beta(L)\propto\frac{1}{L})$ of 1DB.
For the convenience of later analysis, we would also like to calculate the
entropy and the internal energy of the two systems in a quantum isobaric
process. First we consider the 1DB. The entropy expression can be obtained
from the above Eq. (1) quan07 .
$\begin{split}S(L)&=k_{B}\left[\frac{1}{2}+\ln\left(\frac{1}{2}\sqrt{\frac{2mL^{2}}{\pi\hbar^{2}\beta(L)}}\right)\right].\end{split}$
(9)
Through the comparison with the entropy of classical ideal gas fourprocesses ;
atoz , we find that the entropy of classical idea gas reproduces the entropy
(9) of 1DB if we choose the molecule number of the classical ideal gas to be
$N=1$. We plot the entropy-temperature curve (9) of a quantum isobaric process
in Fig. 3. The internal energy of the 1DB during the isobaric process can also
be obtained analytically (the temperature of the heat bath is time-dependent).
$U(L)=-\sum_{n}\frac{e^{-\beta(L)E_{n}(L)}}{Z(L)}E_{n}(L)=\frac{1}{2\beta(L)}.$
(10)
This expression of internal energy verifies the equipartition theorem
fourprocesses , and justifies the result in Ref. quan07 again: the internal
energy of the 1DB depends only on the temperature. From Eqs. (9) and (10) we
see that both the entropy and the internal energy of 1DB have the same form as
that of the classical ideal gas fourprocesses ; atoz if we choose the
molecule number of the classical ideal gas to be $N=1$. Moreover, from Eq. (5)
we know that 1DB has the same equation of state $FL=kT$ as that of the
classical ideal gas $PV=NkT$ except the difference of the particle number.
Thus we conclude that 1DB is the quantum mechanical counterpart of the
classical ideal gas. We would like to mention that in the study of the single-
molecule engine by Szilard and Zurek szilard , they simply employ the equation
of state for the classical ideal gas $PV=Nk_{B}T$ and choose the particle
number to be $N=1$. Nevertheless, this treatment may be questionable because
the equation of state $PV=Nk_{B}T$ stands in the macroscopic and classical
case. When we come to the extreme limit of small system with only a few
degrees of freedom, we must use the quantum mechanical treatment as we present
here. Fortunately, all the treatments of the single molecule engine szilard
by Szilard and Zurek is in accordance with our quantum mechanical treatments.
Thus we say that our discussions lay the foundation for Szilard-Zurek single
molecule engine szilard .
As to the single-mode radiation field, the entropy and the internal energy can
be calculated as that in Ref. scully03
$S(L)=\frac{\left\langle n\right\rangle\hbar\omega}{T}+k\ln(\left\langle
n\right\rangle+1)$ (11) $U(L)=\sum_{n}\frac{e^{-\beta
n\hbar\omega}}{Z(L)}(n+\frac{1}{2})\hbar\omega=(\left\langle
n\right\rangle+\frac{1}{2})\hbar\omega$ (12)
where $\left\langle n\right\rangle=[\exp{(\hbar\omega/kT)}-1]^{-1}$ is the
mean photon number.
It is easy to see that the entropy (11) and the internal energy (12) of a
single mode radiation field have different forms from that of 1DB (9),(10),
and thus from the classical ideal gas. The internal energy (12) of single mode
radiation field depends on both the temperature $\beta$ and the width $L$ of
the potential well, while the internal energy of 1DB (10) depends on $\beta$
only. In addition, the equation of state (7) of the single-mode radiation
field differs from from that (5) of 1DB, and thus from the classical ideal
gas. Based on these observations, we say that the single mode radiation field
has totally different thermodynamic properties from that of classical ideal
gas. It can be inferred that quantum heat engine based on single mode
radiation field can give us new results beyond that of classical ideal gas. As
we mentioned before, the Hamiltonian of the single-mode radiation field is the
same as that of 1DH. Thus all the results about single mode radiation field
are the same as that for 1DH. Alternatively, we can say that 1DH is the
counterpart of single-mode photon gas, in analogy to the fact that 1DB is the
counterpart of classical ideal gas. But it should be mentioned that single-
mode photon gas are still quantum mechanical system, while classical ideal gas
are classical system. In the following we will alternatively use 1DH and
single-mode photon gas.
In addition to our previous studies quan07 , up to now we have extended all
four basic thermodynamic processes to quantum mechanical domain. For a
comparison of quantum thermodynamics processes and their classical
counterparts see Table I.
Table 1: Basic classical thermodynamic processes and their quantum counterparts. Here the classical thermodynamic processes are based on classical ideal gas, while the quantum thermodynamic processes are based on the 1DB. We illustrate the equations of state for the four basic thermodynamic processes, and we also indicate the invariant or varying variables in these processes. Here, we use “VRA” to indicate the invariance of a thermodynamic quantity and “VAR” to indicate it varies or changes. | Isothermal ($T\equiv T_{0}$) | Isochoric ($V\equiv V_{0}$ or $L\equiv L_{0}$) | Isobaric ($P\equiv P_{0}$ or $F\equiv F_{0}$) | Adiabatic ($S\equiv S_{0}$)
---|---|---|---|---
Classical | $P(V)V=const$; VRA: S, V, P; INV: T | $\frac{P(T)}{T}=const$; VRA: S, T, P; INV: V | $\frac{V(T)}{T}=const$; VRA: S, T, V; INV: P, | $P(T)V^{3}(T)=const$; VRA: V, T P; INV: S
Quantum | $F(L)L=const$; VRA: $E_{n}$, $P_{n}$; INV: T | $\frac{F(T)}{T}=const$; VRA: T, $P_{n}$; INV: $E_{n}$ | $\frac{L(T)}{T}=const$; VRA: T, $E_{n}$, $P_{n}$; | $F(T)L^{3}(T)=const$; VRA: $E_{n}$, T; INV: , $P_{n}$
## III QUANTUM BRAYTON CYCLE
Figure 3: Temperature-Entorpy $T-S$ diagram of a quantum Brayton cycle (see
Fig. 4) based on 1DB. In two adiabatic processes, $B\longrightarrow C$ and
$D\longrightarrow A$ the entropy remains a constant.
In the preceding section, we extend the classical isobaric process to quantum
mechanical systems based on the definition of pressure (3). In this section
and the next section, we study two kinds of thermodynamic cycles consisting of
the quantum isobaric process, and compare them with their classical
counterparts. We first consider the quantum Brayton cycle based on 1DB. A
quantum Brayton cycle is a quantum mechanical analogue of the classical
Brayton cycle fourprocesses ; atoz , which consists of two quantum isobaric
processes and two quantum aidabatic processes. Similar to our discussion in
Ref. quan07 , the counterpart of classical adiabatic plus quasistatic process
is quantum adiabatic process adiabatic . In constructing quantum Brayton
cycle, we also requires that i) all the energy level spacing of the work
substance change by the same ratio in the quantum adiabatic process, and ii)
this ratio be equal to the ratio of the temperatures of the two heat baths
just before and after the quantum adiabatic process. It should be mentioned
that in the isothermal process of a Carnot cycle, the temperature of the heat
bath is fixed. However, this is not the case in the isobaric process of a
Bayton cycle. Hence, we cannot simply say that the ratio of the change of the
energy level spacings should be equal to the ratio of the temperatures of the
two heat baths quan07 . For the current example, it can be checked that the
change of the energy level spacings in quantum adiabatic process (see Fig. 3)
$B\rightarrow C$ and $D\rightarrow A$ should be equal to the ratio of
temperatures of the heat baths at $B$ and $C$, or at $D$ and $A$ (because
$\frac{T_{B}}{T_{C}}=\frac{T_{A}}{T_{D}}$). Fortunately, the above condition
i) can be satisfied by some quantum mechanical system, such as 1DB, 1DH, two-
level system, et al, and our study will focus on these systems whose energy
level spacings change in the same ratio in the quantum adiabatic process.
Otherwise, the irreversibility will arise adiabatic . We give a Temperature-
Entropy $T-S$ diagram of the quantum Brayton cycle (See Fig. 3). Through a
standard procedure, we obtain (see Appendix A) the efficiency of the quantum
Brayton cycle based on the 1DB
$\eta_{\mathrm{Brayton}}=1-\left(\frac{F_{0}}{F_{1}}\right)^{\frac{2}{3}},$
(13)
where $F_{0}$ and $F_{1}$ are the pressures of the system during two quantum
isobaric processes (See Fig. 4).
Figure 4: Force-Displacement $F-L$ diagram of a quantum Brayton cycle based on
a single particle in 1DB. $A\longrightarrow B$ represents an isobaric
expansion process with a constant force $F_{1}$; $B\longrightarrow C$
represents an adiabatic expansion process with constant entropy $S_{1}$;
$C\longrightarrow D$ represents an isobaric compression process with constant
pressure $F_{0}$; $D\longrightarrow A$ is another adiabatic compression
process with constant entropy $S_{0}$.
We would like to compare this efficiency of the quantum Brayton cycle (13)
with its classical counterpart. From Eq. (9) we know that in the quantum
adiabatic process ($S=const$), we have $TL^{2}=const$. As a result the
adiabatic exponent $\gamma=3$ is obtained through the comparison with
$TL^{\gamma-1}=const$ for the adiabatic process. Let us recall that the
efficiency of a classical Brayton cycle is
$\eta=1-\left(\frac{F_{0}}{F_{1}}\right)^{1-\frac{1}{\gamma}}$ atoz , where
$\gamma$ is the classical adiabatic exponent. Thus our result (13) bridges the
quantum Brayton cycle and classical Brayton cycle. Hence this justifies that
the definition of pressure (3) for a quantum mechanical system is self-
consistent.
Similarly we obtain we obtain the efficiency of a quantum Brayton cycle based
on 1DH (see Appendix A)
$\eta^{\prime}_{\mathrm{Brayton}}=1-\sqrt{\frac{F_{0}}{F_{1}}}.$ (14)
From Eq. (11) we know that in a quantum adiabatic process $TL=const$. Thus
$\gamma=2$ for 1DH is obtained. It can be seen that the efficiency of a
quantum Brayton cycle obtained here (14) is the same as that of a classical
Brayton cycle. Through the discussion of quantum Brayton cycles based on two
model systems 1DH and 1DB, we see that the definition of pressure (3) for
quantum systems has clear physical implication, and our study bridges the
thermodynamic cycles based on quantum and classical systems.
## IV QUANTUM DIESEL CYCLE
Except for the thermodynamic cycles consisting of two pairs of basic
thermodynamic processes, such as Carnot cycle, Otto cycle quan07 , and Brayton
cycle, there are some interesting thermodynamic cycles consisting of more than
two kinds of thermodynamic processes, such as Diesel cycle. The Diesel cycle
consists of two adiabatic processes, one isobaric processes and one isochcoric
process atoz (see Fig. 5). In order to construct such a quantum Diesel cycle,
we require 1) the quantum adiabatic conditions are satisfied, and 2) all
energy level spacings change in the same ratio in the thermally isolated
process adiabatic . Because this is the quantum counterpart of classical
adiabatic process (thermally isolated plus quasistatic process) adiabatic .
Besides, the ratio of the change of the energy level spacings in the quantum
adiabatic process $D\rightarrow A$ should be equal to the ratio
$\frac{T_{A}}{T_{D}}$ of the temperatures of the heat bath at $A$ and at $D$
(see Fig. 5); the energy level spacing at $C$ should be equal to that at point
$D$ (see Fig. 5).
Figure 5: Force-Displacement $F-L$ diagram of a quantum Diesel cycle based on
1DB and single mode radiation field. $A\longrightarrow B$ represents an
isobaric expansion process with a constant pressure $F_{1}$; $B\longrightarrow
C$ represents an adiabatic expansion process with constant entropy;
$C\longrightarrow D$ represents an isochoric compression process with constant
volume $L_{1}$; $D\longrightarrow A$ is another adiabatic compression process.
In the following we will consider implementing the quantum Diesel cycle in 1DB
and in 1DH. First we consider the 1DB. The efficiency of a quantum Diesel
cycle based on 1DB can be obtained through a straightforward calculation (see
Appendix B)
$\eta_{\mathrm{Diesel}}=1-\frac{1}{3}\frac{r_{E}^{3}-r_{C}^{3}}{r_{E}-r_{C}}=1-\frac{1}{3}(r_{E}^{2}+r_{C}r_{E}+r_{C}^{2}).$
(15)
Here $r_{C}\equiv\frac{L_{2}}{L_{1}}$ (see Fig. 5) and
$r_{E}\equiv\frac{L_{3}}{L_{1}}$ (see Fig. 5) are the compression and
expansion ratios of the volumes. This efficiency for a quantum Diesel cycle
based on 1DB agrees with that of a classical Diesel cycle, too. Through a
similar analysis we obtain the efficiency for a quantum Diesel cycle based on
1DH with the only change of $\gamma$ from 3 in Eq. (15) to 2
$\eta^{\prime}_{\mathrm{Diesel}}=1-\frac{1}{2}\frac{r_{E}^{2}-r_{C}^{2}}{r_{E}-r_{C}}=1-\frac{1}{2}(r_{E}+r_{C}).$
(16)
Table 2: Working efficiencies of typical classical thermodynamic cycles and their quantum counterparts based on different kinds of working substance. It can be seen that 1) except the Carnot cycle, the efficiencies of all the thermodynamic cycles are working substance-dependent, and 2) both quantum thermodynamic cycles and classical thermodynamic cycles have the same efficiency as long as the adiabatic exponent is the same. Adiabatic exponents for monoatomic, diatomic, and polyatomic classical idea gas can be found in adiabatic exponent . Here $T_{C}$ and $T_{H}$ are the temperatures of the cold and hot reservoirs; $V_{0}$ ($L_{0}$, $S_{0}$) and $V_{1}$ ($L_{1}$, $S_{1}$) are the volume (length, area) of the working substance in two isochoric processes; $P_{0}$ ($F_{0}$) and $P_{1}$ ($F_{1}$) are the pressure (force) of the working substance in the two isobaric processes. | Carnot (two isothermal + two adiabatic) | Otto (two isochoric+ two adiabatic) | Brayton (two isobaric + two adiabatic) | Diesel (isochoric + isobaric + two isobaric)
---|---|---|---|---
| $\eta=1-\frac{T_{C}}{T_{H}}$ | $\eta=1-\left(\frac{V_{0}}{V_{1}}\right)^{\gamma-1}$ | $\eta=1-\left(\frac{P_{0}}{P_{1}}\right)^{1-\frac{1}{\gamma}}$ | $\eta=1-\frac{1}{\gamma}\frac{\left(\frac{V_{2}}{V_{1}}\right)^{\gamma}-\left(\frac{V_{3}}{V_{1}}\right)^{\gamma}}{\left(\frac{V_{2}}{V_{1}}\right)-\left(\frac{V_{3}}{V_{1}}\right)}$
| Monoatomic classical idea gas ($\gamma=\frac{5}{3}$) | $\eta=1-\frac{T_{C}}{T_{H}}$ | $\eta=1-\left(\frac{V_{0}}{V_{1}}\right)^{\frac{2}{3}}$ | $\eta=1-\left(\frac{P_{0}}{P_{1}}\right)^{\frac{2}{5}}$ | $\eta=1-\frac{3}{5}\frac{\left(\frac{V_{2}}{V_{1}}\right)^{\frac{5}{3}}-\left(\frac{V_{3}}{V_{1}}\right)^{\frac{5}{3}}}{\left(\frac{V_{2}}{V_{1}}\right)-\left(\frac{V_{3}}{V_{1}}\right)}$
Classical | Diatomic classical idea gas ($\gamma=\frac{7}{5}$) | $\eta=1-\frac{T_{C}}{T_{H}}$ | $\eta=1-\left(\frac{V_{0}}{V_{1}}\right)^{\frac{2}{5}}$ | $\eta=1-\left(\frac{P_{0}}{P_{1}}\right)^{\frac{2}{7}}$ | $\eta=1-\frac{5}{7}\frac{\left(\frac{V_{2}}{V_{1}}\right)^{\frac{7}{5}}-\left(\frac{V_{3}}{V_{1}}\right)^{\frac{7}{5}}}{\left(\frac{V_{2}}{V_{1}}\right)-\left(\frac{V_{3}}{V_{1}}\right)}$
| Polyatomic classical idea gas ($\gamma=\frac{4}{3}$) | $\eta=1-\frac{T_{C}}{T_{H}}$ | $\eta=1-\left(\frac{V_{0}}{V_{1}}\right)^{\frac{1}{3}}$ | $\eta=1-\left(\frac{P_{0}}{P_{1}}\right)^{\frac{1}{4}}$ | $\eta=1-\frac{3}{4}\frac{\left(\frac{V_{2}}{V_{1}}\right)^{\frac{4}{3}}-\left(\frac{V_{3}}{V_{1}}\right)^{\frac{4}{3}}}{\left(\frac{V_{2}}{V_{1}}\right)-\left(\frac{V_{3}}{V_{1}}\right)}$
| Single particle in 1D box ($\gamma=3$) | $\eta=1-\frac{T_{C}}{T_{H}}$ | $\eta=1-\left(\frac{L_{0}}{L_{1}}\right)^{2}$ | $\eta=1-\left(\frac{F_{0}}{F_{1}}\right)^{\frac{2}{3}}$ | $\eta=1-\frac{1}{3}\frac{\left(\frac{L_{2}}{L_{1}}\right)^{3}-\left(\frac{L_{3}}{l_{1}}\right)^{3}}{\left(\frac{L_{2}}{L_{1}}\right)-\left(\frac{L_{3}}{L_{1}}\right)}$
| Single particle in 2D box ($\gamma=2$) | $\eta=1-\frac{T_{C}}{T_{H}}$ | $\eta=1-\frac{S_{0}}{S_{1}}$ | $\eta=1-\left(\frac{P_{0}}{P_{1}}\right)^{\frac{1}{2}}$ | $\eta=1-\frac{1}{2}\left(\frac{S_{2}}{S_{1}}-\frac{S_{3}}{S_{1}}\right)$
| Single particle in 3D box ($\gamma=\frac{5}{3}$) | $\eta=1-\frac{T_{C}}{T_{H}}$ | $\eta=1-\left(\frac{V_{0}}{V_{1}}\right)^{\frac{2}{3}}$ | $\eta=1-\left(\frac{P_{0}}{P_{1}}\right)^{\frac{2}{5}}$ | $\eta=1-\frac{3}{5}\frac{\left(\frac{V_{2}}{V_{1}}\right)^{\frac{5}{3}}-\left(\frac{V_{3}}{V_{1}}\right)^{\frac{5}{3}}}{\left(\frac{V_{2}}{V_{1}}\right)-\left(\frac{V_{3}}{V_{1}}\right)}$
| 1D Single mode photon field ($\gamma=2$) | $\eta=1-\frac{T_{C}}{T_{H}}$ | $\eta=1-\frac{L_{0}}{L_{1}}$ | $\eta=1-\left(\frac{F_{0}}{F_{1}}\right)^{\frac{1}{2}}$ | $\eta=1-\frac{1}{2}\left(\frac{L_{2}}{L_{1}}-\frac{L_{3}}{L_{1}}\right)$
Quantum | 3D Black body radiation field ($\gamma=\frac{4}{3}$) | $\eta=1-\frac{T_{C}}{T_{H}}$ | $\eta=1-\left(\frac{V_{0}}{V_{1}}\right)^{\frac{1}{3}}$ | $\eta=1-\left(\frac{P_{0}}{P_{1}}\right)^{\frac{1}{4}}$ | $\eta=1-\frac{3}{4}\frac{\left(\frac{V_{2}}{V_{1}}\right)^{\frac{4}{3}}-\left(\frac{V_{3}}{V_{1}}\right)^{\frac{4}{3}}}{\left(\frac{V_{2}}{V_{1}}\right)-\left(\frac{V_{3}}{V_{1}}\right)}$
| 1D harmonic oscillator ($\gamma=2$) | $\eta=1-\frac{T_{C}}{T_{H}}$ | $\eta=1-\frac{L_{0}}{L_{1}}$ | $\eta=1-\left(\frac{F_{0}}{F_{1}}\right)^{\frac{1}{2}}$ | $\eta=1-\frac{1}{2}\left(\frac{L_{2}}{L_{1}}-\frac{L_{3}}{L_{1}}\right)$
| 2D harmonic oscillator ($\gamma=\frac{3}{2}$) | $\eta=1-\frac{T_{C}}{T_{H}}$ | $\eta=1-\left(\frac{S_{0}}{S_{1}}\right)^{\frac{1}{2}}$ | $\eta=1-\left(\frac{P_{0}}{P_{1}}\right)^{\frac{1}{3}}$ | $\eta=1-\frac{2}{3}\frac{\left(\frac{S_{2}}{S_{1}}\right)^{\frac{3}{2}}-\left(\frac{S_{3}}{S_{1}}\right)^{\frac{3}{2}}}{\left(\frac{S_{2}}{S_{1}}\right)-\left(\frac{S_{3}}{S_{1}}\right)}$
| 3D harmonic oscillator ($\gamma=\frac{4}{3}$) | $\eta=1-\frac{T_{C}}{T_{H}}$ | $\eta=1-\left(\frac{V_{0}}{V_{1}}\right)^{\frac{1}{3}}$ | $\eta=1-\left(\frac{P_{0}}{P_{1}}\right)^{\frac{1}{4}}$ | $\eta=1-\frac{3}{4}\frac{\left(\frac{V_{2}}{V_{1}}\right)^{\frac{4}{3}}-\left(\frac{V_{3}}{V_{1}}\right)^{\frac{4}{3}}}{\left(\frac{V_{2}}{V_{1}}\right)-\left(\frac{V_{3}}{V_{1}}\right)}$
| spin-1/2 (2-level system) ($\gamma=2$) | $\eta=1-\frac{T_{C}}{T_{H}}$ | $\eta=1-\frac{L_{0}}{L_{1}}$ | $\eta=1-\left(\frac{F_{0}}{F_{1}}\right)^{\frac{1}{2}}$ | $\eta=1-\frac{1}{2}\left(\frac{L_{2}}{L_{1}}-\frac{L_{3}}{L_{1}}\right)$
Before concluding this section, we would like to mention that we can also
discuss the quantum Brayton cycle and the quantum Diesel cycle based on an
arbitrary quantum system, such as the 3D black body radiation field or a
spin-1/2 system in an external magnetic field with the Hamiltonian
$H=\frac{1}{2}B\sigma_{z}$. Here $\sigma_{z}$ is the Pauli matrix and $B$ is
the external magnetic field. It can be seen from the Table II that the
efficiencies for both quantum Carnot cycle and classical Carnot cycle are
always equal to the Carnot efficiency $1-\frac{T_{C}}{T_{H}}$ irrespective of
the properties of the working substance. Different from the Carnot cycle, the
efficiencies of Otto Cycle, Brayton cycle and Diesel cycle are working
substance-dependent (see Table II). More specifically, they depend on the
adiabatic exponent $\gamma$ of the working substance. As long as we get the
adiabatic exponent $\gamma$ of the quantum system, we obtain the explicit
expression of the efficiencies of the quantum thermodynamic cycles by
substituting $\gamma$ into the expression of the efficiencies of the classical
thermodynamics with their adiabatic exponent. For example, for a spin-1/2
system in an external magnetic field, we choose the inverse of the magnetic
field strength as the generalized coordinate $L=\frac{1}{B}$. Then it can be
found that the adiabatic exponent for such a system is $\gamma=2$. As a
result, the efficiencies of a quantum Brayton cycle and a quantum Diesel cycle
based on a spin-1/2 system in an external magnetic field are the same as that
based on 1DH (14), (16). Similarly, the efficiencies for a Brayton cycle and a
Diesel cycle based on 3D radiation field can be obtained straightforwardly by
substituting $\gamma$ with the adiabatic exponent $\frac{4}{3}$ fourprocesses
; atoz for 3D radiation field. In Table II we list the working efficiencies
for several typical thermodynamic cycles based on different kinds of classical
and quantum working substances (based on their adiabatic exponent).
## V CONCLUSIONS AND REMARKS
In summary, in this paper, we study the quantum mechanical analogy of the
classical isobaric process based on a microscopic definition of force. In
studying the thermodynamic properties of a small quantum system, we use a new
pair of conjugate variables $P_{n}$ and $E_{n}$ instead of the usual
thermodynamic variables $P$ and $V$ or $T$ and $S$ conjugate . The general
expression of force for an arbitrary quantum system
$F=-\sum_{n}P_{n}\frac{dE_{n}(L)}{dL}$ is found. It can be checked that this
expression is in accordance with the force
$F=-\left(\frac{\partial\mathbb{F}}{\partial L}\right)_{T}$ in statistical
mechanics if the quantum system is in thermal equilibrium with a heat bath. In
addition we clarify the relation between adiabatic process (thermally isolated
plus quasistatic process) in classical systems and quantum adiabatic process
in quantum systems, and we find that all energy level spacings change in the
same ratio in the quantum adiabatic process is essential in simulating the
classical adiabatic process. Otherwise irreversibility will arise adiabatic .
Based on quantum isobaric processes, we make quantum mechanical extension of
some typical thermodynamic cycles. The properties of these quantum
thermodynamic processes and cycles are clarified, and we bridges the quantum
thermodynamic cycles and their classical counterpart. The quantum heat engines
and their classical counterparts have the same efficiencies as long as their
working substance has the same adiabatic exponent. The definitions of force
and work for a single-particle quantum system may have important application
in the experimental exploration of nonequilibrium thermodynamics in small
quantum systems, such as quantum Jarzynski equality and quantum Crooks
Fluctuation Theorem phystoday05 ; quan08 . Though the working substance of
quantum heat engines deviates from thermodynamic limit, we reproduce the
efficiency of classical heat engines. Hence our study lay the concrete
foundation for Szilard-Zurek single molecule engine. Moreover, we found the
close relation between classical ideal gas and 1DB, and between single-mode
photon gas and 1DH.
Before concluding this paper, we would like to mention that in our current
study we focus on the quantum single-particle system, and its related quantum
mechanical generalization of heat, work, pressure, and we regain the results
of classical thermodynamic processes and cycles. We also notice some studies
about quantum heat engines with quantum many body system as the working
substance manybody . For quantum many body system, e.g., ideal bosonic gas or
ideal fermionic gas, the mechanical variable, such as heat work, pressure, are
well defined and their equation of state as well as their expression of
internal energy pathria deviate from that of the classical ideal gas. As a
result, the properties of quantum thermodynamic cycles based on the quantum
many-body system deviate from that of classical ideal gas due to quantum
degeneracy. Finally, similar to the discussion about finite-power Carnot
engine curzon , we can discuss about finite-time quantum Brayton cycle and
quantum Diesel cycle. Finite-power analysis of Brayton cycle and Diesel cycle
will be given later.
## VI acknowledgments
This work is supported by U.S. Department of Energy through the LANL/LDRD
Program. The author thanks Prof. P. W. Milonni for helpful discussions and
Prof. J. Q. You for the hospitality extended to him during his visit to Fudan
university.
## Appendix A OPERATION EFFICIENCY OF QUANTUM BRAYTON CYCLE
According to the definition of heat exchange (1) in the quantum mechanical
system, we obtain the heat absorbed by the system from a time-dependent heat
bath during the quantum isobaric expansion process $A\longrightarrow B$ (see
Fig. 3 and Fig. 4)
$\begin{split}{\mathchar
22\relax\mkern-11.0mud}Q_{AB}=&\int_{L_{A}}^{L_{B}}\left[\sum_{n}E_{n}(L)\frac{dP_{n}(L)}{dL}\right]dL\\\
=&\sum_{n}\int_{L_{A}}^{L_{B}}\left[\left[E_{n}(L)P_{n}(L)\right]^{\prime}-\frac{dE_{n}(L)}{dL}P_{n}(L)\right]dL\\\
=&\sum_{n}[E_{n}(L_{B})P_{n}(L_{B})-E_{n}(L_{A})P_{n}(L_{A})]\\\
&+\int_{L_{A}}^{L_{B}}F(L)dL\\\
=&\frac{1}{2}[F_{1}L_{B}-F_{1}L_{A}]+F_{1}(L_{B}-L_{A})\\\
=&\frac{3}{2}F_{1}(L_{B}-L_{A}).\end{split}$ (1)
In obtaining the above result we have used Eq. (5) and Eq. (10). Similarly, we
obtain the heat released to the time-dependent entropy sink
${\mathchar 22\relax\mkern-11.0mud}Q_{CD}=\frac{3}{2}F_{0}(L_{C}-L_{D}).$ (2)
Hence, the efficiency of the quantum Brayton cycle based on a 1DB can be
expressed as
$\eta_{\mathrm{Brayton}}=1-\frac{F_{0}(L_{C}-L_{D})}{F_{1}(L_{B}-L_{A})}.$ (3)
Due to the equation of motion (5) and the expression of the internal energy
(10), we have $F_{1}\times L_{B}/2=U(L_{B})$, $F_{0}\times L_{C}/2=U(L_{C})$.
In addition to the relation of the internal energies in the quantum adiabatic
process $B\longrightarrow C$
$\frac{U(L_{B})}{U(L_{C})}=\left(\frac{L_{C}}{L_{B}}\right)^{2},$ (4)
we have
$\frac{F_{1}}{F_{0}}=\left(\frac{L_{C}}{L_{B}}\right)^{3}$ (5)
for the quantum adiabatic process $B\longrightarrow C$. Through a similar
analysis we obtain
$\frac{F_{1}}{F_{0}}=\left(\frac{L_{D}}{L_{A}}\right)^{3}$ (6)
for another quantum adiabatic process $D\longrightarrow A$. Based on all the
above results (A3), (A5), and (A6), we obtain the efficiency of the quantum
Brayton cycle based on the 1DB
$\eta_{\mathrm{Brayton}}=1-\left(\frac{F_{0}}{F_{1}}\right)^{\frac{2}{3}}.$
(7)
In the following we consider a quantum Brayton cycle based on 1DH. Similar to
the above analysis, we calculate the heat absorbed by the system during the
quantum isobaric expansion process $A\longrightarrow B$ (see Fig. 3)
$\begin{split}{\mathchar
22\relax\mkern-11.0mud}Q_{AB}=&\int_{L_{A}}^{L_{B}}\left[\sum_{n}E_{n}(L)\frac{dP_{n}(L)}{dL}\right]dL\\\
=&[U(L_{B})-U(L_{A})]+\int_{L_{A}}^{L_{B}}Fd(L)\\\
=&\left(\frac{\hbar\omega_{B}}{e^{\beta(L_{B})\hbar\omega_{B}}-1}+\frac{\hbar\omega_{B}}{2}\right)\\\
&-\left(\frac{\hbar\omega_{A}}{e^{\beta(L_{A})\hbar\omega_{A}}-1}+\frac{\hbar\omega_{A}}{2}\right)+F_{H}(L_{B}-L_{A})\\\
=&F_{1}(L_{B}-L_{A}),\end{split}$ (8)
where we have used the relations (7) and (12) in the quantum isobaric process
($A\longrightarrow B$). Similarly, we obtain the heat released to the entropy
sink in another quantum isobaric process $C\longrightarrow D$
${\mathchar 22\relax\mkern-11.0mud}Q_{CD}=F_{0}\left(L_{C}-L_{D}\right).$ (9)
The efficiency of the quantum Brayton cycle based on a 1DH can be expressed as
$\eta^{\prime}_{\mathrm{Brayton}}=1-\frac{F_{0}(L_{C}-L_{D})}{F_{1}(L_{B}-L_{A})}.$
(10)
From Eqs. (7) and (12) we have $F_{1}\times L_{B}=U(L_{B})$ and $F_{0}\times
L_{C}=U(L_{C})$. In addition to the relation of internal energy in the quantum
adiabatic process
$\frac{U(L_{B})}{U(L_{C})}=\frac{L_{C}}{L_{B}},$ (11)
we have
$\frac{F_{1}}{F_{0}}=\left(\frac{L_{C}}{L_{B}}\right)^{2}.$ (12)
Hence, from Eqs. (A10) and (A12) we obtain the efficiency of a quantum Brayton
cycle based on 1DH
$\eta^{\prime}_{\mathrm{Brayton}}=1-\sqrt{\frac{F_{0}}{F_{1}}}.$ (13)
## Appendix B OPERATION EFFICIENCY OF QUANTUM DISEL CYCLE
For a quantum Diesel cycle (see Fig. 5), the input energy in the quantum
isobaric process $A\rightarrow B$ and the output energy in the quantum
isochoric process $C\rightarrow D$ can be calculated as
$\begin{split}Q_{in}&=C_{P}(T_{B}-T_{A}),\\\
Q_{out}&=C_{V}(T_{C}-T_{D}),\end{split}$ (14)
where $C_{P}$ and $C_{V}$ are the heat capacity at constant pressure and
constant volume respectively; $T_{A}$, $T_{B}$, $T_{C}$, and $T_{D}$ are the
temperatures of the system at different points of the Diesel cycle (see Fig.
5). Thus the efficiency of the quantum Diesel cycle can be expressed in terms
of heat capacities and temperatures
$\eta=\frac{Q_{in}-Q_{out}}{Q_{in}}=1-\frac{C_{V}(T_{C}-T_{D})}{C_{P}(T_{B}-T_{A})}.$
(15)
It is convenient to express this efficiency (B2) in terms of compression
ration $r_{C}\equiv\frac{L_{2}}{L_{1}}$ (see Fig. 5) and the expansion ratio
$r_{E}\equiv\frac{L_{3}}{L_{1}}$ (see Fig. 5) of the volumes. Now using the
equation of state $FL=kT$ (5) and $\frac{C_{P}}{C_{V}}=\gamma=3$ for 1DB, the
efficiency (B2) can be rewritten as
$\eta=1-\frac{1}{3}\frac{(F_{C}L_{C}-F_{D}L_{D})}{(F_{B}L_{B}-F_{A}L_{A})}.$
(16)
By utilizing the facts $L_{C}=L_{D}=L_{1}$ and $F_{A}=F_{B}=F_{1}$ (see Fig.
5), we further simplify the Eq. (B3) to
$\eta=1-\frac{1}{3}\frac{L_{1}(F_{C}-F_{D})}{F_{1}(L_{B}-L_{A})}=1-\frac{1}{3}\frac{(\frac{F_{C}}{F_{1}}-\frac{F_{D}}{F_{1}})}{(r_{E}-r_{C})}.$
(17)
Finally by making use of the adiabatic condition $FL^{3}=const$ for 1DB in the
quantum adiabatic process, we obtain
$\begin{split}\frac{F_{C}}{F_{1}}&=\left(\frac{L_{3}}{L_{1}}\right)^{3}=r_{E}^{3},\\\
\frac{F_{D}}{F_{1}}&=\left(\frac{L_{2}}{L_{1}}\right)^{3}=r_{C}^{3}.\end{split}$
(18)
Substituting Eq. (B5) into Eq. (B4), the efficiency of a quantum Diesel cycle
based on 1DB can be written as
$\eta_{\mathrm{Diesel}}=1-\frac{1}{3}\frac{r_{E}^{3}-r_{C}^{3}}{r_{E}-r_{C}}=1-\frac{1}{3}(r_{E}^{2}+r_{C}r_{E}+r_{C}^{2}).$
(19)
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|
arxiv-papers
| 2008-11-17T16:48:12 |
2024-09-04T02:48:58.842396
|
{
"license": "Public Domain",
"authors": "H. T. Quan",
"submitter": "Haitao Quan",
"url": "https://arxiv.org/abs/0811.2756"
}
|
0811.2828
|
]Consejo Nacional de Invesigaciones Cientificas y Tecnicas,CONICET. ]Consejo
Nacional de Invesigaciones Cientificas y Tecnicas,CONICET.
]Consejo Nacional de Invesigaciones Cientificas y Tecnicas, CONICET.
# Transport and dynamical properties of inertial ratchets
M.F.Carusela [ flor@ungs.edu.ar Instituto de Ciencias, Universidad de
General Sarmiento, J.M.Gutierrez 1150, (1613) Los Polvorines, Buenos Aires,
Argentina. A.J.Fendrik [ fendrik@df.uba.ar Departamento de Física
J.J.Giambiagi, Facultad de Ciencias Exactas y Naturales. Universidad de Buenos
Aires.(1428) Buenos Aires. Argentina. L.Romanelli [ lili@ungs.edu.ar
Instituto de Ciencias, Universidad de General Sarmiento, J.M.Gutierrez 1150,
(1613) Los Polvorines, Buenos Aires, Argentina.
###### Abstract
In this paper we discuss the dynamics and transport properties of a massive
particle, in a time dependent periodic potential of the ratchet type, with a
dissipative environment. The directional currents and characteristics of the
motion are studied as the specific frictional coefficient varies, finding that
the stationary regime is strongly dependent on this parameter. The maximal
Lyapunov exponent and the current show large fluctuations and inversions,
therefore for some range of the control parameter, this inertial ratchet could
originate a mass separation device. Also an exploration of the effect of a
random force on the system is performed.
###### pacs:
05.60.Cd, 05.40.Ca
## I Introduction
Transport phenomena play a crucial role in many problems from physics, biology
and social science. In particular, there have been an increasing interest in
transport properties of the Brownian ratchets (BR), devices out of thermal
equilibrium in which a nonzero net drift velocity may be obtained from
fluctuations interacting with broken symmetry structuresdenisov ; flach .
These devices that where first proposed by Smoluchowsky smolu and later
discussed by Feynmann feyn have deserved a great deal of attention in the
literature (for a review see Ref.reimann ). There is a wide diversity of areas
in which BR’s are applied, for instance to the study of molecular motors
astumian , the description of ion channels and molecular transport within
cells bier , the treatment of Parrondo’s paradoxical games parrondo ; harmer ,
optical ratchets and directed motion of laser cooled atoms mennerat and
coupled Josephson junctions zapata . In particular, these nonequilibrium
models become especially interesting due to their technological applications
on nanoscales and microscales ajdari ; kaplan such as microscopic particle
separation ertas ; duke ; lindner .
In these models a net transport can be induced with fluctuations associated
with an additive force and noise, but also without noise, in overdamped popes
or underdamped systems mateos ; hangui . In particular, inertial ratchets may
exhibit a very complex and rich dynamics, both regular and chaotic motion.
Moreover, they can display multiple reversal currents where their directions
depend on the inertia term and the amount of friction.
In this paper, the problem of transport in periodic asymmetric potential of
the ratchet type is addresed. We study a model with an underdamped ratchet
under the influence of spatio-temporal external perturbations without and with
noise. Other authors, have studied the reversal current as a function of the
intensity of an external driving mateos ; hangui ; larrondo . One of the facts
of this paper is to reveal the current reversal as a function of the specific
frictional coefficient that provides a mass separator device.
The present work is organized as follows: in Section II we introduce the model
under study; in Section III the dynamical study of the system in particular in
the transport regime is presented. Section IV is devoted to the study of the
role of noise to modify the characteristis of the dynamics and to induce the
transport current . Finally in Section V we summarize our results. Some of
technical details are given in an Appendix.
## II Anular inertial ratchet
We consider non interactive massive ($m$) particles placed in a one-
dimensional ring of radius $R=1$, inmmersed in a dissipative environment,
under a periodic potential and driven by an external force. The time evolution
is given by the equation:
$m\ddot{x}=-\mu\dot{x}-\frac{\partial V_{\alpha}(x)}{\partial x}+F^{dr}(x,t).$
(1)
In Eq.[1], $0\leq x(t)\leq 2\pi$ represents the coordinate of the particle,
$\mu$ the damping coefficient and $V(x)_{\alpha}$ is a one dimensional,
periodic potential given by
$V_{\alpha}(x)=\left\\{\begin{array}[]{ll}V_{o}\cos\left(\frac{\pi(\alpha+1)x}{a\alpha}\right),\;\mbox{for}\;0\leq
x\leq\frac{a\alpha}{(\alpha+1)}\\\
-V_{o}\cos\left(\frac{\pi(\alpha+1)x}{a}-\alpha\pi\right),\;\mbox{for}\;\frac{a\alpha}{(\alpha+1)}\leq
x\leq a,\end{array}\right.$ (2)
fulfilling the periodicity condition $V_{\alpha}(x+a)=V_{\alpha}(x)$ where
$a=2\pi/N$ being $N$ the number of wells (or sites) along the ring. The
parameter $\alpha$ ($\alpha>0$) controls the left-right asymmetry of the
potential. For $\alpha>1(<1)$ the minimum in each well of the ratchet is
displaced towards the right (left) while $\alpha=1$ corresponds to left-right
symmetric potential. Figure 1(a) displays this static potential for $N=4$ and
$\alpha=5$.
Figure 1:
(a) Static potential $V_{\alpha}(x)$ with $\alpha=5$ as a function of the
coordinate $x$. (b) Total potential $V_{\alpha}(x)+V^{dr}(x,t)$ with
$\varepsilon<\varepsilon_{c}$ for $t=\pi/2\omega$ as a function of the
coordinate $x$.
Particles are driven by the external periodic driving force $F^{dr}(x,t)$. We
take this to be the gradient of a time dependent potential with a spatial
periodicity that is twice the one of $V_{\alpha}$ in order that consecutive
wells alternate in time as absolute minima. This condition restricts $N$ to be
even. We therefore consider
$F^{dr}(x,t)=-\varepsilon\frac{\partial V^{dr}(x,t)}{\partial
x}=-\varepsilon\sin(\omega t)\frac{d\sin(Nx/2)}{dx},$ (3)
that represents a longitudinal stationary wave along the ring. Time is
measured in units of the period $\tau=2\pi/\omega$ of the external driving and
$\varepsilon$ is the coupling strength. There is a critical value
$\varepsilon=\varepsilon_{c}$ such that if $\varepsilon<\varepsilon_{c}$ the
whole potential ($V_{\alpha}(x)+V^{dr}(x,t)$) always has a minimum per site.
For $\varepsilon>\varepsilon_{c}$ the potential may loose minima corresponding
to alternate sites as the time varies. Figure 1(b) displays the whole
potential at $\tau/4$ for $N=4$ and $\alpha=5$ and
$\varepsilon<\varepsilon_{c}$.
According the values of the parameters and the initial conditions the motion
would be bounded (libration near the sites) or unbounded (rotations around the
ring).
To obtain a directional current in the unbounded dynamics, it is useful to
study the symmetries of the equation of motion Eq.[1]. For hamiltonian systems
(i.e. $\mu=0$ ) with $\alpha=1$, there are two symmetries that prevent
directional transport:
$\displaystyle S_{1}:x\rightarrow
x+(2k+1)\frac{2\pi}{N},\;t\rightarrow-t,\;v\rightarrow-v;$ (4) $\displaystyle
S_{2}:x\rightarrow(2k+1)\frac{2\pi}{N}-x,\;t\rightarrow t,\;v\rightarrow-v.$
(5)
$S_{1}$ symmetry leaves invariant Eq.[1] while $S_{2}$ changes the sign the
whole expression. In accord to both symmetries, for each solution with
velocity $v$ will be another solution with value $-v$. If $\alpha\not=1$
$S_{2}$, symmetry is removed but $S_{1}$ still holds. When $\mu>0$ the only
symmetry removed is $S_{1}$. Therefore, for obtaining directional transport
the values of $\alpha$ and $\mu$ have to be $\alpha\not=1$ and $\mu>0$. Taking
this into account, we claim that the current is induced by damping. In the
overdamped limit, such that the inertial term can be dropped out, the
transport phenomena in the ratchet described by Eq.[1] (setting $m=0$) has
been well understood nos1 . To determine inertial effects on the directional
current, we will study the characteristics of the dynamics as a function of
the mass.
For practical reasons, we have numerically solved Eq.[1] using the Fourier
expansion of the potencial $V_{\alpha}(x)$ Eq.[2] up to 20-th order rather
than the original one (see appendix).
## III Transport and dynamics.
In this section we will discuss the characteristics of the directional
transport and the dynamics of the ratchet as a function of the specific
frictional coefficient $\mu/m$. Let us consider the dimensionless version of
Eq.[1] as:
$\ddot{x}=\beta\left[-\dot{x}+\gamma\left(F_{\alpha}(Nx)-\frac{\varepsilon^{\prime}}{2}\cos{(Nx/2)}\sin{(2\pi
t)}\right)\right],$ (6)
where $\beta=\frac{2\pi\mu}{m\omega}$ is the specific frictional parameter,
$\gamma=\frac{2\pi V_{o}N}{\mu\omega}$ is the ratio between the external force
and the dissipation, $F(Nx)=-\frac{1}{NV_{o}}\frac{\partial
V_{\alpha}(Nx)}{\partial x}$ and $\varepsilon^{\prime}=\varepsilon/V_{o}$ (in
the following we drop the prime).
All the calculations that we report where made for $N=4$, $V_{o}=10$,
$\omega=6$, $\mu=1$ (that is $\gamma=40\pi/3$) and $\varepsilon=6.5$ greater
than $\varepsilon_{c}=6.26007$. We fix all the parameters except $m$,
therefore we study the system as long as $\beta$ is changing.
We study the dynamical behavior of the system in the stationary regime
starting from an ensamble of one hundred random initial conditions
($N_{in}=100$) for each $\beta$ . We have calculated the the mean velocity
$\bar{v}$, measure in site per period of the driving force, for each initial
condition is:
$\bar{v}=\left(\frac{N}{2\pi}\right)\lim_{t\to\infty}\frac{(x(t)+2\pi k)}{t},$
(7)
where $k$ is the winding number (that is the number of rotation that the
particle makes around the ring). Then, the average ensamble velocity is given
by:
$\langle\bar{v}\rangle=\frac{1}{N_{in}}\sum_{j=1}^{N_{in}}\bar{v}_{j},$ (8)
and its fluctuations:
$\sigma_{\langle\bar{v}\rangle}=\sqrt{\frac{1}{N_{in}}\sum_{j=1}^{N_{in}}(\bar{v}_{j}-\langle\bar{v}\rangle)^{2}}.$
(9)
In the same way, the average maximum Lyapunov exponent is:
$\langle L_{max}\rangle=\frac{1}{N_{in}}\sum_{j=1}^{N_{in}}L_{max}^{j},$ (10)
where $L_{max}^{j}$ corresponds to the maximum Lyapunov exponent for the
$j-th$ initial condition (calculated by algorithm given in Ref.lyapurui ) and
the fluctuation:
$\sigma_{\langle
L_{max}\rangle}=\sqrt{\frac{1}{N_{in}}\sum_{j=1}^{N_{in}}(L_{max}^{j}-\langle
L_{max}\rangle)^{2}}.$ (11)
In Figure 2 the magnitudes defined above are displayed. By inspecting this
figure (a,b,c, and d), we found diverse and complementary information that can
be extract from them. From Figure 2(a), we observe the average of the maximum
Lyapunov exponent belonging to the mentioned stationary orbits. These
exponents have positive and negative values. The negative values, as usual,
would be related with regular behavior meanwhile the positive with irregular
one, this assert holds when the dispersion shown in Fig.2(b) remains small.
However from the same figure, we found coexistence of different attractors. We
discuss some of them, indicated by the arrows in Fig.3. The average ensamble
velocity is shown in Fig.2(c) while in Fig.2(d) its dispersion. It can be
observe for $\beta>5$ an average current $\langle\bar{v}\rangle_{max}=2$ is
established with zero dispersion. We remark that in such limit, the system is
overdamped and the inertial term is negligeable for studying the current (see
ref.nos1 ) althougth its effect is present in the Lyapunov exponent as can be
seen from its fluctuations above $\beta=5$ (Fig.3) related to the intrawell
dynamics. For lower values of $\beta$, a complex behavior is observed: the
values of the current fluctuates from $-2$ to $2$ in some regions while it
vanishes in others. The fluctuations displayed in the figures do not
correspond to a statistical effect but to the complex structure of the dynamic
(i.e. appearance or desappearance of diferent kind of orbits), what is
therefore worth to a more detailed analysis. We varied $\beta$ with a smaller
step in order to get a zoom in a particular range as can be seen in Fig.3. The
arrows show particular values where there are coexistence of attractors. The
label A correspond to a chaotic orbit (with almost no transport) together with
a regular one whose current is $\bar{v}=-4/3$. B is a region where arise a
regular orbit of current $\bar{v}=2$ meanwhile the chaotic orbit persists near
to $\beta=1.49$. In C three regular orbits with current $\bar{v}=2$,
$\bar{v}=-2$ and $\bar{v}=0$ are present. The label D corresponds to
coexistence of two regular orbits of currents $\bar{v}=-2$ and $\bar{v}=2/3$
respectively.
Figure 2: Average of the maximum Lyapunov exponent $\langle L_{max}\rangle$
(a), dispersion of the maximum Lyapunov exponent $\sigma_{L}$ (b), average
velocity $\langle\bar{v}\rangle$ (c) and dispersion of average velocity
$\sigma_{\langle v\rangle}$ (d) as a function of the dissipation $\beta$. The
velocity is measured in terms of sites per period of the driven force. For
sake of clarity in the four graphs the points are linked by lines. The
vertical dashed lines limit the region shown in Fig.3. Figure 3: Blow up of
Fig. 2. Figure 4: Projection of the trajectory in the phase space and the
position of the particle versus time for differents values of $\beta$. (a)
Regular orbit with mean velocity $\bar{v}=2$. (b) Same as above with
$\bar{v}=-2$. (c) $\bar{v}=-4/3$. (d) $\bar{v}=2/3$ and (e) chaotic behavior
with $\bar{v}=0$.
Figure 4 displays the position of the particle as a function of time and the
projection of the trajectory onto the phase space. The cases shown are related
to those cases discussed in the text.
Figure 5: Histogram of the maximum Lyapunov exponent (left panel) and
histogram of the mean velocity $\bar{v}$ (right panel) for different values of
$\beta$. All the calculations where obtained with an ensamble of 100 random
initial conditions. Each row (from top to bottom) corresponds to the regions
indicated by arrows A, B, C and D in Fig. 3.
To stress the coexistence of attractors for a given damping value, as
discussed above, the histograms of the maximun Lyapunov exponent
($n_{L_{max}}$) and the mean velocity ($n_{\bar{v}}$) are displayed in Fig.5.
The highest value of $n_{L_{max}}$ in the row (c) is the sum of the values of
$n_{\bar{v}}$ for $\bar{v}=2$ and $\bar{v}=-2$ because both orbits have the
same maximal Lyapunov for this $\beta$.
It is interesting to note that the current $\langle\bar{v}\rangle$ in Fig.2(c)
shows abrupt inversions as a $\beta$ changes in a similar way to other
ratchets when the parameters varies. As the variation of $\beta$ depends only
on the mass $m$, similar particles but different masses will have different
velocities. Therefore, this behavior originates a mass separation device.
## IV The effect of noise
We now turn to study of transport current assisted by noise in presence of an
external random force $\xi(t)$ fulfilling
$\langle\xi(t)\xi(t^{\prime})\rangle=\sigma^{2}\delta(t-t^{\prime})$. We solve
the following stochastic differential equation:
$\ddot{x}=\beta\left[-\dot{x}+\gamma\left(F_{\alpha}(Nx)-\frac{\varepsilon^{\prime}}{2}\cos{(Nx/2)}\sin{(2\pi
t)}\right)+\xi(t)\right].$ (12)
In the noiseless overdamped limit ($m\rightarrow 0$), the motion is bounded
($\bar{v}=0$) or unbounded ($\bar{v}\neq 0$) according to
$\varepsilon<\varepsilon_{c}$ or $\varepsilon>\varepsilon_{c}$. In both cases
the stationary motion is a single regular orbit robust under slight variation
of the coupling. As it was shown in ref. nos1 the addition of an appropriate
noise to the bounded regular orbit can induce a transition to the unbounded
motion ($\bar{v}\neq 0$) rather than its destruction. In other words, the
effect of noise on the bounded motion can mimic the increase of coupling
$\varepsilon$ from a lower to a higher value than the critical, therefore we
will have a brownian ratchet whose maximal current is near to those of the
unbounded orbit.
This scope would be quite different when the inertial term is non negligeable
since the stationary unbounded motion is now more complex due to the fact that
small variation in the coupling may introduce or destroy orbits larrondo . On
the other hand in the inertial system the bounded and the unbounded motion are
separated by a fuzzy region of $\varepsilon$ where there is coexistence of
both types of motion.
Unlike the overdamped system the mere addition of noise to the bounded motion
does not lead to obtain transport although it depends onto the robustness and
the mean velocity of the unbounded orbits in the chosen parameter region
($\beta,\gamma,\varepsilon$) therefore it is not easy to predict when there
will be directional currents. As an example, in Fig.6(a), we show the current
obtained as a function of noise added to a bounded system
($\varepsilon=4.1,\beta=1.5708$). In this case, the maximum value of the
velocity corresponds to an optimal noise (near $\sigma=3.5$) but its value is
far from the maximum value which is reached in the overdamped system.
Now, we add the random force to a system where should have transport even in
absence of noise (as that shown in 3). The effect of noise will strongly
depend upon the parameter $\beta$ because, as we have already seen, it
determines the properties of the transport in the stationary regime (current
of the single orbits, coexistence of orbits and their stability etc.). Let us
consider the deterministic system whose $\beta=1.987$ is indicated by the
arrow E in the Fig.3. There are coexistence of two orbits. One of them, has
current $\bar{v}=-2$ and slightly negative maximal Lyapunov exponent and the
other one has current $\bar{v}=-1$ and slightly positive maximal Lyapunov
exponent. In the first row of Fig. 7 the histograms of both are displayed. The
mixing of two orbits ocurs in such a way that the total average current and
the average maximal Lyapunov are $\langle\bar{v}\rangle=0.34$, $\langle
L_{max}\rangle=0.013$ respectively.
When the random force is added, for an optimal noise ($\sigma=0.3$), the
stationary orbits have $\bar{v}=-2$ and slightly negative maximal Lyapunov
exponents. This can be observed in Fig.6(b) and (c) where
$\langle\bar{v}\rangle$ and $\langle L_{max}\rangle$ as a function of noise
$\sigma$ are displayed. For sake of clarity, in the second row of Fig.7 the
histograms of $\bar{v}=-2$ and $\langle L_{max}\rangle$ (see Ref.lyapurui )
are displayed when the noise is near of the optimal one.
Figure 6: a) Average velocity $\langle\bar{v}\rangle$ vs. $\sigma$. The
chosen parameters are $\varepsilon=4.1$, $\omega=6$, $\beta=1.5708$. The
calculations where obtained with an ensamble of 200 random initial conditions
for each value of $\sigma$. b) Average velocity $\langle\bar{v}\rangle$ vs.
$\sigma$. The chosen parameters are $\varepsilon=6.5$, $\omega=6$,
$\beta=1.987$. c) Average of the maximal Lyapunov $\langle L_{max}\rangle$ as
a function of $\sigma$. The parameters are the same as in b). Figure 7: The
upper row displays the histograms of the mean velocity $\bar{v}$ and maximal
Lyapunov exponent $L_{max}$ for deterministic systems for $\beta=1.987$,
$\varepsilon=6.5$. The lower row shows the same histograms when the random
force $\sigma=0.2$ is added.
## V Summary and conclusions
In this work we have studied the dynamics in the stationary regime of an
inertial ratchet as a function of the specific frictional coefficient
($\beta$). This dynamics is strongly dependent on such coefficient since in
some regions slight variation of this parameter can destroy stationary orbits
and/or it can create new ones. There are regions where two or three orbits of
different characteristics coexist. This can be verify from the fluctuations of
the maximal average Lyapunov exponent as long as the respectively histograms.
This fact is reflected in the transport properties of the system, since the
mean velocity of the orbits involved, should be quite different not only in
their magnitude although in their directions. The large fluctuations and
inversions showed in the average velocity when $\beta$ varies are the
manifestation of such phenomenum. This property could be useful to build a
mass separation device.
We have also explored the effect of noise in the system when a random force is
added.
In the overdamped limit, when the deterministic dynamics is robust enough
against noise it is possible to obtain a directional current assisted by noise
by adding a random force to the system when its motion is bounded. In other
words, the presence of noise in moderate amounts induces the transitions
bounded-unbounded (libration-rotation) before the destruction of the
deterministic dynamics structure. If the inertial term becomes relevant, our
results show that it is possible to obtain this effect but with a maximal
current lesser than the ideal. This behavior may be due to two different
causes: 1) The presence of the inertial term makes the deterministic dynamics
less robust against the noise 2) The unbounded orbits where the system access
by adding the noise have smaller values of $\bar{v}$ than the ideal $\bar{v}$
for the choosen parameters. This could be decided with a more exhaustive
analysis which is beyond the scope of the present work.
Next, we explore the inclusion of a random force in the deterministic system
where the dynamics is already unbounded. We have consider a region where there
are coexistence of two stationary orbits with different characteristics (in
particular their $\bar{v}$ have opposite sign such that the average
$\langle\bar{v}\rangle$ is slightly positive). We observe, when noise is
added, one orbit is not robust and it dessapears while the other one (with
$\bar{v}=-2$) persists. Then, the current of the system, not only has an
inversion but increases its absolute value. We wish to emphasize that noise,
in this case, stabilizes the dynamics as can be seen from the reduction of
average maximal Lyapunov exponent $\langle L_{max}\rangle$. This is due to the
fact that the destroyed orbit has positive Lyapunov exponent.
Another possible effect of noise when it is added to the system in a region of
$\beta$ with coexistence of orbits, is to induce transitions between them
(provided that both orbits are robust against noise). We have not observed
this but a more comprehensive study should be make.
###### Acknowledgements.
The authors wishes to acknowledge CONICET (PIP6124) for financial assistance.
*
## Appendix A
Taking in mind the future study of the quantum analogue of this kind of
ratchet using the Bloch-Floquet formalism, we have considered the Fourier
expansion of the Potential Eq.[2] rather than its exact expression. That is,
for $N$ sites, $a=\frac{2\pi}{N}$ and $0\leq x\leq 2\pi$:
$V_{\alpha}(x)=V_{o}\sum_{l=1}\left[a_{l}(\alpha)cos(Nlx)+b_{l}(\alpha)sin(Nlx)\right].$
(13)
The coeficients $a_{l}(\alpha)$ and $b_{l}(\alpha)$ have annalytical
expressions given by:
$\displaystyle a_{l}(\alpha)=F_{l}(\alpha)\sin\left(\frac{2\alpha
l\pi}{1+\alpha}\right),$ $\displaystyle
b_{l}(\alpha)=F_{l}(\alpha)\left[1+\cos\left(\frac{2\alpha
l\pi}{1+\alpha}\right)\right];$ (14)
where
$F_{l}(\alpha)=\frac{4l(\alpha-1)(\alpha+1)^{3}}{[(\alpha+1)^{2}-(2l)^{2}][(2\alpha
l)^{2}-(\alpha+1)^{2}]\pi}\;.$ (15)
For $(2l-1)=\alpha$ or $1/(2l-1)=\alpha$ these expessions are undeterminated
but taking the appropiate limit, we obtain:
$\displaystyle a_{l}(\alpha)=\frac{1}{2l},$ $\displaystyle b_{l}(\alpha)=0.$
(16)
As a mesure of the difference of expansion Eq.[13] up to 20 th order from the
exact potential Eq.[2] we consider
$\sigma=\frac{1}{V_{o}}\sqrt{\frac{1}{2\pi}\int_{0}^{2\pi}(V_{apr}-V_{exact})^{2}dx}=6\times
10^{-4}.$ (17)
In Figure 1(a), the exact potential is undistinguishable from its Fourier
expansion.
## References
* (1) S.Denisov, L.Morales-Molina, S.Flach and P.Hänggi, Phys. Rev. A 75, 063424 (2007).
* (2) S.Flach, O.Yevtushenko and Y.Zolotaryuk, Phys. Rev. Lett. 84, 2358 (2000).
* (3) M. von Smoluchowski Phys. Zeitschrift XIII, 1069 (1912)
* (4) R.P.Feynamnn, R.B. Leighton and M. Sands The Feynamnn Lectures in Physics 1 46.1 Addison-Wesley (1963)
* (5) P.Reimann, Phys.Rep. 361, 57-265 (2002)
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* (7) M.Bier, Contemp. Phys. 38, 371-379 (1997); M.Bier, Stochastic Dynamics 38681-87 Springer, Berlin 1997
* (8) J.M. Parrondo, Borja Jimenez de Cisneros tranlated by P.Amengual J.Sotelo anb D.Abbott; cond-mat/0309053 v2 (2003)
* (9) G.P. Harmer and D. Abbott, Statistical Science, 14, 206, (1999)
* (10) C.Mennerat-Robilliard, D.Lucas, S.Guibal, J.Tabosa, C.Jurczak, J.Y.Courtois, G.Grynberg, Phys. Rev. Lett.82, 851 (1999)
* (11) I.Zapata, R.Bartussek, F.Sols, P.Hänggi, Phys. Rev. Lett.77, 2292 (1996)
* (12) J.Rousselet, L.Salome, A.Ajdari and J.Prost, Nature(London) 370, 446 (1994)
* (13) L.P.Faucheux, L.S.Bourdieu, P.D.Kaplan and A.J.Libchaber, Phys. Rev. Lett 74, 1504 (1995)
* (14) D.Ertas, Phy. Rev. Lett. 80, 1548 (1998)
* (15) T.A.J.Duke, R.H.Austin, Phys. Rev. Lett.80, 1552 (1998)
* (16) B.Lindner, L.Schimansky-Geier, P.Reimann and P.Hänggi, AIP Conference Proc. 411, 309 (1997).
* (17) M.N.Popescu, C.M.Arizmendi, A.L.Salas-Brito, F.Family, Phys. Rev. Lett 85, 3321 (2000)
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* (19) P.Jung, J.G.Kissner, P.Hänggi, Phys. Rev. Lett 76, 3436 (1996)
* (20) H.A.Larrondo, Fereydoon Family,C.M.Arizmendi, Physica A 303, 67 (2002)
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|
arxiv-papers
| 2008-11-18T15:48:21 |
2024-09-04T02:48:58.852781
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M.F.Carusela, A.J.Fendrik, L.Romanelli",
"submitter": "Mar\\'ia Florencia Carusela",
"url": "https://arxiv.org/abs/0811.2828"
}
|
0811.2840
|
# Stellar Velocity Profiles and Line-Strengths out to Four Effective Radii in
the Early-Type Galaxy NGC 3379
A. Weijmans1, M. Cappellari2, P.T. de Zeeuw3,1, E. Emsellem4,
J. Falcón-Barroso5, H. Kuntschner3, R.M. McDermid6,
R.C.E. van den Bosch7, and G. van de Ven8
###### Abstract
We describe a new technique to measure stellar kinematics and line-strengths
at large radii in nearby galaxies. Using the integral-field spectrograph
SAURON as a ’photon-collector’, we obtain spectra out to four effective radii
($R_{e}$) in the early-type galaxy NGC 3379. By fitting orbit-based models to
the extracted stellar velocity profile, we find that $\sim$ 40% of the total
mass within 5 $R_{e}$ is dark. The measured absorption line-strengths reveal a
radial gradient with constant slope out to 4 $R_{e}$.
1Sterrewacht Leiden, Leiden University, Postbus 9513, 2300 RA Leiden, the
Netherlands [weijmans@strw.leidenuniv.nl]
2Sub-Department of Astrophysics, University of Oxford, Denys Wilkinson
Building, Keble Road, Oxford OX1 3RH, UK
3ESO, Karl-Schwarzschild-Str 2, 85748 Garching, Germany
4CRAL, University of Lyon, 9 Avenue Charles André, 69230 Saint Genis Laval,
France
5ESTEC, Postbus 299, 2200 AG Noordwijk, the Netherlands
6Gemini Observatory, Northern Operations Centre, 670 N. A’ohoku Place, Hilo,
Hawaii 96720 USA
7McDonald Observatory, University of Texas, Austin, TX 78712, USA
8IAS, Einstein Drive, Princeton, NJ 08540, USA
## 1\. Introduction
Although the presence of dark matter dominated haloes around spiral galaxies
is well established (see e.g. van Albada et al. 1985), little is known about
the dark haloes around early-type galaxies. Large regular H i discs or rings,
whose kinematics are often used to constrain the properties of dark haloes,
are rare in these systems (though see Franx, van Gorkom, & de Zeeuw 1994;
Weijmans et al. 2008), so that we are forced to use other tracers of the
gravitational potential.
Here we use stellar kinematics obtained with the integral-field spectrograph
SAURON (Bacon et al. 2001) at large radii in the elliptical (E1) galaxy NGC
3379 to model its dark halo. This galaxy is of intermediate luminosity
($M_{B}=-20.57$) and has a half-light or effective radius $R_{e}$ of 42
arcsec, which corresponds to 2.1 kpc at its distance of 10.3 Mpc.
## 2\. Method
We centred SAURON at 2.6 and 3.5 $R_{e}$ on both sides of the nucleus of NGC
3379, close to its major axis (see Fig. 1, left panel). A single spectrum of
one lenslet is dominated by noise at these large radii, but adding all spectra
of all lenslets together we obtained in three out of our four fields
sufficient signal-to-noise to measure the stellar absorption line-strengths
and the line-of-sight velocity distribution (LOSVD) up to the fourth Gauss-
Hermite moment $h_{4}$. This last parameter is necessary to break the well-
known mass-anisotropy degeneracy when modeling the dark halo (e.g. Gerhard
1993).
|
---|---
Figure 1.: Left: Positions of our observed fields in NGC 3379. The red boxes
denote each SAURON field-of-view. The skylenslets (red short thick lines) are
aligned with the long side of the SAURON field, at a distance of two
arcminutes. The dashed line denotes the major axis of NGC 3379. The underlying
$V$-band image was obtained with the 1.3-m McGraw-Hill Telescope at MDM
Observatory. Right: LOSVD of NGC 3379 out to 4 $R_{e}$. The black stars are
long-slit data from Statler & Smecker-Hane (1999) and the central red dots are
SAURON observations obtained in the original survey (Emsellem et al. 2004).
The red dots at large radii are our new observations, and double the spatial
extend of the data.
## 3\. Results
We measured the LOSVD using the penalized pixel fitting method (pPXF) by
Cappellari & Emsellem (2004). The resulting LOSVD (Fig. 1, right panel) shows
a smooth continuation of existing stelllar kinematic measurements (Statler &
Smecker-Hane 1999).
We use a Schwarzschild model (van den Bosch et al. 2008; van de Ven, de Zeeuw,
& van den Bosch 2008) to fit our measurements, including the central SAURON
field of the original survey (Emsellem et al. 2004) and the long-slit data of
Statler & Smecker-Hane (1999). The black hole mass and the (nearly
axisymmetric) shape of the stellar distribution of NGC 3379 are taken from van
den Bosch (2008). We model the spherical dark halo with an NFW profile
(Navarro, Frenk, & White 1996) with a concentration $c=10$ (Bullock et al.
2001). Our best-fit model is shown in Fig. 2, and has a total halo mass
$M_{200}$ of $1.0\times 10^{12}$ $M_{\odot}$, which corresponds to 10 times
the total stellar mass of NGC 3379.
We obtained line-strength measurements following the procedures outlined in
Kuntschner et al. (2006). We find that the slope of the line-strength
gradients remains constant out to at least 4 $R_{e}$, although our values of
Fe5015 are not consistent with this trend (Fig. 3). Plotting our measurements
on the stellar population models of Thomas, Maraston, & Bender (2003), we find
that the stellar halo population is old ($\sim$ 12 Gyr) and metal-poor (below
20% solar).
Figure 2.: Best-fit model (solid line) overplotted on datapoints. The blue stars are (symmetrized) long-slit data from Statler & Smecker-Hane (1999) and the red dots are our datapoints (horizontal error bars indicate the width of the SAURON field-of-view). Also shown are a model without a dark halo (dashed line) and a model with a ten times too massive halo (dot-dashed line). These models do not fit the observed dispersion and $h_{4}$ profiles. |
---|---
Figure 3.: Left: Line-strength gradients (from top to bottom: H$\beta$, Fe5015
and Mg $b$, in magnitudes) out to 4 $R_{e}$ in NGC 3379. Black triangles
denote SAURON data from the original survey (Kuntschner et al. 2006), and red
dots are our new measurements. Right: H$\beta$ index against [MgFe50]′,
overplotted on the stellar population models of Thomas et al. (2003). Black
dots show measurements from the SAURON survey, while the coloured dots are
averaged along isophotes (see inset for colour coding). The black filled
squares are our measurements at large radii.
## 4\. Conclusion
We showed that by using SAURON as a ’photon collector’, we can measure both
the stellar velocity profile and absorption line-strengths out to large radii
in early-type galaxies. We presented our measurements of NGC 3379 and modeled
its dark halo. In our best-fit model, 41% of the total mass within 5 $R_{e}$
is dark. We will present more elaborate modeling of the dark halo of NGC 3379
and comparisons with literature values in a forthcoming paper (Weijmans et al.
in preparation), as well as a comparable dataset and analysis for the
elliptical galaxy NGC 821.
### Acknowledgments.
It is a pleasure to thank the organisers for a stimulating and fruitful
conference. We gratefully acknowledge Chris Benn, Eveline van Scherpenzeel,
Richard Wilman and the ING staff for support on La Palma.
The SAURON observations were obtained at the William Herschel Telescope,
operated by the Isaac Newton Group in the Spanish Observatorio del Roque de
los Muchachos of the Instituto de Astrofísica de Canarias.
## References
* Bacon et al. (2001) Bacon, R., et al. 2001, MNRAS, 326, 23
* Bullock et al. (2001) Bullock, J.S., Kolatt, T.S., Sigad, Y., Somerville, R.S., Kravtsov, A., Klypin, A.A., Primack, J.R., & Dekel, A. 2001, MNRAS, 321, 559
* Cappellari & Emsellem (2004) Cappellari, M., & Emsellem, E. 2004, PASP, 116, 138
* Emsellem et al. (2004) Emsellem, E., et al. 2004, MNRAS, 352, 721
* Franx et al. (1994) Franx, M., van Gorkom, J.H., & de Zeeuw, P.T. 1994, ApJ, 436, 642
* Gerhard (1993) Gerhard, O.E. 1993, MNRAS, 265, 213
* Kuntschner et al. (2006) Kuntschner, H., et al. 2006, MNRAS, 369, 497
* Navarro et al. (1996) Navarro, J.F., Frenk, C.S., & White, S.D. 1996, ApJ, 462, 563
* Statler & Smecker-Hane (1999) Statler, T.S., & Smecker-Hane, T. 1999, AJ, 117, 839
* Thomas et al. (2003) Thomas, D., Maraston, C., & Bender, R. 2003, MNRAS, 339, 897
* van Albada et al. (1985) van Albada, T.S., Bahcall, J.N., Begeman, K., & Sancisi, R. 1985, ApJ, 295, 305
* van den Bosch (2008) van den Bosch, R.C.E. 2008, PhD thesis, Leiden University
* van den Bosch et al. (2008) van den Bosch, R.C.E., van de Ven, G., Verolme, E.K., Cappellari, M., & de Zeeuw, P.T. 2008, MNRAS, 385, 647
* van de Ven et al. (2008) van de Ven, G., de Zeeuw, P.T., & van den Bosch, R.C.E. 2008, MNRAS, 385, 614
* Weijmans et al. (2008) Weijmans, A., Krajnović, D., van de Ven, G., Oosterloo, T.A., Morganti, R., & de Zeeuw, P.T. 2008, MNRAS, 383, 1343
|
arxiv-papers
| 2008-11-18T06:04:40 |
2024-09-04T02:48:58.859683
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Weijmans (1), M. Cappellari (2), P.T. de Zeeuw (3,1), E. Emsellem\n (4), J. Falcon-Barroso (5), H. Kuntschner (3), R.M. McDermid (6), R.C.E. van\n den Bosch (7) and G. van de Ven (8) ((1) Sterrewacht Leiden, (2) Oxford, (3)\n ESO, (4) CRAL/Observatoire de Lyon, (5) ESTEC, (6) Gemini Observatory, (7)\n University of Texas at Austin, (8) IAS Princeton)",
"submitter": "Anne-Marie Weijmans",
"url": "https://arxiv.org/abs/0811.2840"
}
|
0811.2918
|
11institutetext: Institut für Physik, Universität Augsburg, D-86135 Augsburg,
Germany
Department of Mathematical Sciences, Loughborough University, Loughborough,
Leicestershire, LE11 3TU, UK
Nonlinearity, bifurcation, and symmetry breaking Drops and bubbles Liquid-
solid interfaces
# Depinning of three-dimensional drops from wettability defects
Ph. Beltrame 11 P. Hänggi 11 U. Thiele 221122
###### Abstract
Substrate defects crucially influence the onset of sliding drop motion under
lateral driving. A finite force is necessary to overcome the pinning influence
even of microscale heterogeneities. The depinning dynamics of three-
dimensional drops is studied for hydrophilic and hydrophobic wettability
defects using a long-wave evolution equation for the film thickness profile.
It is found that the nature of the depinning transition explains the
experimentally observed stick-slip motion.
###### pacs:
47.20.Ky
###### pacs:
47.55.D-
###### pacs:
68.08.-p
## 1 Introduction
Drops sliding along a solid substrate under the influence of a lateral force
are a very common physical phenomenon. The force might be gravity for drops on
an inclined or vertical wall, centrifugal forces for drops on a rotating disk
or external shear for drops in an ambient flow. Note that lateral gradients in
wettability, temperature or electrical fields can as well drive sliding
motion. For smooth homogeneous substrates an arbitrarily small driving force
results in drops that move with constant velocity and shape [1, 2, 3]. Larger
driving forces may lead to shape instabilities, e.g., trailing cusps may
evolve that periodically emit small satellite drops [1, 4].
Real substrates, however, are normally not smooth. They are rough or have
local chemical or topographical defects. Even microscopic defects can have a
strong influence on the drop dynamics. The heterogeneities may cause stick-
slip motion [5, 6] or roughening [7, 8] of moving contact lines, and are
thought to be responsible for contact angle hysteresis [9, 10, 11, 12]. Note
that a local variation of the driving force (e.g., electrostatic field or
temperature gradient) may play the same role as a substrate defect.
The present paper focuses on the depinning of three-dimensional drops from
hydrophobic and hydrophilic line defects that pin them at their front and
back, respectively: A hydrophobic defect is less wettable for the drop that
therefore has to be forced to pass it. On the contrary, a hydrophilic defect
is more wettable and the drop has to be forced to leave it as sketched in Fig.
1. A recent theoretical study of the depinning dynamics of less realistic two-
dimensional drops employs lubrication approximation and finds stick-slip
motion beyond depinning [13, 14].
Figure 1: Sketch of the three-dimensional geometry of the problem: a drop on
a heterogeneous substrate and under a driving force $\mu$ along the
x-direction. Thereby the heterogeneous wettability is assumed to depend on the
x-direction spatial direction only.
The present work is based on a thin film evolution equation in long-wave
approximation [15, 16] that incorporates wettability in the form of an
additional pressure term – the so-called disjoining pressure [9]. It models
the effective molecular interactions between the substrate and the free
surface of the liquid, e.g., long-range apolar van der Waals interactions and
short-range polar electrostatic or entropic interactions [17]. With the proper
choice of terms such a disjoining pressure describes the behaviour of drops of
partially wetting liquid with a small equilibrium contact angle that coexist
with an ultra-thin precursor film. An advantage of such a model is the absence
of a contact line singularity. Note that although only small contact angles
and small driving forces are compatible with the long-wave approximation
results are often qualitatively correct for more general conditions.
Incorporating wettability in the form of a disjoining pressure allows to study
the influence of chemical substrate heterogeneities or defects by a spatial
modulation of the involved material parameters. For dewetting thin films
without lateral driving this is done in [18, 19].
The analysis of the two-dimensional problem in Refs. [13, 14] consists of a
study of (i) steady drops and their stability based on continuation techniques
for ordinary differential equations [20] and (ii) time-periodic solutions
sliding over a regular array of defects based on ’standard’ time-stepping
schemes. The here presented study of the three-dimensional case is based on
recently developed effective algorithms for both, the continuation of pinned
steady drops described by a partial differential equation and the time
simulation of the dynamics beyond depinning [21].
## 2 Model and numerical method
We consider a liquid layer or drop on an inhomogeneous two-dimensional solid
substrate as sketched in Fig. 1. The liquid partially wets the substrate (with
a small equilibrium contact angle) and is subject to a small constant lateral
force $\mu$ that acts in the $x-$direction. Employing the long-wave or
lubrication approximation the dimensionless evolution equation for the film
thickness profile $h(x,y,t)$ is derived from the Navier-Stokes equations,
continuity and boundary conditions (no-slip at substrate, force equilibria at
free surface) [15, 16, 13]. It reads
$\partial_{t}h=-\nabla\cdot\left\\{m(h)\left[\nabla\left(\Delta
h+\Pi(h,x)\right)+\mu\mathbf{e_{x}}\right]\right\\},$ (1)
where $\nabla=(\partial_{x},\partial_{y})$ and
$\Delta=\partial_{xx}^{2}+\partial_{yy}^{2}$ are the planar gradient and
Laplace operator, respectively. The mobility function $m(h)=h^{3}$ corresponds
to Poiseuille flow and $\Delta h$ represents the Laplace pressure
(capillarity). The disjoining pressure $\Pi(h,x)$ models the position-
dependent wetting properties that in the case of transverse line defects only
depend on the streamwise direction. The literature discusses a plethora of
different functional forms for $\Pi(h)$ [9, 22]. Most model the presence of an
ultra-thin precursor film of about 1-10 nm thickness and thereby avoid a
’true’ film rupture. We employ long-range apolar van der Waals interactions
combined with a short-range polar interaction [9, 17, 23]
$\Pi(h,x)=\frac{b}{h^{3}}-\left(1+\epsilon\xi(x)\right)e^{-h},$ (2)
where $\epsilon$ and $\xi(x)$ are the amplitude and profile of the
heterogeneity, respectively. To model a localized defect $\xi(x)$ is based on
Jacobi elliptic functions as described in [13, 14]. Typical examples can be
seen below in Fig. 2. The amplitude $\epsilon$ represents the wettability
contrast. For $\epsilon<0$ [$\epsilon>0$] the defect is less [more] wettable
than the surrounding substrate, i.e., the defect is hydrophobic [hydrophilic].
Based on the Jacobi functions we study drops on a periodic array of defects.
The period $L_{x}$ is chosen sufficiently large to avoid interactions between
subsequent drops/defects. The imposed spatial periodicity allows to
characterize stick-slip motion by its period in time.
Based on Eq. (1) with (2) the depinning behaviour in the three-dimensional
(3d) case is analysed following the methodology used in [13, 14] for the two-
dimensional (2d) case. Steady-state solutions (pinned drops) and their
stability are determined using continuation techniques and the stick-slip
motion beyond the depinning threshold is analysed using time-stepping
algorithms. In the 2d case an explicit scheme suffices for the latter. The
continuation can be performed using the package AUTO [24] as the underlying
equation corresponds to a system of ODE’s [16]. In the 3d case an effective
and exact time simulation of Eq. (1) is challenging and leads to a number of
numerical problems [25, 4, 26]. Here we employ a recently developed approach
[21] based on exponential propagation [27]. It allows for a very good estimate
of the optimal timestep for the different regimes of the dynamics. This is of
paramount importance as close to the depinning transition it needs to be
varied over many orders of magnitude. The second advantage lies in the
possibility to adapt the time-stepping scheme in a way that it can be used to
continue the branches of steady drop states and to determine their stability.
For details see Ref. [21].
## 3 Depinning of 2d drops
Figure 2: Selected drop profiles (top row) and corresponding bifurcation
diagrams (bottom row) for localized hydrophilic ($\epsilon=0.3$, left column)
and hydrophobic ($\epsilon=-0.3$, right column) defect in the 2d case. (a) and
(b) give steady drop profiles for several driving forces $\mu\geq 0$ as given
in the legend. For $\mu=4\cdot 10^{-3}$ stable (solid line with symbol “s”)
and unstable (dotted line with symbol “u”) steady drops are represented. The
lower part of the panels gives the heterogeneity profile $\xi(x)$. (c) and (d)
characterize branches of steady drop solutions by the dependence of their
$L^{2}$ norm ($||\delta h||=\sqrt{\int_{0}^{L}(h-\bar{h})^{2}dx/L}$) on the
lateral driving force $\mu$ for various defect strength $\epsilon$ as given in
the legend. Dashed lines indicate unstable solutions. Domain length, volume
and resulting drop height on a homogeneous substrate are $L_{x}=40$, $V=66$
and $h_{max}=4.0$, respectively.
Before focusing on the 3d case we shortly present results for 2d using
equivalent parameter values to allow for a qualitative and quantitative
comparison. Without lateral force ($\mu=0$) there exists a unique stable drop
solution for each wettability contrast $\epsilon$. The drop sits on top of a
hydrophilic defect (dashed line in Fig. 2(a)) or in the middle between
hydrophobic defects (dashed line in Fig. 2(b)). Note that other steady
solutions may exist that are normally unstable. For an in-depth study of
solutions on a horizontal substrate (for another $\Pi(h)$) see [19].
Increasing the lateral driving force $\mu$ from zero the drop does not start
to slide as for the homogeneous substrate, but remains pinned by the defect. A
hydrophobic defect blocks the drop at the front, it becomes compressed and
heightens (see Fig. 2(b)) until it finally depins. This can best be seen in
the bifurcation diagram Fig. 2(d) where the norm of stable and unstable steady
solutions is shown in dependence of $\mu$ for several wettability contrasts.
The norm of the stable drop solution first increases, then decreases slightly
and the branch annihilates with an unstable one at $\mu_{c}$. In contrast, a
hydrophilic defect holds a drop at its back, with increasing driving it
becomes stretched and lower (see Fig. 2(a)) until it finally depins. The
accompanying bifurcation diagram (Fig. 2(c)) shows that the norm decreases
till the branch annihilates with an unstable one, e.g., for $\epsilon=-0.3$ at
$\mu_{c}\approx 0.005$.
Beyond the critical value $\mu_{c}$, there exists in both cases only a single
branch of steady solutions that are all linearly unstable. Its norm approaches
zero with increasing $\mu$ (not shown) indicating that it corresponds to
slightly modulated film solutions. This state being unstable, a time-dependent
state is expected that corresponds to sliding drops. In the 2d case such
solutions where discussed in Ref. [13]. Related solutions and the character of
the depinning transition for the 3d case will be discussed next.
## 4 Depinning of 3d drops
Figure 3: Bifurcation diagram for drops pinned by a hydrophilic line defect of
strength $\epsilon=-0.3$. Shown is the $L^{2}$ norm $||\delta h||$ of steady
solutions in dependence of the lateral driving force $\mu$. The branch of
stable pinned drops corresponds to the solid line whereas unstable solutions
are given as dotted lines. Beyond the depinning bifurcation, triangles
represent the time-averaged $L^{2}$ norm of time-periodic solutions that
correspond to sliding drops performing a stick-slip motion. The domain size is
$40\times 40$. Crosses indicate profiles given in Fig. 4. The inset gives for
the stick-slip motion the dependence of the time-period on $\mu-\mu_{c}$. The
straight line corresponds to a power law with exponent $-1/2$.
We consider now the full 3d geometry as sketched in Fig. 1. In particular, we
look at hydrophilic and hydrophobic line defects that lie orthogonal to the
direction of the driving force. In the present 3d setting one can re-interpret
the findings for 2d drops as referring to the depinning of a liquid ridge from
a line defect assuming that the transverse translational symmetry is not
broken in the depinning process.
To compare the depinning of such a ridge and the one of a true 3d drop we use
the continuation and time-stepping techniques outlined above. Furthermore all
parameters with the exception of the drop volume are chosen as in the 2d case.
For the latter we use a value such that the maximal drop height on a
homogeneous substrate without driving force ($\mu=0$) is equal to the one of
the ridge.
Figure 4: Shown are contours of steady drop solutions for a hydrophilic defect
for $\mu=3.5\cdot 10^{-3}$. Profiles from left to right correspond to crosses
in Fig. 3 from high to low norm. The left panel presents the stable pinned
drop. The thin horizontal line marks the wettability maximum. The remaining
parameters are as in Fig. 3. Figure 5: Shown are snapshots of drop profiles at
different stages of a stick-slip cycle (at times given below the individual
panels) for a drop depinning from a hydrophilic line defect (marked by the
horizontal line). The chosen driving $\mu=5.193\cdot 10^{-3}$ is still close
to the critical $\mu_{c}$. Color code and remaining parameters are as in 3.
Figure 6: Bifurcation diagram for drops pinned by a hydrophobic line defect of
strength $\epsilon=0.3$. The presented norms, line styles, symbols, domain
size and inset are as in Fig. 3. Corresponding profiles are given in Fig. 7.
The straight line in the inset corresponds to a power law with exponent
$-1/4$. Figure 7: Shown are contours of steady drop solutions for a
hydrophobic defect for $\mu=5.7\cdot 10^{-3}$. Profiles from left to right
correspond to crosses in Fig. 6 from high to low norm. The left panel presents
the stable pinned drop. The horizontal line marks the wettability minimum. The
remaining parameters are as in Fig. 6. Figure 8: Shown are snapshots of drop
profiles at different stages of a stick-slip cycle for a depinned drop near
the depinning bifurcation (at $\mu=7.898\cdot 10^{-3}$ and times as given
below the individual panels) for a hydrophobic line defect (marked by the
horizontal line). Color code and remaining parameters are as in 6. Figure 9:
Shown are contours of steady rivulet solutions for large driving force
$\mu=0.05$ for (a) hydrophilic and (b) hydrophobic line defects. The remaining
parameters are as in Figs. 3, and 6, respectively. The horizontal line marks
the extrema of the wettability profile.
Fig. 3 shows the bifurcation diagram for a single drop on a square domain. The
stable drop is pinned at its back by the hydrophilic line defect with
$\epsilon=-0.3$. On the horizontal substrate ($\mu=0$) the drop sits
symmetrically on the defect its contour being an ellipse with the long axis on
the defect (not shown). When increasing $\mu$ the drop moves to the downstream
side of the defect where it is retained by the high wettability patch below
its tail. With increasing $\mu$ it stretches downstream but is compressed
transversally. The combined effect of the two processes leads contrary to the
2d case to an increase of the norm. The stable branch loses its stability via
a saddle-node bifurcation at the critical driving $\mu_{c}=5.193\cdot
10^{-3}$. The continuing unstable branch is turned towards smaller $\mu$. It
turns back again at a further saddle-node bifurcation and the resulting ’low-
norm’ branch then continues towards large $\mu$. A selection of steady stable
and unstable drop solutions corresponding to the crosses in Fig. 3 is given in
Fig. 4. The left panel corresponds to the pinned stable drop described above,
the middle panel represents an unstable drop that one could call “at
depinning”: it has an oval front shape and is connected to the hydrophilic
patch by a thin bridge that almost looks cusp-like and seems to be at the
point of breaking. Physically it corresponds to a threshold solution: If it is
moved a bit upstream [downstream] it retracts [slips to the next defect] and
converges to the stable drop solution. The left panel, finally, gives the
unstable solution of lowest norm. It resembles two drops joined by a thin
thread with the smaller one sitting on the heterogeneity. The character of
solutions on this branch at large $\mu$ is discussed below.
For $\mu>\mu_{c}$ no steady stable solutions exist and we expect the system to
exhibit a time-dependent behaviour. In particular, we expect in the present
spatially periodic setting that drops depin from one hydrophilic defect and
slide to the next one. There they do not stop but only slow down as the defect
tries to retain them. We probe this behaviour using a time-stepping algorithm.
The time-averaged norm for several $\mu$ is given in Fig. 3 and one can well
appreciate that the corresponding solution branch emerges from the saddle-node
bifurcation at $\mu_{c}$ indicating that it is actually a Saddle Node Infinite
PERiod (SNIPER) bifurcation. This is furthermore corroborated by the square
root dependence of the inverse time-period (mean sliding speed) on
$\mu-\mu_{c}$ that is given in the inset of Fig. 3 [28, 13]. An example of a
time series of snapshots for a stick-slip motion of a single drop is given in
Fig. 5. Note that the times at which the snapshots are taken are not
equidistant. It takes the drop about 25000 time units to slowly stretch away
from the defect (snapshot 1 to 2). Then within 500 units it depins and slides
to the next defect (snapshot 2 to 5), where it needs another 25000 units to
reach an identical state as in snapshot 1 (snapshot 5 to 6). The depinning
itself resembles a pinch-off event at a water tap: the bridge between drop and
a ’reservoir’ on the hydrophilic stripe becomes thinner until it snaps. Once
the main drop slides a small drop remains behind on the defect. All together
for the chosen value of $\mu$ the ratio of stick/stretch and slip phase is
about $50:1$. The ratio diverges when approaching the bifurcation.
Next we discuss the case of a hydrophobic defect. Fig. 6 is the corresponding
bifurcation diagram. It shows as solid line stable solutions corresponding to
single drops blocked at their front by the line defect with $\epsilon=0.3$.
Dashed lines indicate unstable steady solutions. The general behaviour
resembles strongly the related 2d case and as well the hydrophilic case. In
particular, does the norm of the stable solutions increase with increasing
$\mu$ as the drop is increasingly pushed against the defect and becomes
therefore steeper. The drop itself becomes more oval as its transverse width
increases but the streamwise one decreases. An example of such a stable steady
drop can be seen in the left panel of Fig. 7. The two other panels represent
the two unstable solutions that exist for identical $\mu$ (crosses in Fig. 6).
Both unstable drops are situated mainly upstream of the defect but have
downstream protrusions of different length and strength that reach the
substrate beyond the defect. The drop on the middle branch corresponds to a
threshold solution as in the hydrophilic case.
Time simulations indicate that depinning occurs again via a sniper
bifurcation, i.e., a branch of time-periodic solutions emerges from the
saddle-node at $\mu_{c}$. However, the time-period does not diverge as
$(\mu-\mu_{c})^{-1/2}$ but rather with the power $-1/4$ (inset of Fig. 6). An
example of a time series of snapshots for a stick-slip motion of a depinned
drop is given in Fig. 8. The drop needs about 1600 time units to slowly let a
’protrusion’ creep over the defect (snapshot 1 to 2). Then within 400 units it
depins and slides to the next defect (snapshot 2 to 5), where it needs another
1200 units to reach the state as in snapshot 1 (snapshot 5 to 6). Then the
cycle starts again. All together for the chosen value of $\mu$ the ratio of
stick and slip phase is about $7:1$. Once the drop is depinned a small drop is
retained behind the defect (snapshot 4).
Comparing the 2d and 3d cases we find that the depinning behaviour for drops
of equal height agrees qualitatively, but quantitatively there is a small
systematic difference. In both cases we find depinning transitions via a
sniper bifurcation at a critical driving $\mu_{c}$. However, in 3d $\mu_{c}$
is about 10-15% larger than the one in 2d. This is a result of the smaller
mass per lateral length the 3d drop has as compared to the ridge. Such an
effect increases $\mu_{c}$ as it implies a smaller “effective 2d loading” in
the 3d case. Actually, from the dependency on loading in 2d (see [13]) one
would expect an even larger difference. The reason for the small increase may
be the additional degree of freedom that a true 3d drop has as compared to a
translationally invariant ridge. It allows the drop to ’probe’ the barrier
locally by an advancing protrusion (in the hydrophobic case) or by thinning
its backward bridge to the defect (in the hydrophilic case). It can therefore
use a pathway of morphological changes for depinning that a 2d drop is not
able to use. Note that individual stations of this pathway that can be seen in
Figs. 5 and 8 do very much resemble the unstable steady solutions presented in
Figs. 4 and 7, respectively. This indicates that the steady solutions that
exist below $\mu_{c}$ are still present in the phase space as ’ghost
solutions’ [28] and can be seen in the course of the time periodic motion
beyond $\mu_{c}$. When we have discussed that the depinning behaviour for
$\epsilon=\pm 0.3$ in 2d and 3d is very similar we have focused on the branch
of pinned drops only. Note that the connectivity of the unstable branches is
not that similar. In this respect the 3d case resembles the 2d case at a
larger contrast $\epsilon$.
Finally, we discuss the character of the single remaining steady state
solution for large $\mu$. Comparing bifurcation diagrams for large increasing
$\mu$ (not shown) one notes that in the 2d case the norm approaches zero and
the solutions resemble slightly modulated films. In contrast, in 3d the norm
approaches a finite value, i.e., there remains a non-trivial large amplitude
structure. The character of this structure can be appreciated in Fig. 9.
Equally for hydrophilic as for hydrophobic defects one finds a rivulet with
drop-like transverse cross sections and comparatively small variation in
streamwise direction. Increasing $\mu$ the streamwise modulation becomes even
smaller and the thickness profiles in the transverse direction are very close
to steady 2d drops on horizontal substrates of corresponding wettability. The
rivulet is linearly unstable below a large finite driving $\mu_{r}$. There it
stabilizes via a Hopf bifurcation that as well forms the end point of the
branch of time-periodic solutions.
## 5 Conclusion
We have studied depinning three-dimensional drops under lateral driving for
localized hydrophobic and hydrophilic line defects employing on the one hand
continuation techniques to obtain steady-state solutions (pinned drops and
rivulets) and their stability and on the other hand a time-stepping algorithm
to study the dynamics of the stick-slip motion beyond depinning. We have found
that for the studied parameter range the depinning behavior is qualitatively
similar in 2d and 3d: Drops are pinned up to a critical driving $\mu_{c}$
where they depin via a sniper bifurcation. Quantitatively there exists a small
systematic difference – the 3d $\mu_{c}$ is slightly larger than the 2d one.
Our interpretation is that the difference results mainly from a lower
“effective 2d loading” in the 3d case but is has as well to be taken into
account that the 3d drop has additional degree of freedom enabling it to
employ pathways of morphological changes for depinning that a 2d drop is not
able to access.
The sniper bifurcation is in the hydrophilic case characterized by a square-
root power law dependence of the inverse time scale of depinning on the
distance from threshold $\mu-\mu_{c}$. Beyond $\mu_{c}$ the unsteady motion
resembles the stick-slip motion observed in experiment: The advance of the
drop is extremely slow when it ’creeps away’ from the hydrophilic region, and
very fast once the thread connecting the back of the drop to the defect has
broken and the drop slides to the next defect. The difference in time scales
for the stick and the slip phase diverges when approaching $\mu_{c}$. For a
hydrophilic defect, however, we have found a degenerate sniper bifurcation as
the power law has an exponent of about 1/4. Re-considering the 2d case we
found that there as well in the hydrophobic case the power is about 1/3, i.e.,
it differs from the expected 1/2 (cf. [13]). This may result from a degeneracy
of the problem that has, however, still to be identified.
Note that for hydrophobic defects of large strength depinning may occur at
very large driving via a Hopf instead of a sniper bifurcation [14]. Then
depinning is caused by the flow in the wetting layer. For realistic forces the
effect can not be observed for partially wetting nano- or micro-drops on an
incline or rotating disc and we have not considered the parameter regime here.
Note, however, that micro-drops of dielectric liquids generated by an electric
field in a capacitor can coexist with a thick wetting layer of 100nm to
1$\mu$m stabilized by van der Waals interaction [29, 30]. In such a setting
both depinning mechanisms should be observable using gravity as the driving
force (see appendix of [13]).
###### Acknowledgements.
We acknowledge support by the EU [MRTN-CT-2004005728 PATTERNS] and the DFG
[SFB 486, B13].
## References
* [1] Podgorski T., Flesselles J.-M. Limat L. Phys. Rev. Lett. 872001036102\.
* [2] Thiele U., Velarde M. G., Neuffer K., Bestehorn M. Pomeau Y. Phys. Rev. E 642001061601.
* [3] Snoeijer J. H., Le Grand N., Limat L., Stone H. A. Eggers J. Phys. Fluids 192007042104.
* [4] Thiele U., Neuffer K., Bestehorn M., Pomeau Y. Velarde M. G. Colloid Surf. A 206200287.
* [5] Schäffer E. Wong P. Z. Phys. Rev. Lett. 8019983069\.
* [6] Tavana H., Yang G. C., Yip C. M., Appelhans D., Zschoche S., Grundke K., Hair M. L. Neumann A. W. Langmuir 222006628 36.
* [7] Golestanian R. Raphaël E. Europhys. Lett. 552001228\.
* [8] Robbins M. O. Joanny J. F. Europhys. Lett. 31987729.
* [9] de Gennes P.-G. Rev. Mod. Phys. 571985827.
* [10] Leger L. Joanny J. F. Rep. Prog. Phys. 551992431.
* [11] Quéré D., Azzopardi M. J. Delattre L. Langmuir 1419982213\.
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* [13] Thiele U. Knobloch E. New J. Phys. 82006313, 1.
* [14] Thiele U. Knobloch E. Phys. Rev. Lett. 972006204501.
* [15] Oron A., Davis S. H. Bankoff S. G. Rev. Mod. Phys. 691997931\.
* [16] Kalliadasis S. Thiele U. (Editors) Thin Films of Soft Matter (Springer, Wien / New York) 2007 CISM 490.
* [17] Sharma A. Langmuir 91993861.
* [18] Konnur R., Kargupta K. Sharma A. Phys. Rev. Lett. 842000931\.
* [19] Thiele U., Brusch L., Bestehorn M. Bär M. Eur. Phys. J. E 112003255.
* [20] Doedel E., Keller H. B. Kernevez J. P. Int. J. Bifurcation Chaos 11991493.
* [21] Beltrame P. Thiele U. submitted 2008.
* [22] Teletzke G. F., Davis H. T. Scriven L. E. Rev. Phys. Appl. 231988989\.
* [23] Thiele U., Velarde M. G. Neuffer K. Phys. Rev. Lett. 872001016104\.
* [24] Doedel E., Paffenroth R., Champneys A., Fairgrieve T., Kuznetsov Y., Sandstede B. Wang X. Tech. Rep. Caltech (2001).
* [25] Bertozzi A. L., Grün G. Witelski T. P. Nonlinearity 1420011569\.
* [26] Becker J. Grün G. J. Phys.: Condens. Matter 172005S291.
* [27] Friesner R. A., Tuckerman L. S., Dornblaser B. C. Russo T. V. J. Sci. Comp. 41989327.
* [28] Strogatz S. H. Nonlinear Dynamics and Chaos (Addison-Wesley) 1994\.
* [29] Lin Z., Kerle T., Baker S. M., Hoagland D. A., Schäffer E., Steiner U. Russell T. P. J. Chem. Phys. 11420012377.
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|
arxiv-papers
| 2008-11-18T14:30:36 |
2024-09-04T02:48:58.865750
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ph. Beltrame, P. H\\\"anggi and U. Thiele",
"submitter": "Philippe Beltrame",
"url": "https://arxiv.org/abs/0811.2918"
}
|
0811.3205
|
Star Formation from Spitzer (Lyman) to Spitzer (Space Telescope) and Beyond
A summary of Symposium 9, JENAM 2008
> held in Vienna, 10-12 September 2008, and convened by João Alves (Calar Alto
> Observatory) and Virginia Trimble (U. California Irvine and Las Cumbres
> Observatory)
The confluence of the 400th anniversary of astronomical telescopes, the
completion of the basic, cold, 5-year mission of the Spitzer Space Telescope,
and the near-certain advent of JWST, ALMA, and extremely large, ground-based
telescopes seemed to invite a symposium to investigate the past, present, and
future of star formation studies. While this summary attempts to mention
everybody, with at least one significant idea from each speaker, including the
one-minute poster presentations, it will surely fail. The sessions were
expertly chaired by L. Woltjer, C. Cesarsky (also involved in the ESO event),
J. Andersen, and H.-M. Maitzen.
The Symposium started with two historical introductions (V. Trimble & B.G.
Elmegreen), addressing, first, the very long time required for astronomers all
to agree, only after 1950, that star formation is an on-going process, not
something that happened long ago (whether 107, 101Γ, or 1012 years ago) when
the universe was very different, and second, the vital roles of Lyman Spitzer,
his immediate predecessors, colleagues, and students, in establishing the
existence and properties of interstellar matter, from which stars could form,
and the processes that would allow them to do so. Remarkably, Spitzer was
never interested in the idea of cold molecular hydrogen as the raw material of
star formation and came rather late to the idea of turbulence as an important
process.
We follow the “seven simplest lessons from 60 years of star formation”, as
outlined by J. Alves, as a logical order to this summary, and invite you to
keep an eye out for some of the topics of on-going dispute, including (a)
whether the initial mass function (IMF) is universal, what determines it, and
whether it is closely related to the mass distribution of dense cores in pre-
stellar clouds (Core Mass Function or CMF), (b) whether triggering is
important, (c) whether massive stars form the same way as ones that can remain
below Eddington luminosity throughout the process, (d) environmental effects
and the role of binaries, (e) how brown dwarfs form, and (f) how (in)efficient
is star formation, and why. And so on to the seven “certainties”, keeping in
mind that Z is metallicity and z is redshift.
1. 1.
Stars form continually in the cold interiors of dark molecular clouds (if you
doubt this, please leave the room). Multiwavelength studies of specific
regions persuaded us all to remain (I. Zinchenko, on S76E, with triggering by
HII expansion; M. Rengel on the second class 0 source in Lupus 3, indicating
these live only 104 yr; P. Persi on a new SF site NGC 6334 IV (MM3); and
Nakajima, also on the Lupus 3 region).
2. 2.
Star formation is inefficient, meaning that, if you look at a particular mass
of cool, dense molecular-gas, the fraction of it turned into stars in a
dynamical time is typically a few percent (J. Silk), though larger values are
possible in bound clouds (I. Bonnell) and very different numbers probably
describe star formation in galaxies very unlike the Milky Way and at large z
(E. Grebel).
3. 3.
Most stars form in groups of 10 - 106. Cluster environments can enhance disk
accretion onto planetary cores (S. Pfalzner). Brown dwarfs are more spread out
than stars (S. Schmeja), though, like the evidence for mass segregation as
clusters age, this surely has some contribution from source confusion in dense
centers.
4. 4.
There is a characteristic product, a log-normal IMF peaking at 0.2-0.3 M⊙
though this too could have been very different long ago and far away (Grebel).
Also, low mass stars are single (R. Jayawardhana on Cha I and Upper Sco, also
providing a candidate for the first directly-imaged exoplanet), in contrast to
Herbig AeBe stars, most of which are binaries, their disks aligned with their
orbit planes (R. Ooudmaijer).
5. 5.
Feedback processes are ubiquitous and important. There are jets at all
wavelengths (K. Stapelfeldt on numerous new Herbig-Haro objected detected by
SST), the need for ongoing supernovae to keep star formation down to the
observed 2% (J. Silk), and perhaps even massive star feedback to form clusters
(J. Alves).
6. 6.
Stars form with and from accretion disks across the full mass range from BDs
to OBs, and there is a definite time sequence over which the disks disappear
(I. Tsukugoshi on T Tauri stars). There are also evolutionary sequences in
maser type, radio emission, and SED shapes (R. Oudmaijer). Whole clusters also
evolve (S. Schmeja) from hierarchical to centrally condensed structures.
7. 7.
Nature does some “pre-packaging”, so that the distribution of core masses, the
CMF, has the same shape as the IMF (though shifted to larger masses) and must
somehow give rise directly to the IMF (J. Alves). This was perhaps the topic
of greatest dispute among the “certainties”. Several speakers asked whether
the CMF predicts the IMF (R. Kawabe reporting several AzTEC/ASTE surveys; R.
Smith noting that different methods yield different observed CMFs; P.
Hennebelle remarking on the range of relevant processes, with outflows,
accretion, and turbulence of comparable importance; and S. Dib suggesting that
the transformation from CMF to IMF is a function of environment), I. Bonnell
firmly denied a directly link between CMF and IMF once one allows for
continuing fragmentation as well as core0 accretion.
Not yet at the level of eternal verities are the primacy of massive stars in
the formation process (with disk accretion, competitive accretion, and stellar
collisions and mergers in environments of increasing density, according to R.
Klein, and the private opinion of VT) the need for all the processes you can
think of (gravity, angular momentum transfer, magnetic fields, accretion,
turbulence, feedback - this is either the good news or the bad news, depending
on how you feel about programming). But the probability that there is no
further missing physics counts as good news.
Then came four outstanding review talks, two from observers two from theorists
(and if you are organizing a seminar series this year, try to get at least one
of these speakers!). First, K. Stapelfeldt provided an overview of the Spitzer
mission, the five-year cold part of which is essentially over, but a two-year
“warm” extension, during which the two shorter wavelengths will still be
usable, has been approved. SST is currently about 1 AU from Earth, drifting
backwards, and eventually will not be able to turn in the right direction and
send us data.
Among the discoveries important for star formation have been,
* •
70% of infrared dark clouds have embedded protostars (and those that do not
could have BDs or might eventually disrupt)
* •
at least one region has remarkably gray dust with A24μm/AK $=$0.44 there is
spectroscopic evidence for many kinds of grains, including large ice-mantled
ones
* •
water is found in many places as vapor or ice; there is also acetylene
* •
the statistics of class 0, I and II sources are not quite as expected
* •
disks with central holes, perhaps due to planets, are fairly common
* •
protostellar disks last 107 years and debris disks 108 years; debris disks
imply that agglomeration has proceeded at least as far as planetesimals,
comets, and asteroids
Second, E. Grebel absolutely blasted through the very different contexts in
which star formation occurs, from starbursts down to dwarf galaxies, pointing
out the different rates, patterns, efficiencies, and probably IMFs, and the
evidence for different modes in common galaxy types, as observed or as
inferred from the resultant star populations. Continuous, episodic, or one-
shot star formation occurs depending on gas content, mass density of the
galaxy, and interactions or accretion. Some other points she made (far from a
complete list) include,
* •
stars are now forming in S and Irr galaxies, in galactic centers, and in
interacting galaxies. Star bursts process 100 M⊙/yr and ULIRGs up to 1000
M⊙/yr
* •
typical spirals form 20 M⊙/yr, much larger than the Milky Way value of 1-3
M⊙/yr
* •
for many gE’s the rate is roughly 0 M⊙/yr, but about 1/3 have evidence
(including Galex UV colors) for active rather than passive evolution, that is
some on-going star formation
* •
field gE’s have their oldest stars about 2 Gyr younger than cluster gE’s
* •
E+A galaxies indicate cessation of star formation at a definite time in the
past
* •
the Milky Way has a number of discrete stellar populations, distinguished by
age and Z, including globular clusters (not themselves all the same) two sorts
of field halo stars, two sorts of disk stars, and a bulge
* •
there was a time gap between the end of halo and beginning of disk star
formation in the MW which is not understood; the bulge stars are mostly older
than 10 Gyr and have [Fe/H] across the range -2.0 to +0.5
* •
most large galaxies show age and metallicity gradient
* •
it is not clear whether Irr galaxies have massive halos; the star velocity
dispersion is close to rotation speed, and HI tends to be spherical (consider
maps of LMC)
* •
IR galaxies host 10-20% of current star formation
* •
there are tidal tail galaxies and BCDs (with HI and star formation
concentrated at their centers)
* •
dwarf galaxy SF is very inhomogeneous, and you can see pollution by single SNe
as scatter in relative abundances
* •
the ratio of s to r products is an age indicator
* •
winds are important
* •
star formation in the outskirts of S’s is not understood
Third, J. Silk described the multitude of physical processes that must be
considered in theories of star formation, the evidence for them, and some of
the outstanding questions. Key issues include the IMF, star formation
efficiency, turbulence, quenching, and triggering. Among the points he made
were,
* •
the IMF is not necessarily constant, and if it was top heavy at large z, this
will affect the SFR(z) you derive from any tracer
* •
the mass assembly history derived from SST and star formation histories
derived by other methods disagree at z$=3-4$; differences in stellar M/L (the
IMF) are a likely cause
* •
core velocities are mildly supersonic in the $\rho$ Oph region; more
generally, porosity of the ISM is self-regulated, so that star bursts have
high turbulence and low porosity, while quenching occurs with low turbulence
and high porosity
* •
the percentage of gas in GMCs is also regulated by turbulence
* •
quenching is due to different processes on different scales and in galaxies of
different masses, for instance fountain and outflows on large scales in normal
galaxies, but BH accretion, jets, and radiation in AGNs, whose activity is
quenched at the same time, corresponding to the well-known black hole$-$bulge
relation
* •
triggering is seen on assorted scales but is not universal
* •
AGNs can also enhance star formation by compressing gas, and the SFR depends
on interactions between hot and cold gas
* •
downsizing means both that big halos formed first and that the ratio of
(SFR)/M(already in stars) declines toward the present from z=2.5. The process
is perhaps magnetically regulated.
Fourth, the primary discussion of star formation calculated from numerical
simulations came from I. Bonnell, for whom the key questions are the why’s of
star masses and the IMF, of inefficiency, of clusters vs. distributed SF, and
the how of core properties giving rise to star masses. On this last point, he
firmly concluded that, because of on-going accretion plus fragmentation, it is
very unlikely that there is a 1:1 relation between core mass and stellar mass.
Initial conditions are obviously important for these simulations, so that the
SST survey of GMCs (the stage where $\rho=$10-17⋯-21 g/cm3) is vital input.
Other things that matter include binaries and disks. Most star formation
occurs in bound structures, where low mass stars and BDs form from gas falling
into the cluster, while high mass stars result from rapid accretion (slowed
but not stopped by feedback) in incipient cluster cores. Bound gas clouds have
SFE around 15% vs. 3% for unbound clouds.
Several of the shorter contributions were of direct relevance to these issues,
for instance high resolution mapping of Av in Barnard 59 as a probe of SF
efficiency (C. Roman), the need (in calculations) for external confining
pressure to keep gas together and allow small length-scale fluctuations to
grow (J. Dale), the dominance of small separations and mass ratios near one
for low-mass binaries (R. Jayawardhana), and the significantly larger
luminosities of ultracompact Hii regions compared to massive YSOs (R.
Oudmaijer),
And the future came at the end. We heard about several ongoing and upcoming
projects, including,
* •
the APEX, Atacama Pathfinder, which sees known SF regions, starless cores, hot
molecular cores, IRAS sources, embedded clusters) and CH30H maser sources, for
which follow-up searches with Effelsberg, IRAM, and Mopra yielded only one
non- detection, a planetary nebula! (F. Schiller)
* •
SOFIA is coming, with a call for proposals due in December 2008 (M.
Hannebush), and more about SOFIA from R. Klein, who pointed out that one of
its major goals is to identify the dominant formation mechanisms for massive
stars, though he left the impression that everything that anybody has
suggested happens somewhere.
* •
an all-sky map of Galacic GMCs now in progress, derived from 2MASS extinction
measurements (J. Rowles)
* •
a concept study for a 4-meter space telescope usable from mid UV to near IR
(R. Jansen)
* •
a survey of Gould’s belt (the diffuse material primarily, not the OB star)
with HARP on the JCMT; and SCUBA-2 is coming in 2009 (J. Hatchell)
* •
ALMA, for which L. Testi described the science goals, required capabilities
(in terms of mm/submm resolution of 0.1′′ and sensitivity sufficient to map CO
and [CI] over the entire Milky Way), and timeline. But, he said, it will
neither image exoplanets “nor solve the star formation problem” (partly, one
suspects, because it is a little difficult to decide just what “the” star
formation problem is).
Our grandest view of the future came from M. McCaughrean who emphasized the
facilities that will become available over the next decade or two, including
ALMA, the large, ground-based E-ELT (plus the TMT and GMT), radio facilities
like e-MERLIN, LOFAR, and SKA, and, in space, the upgraded HST,
Herschel,Sofia, Gaia, and Kepler. But, he concluded, the most important new
facility will be JWST, with a five-year mission promised and the potential for
another five years before gases and such run out. He indicated that the single
most important thing it has to offer is greatly improved angular resolution
and that, similarly in planning the new, large ground-based telescopes, the
best possible angular resolution is more important than pushing into the
thermal infrared. Goals are 0.01-0.1′′, though one dan make this sound more
impressive by speaking of 10-100 miliarcseconds. Some of these facilities will
return data by the Tera- and PetaByte, so that improved capacity for number
receiving, storing, processing, and crunching will also be vital. An
interesting case (not mentioned) is LSST, where the decision has to be made
just how much raw data can be kept, so that, for instance, if a flare occurs
in a star formation region somewhere far away, one can go back over the past
years’ images, where the source may have been a two-sigma, three photon
smudge, and determine how bright and how variable it was previously.
João Alves (Calar Alto Observatory)
Virginia Trimble (Univ. of California Irvine and Las Cumbres Observatory)
|
arxiv-papers
| 2008-11-19T21:00:38 |
2024-09-04T02:48:58.877569
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Joao Alves, Virginia Trimble",
"submitter": "Joao Alves",
"url": "https://arxiv.org/abs/0811.3205"
}
|
0811.3364
|
# Quantum control gates with weak cross-Kerr nonlinearity
Qing Lin qlin@mail.ustc.edu.cn College of Information Science and
Engineering, Huaqiao University (Xiamen), Xiamen 361021, P.R.China. Jian Li
Department of Physics, Southeast University, Nanjing 211189, P. R. China
###### Abstract
In this paper, with the weak cross-Kerr nonlinearity, we first present a
special experimental scheme called controlled-path gate with which the
realization of all possible bipartite positive-operator-value measurements of
two-photon polarization states may be nearly deterministic. Following the same
technique, the realization of quantum control gates, including the controlled-
NOT gate, Fredkin gate, Toffoli gate, arbitrary controlled-U gate, and even
arbitrary multi-controlled-U gate, are proposed. The corresponding
probabilities are 1/2, 1/8, 2/23, etc. respectively. Only the coherent states
are required but not any ancilla photons, and no coincidence measurement are
required which results in these gates are scalable. The structures of these
gates are very simple, and then we think they are feasible with the current
experimental technology in optics.
###### pacs:
03.67.Lx, 42.50.Ex
## I Introduction
In the quantum computation, quantum control gates play a very important role.
It was proven that two-qubit unitary gates and single-qubit gates are
sufficient for universal quantum computation Nielsen . In linear optics, many
schemes are provided for the realization of two-qubit unitary gates, for
example, controlled-NOT (CNOT) gates CNOT or controlled-phase gates CP .
However, some of these gates work on the coincidence basis which results in
these gates are not scalable, i.e., these gates can not be used to realize
multi-qubit gates and then the universal computation. Moreover, all these
gates are probabilistic which result in the probability of the realization of
universal computation may be tiny, for the reason that many two-qubit unitary
gates required. For example, quantum Fredkin gate can be constructed by five
CNOT gates and some single-qubit gates Smolin , and the probability of CNOT
gate is only $1/4$ in linear optics CNOT , then the probability of Fredkin
gate is $4^{-5}=9.8\times 10^{-4}$. To avoid the inefficient, more efficient
even deterministic gates must be looked for. Fortunately, with the weak cross-
Kerr nonlinearity, a parity projector Barrett and a deterministic CNOT gate
Nemoto had been proposed, and then the universal computation can be realized
deterministic in principle. However, the universal computation and even a
multi-qubit gate may be need too many CNOT gates, then the structure may be
too complex to be realized in optics. Alternatively, it is interesting to look
for some multi-qubit gates with simple structure even though the probability
is not unit. In this paper, we will present the quantum control gates with
very simple structure, and we think these gates may be more feasible with the
current experimental technology.
This paper is organized as follows. In sec.II, we first propose a scheme of a
gate we call it controlled-path (C-path) gate with the weak cross-Kerr
nonlinearities, and then use this gate to realize all possible bipartite
positive-operator-value measurements (POVMs) of two-photon polarization
states. In addition, this technique is developed to realize the CNOT gate,
Fredkin gate, Toffoli gate, controlled-U (CU) gate and even multi-controlled-U
(MCU) gate. Sec.III is for conclusion remarks.
## II Quantum control gate
Before we outline our schemes of quantum control gates, we briefly review the
useful weak cross-Kerr nonlinearity which has been used in Refs. Barrett ;
Nemoto ; Spiller ; Kok . Suppose a non-linear weak cross-Kerr interaction
between a signal state (photonic qubit)
$\left|\psi\right\rangle=c_{1}\left|0\right\rangle+c_{2}\left|1\right\rangle+c_{2}\left|2\right\rangle$
and a coherent state $\left|\alpha\right\rangle$. After the evolution, the
output state is,
$\left|\psi\right\rangle\left|\alpha\right\rangle\rightarrow
c_{1}\left|0\right\rangle\left|\alpha\right\rangle+c_{2}\left|1\right\rangle\left|\alpha
e^{i\theta}\right\rangle+c_{2}\left|2\right\rangle\left|\alpha
e^{i2\theta}\right\rangle,$ (1)
where $\theta$ is induced by the nonlinearity. Through a general homodyne-
heterodyne measurement of the phase of the coherent state, the signal state
$\left|\psi\right\rangle$ will be projected into a definite number state or
superposition of number states. Because the measurement can be performed with
high fidelity, the projection is nearly deterministic. This technique has
first been used to realize a parity projector Barrett , and then a CNOT
gateNemoto . It provides a new route to new quantum computation Spiller . The
requirement for this technique is $\alpha\theta>1$ Spiller , where $\alpha$ is
the amplitude of the coherent state. Even with the weak nonlinearity ($\theta$
is small), this requirement can be satisfied with large amplitude of the
coherent state, then this requirement may be feasible with current
experimental technology. Our schemes of quantum control gates also work with
the weak cross-Kerr nonlinearity.
### II.1 C-path gate
Firstly, we discuss the C-path gate. Here, we use the polarization of photons
as qubit and define the horizontally (vertically) linear polarization
$\left|H\right\rangle$($\left|V\right\rangle$) as the qubit
$\left|0\right\rangle$($\left|1\right\rangle$). Consider a two-qubit initially
prepared in the state
$\left|\Psi\right\rangle=\alpha\left|H\right\rangle_{1}\left|H\right\rangle_{2}+\beta\left|H\right\rangle_{1}\left|V\right\rangle_{2}+\gamma\left|V\right\rangle_{1}\left|H\right\rangle_{2}+\delta\left|V\right\rangle_{1}\left|V\right\rangle_{2}$,
where
$\left|\alpha\right|^{2}+\left|\beta\right|^{2}+\left|\gamma\right|^{2}+\left|\delta\right|^{2}=1$.
In a C-path gate, the paths of the first photon are controlled by the second
photon. The experimental setup is shown in Fig.1. The control photon is
transmitted through a balanced Mach-Zehnder (M-Z) interferometer formed by two
polarizing beam splitters (PBS1, PBS2) which let the photon
$\left|H\right\rangle$ be passed and the photon $\left|V\right\rangle$ be
reflected, while the target photon is injected into a 50:50 beam splitter
(BS). The two photons combined with a coherent state
$\left|\alpha\right\rangle$ interact with the cross-Kerr nonlinearities, such
that a phase shift will be induced in the coherent state. Suppose the control
photon induces a controlled phase shift $-\theta$, while the target photon
induces a controlled phase shift $\theta$, then the input state
$\left|\Psi\right\rangle\left|\alpha\right\rangle$ will evolve to the follows:
$\displaystyle\frac{1}{\sqrt{2}}\left(\alpha\left|H\right\rangle_{1}^{S_{1}}\left|H\right\rangle_{2}+\beta\left|H\right\rangle_{1}^{S_{2}}\left|V\right\rangle_{2}+\gamma\left|V\right\rangle_{1}^{S_{1}}\left|H\right\rangle_{2}+\delta\left|V\right\rangle_{1}^{S_{2}}\left|V\right\rangle_{2}\right)\left|\alpha\right\rangle$
$\displaystyle+\frac{1}{\sqrt{2}}\left(\alpha\left|H\right\rangle_{1}^{S_{2}}+\gamma\left|V\right\rangle_{1}^{S_{2}}\right)\left|H\right\rangle_{2}\left|\alpha
e^{-i\theta}\right\rangle+\frac{1}{\sqrt{2}}\left(\beta\left|H\right\rangle_{1}^{S_{1}}+\delta\left|V\right\rangle_{1}^{S_{1}}\right)\left|V\right\rangle_{2}\left|\alpha
e^{i\theta}\right\rangle,$ (2)
where the superscripts $S_{1}$, $S_{2}$ denote the paths of the first photon.
Through a general homodyne-heterodyne measurement (X homodyne measurement),
the two-photon state will be projected into the following state,
$\alpha\left|H\right\rangle_{1}^{S_{1}}\left|H\right\rangle_{2}+\beta\left|H\right\rangle_{1}^{S_{2}}\left|V\right\rangle_{2}+\gamma\left|V\right\rangle_{1}^{S_{1}}\left|H\right\rangle_{2}+\delta\left|V\right\rangle_{1}^{S_{2}}\left|V\right\rangle_{2}.$
(3)
Here we only retain the case that no phase shift induced in the coherent
state, and the success probability is $P_{succ}^{CP}=1/2$. If a switch (S)
which will exchange the two photons and a phase shift conditionally controlled
by the homodyne detection through a classical feedforward are applied, this
C-path gate is nearly deterministic, i.e., $P_{succ,\max}^{CP}=1$. By the same
way, one can implement a multi-controlled-path gate in which multiple qubits
control the paths of the other qubits.
This C-path gate is very useful in the quantum computation for the reason that
many quantum control gates (for example, CNOT gate, Fredkin gate, etc.) can be
realized by some operations performed in the different paths of the target
photons. These schemes of quantum control gates will be discussed in the
following. Now we discuss the first use of this control-path gate. If we place
a half wave plate (HWP, set at 22.5∘-Hadamard gate) in the path of the control
photon which is shown in the dashed line of the Fig.1, the following state can
be achieved,
$\frac{1}{\sqrt{2}}\left(\alpha\left|H\right\rangle_{1}^{S_{1}}+\beta\left|H\right\rangle_{1}^{S_{2}}+\gamma\left|V\right\rangle_{1}^{S_{1}}+\delta\left|V\right\rangle_{1}^{S_{2}}\right)\left|H\right\rangle_{2}+\frac{1}{\sqrt{2}}\left(\alpha\left|H\right\rangle_{1}^{S_{1}}-\beta\left|H\right\rangle_{1}^{S_{2}}+\gamma\left|V\right\rangle_{1}^{S_{1}}-\delta\left|V\right\rangle_{1}^{S_{2}}\right)\left|V\right\rangle_{2}.$
(4)
If the detection of the control photon infers its polarization is
$\left|H\right\rangle$, the initial state $\left|\Psi\right\rangle$ has been
transferred onto the following state of a single photon in the Hilbert space
of its polarization and path states,
$\left|\Phi\right\rangle=\alpha\left|HS_{1}\right\rangle+\beta\left|HS_{2}\right\rangle+\gamma\left|VS_{1}\right\rangle+\delta\left|VS_{2}\right\rangle.$
(5)
The success probability is $P_{succ}^{CT}=1/2$. If a classical feedforward
phase shift $\pi$ is induced to the path $S_{2}$ when the detection infers the
polarization of the control photon is $\left|V\right\rangle$, the success
probability will increase to $1$.
The transformation $\left|\Psi\right\rangle\rightarrow\left|\Phi\right\rangle$
is crucial for the realization of all possible bipartite POVMs of two-photon
polarization states in Ref. Ahnert . In their scheme, a special three-photon
entangled state created by a quantum Fredkin gate and a teleportation process
of five photons are required for this transformation. It is evident that our
scheme is better than their scheme in the amount of resource, the complexity
of the operations, and the great advantage of our scheme is the success
probability is nearly unity which makes the realization of all possible
bipartite POVMs of two-photon polarization states nearly deterministic.
### II.2 CNOT gate
Secondly, we discuss the CNOT gate. Suppose two photons initially prepared in
the state $\left|\Psi\right\rangle$, and the CNOT gate can be described by the
following transformation,
$\left|\Psi\right\rangle\rightarrow\alpha\left|H\right\rangle_{1}\left|H\right\rangle_{2}+\beta\left|H\right\rangle_{1}\left|V\right\rangle_{2}+\gamma\left|V\right\rangle_{1}\left|V\right\rangle_{2}+\delta\left|V\right\rangle_{1}\left|H\right\rangle_{2},$
(6)
The experimental setup is shown in Fig.2, here the first photon is the control
photon which is transmitted through a balanced M-Z interferometer formed by
two PBSs (PBS1, PBS2), while the target photon is also transmitted through a
balanced M-Z interferometer formed by two BSs (BS1, BS2) whose transmissivity
(reflectivity) are $T_{1},T_{2}$ ($R_{1},R_{2}$) respectively. A single-photon
operation $\sigma_{x}$ is performed in one arm. With the cross-Kerr
nonlinearities and a X homodyne measurement associated with the classical
feedforward, the following states can be achieved in the output,
$\sqrt{T_{1}R_{2}}\left(\alpha\left|H\right\rangle_{1}\left|H\right\rangle_{2}+\beta\left|H\right\rangle_{1}\left|V\right\rangle_{2}\right)+\sqrt{R_{1}T_{2}}\left(\gamma\left|V\right\rangle_{1}\left|V\right\rangle_{2}+\delta\left|V\right\rangle_{1}\left|H\right\rangle_{2}\right),$
(7)
or
$\sqrt{R_{1}T_{2}}\left(\alpha\left|H\right\rangle_{1}\left|H\right\rangle_{2}+\beta\left|H\right\rangle_{1}\left|V\right\rangle_{2}\right)+\sqrt{T_{1}R_{2}}\left(\gamma\left|V\right\rangle_{1}\left|V\right\rangle_{2}+\delta\left|V\right\rangle_{1}\left|H\right\rangle_{2}\right).$
(8)
Compared with the Eq. (6), it is immediately to find that the CNOT operation
is completed when the condition $\sqrt{T_{1}R_{2}}=\sqrt{R_{1}T_{2}}$ is
satisfied, and the success probability $P_{succ}^{CNOT}=2T_{1}R_{2}$. It is
easy to find that the maximum success probability is
$P_{succ,\max}^{CNOT}=1/2$ when $T_{1}=R_{2}=1/2$. Compared with the scheme
proposed by Nemoto et al Nemoto , our scheme is probabilistic but no ancilla
photons are required.
### II.3 Fredkin gate
Thirdly, we discuss the Fredkin gate which is also called controlled-swap
gate. Consider a single photon (control photon) in the state
$\left|\psi\right\rangle=\alpha\left|H\right\rangle+\beta\left|V\right\rangle$
($\left|\alpha\right|^{2}+\left|\beta\right|^{2}=1$), and two photons (target
photons) in the state
$\left|\phi\right\rangle=p_{1}\left|\Psi^{+}\right\rangle+p_{2}\left|\Psi^{-}\right\rangle+p_{3}\left|\Phi^{+}\right\rangle+p_{4}\left|\Phi^{-}\right\rangle$
($\underset{i}{\sum}\left|p_{i}\right|^{2}=1$), where
$\left\\{\left|\Psi^{\pm}\right\rangle,\left|\Phi^{\pm}\right\rangle\right\\}$
are the Bell states. A Fredkin gate can be described by the following
transformation,
$\displaystyle\left|\psi\right\rangle\left|\phi\right\rangle$
$\displaystyle\rightarrow\alpha\left|H\right\rangle\left(p_{1}\left|\Psi^{+}\right\rangle+p_{2}\left|\Psi^{-}\right\rangle+p_{3}\left|\Phi^{+}\right\rangle+p_{4}\left|\Phi^{-}\right\rangle\right)$
$\displaystyle+\beta\left|V\right\rangle\left(p_{1}\left|\Psi^{+}\right\rangle-
p_{2}\left|\Psi^{-}\right\rangle+p_{3}\left|\Phi^{+}\right\rangle+p_{4}\left|\Phi^{-}\right\rangle\right),$
(9)
that is, if the control photon is in the state $\left|H\right\rangle$, the
target two photons are unchanged; while the control photon is in the state
$\left|V\right\rangle$, a swap operation is implemented to the target two
photons. For the reason that, only the singlet state
$\left|\Psi^{-}\right\rangle$ is antisymmetric while the other three Bell
states are symmetric, the swap operation only results in a phase shift $\pi$
to the state $\left|\Psi^{-}\right\rangle$ while the other states unchanged.
Our scheme of the Fredkin gate is shown in Fig.3. The control photon is
transmitted through a balanced M-Z interferometer formed by two PBSs (PBS1,
PBS2), while the two target photons are transmitted through a balanced M-Z
interferometer formed by two BSs (BS1, BS2 or BS3, BS4) whose transmissivity
(reflectivity) are $T_{1},T_{2}$ or $T_{3},T_{4}$ ($R_{1},R_{2}$ or
$R_{3},R_{4}$) respectively. In addition, a balanced M-Z interferometer (in
the dashed line of Fig.3) formed by two BSs (BS5, BS6) associated with a phase
shift $\pi$ in one arm is required. The Hong-Ou-Mandel interference in this
M-Z interferometer yields the following transformation Hong :
$\left|\Psi^{-}\right\rangle\rightarrow-\left|\Psi^{-}\right\rangle;\text{
}\left|\Psi^{+}\right\rangle\left(\left|\Phi^{\pm}\right\rangle\right)\rightarrow\left|\Psi^{+}\right\rangle\left(\left|\Phi^{\pm}\right\rangle\right).$
(10)
Compared with the above two schemes, we change the phase shift induced by the
control photon to be $-2\theta$, while the phase shift is $\theta$ for the two
target photons. If the cross-Kerr nonlinearities are used and we retain the
case that no phase shift induced in the coherent state, we will achieve the
following state in the output:
$\displaystyle\sqrt{T_{1}R_{2}R_{3}T_{4}}\alpha\left|H\right\rangle\left(p_{1}\left|\Psi^{+}\right\rangle+p_{2}\left|\Psi^{-}\right\rangle+p_{3}\left|\Phi^{+}\right\rangle+p_{4}\left|\Phi^{-}\right\rangle\right)$
$\displaystyle+\sqrt{R_{1}T_{2}T_{3}R_{4}}\beta\left|V\right\rangle\left(p_{1}\left|\Psi^{+}\right\rangle-
p_{2}\left|\Psi^{-}\right\rangle+p_{3}\left|\Phi^{+}\right\rangle+p_{4}\left|\Phi^{-}\right\rangle\right).$
(11)
Compared with the Eq.(9), the Fredkin gate is realized when the condition
$\sqrt{T_{1}R_{2}R_{3}T_{4}}=\sqrt{R_{1}T_{2}T_{3}R_{4}}$ is satisfied, then
the success probability is $P_{succ}^{Fredkin}=T_{1}R_{2}R_{3}T_{4}$. Hence
the maximum success probability is $P_{succ}^{Fredkin}=1/16$ when
$T_{1}=R_{2}=R_{3}=T_{4}=1/2$. Moreover, if a M-Z interferometer which is same
to the M-Z interferometer in the dashed line associated with a phase shift
$\pi$ conditionally controlled by the homodyne detection (the phase of the
coherent state is $\pm 2\theta$) through a classical feedforward, is
implemented in the outputs of BS2 and BS4, the probability may be
$P_{succ,\max}^{Fredkin}=1/8$.
Now we compare our scheme of Fredkin gate with the previous schemes. In 1989,
Milburn used the cross-Kerr nonlinearities to realize the Fredkin gate Milburn
, however, its cross-Kerr nonlinearities operate in single photon level, so it
requires huge nonlinearities which is a great challenge for the current
experimental technology. In linear optics, two types of Fredkin gate, heralded
gate and post-selected gate, had been proposed Gong ; Fiurasek1 ; Fiurasek2 .
Exclusive of the requirement of ancilla photons and small probability, the
shortcomings of these gates are obvious. The heralded Fredkin gates require
single-photon detectors which is also a great challenge for the current
technology, and the post-selected Fredkin gates work on the coincidence basis
which results in these gates are not scalable. Compared with these schemes,
only the coherent states are required in our scheme, and the structure is so
simple that we think it is feasible with the current technology.
### II.4 Toffoli gate, CU gate and MCU gate
A little change that a CNOT gate or arbitrary two-qubit unitary gate replaces
the setups in the dashed line of the Fig.3, associated with appropriate
transmissivities of the four beam splitters, is enough for the realization of
the Toffoli gate or the CU gate. In the following, we calculate the
probability of Toffoli gate and the CU gate. For the Toffoli gate, two
coherent states are required because a CNOT gate is included in this scheme.
Consider a single photon (control photon) in the state
$\left|\psi\right\rangle=\alpha\left|H\right\rangle+\beta\left|V\right\rangle$
($\left|\alpha\right|^{2}+\left|\beta\right|^{2}=1$), and two photons (target
photons) in the state
$\left|\phi\right\rangle=q_{1}\left|HH\right\rangle+q_{2}\left|HV\right\rangle+q_{3}\left|VH\right\rangle+q_{4}\left|VV\right\rangle$
($\underset{i}{\sum}\left|q_{i}\right|^{2}=1$). Suppose that the
transmissivities (reflectivities) of the four BSs are
$T_{1},T_{2},T_{3},T_{4}$ ($R_{1},R_{2},R_{3},R_{4}$) respectively, now the
modified scheme of the Fredkin gate will evolve the initial state
$\left|\psi\right\rangle\left|\phi\right\rangle$ to the follows (here we also
retain the case that no phase shift induced in the coherent state),
$\displaystyle\sqrt{T_{1}R_{2}R_{3}T_{4}}\alpha\left|H\right\rangle\left(q_{1}\left|HH\right\rangle+q_{2}\left|HV\right\rangle+q_{3}\left|VH\right\rangle+q_{4}\left|VV\right\rangle\right)$
$\displaystyle+\frac{1}{\sqrt{2}}\sqrt{R_{1}T_{2}T_{3}R_{4}}\beta\left|V\right\rangle\left(q_{1}\left|HH\right\rangle+q_{2}\left|HV\right\rangle+q_{3}\left|VV\right\rangle+q_{4}\left|VH\right\rangle\right),$
where the coefficient $1/\sqrt{2}$ is induced by the CNOT gate. The Toffoli
gate is completed when the condition
$\sqrt{T_{1}R_{2}R_{3}T_{4}}=\frac{1}{\sqrt{2}}\sqrt{R_{1}T_{2}T_{3}R_{4}}$ is
satisfied. The success probability is
$P_{succ}^{Toffoli}=T_{1}R_{2}R_{3}T_{4}$. Choose
$T_{1}=R_{2}=R_{3}=T_{4}=\frac{1}{\sqrt[4]{2}+1}$, the success probability may
be $P_{succ}^{Toffoli}\doteq\frac{1}{23}$. Similarly, a CNOT gate conditional
controlled by the homodyne detection (the phase of the coherent state is $\pm
2\theta$) through a classical feedforward is implemented in the outputs of BS2
and BS4, the probability may be $P_{succ,\max}^{Toffoli}=\frac{2}{23}$.
In linear optics, two types of Toffoli gate, heralded gate and post-selected
gate, had been proposed Fiurasek2 ; Ralph . Similarly, exclusive of the
requirement of ancilla photons and small probability, the uses of single-
photon detectors and the coincidence measurement limit their use in the
universal computation. These shortcomings are not exist in our scheme, and the
simple structure makes it much feasible with current technology.
The realization of CU gate is similar, and the success probability is
determined by the probability of the arbitrary unitary gate (suppose as $1/p$)
which can be realized by some CNOT gates and single-qubit gates, and the
transmissivities of the four beam splitters. The condition for the CU gate is
$\sqrt{T_{1}R_{2}R_{3}T_{4}}=\frac{1}{\sqrt{p}}\sqrt{R_{1}T_{2}T_{3}R_{4}}$.
Also choose $T_{1}=R_{2}=R_{3}=T_{4}=\frac{1}{\sqrt[4]{p}+1}$, the success
probability of CU gate may be
$P_{succ}^{CU}=\left(\frac{1}{\sqrt[4]{p}+1}\right)^{4}$, and it may be
$P_{succ,\max}^{CU}=2\left(\frac{1}{\sqrt[4]{p}+1}\right)^{4}$ with some
additional setups similar to the Toffoli gate. In addition, it is
straightforward to develop this technique to the realization of MCU gate which
is shown in Fig.4. The realization is described in the following. The control
photons are all transmitted through a balanced M-Z interferometer formed by
two PBSs respectively, while the target photons are all transmitted through a
balanced M-Z interferometer formed by two BSs respectively. Next, similar to
the setups in the dashed line of Fig.3, in one arm of all the M-Z
interferometers form by the BSs, we implement a multi-qubit unitary gate which
can be realized by the quantum control gates described above. Assisted by some
coherent states and the weak cross-Kerr nonlinearity, the MCU gate can be
realized associated with the appropriate transmissivities of the BSs. Compared
with the realization of MCU gate with many CNOT gates and single-qubit gates,
our scheme can reduce the complexity of the realization greatly.
## III Conclusion
In this paper, with the weak cross-Kerr nonlinearity, we first present a
special experimental scheme called C-path gate with which the realization of
all possible bipartite POVMs of two-photon polarization states can be simpler
and nearly deterministic. Following the same technique, the schemes of the
realization of quantum control gates have been proposed, including the CNOT
gate $\left(1/2\right)$, Fredkin gate $\left(1/8\right)$, Toffoli gate
$\left(2/23\right)$, CU gate and even MCU gate. All these gates are scalable
with the certain probabilities which are larger than those gates in linear
optics. Less resource are required and the structures of these gates are so
simple that we think they are feasible with current technology and may be
useful for the realization of universal computation in optics.
###### Acknowledgements.
The author would like to thank Dr Pieter Kok and Bill Munro for their helpful
discussions. The author Qing Lin was funded by the HuaQiao University
Foundation, China (Grant No. 07BS406).
## References
* (1) M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge) (2000).
* (2) T. B. Pittman, M. J. Fitch, B. C Jacobs and J. D. Franson, Phys. Rev. A, 68, 032316 (2003); J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph and D. Branning, Nature (London), 426, 264 (2003); Z. Zhao, A. N. Zhang, Y. A. Chen, H. Zhang, J. F. Du, T. Yang and J. W. Pan, Phys. Rev. Lett. 94, 030501 (2005); R. Okamoto, H. F. Hofmann, S. Takeuchi and K. Sasaki, Phys. Rev. Lett. 95, 210506 (2005); X. H. Bao, T. Y. Chen, Q. Zhang, J. Yang, H. Zhang, T. Yang and J. W. Pan, Phys. Rev. Lett. 98, 170502 (2007).
* (3) N. K. Langford, T. J. Weinhold, R. Prevedel, K. J. Resch, A. Gilchrist, J. L. O’Brien, G. J. Pryde and A. G. White, Phys. Rev. Lett. 95, 210504 (2005); N. Kiesel, C. Schmid, U. Weber, R. Ursin and H. Weinfurter, Phys. Rev. Lett. 95, 210505 (2005).
* (4) J. A. Smolin and D. P. DiVincenzo, Phys. Rev. A, 53, 2855 (1996).
* (5) S. D. Barrett, P. Kok, K. Nemoto, R. G. Beausoleil, W. J. Munro and T. P. Spiller, Phys. Rev. A, 71, 060302(R) (2005).
* (6) K. Nemoto and W. J. Munro, Phys. Rev. Lett. 93, 250502 (2004).
* (7) T. P. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock and G. J. Milburn, New J. Phys. 8, 30 (2006).
* (8) P. Kok, Phys. Rev. A 77, 013808 (2008).
* (9) S. E. Ahnert and M. C. Payne, Phys. Rev. A 73, 022333 (2006).
* (10) G. J. Milburn, Phys. Rev. Lett. 62, 2124 (1989).
* (11) Y. X. Gong, G. C. Guo and T. C. Ralph, Phys. Rev. A, 78, 012305 (2008).
* (12) J. Fiurášek, Phys. Rev. A, 78, 032317 (2008).
* (13) J. Fiurášek, Phys. Rev. A, 73, 062313 (2006).
* (14) C. K. Hong, Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987).
* (15) T. C. Ralph, K. J. Resch and A. Gilchrist, Phys. Rev. A 75, 022313 (2007).
Figure 1: Controlled-path gate with weak cross-Kerr nonlinearity. Assisted by
the switch (S) and the phase shift conditional controlled by the homodyne
detection through a classical feedforward, this gate is nearly deterministic.
If the setups in the dashed line are used, this gate can be used to realize
all possible bipartite positive-operator-value measurements of two-photon
polarization states nearly determinately. Figure 2: CNOT gate with the weak
corss-Kerr nonlinearity. Assisted by a classical feedforward, this gate can be
implemented with the probability 1/2. Figure 3: Fredkin gate with the weak
cross-Kerr nonlinearities. The setups in the dashed line will complete the
transformation
$\left|\Psi^{-}\right\rangle\rightarrow-\left|\Psi^{-}\right\rangle;\left|\Psi^{+}\right\rangle\left(\left|\Phi^{\pm}\right\rangle\right)\rightarrow\left|\Psi^{+}\right\rangle\left(\left|\Phi^{\pm}\right\rangle\right).$
Associated with the nonlinearities and appropriate transmissivities of four
beam splitters, the Fredkin gate is realized with the probability 1/16. If
some additional setups are used, the probability will increase to 1/8. For
details, see text. Figure 4: Multi-control-U gate.
|
arxiv-papers
| 2008-11-20T15:59:25 |
2024-09-04T02:48:58.888324
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Qing Lin and Jian Li",
"submitter": "Lin Qing",
"url": "https://arxiv.org/abs/0811.3364"
}
|
0811.3459
|
# Smoothing out Negative Tension Brane
Kin-ya Oda∗, Takao Suyama† and Naoto Yokoi‡
∗Physics Department, Osaka University, Osaka 564-0063, Japan
E-mail: odakin@phys.sci.osaka-u.ac.jp
†Center for Theoretical Physics, Seoul National University, Seoul 151-747,
Korea
E-mail: suyama@phya.snu.ac.kr
‡Institute of Physics, University of Tokyo, Tokyo 153-8902, Japan
E-mail: nyokoi@hep1.c.u-tokyo.ac.jp
(March 6, 2009)
###### Abstract
We propose an extension of the five dimensional gravitational action with an
external source in order to allow arbitrary smoothing of the negative tension
brane in the Randall-Sundrum model. This extended action can be derived from a
model with an auxiliary four form field coupled to the gravity. We point out a
further generalization of our model in relation to tachyon condensation. A
possible mechanism for radion stabilization in our model is also discussed.
OU-HET-614/2008
SNUTP08-010
UT-Komaba/08-19
RIKEN-TH-100
## 1 Introduction
The huge hierarchy between the weak and Planck scales has been one of the most
important issues in the theory of elementary particles. The braneworld with
the warped compactification, which is proposed by Randall and Sundrum, is an
attractive model to explain the hierarchy between the weak and Planck scales
[1]. Even though the hierarchy problem is resolved in the Randall-Sundrum
scenario, we still need a fine tuning among the bulk cosmological constant and
the brane tensions in order to acquire the observed four-dimensional
cosmological constant $\Lambda_{4}\lesssim 10^{-120}M_{P}^{4}$, where $M_{P}$
is the four-dimensional Planck scale determined by the Newton’s law of
gravitation in four dimensions. This fine tuning requires that one of the
branes should have a negative tension. Although the negative tension brane
appears to lead to some pathology in general relativity [2] (see also [3]),
detailed analyses have revealed that the four-dimensional effective theory on
the negative tension brane can be consistent with the standard Einstein
gravity [4, 5] as far as the moduli for the compactification radius, the
radion, is fixed by some mechanism, such as the one proposed by Goldberger and
Wise [6].
Recent progress in string theory has revealed that the compactified space is
generically warped under the presence of background fluxes [7], and much
effort to realize Randall-Sundrum scenario in superstring theory has been
made, see e.g. [8] and references therein. The realization of Randall-Sundrum
scenario in superstring theory naturally requires the supersymmetric extension
of the Randall-Sundrum model in five-dimensional effective supergravity.
So far, two ways to supersymmetrize the Randall-Sundrum model are proposed in
[9, 10] and in [11], respectively. The former involves a kinky gauge coupling
which has position dependence like a step function in the extra dimension.
Especially the multiple of the step function and its derivative, the delta
function, vanishes everywhere. The latter does not involve the position
dependent gauge coupling but it implicitly assumes that the multiple of a step
function and a delta function takes a finite non-zero value on the branes [12,
13].111We thank Y. Sakamura for pointing out this issue.
We want to have a way to regularize, i.e. smooth out, the step function and
the delta function in the Randall-Sundrum geometry in particular on the brane.
The step function would be realized as a kink solution of a bulk field. The
positive tension brane has been smoothed out by introducing a bulk scalar
field with solitonic kink solutions [14]. It has been shown that the negative
tension brane cannot be smoothed by a bulk scalar field without an instability
of the spacetime [15, 16, 17, 18].
Here we propose an extension of the five-dimensional Einstein-Hilbert action
that allows the smoothing of the negative tension brane. In our regularized
model, the multiple of the step function and the delta function vanishes on
the branes in the singular limit, in agreement with [9, 10]. Our model might
provide an insight on the supersymmetrization of the Randall-Sundrum model
without resorting to singular configurations.
Besides the somewhat technical motivation presented above, our study can be
placed on more general physical ground. Singular objects such as D-branes in
string theory are often interpreted as solitonic smooth objects in dynamical
or quantum mechanical analysis. Our regularization could be a useful tool for
these analyses in the braneworld models.
In the next section, we describe our model of the extension of five-
dimensional Einstein gravity. In Section 3, we present a possible embedding of
our model into the standard five-dimensional gravity with a four-form field.
Further extension with a coupling between the four-form and a scalar source is
also shown, inspired by the tachyon condensation in string theory. In Section
4, we show that the radion can be stabilized by a natural extension of the
Goldberger-Wise mechanism. In the last section, we provide summary and
discussions.
## 2 Modified Gravitational Action
Let us first briefly review the Randall-Sundrum model [1]. The fifth dimension
$y$ is compactified on $S^{1}/Z_{2}$ by the identifications $y\sim y+2L$ and
$-y\sim y$. When we restrict it to $-L<y\leq L$, the gravitational part of the
action reads
$\displaystyle S_{0}$ $\displaystyle={M_{*}^{3}}\int
d^{5}x\sqrt{-g}\left({R\over
2}-\Lambda-\lambda_{0}\delta(y)-\lambda_{L}\delta(y-L)\right),$ (1)
where $M_{*}^{3}=1/8\pi G$ with $G$ being the higher dimensional Newton
constant.222When we expand around the flat background
$g_{MN}=\eta_{MN}+h_{MN}$, the graviton $h_{MN}$ is canonically normalized
with the unit $32\pi G=1$. Note that this action does not have the full five-
dimensional diffeomorphism invariance, but the only four-dimensional
diffeomorphism invariance and reparametrization invariance with respect to the
compact direction $y$.
The Randall-Sundrum geometry is a slice of the five dimensional anti de Sitter
space (AdS5):
$\displaystyle ds^{2}$
$\displaystyle=e^{-2k|y|}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2},$ (2)
where $k$ is its curvature scale. The metric (2) becomes a solution to the
five dimensional Einstein equation $\delta S/\delta g^{MN}=0$ if the five
dimensional bulk cosmological constant $\Lambda$ and the brane tensions
$\lambda_{0}$ and $\lambda_{L}$ are fine-tuned to be
$\displaystyle\Lambda=-k\lambda_{0}=k\lambda_{L}=-6k^{2}<0.$ (3)
Physically the condition (3) implies that the ultraviolet (UV) brane at $y=0$
has the positive tension while the infrared (IR) brane at $y=L$ has the
negative one. In [14] it has been shown that the positive tension brane can be
smoothed out by introducing a bulk scalar having a kink profile.
There have been attempts to smooth out the negative tension brane by
introducing a so-called ghost scalar, which is a propagating degree of freedom
having a wrong-sign kinetic term [15, 16, 17]. Obviously there are resulting
difficulties both at the classical and quantum levels. Here we attempt to
smooth out the negative tension brane in a different way. A possible
resolution to the wrong sing kinetic term is shown in the next section.
Now we propose a modification of the action (1) preserving the same symmetry,
that is the four dimensional diffeomorphisms and the reparametrization of the
fifth coordinate $y$:
$\displaystyle S_{g}$ $\displaystyle=M_{*}^{3}\int
d^{5}x\sqrt{-g}\left({R\over
2}-\Lambda\varepsilon(y)^{2}-\lambda{\varepsilon^{\prime}(y)\over
2\sqrt{g_{yy}}}\right),$ (4)
where $\varepsilon(y)$ is an arbitrary $Z_{2}$ odd periodic function:
$\displaystyle\varepsilon(y+2L)$ $\displaystyle=\varepsilon(y),$
$\displaystyle\varepsilon(-y)$ $\displaystyle=-\varepsilon(y),$ (5)
and $\varepsilon^{\prime}(y)=\partial_{y}\varepsilon(y)$.
If we again fine tune the cosmological constant and the brane tension:
$\displaystyle\Lambda=-k\lambda=-6k^{2}<0,$ (6)
the Einstein equation
$\displaystyle R_{MN}-{g_{MN}\over
2}R+\Lambda\varepsilon(y)^{2}g_{MN}+\lambda{\varepsilon^{\prime}(y)\over
2\sqrt{g_{yy}}}\left(g_{MN}-{g_{yM}g_{yN}\over g_{yy}}\right)$
$\displaystyle=0$ (7)
has a solution:
$\displaystyle ds^{2}=e^{-2\sigma(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2},$
(8)
where
$\displaystyle\sigma(y)$
$\displaystyle=k\int^{y}d\tilde{y}\,\varepsilon(\tilde{y}).$ (9)
We note that the equation of motion of $\varepsilon(y)$:
$2\Lambda\varepsilon(y)=\frac{\lambda}{2\sqrt{-g}}\partial_{y}\left(\frac{\sqrt{-g}}{\sqrt{g_{yy}}}\right),$
(10)
is automatically satisfied due to Eqs. (6) and (9).
Now we can have an arbitrary shape of the $Z_{2}$ even periodic function
$\sigma(y)$ by a proper choice of $\varepsilon(y)$. A choice of the external
source $\varepsilon(y)$ gives a corresponding gravitational action. Any choice
of $\varepsilon(y)$ can satisfy the Einstein equation as well as the equation
of motion of $\varepsilon(y)$ itself, with the tuning (6). One might wonder
what is the source of such a large symmetry. We come back to this point in the
next section.
When we take $\varepsilon(y)$ to be the step function:
$\displaystyle\varepsilon(y)$ $\displaystyle=\begin{cases}1&(0<y<L),\\\
-1&(-L<y<0),\end{cases}$ (11)
we recover the original setup (1) with $\lambda=\lambda_{0}=-\lambda_{L}$. The
point here is that a continuous deformation of the negative tension brane
becomes possible. For instance, we can take333See e.g. footnote 5 in Ref.
[19].
$\displaystyle\varepsilon(y)$ $\displaystyle=-\tanh[\beta(y+L)]+\tanh\beta
y-\tanh[\beta(y-L)]$ (12)
for $-L\leq y\leq L$.444It is straightforward to make (12) symmetric under the
translation $y\rightarrow y+2L$ by adding terms at other integer multiples of
$L$. In the $\beta\to\infty$ limit, $\varepsilon(y)$ in (12) goes back to the
step function (11). For (12), the $\sigma(y)$ is obtained as555The value of
the constant term can be varied by rescaling $x^{\mu}$.
$\displaystyle\sigma(y)$ $\displaystyle={k\over\beta}\log{\cosh\beta
y\over\cosh[\beta(y-L)]\cosh[\beta(y+L)]}+2.$ (13)
As an example, the plot for $\beta=50$ is shown in Fig. 1. Note that this
gives a smooth regularization of the Randall-Sundrum geometry which is fully
consistent with the bulk Einstein equation.
Figure 1: Smoothed functions $\sigma(y)$ (left),
$\varepsilon(y)=\sigma^{\prime}(y)/k$ (center), and $\varepsilon^{\prime}(y)$
(right) for $\beta=50$.
In our formulation, the multiple of the (regularized) step and delta functions
is always zero on the branes,
$\displaystyle\varepsilon(y)\varepsilon^{\prime}(y)=0\qquad\text{at $y=0,L$}.$
(14)
Therefore, it may be utilized as a tool to supersymmetrize the Randall-Sundrum
model in the formulation of [9, 10], but not directly for [11].
## 3 Four Form Field and Tachyon Condensation
So far, the action (4) is given just by hand. One might consider that this is
too ad hoc. Here we argue that the action (4) could be regarded as an
effective theory of an underlying microscopic theory.
Let us consider the following five dimensional gravitational action coupled to
a four-form gauge field $A_{4}$:
$\displaystyle S$ $\displaystyle={M_{*}^{3}\over
2}\int\left(R\wedge*1+F_{5}\wedge*F_{5}\right),$ (15)
where $F_{5}=dA_{4}$ and $*$ is the Hodge dual in five dimensions. Note that
the gauge kinetic term has a wrong sign here. This does not lead to an
immediate inconsistency because $A_{4}$ has no physical degree of freedom in
five dimensions. Such a wrong-sign “auxiliary” gauge field has also been
introduced by Turok and Hawking for an inflation model inspired by the
M-theory [20].
Generically one can write a five form field in five dimensions as
$\displaystyle F_{5}$ $\displaystyle=f(x)\sqrt{-g}\,d^{5}x,$
$\displaystyle*F_{5}$ $\displaystyle=-f(x),$ (16)
where $d^{5}x=dx^{0}\wedge\dots\wedge dx^{3}\wedge dy$ and $f(x)$ is a scalar
function. The action in terms of $f(x)$ reads
$\displaystyle S$ $\displaystyle={M_{*}^{3}\over
2}\int\sqrt{-g}\,d^{5}x\left(R-f^{2}\right)={M_{*}^{3}\over
2}\int\sqrt{-g}\,d^{5}x\left(R+\varphi^{2}-2\varphi f\right),$ (17)
where we have introduced another auxiliary field $\varphi$ in the last step.
The gauge field $A_{4}$ has five independent component fields. Assuming the
invariance under the four dimensional diffeomorphisms and the
reparametrization of $y$, it would be natural to expect that the components of
$A_{4}$ except for $A_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}$, where
$\mu_{i}=0,\cdots,3$, are irrelevant, and therefore they could be ignored from
the beginning. For the remaining component, we make the following field
redefinition:
$\displaystyle A_{4}$ $\displaystyle=a(x)\sqrt{-g^{(4)}}\,d^{4}x,$ (18)
where $a(x)$ is a scalar function and $d^{4}x=dx^{0}\wedge\dots\wedge dx^{3}$.
In terms of $a(x)$, the field strength becomes
$\displaystyle f$
$\displaystyle={1\over\sqrt{-g}}\partial_{y}\left(a(x)\sqrt{-g^{(4)}}\right).$
(19)
Putting this into the action, we obtain
$\displaystyle S$ $\displaystyle={M_{*}^{3}\over 2}\int
d^{5}x\sqrt{-g}\left(R+\varphi^{2}+{2a(x)\over\sqrt{g_{yy}}}\partial_{y}\varphi\right),$
(20)
where we have performed the partial integration. If we could choose
$\varphi(x)$ and $a(x)$ as
$\displaystyle\varphi(x)$ $\displaystyle=\sqrt{\Lambda}\,\varepsilon(x),$
$\displaystyle a(x)$ $\displaystyle={\lambda\over 4\sqrt{\Lambda}},$ (21)
we recover the extended action (4).
Now it is obvious why $\varepsilon$ can be an arbitrary function, that is, why
$\delta S/\delta\varepsilon=0$ is always satisfied because $\varepsilon$ is
nothing but an auxiliary field $\varphi$. The fine tuning of the cosmological
constant and the brane tension is done by fixing the constant value of the
scalar field $a(x)$. We note that $a(x)$ must be treated as a genuinely
external source, as long as the action (15) is regarded as the microscopic
action of our theory, because the equation of motion for $a(x)$ leads to the
unwanted condition $\varepsilon^{\prime}(y)=0$ which does not allow a kink-
like solution.
It is interesting to observe that the action (15) can be rewritten as (20)
which is similar to (4). There might be a more fundamental theory whose action
includes the terms in (15) so that our action (4) can be regarded as (a part
of) its effective action. Then the resultant equation of motion of $a(x)$
would be more complicated, hopefully allowing a non-trivial solution for
$\varphi(x)$. The constant $a(x)$ solution should be consistent with the
equations of motion of the fundamental theory.
In the more fundamental theory, the profile of $\varepsilon(y)$, which has
been completely arbitrary so far, might be determined by some mechanism. For
example, suppose that there is the following term of the Chern-Simons type in
the action of the underlying theory:
$\displaystyle S_{CS}$ $\displaystyle=\int dT\wedge A_{4},$ (22)
where $T$ is a $Z_{2}$-odd scalar source. Due to this term, the equation of
motion of $a(x)$ can relate $\varphi(x)$ to $T(x)$. As a result, if $T(x)$ has
a kink-like profile, so would $\varphi(x)$. Note that this term is
topological, i.e. independent of metric and does not change the Einstein
equation. We also note that this extra term often appears in string theory in
the context of the tachyon condensation where the $T(x)$ is indeed identified
as a tachyon field, see [21] for a review and references therein. Further
investigation of a possible underlying theory that can lead to our action is
left for future research.
## 4 Introducing bulk scalar field
In general a bulk scalar field plays a crucial role in the radion
stabilization [6, 14]. To see the dependence on radius $L$ explicitly in our
setup, we rewrite our background metric (2) as
$\displaystyle ds^{2}$
$\displaystyle=e^{-2kL\alpha(\theta)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+L^{2}d\theta^{2},$
(23)
where $\alpha(\theta)=\alpha(\theta+2)$ is an arbitrary dimensionless periodic
function. One can check that the equation of motion for the metric is
satisfied for an arbitrary $L$, which is promoted to a massless field, radion,
in the perturbation analysis.
In our setup having symmetries of four dimensional diffeomorphisms and
reparametrization of the fifth coordinate, a bulk scalar field $\Phi$ can
couple to the gravity in the following form
$\displaystyle S$ $\displaystyle=M_{*}^{3}\int d^{5}x\sqrt{-g}\left[{R\over
2}-\Lambda(\Phi)\varepsilon^{2}-\lambda(\Phi){\partial_{5}\varepsilon\over
2\sqrt{g_{55}}}-{g^{MN}\over
2M_{*}^{3}}\partial_{M}\Phi\partial_{N}\Phi\right],$ (24)
where the cosmological constant $\Lambda$ and the brane tension $\lambda$ are
now promoted to functions of $\Phi$. The mass dimensions are: $[\Phi]=3/2$,
$[\Lambda]=2$, and $[\lambda]=1$. Assuming that the vacuum expectation value
is much smaller than the Planck scale $\Phi^{2}\ll M_{*}^{3}$, we can expand
in terms of $(\Phi^{2}/M_{*}^{3})$
$\displaystyle\Lambda(\Phi)$ $\displaystyle=M_{*}^{2}\left(c_{1}+{c_{2}\over
2}{\Phi^{2}\over M_{*}^{3}}+{c_{3}\over 4!}\left({\Phi^{2}\over
M_{*}^{3}}\right)^{2}+\cdots\right),$ $\displaystyle\lambda(\Phi)$
$\displaystyle=M_{*}\left(d_{1}+{d_{2}\over 2}{\Phi^{2}\over
M_{*}^{3}}+{d_{3}\over 4!}\left({\Phi^{2}\over
M_{*}^{3}}\right)^{2}+\cdots\right),$ (25)
where we have assumed the invariance under the flip $\Phi\rightarrow-\Phi$ for
simplicity.
To allow the solution of the form of the generalized step function
$\varepsilon(\theta)=\alpha^{\prime}(\theta)$, we set the first order
constants to be $c_{1}=-6(k/M_{*})^{2}$ and $d_{1}=6(k/M_{*})$. Note that the
fine tuning of the first order constants $c_{1}$ and $d_{1}$ corresponds to
the fine-tuning of the cosmological constant, which is inevitable for all the
versions of Randall-Sundrum type models so far.
The resultant bulk scalar field equation is
$\displaystyle\Phi^{\prime\prime}(\theta)-4kL\alpha^{\prime}(\theta)\Phi^{\prime}(\theta)-L^{2}M_{*}^{3}\left[\alpha^{\prime}(\theta)^{2}{\partial\Lambda\over\partial\Phi}+{\alpha^{\prime\prime}(\theta)\over
2L}{\partial\lambda\over\partial\Phi}\right]=0.$ (26)
Once we find the solution to the field equation (26) for $\Phi$, generically a
potential for the radion $L$ in the four-dimensional effective theory is
generated as
$\displaystyle V(L)$
$\displaystyle=LM_{*}^{3}\int_{-1}^{1}d\theta\,e^{-4kL\alpha(\theta)}\left(\alpha^{\prime}(\theta)^{2}\Lambda(\Phi)+{\alpha^{\prime\prime}(\theta)\over
2L}\lambda(\Phi)+{1\over 2L^{2}M_{*}^{3}}\Phi^{\prime}(\theta)^{2}\right).$
(27)
Since, in general, the equation (26) is highly non-linear, we cannot expect to
have an analytic solution.666Of course, it is straightforward to solve the
differential equations numerically. However, for free scalar field, one can
obtain an analytic solution for a certain choice of parameters as we show
below. By an ansatz $\Phi(\theta)=v^{3/2}e^{\sigma(\theta)}$, the field
equation (26) reads
$\displaystyle\sigma^{\prime\prime}(\theta)+\sigma^{\prime}(\theta)^{2}-4kL\,\alpha^{\prime}(\theta)\sigma^{\prime}(\theta)-\left(LM_{*}\right)^{2}\left[c_{2}+\cdots\right]\alpha^{\prime}(\theta)^{2}-{LM_{*}\over
2}\left[d_{2}+\cdots\right]\alpha^{\prime\prime}(\theta)=0,$ (28)
where terms denoted by $\cdots$ are $O(v^{3}/M_{*}^{3})$ and neglected in the
following. The parameter $v$ becomes the vacuum expectation value of $\Phi$ at
the positive tension brane in the limit (11) with the normalization
$\alpha(0)=0$. Here, we set the second order constants $c_{2}={d_{2}\over
4}\left(d_{2}-{8k\over M_{*}}\right)$ so that we can have an analytic solution
$\displaystyle\sigma(\theta)$ $\displaystyle={d_{2}\over
2}LM_{*}\alpha(\theta).$ (29)
There should be a more general solution to the equation (26), since it is the
second order differential equation while the solution specified by (29) has
only one integration constant $v$. To examine the general solution, it is
convenient to define
$\Psi(\theta)=e^{-d_{2}LM_{*}\alpha(\theta)/2}\Phi(\theta)$. The equation for
$\Psi(\theta)$ is reduced to
$\Psi^{{}^{\prime\prime}}(\theta)+\left(d_{2}M_{*}-4k\right)L\alpha^{{}^{\prime}}(\theta)\Psi^{{}^{\prime}}(\theta)=0.$
(30)
Obviously, $\Psi(\theta)=\textrm{const.}$ is a trivial solution. The above
equation can be integrated easily and the general solution becomes
$\Psi(\theta)=A+B\int^{\theta}d\tilde{\theta}\exp\left(\left(d_{2}M_{*}-4k\right)L\alpha(\tilde{\theta})\right).$
(31)
Since the integral in the second term does not satisfy periodicity condition
$\Psi(\theta+2)=\Psi(\theta)$, one has to put $B=0$. This proves that the
solution (29) gives the general periodic solution to Eq. (26).
Using this solution, we have the effective potential for the radius $L$ up to
quartic order ($\sim\Phi^{4}$):
$\displaystyle V(L)$ $\displaystyle={Lv^{6}\over
24M_{*}}\left(c_{3}-d_{3}\tilde{d}_{2}\right)\int_{-1}^{1}d\theta\,\alpha^{\prime}(\theta)^{2}e^{2\tilde{d}_{2}LM_{*}\alpha(\theta)},$
(32)
where we have defined $\tilde{d}_{2}\equiv d_{2}-{2k\over M_{*}}$. Note that
$c_{1}$ and $d_{1}$ terms are being understood to cancel the gravitational
part and hence omitted and that the quadratic $\Phi^{2}$ terms vanish too.
Since all the non-linear terms are suppressed by $v^{3}/M_{*}^{3}$, our free
field solution and effective potential correspond to the leading
approximations in the $v^{3}/M_{*}^{3}$ expansion.
In Fig. 2, we plot the potential $V(L)$ with the regularized profile (13) for
$\beta=50$ and the parameters
$\tilde{d}_{2}=-1,~{}c_{3}-d_{3}\tilde{d}_{2}=-0.1$ as an illustration.
Figure 2: Sample radion potential being rescaled as $V(LM_{*})/{v^{6}\over
24M_{*}^{2}}$ vs. dimensionless redius $LM_{*}$ for parameters
$\tilde{d}_{2}=-1$ and $c_{3}-d_{3}\tilde{d}_{2}=-0.1$.
We see that the non-trivial potential for $L$ is generated as the first
approximation in the $v^{3}/M_{*}^{3}$ expansion. Several comments are in
order:
* •
The resultant potential minimum value is negative here and we have to fine-
tune the cosmological constant again by shifting $c_{1},d_{1},c_{2},d_{2}$.
* •
In the singular limit, $\alpha(\theta)\rightarrow|\theta|$, the effective
potential becomes a monotonically decreasing function of $L$, and does not
have a minimum. The non-trivial minimum is generated due to the smoothness of
the branes.
* •
The original Goldberger-Wise setup [6] corresponds to $c_{3}=0$, where it has
been crucial to have different (positive) brane potentials from each
other.777The different potentials give the different boundary conditions on
the bulk scalar field that lead to the non-trivial wave function profile in
the extra dimension. Then the kinetic term balances the bulk mass term in the
total energy and serves a “repulsive force” to keep a finite radius for the
extra dimension, see e.g. Ref. [22]. However, in our action (24), the brane
potential is given by a single function $\lambda(\Phi)$ and cannot be
independent of each other even in the singular limit. In particular, the sign
of quartic coupling in the brane potential is flipped by the factor
$\partial_{5}\varepsilon\propto\left[\delta(y)-\delta(y-L)\right]$ in the
step-function limit (11) in Eq. (24), while it should be positive on the both
branes in the Goldberger-Wise setup. It would also be interesting to realize
the Goldberger-Wise mechanism by introducing another bulk scalar field in our
setup.
* •
The study including the higher order non-linear terms in the equation of
motion for $\Phi$ would require a detailed numerical analysis, which we leave
for future research. Also, the back-reaction from $\Phi$ to the metric is
neglected at this order. For a fixed four form field, we expect that
gravitational instability would be absent after the stabilization of the
radion. See also Refs. [23, 24, 25] for related issues.
## 5 Summary and Discussions
We have proposed the modification of the five dimensional gravitational action
that allows an arbitrary smoothing of the negative tension brane in the
context of the Randall-Sundrum I brane world. This can be viewed not only as a
possible regularization but also as arising from the high energy theory with
the four form auxiliary gauge field. It would be interesting to investigate
possible connection of our mechanism to the four form mechanism in the five
dimensional supergravity [12, 13].
The application of our regularized $Z_{2}$-odd function $\varepsilon(y)$ to
the $Z_{2}$-odd graviphoton gauge coupling might be possible in the context of
the supersymmetric Randall-Sundrum model [9, 10] with
$\varepsilon(y)\varepsilon^{\prime}(y)=0$ on the branes, while the version
[11] with $\varepsilon(y)\varepsilon^{\prime}(y)\neq 0$ would require further
elaboration.
We have discussed that the profile of the four form field can arise from the
tachyon condensation in string theory. In the point of view of tachyon
condensation, it would be natural to expect the following scenario to be
realized: The five-dimensional spacetime is filled with an unstable D-brane on
which there exists a tachyon $T$. If $T$ forms a kink, then the kink would be
regarded as a stable D3-brane to which $A_{4}$ couples. In other words, the
braneworld is created dynamically. However, it seems strange that, in this
scenario, an anit-kink corresponds to a negative-tension brane since it is
usually regarded as just an anti-brane with a positive tension. It would be
worth studying this issue further.
Our model does not allow the Goldberger-Wise mechanism for the radion
stabilization. However we show that the introduction of the bulk scalar field
and its natural embedding into our extended action can stabilize the radion.
The stability under gravitational perturbation of the system would be worth
investigating.
## Acknowledgment
We thank A. Miwa, Y. Sakamura and T. Shiromizu for useful comments. We
acknowledge the YITP workshop YITP-W-06-11 on “String Theory and Quantum Field
Theory,” Kyoto, Japan (2006), where partial results of this work is presented.
This work was initiated in the Theoretical Physics Laboratory, RIKEN. The
research of N.Y. is supported in part by the JSPS Research Fellowships for
Young Scientists. The work of K.O. is partly supported by Scientific Grant by
Ministry of Education and Science (Japan), Nos. 19740171, 20244028, and
20025004. The work of T.S. is supported in part by the Korea Research
Foundation Leading Scientist Grant (R02-2004-000-10150-0), Star Faculty Grant
(KRF-2005-084-C00003) and the Korea Research Foundation Grant, No.
KRF-2007-314-C00056.
## References
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* [2] T. Shiromizu, K. i. Maeda and M. Sasaki, “The Einstein equations on the 3-brane world,” Phys. Rev. D 62 (2000) 024012 [arXiv:gr-qc/9910076].
* [3] A. N. Aliev and A. E. Gumrukcuoglu, “Gravitational field equations on and off a 3-brane world,” Class. Quant. Grav. 21 (2004) 5081 [arXiv:hep-th/0407095].
* [4] S. Kanno and J. Soda, “Radion and holographic brane gravity,” Phys. Rev. D 66 (2002) 083506 [arXiv:hep-th/0207029].
* [5] T. Shiromizu and K. Koyama, “Low energy effective theory for two brane systems: Covariant curvature formulation,” Phys. Rev. D 67 (2003) 084022 [arXiv:hep-th/0210066].
* [6] W. D. Goldberger and M. B. Wise, “Modulus stabilization with bulk fields,” Phys. Rev. Lett. 83, 4922 (1999) [arXiv:hep-ph/9907447].
* [7] S. B. Giddings, S. Kachru and J. Polchinski, “Hierarchies from fluxes in string compactifications,” Phys. Rev. D 66, 106006 (2002) [arXiv:hep-th/0105097].
* [8] F. Brummer, A. Hebecker and E. Trincherini, “The throat as a Randall-Sundrum model with Goldberger-Wise stabilization,” Nucl. Phys. B 738 (2006) 283 [arXiv:hep-th/0510113].
* [9] T. Gherghetta and A. Pomarol, “Bulk fields and supersymmetry in a slice of AdS,” Nucl. Phys. B 586, 141 (2000) [arXiv:hep-ph/0003129].
* [10] A. Falkowski, Z. Lalak and S. Pokorski, “Supersymmetrizing branes with bulk in five-dimensional supergravity,” Phys. Lett. B 491 (2000) 172 [arXiv:hep-th/0004093].
* [11] R. Altendorfer, J. Bagger and D. Nemeschansky, “Supersymmetric Randall-Sundrum scenario,” Phys. Rev. D 63 (2001) 125025 [arXiv:hep-th/0003117].
* [12] E. Bergshoeff, R. Kallosh and A. Van Proeyen, “Supersymmetry in singular spaces,” JHEP 0010 (2000) 033 [arXiv:hep-th/0007044].
* [13] T. Fujita, T. Kugo and K. Ohashi, “Off-shell formulation of supergravity on orbifold,” Prog. Theor. Phys. 106 (2001) 671 [arXiv:hep-th/0106051].
* [14] O. DeWolfe, D. Z. Freedman, S. S. Gubser and A. Karch, “Modeling the fifth dimension with scalars and gravity,” Phys. Rev. D 62 (2000) 046008 [arXiv:hep-th/9909134].
* [15] R. Koley and S. Kar, “Bulk phantom fields, increasing warp factors and fermion localisation,” Mod. Phys. Lett. A 20 (2005) 363 [arXiv:hep-th/0407159].
* [16] M. Pospelov, “Ghosts and tachyons in the fifth dimension,” Int. J. Mod. Phys. A 23 (2008) 881 [arXiv:hep-ph/0412280].
* [17] N. J. Nunes and M. Peloso, “On the stability of field-theoretical regularizations of negative tension branes,” Phys. Lett. B 623 (2005) 147 [arXiv:hep-th/0506039].
* [18] R. Koley and S. Kar, “A novel braneworld model with a bulk scalar field,” Phys. Lett. B 623 (2005) 244 [Erratum-ibid. B 631 (2005) 199] [arXiv:hep-th/0507277].
* [19] K. Oda and A. Weiler, “Wilson lines in warped space: Dynamical symmetry breaking and restoration,” Phys. Lett. B 606 (2005) 408 [arXiv:hep-ph/0410061].
* [20] N. Turok and S. W. Hawking, “Open inflation, the four form and the cosmological constant,” Phys. Lett. B 432, 271 (1998) [arXiv:hep-th/9803156].
* [21] A. Sen, “Non-BPS states and branes in string theory,” arXiv:hep-th/9904207.
* [22] C. Csaki, “TASI lectures on extra dimensions and branes,” arXiv:hep-ph/0404096.
* [23] J. L. Lehners, P. Smyth and K. S. Stelle, “Stability of Horava-Witten spacetimes,” Class. Quant. Grav. 22 (2005) 2589 [arXiv:hep-th/0501212].
* [24] D. Maity, S. SenGupta and S. Sur, “Stability analysis of the Randall-Sundrum braneworld in presence of bulk scalar,” Phys. Lett. B 643 (2006) 348 [arXiv:hep-th/0604195].
* [25] D. Maity, S. SenGupta and S. Sur, “The role of higher derivative bulk scalar in stabilizing a warped spacetime,” arXiv:hep-th/0609171.
|
arxiv-papers
| 2008-11-21T03:24:15 |
2024-09-04T02:48:58.897171
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kin-ya Oda, Takao Suyama, and Naoto Yokoi",
"submitter": "Kin-ya Oda",
"url": "https://arxiv.org/abs/0811.3459"
}
|
0811.3711
|
# Deformations of symplectic vortices
Eduardo Gonzalez Department of Mathematics University of Massachusetts Boston
100 William T. Morrissey Boulevard Boston, MA 02125 eduardo@math.umb.edu and
Chris Woodward Mathematics-Hill Center, Rutgers University, 110 Frelinghuysen
Road, Piscataway, NJ 08854-8019, U.S.A. ctw@math.rutgers.edu
###### Abstract.
We prove a gluing theorem for a symplectic vortex on a compact complex curve
and a collection of holomorphic sphere bubbles. Using the theorem we show that
the moduli space of regular stable symplectic vortices on a fixed curve with
varying markings has the structure of a stratified-smooth topological
orbifold. In addition, we show that the moduli space has a non-canonical
$C^{1}$-orbifold structure.
Partially supported by NSF grant DMS060509
###### Contents
1. 1 Introduction
2. 2 Deformations of holomorphic curves
3. 3 Deformations of holomorphic maps from curves
4. 4 Deformations of symplectic vortices
## 1\. Introduction
In this paper we generalize the following result on existence of universal
deformations for stable (pseudo-)holomorphic maps. Let $(X,\omega)$ be a
compact symplectic manifold equipped with a compatible almost complex
structure $J$, and $({\Sigma},{j})$ a compact nodal complex curve. A map
${u}:{\Sigma}\to X$ is holomorphic if
$\overline{\partial}{u}:=J_{{u}}\circ{\operatorname{d}}{u}-{\operatorname{d}}{u}\circ{j}=0$
on each component of ${\Sigma}$. One naturally has the notion of a stratified-
smooth family of holomorphic maps, and hence the notion of a deformation,
namely the germ of a family around the central fiber together with an
isomorphism of the central fiber with the given map. Recall that a deformation
is universal if any other deformation is obtained from it by pullback, in a
unique way, by a map of parameter spaces. A holomorphic map ${u}:{\Sigma}\to
X$ is regular if the linearized Cauchy-Riemann operator is surjective. The
following theorem is the result of the well-known gluing construction for
holomorphic maps, c.f. Ruan-Tian [21] or the text McDuff-Salamon [15, Chapter
10] in the case of genus zero:
###### Theorem 1.0.1.
A regular holomorphic map ${u}:{\Sigma}\to X$ admits a stratified-smooth
universal deformation iff it is stable.
The construction of the universal deformation proceeds via the implicit
function theorem. For each element in the infinitesimal deformation space of
the stable map one first produces an approximate solution and then applies the
implicit function theorem to find an exact solution. Unfortunately one uses a
different Sobolev space for each “gluing parameter” controlling the domain,
which means that it is rather tricky to show that each nearby stable
holomorphic map occurs only once in the resulting family. A slightly jazzed up
version of the above theorem implies that the gluing construction gives rise
to orbifold charts on the regular locus of the moduli space of stable
holomorphic maps. Uniqueness of the universal deformations implies that the
smooth structures on each stratum are independent of the Sobolev spaces used
in the implicit function theorem. One can make these charts $C^{1}$-compatible
by suitable choices of gluing profiles, that is, coordinates on the local
deformation spaces; however the $C^{1}$-structure on the moduli space is not
canonical. The first part of the paper contains an exposition of the above
theorem, which is rather scattered in the literature.
The main result of the paper is a generalization of the theorem above to
certain gauged (pseudo)holomorphic maps, namely symplectic vortices as
introduced by Mundet [16] and Cieliebak, Gaio and Salamon, see [5]. Let $G$ be
a compact Lie group and $X$ a Hamiltonian $G$-manifold equipped with a moment
map $\Phi:X\to\mathfrak{g}^{*}$ and an invariant almost complex structure $J$.
Let $\Sigma$ be a compact smooth holomorphic curve with complex structure $j$
and equipped with an area form $\operatorname{Vol}_{\Sigma}$. A gauged
holomorphic map with values in $X$ consists of a smooth principal $G$-bundle
$P\to\Sigma$, a connection $A$ on $P$, and a smooth section $u:\Sigma\to
P(X):=P\times_{G}X$ such that $\overline{\partial}_{A}u=0$ where
$\overline{\partial}_{A}$ is defined using the splitting given by the
connection $A$ and the complex structures $J,j$. Let
$F_{A}\in\Omega^{2}(\Sigma,P(\mathfrak{g}))$ denote the curvature of $A$ and
$P(\Phi):P(X)\to P(\mathfrak{g})$ the map induced by $\Phi$. The space of
gauged holomorphic maps admits a formal symplectic structure depending on a
choice of invariant metric on $\mathfrak{g}$ so that the action of the group
of gauge transformations is formally Hamiltonian. A symplectic vortex is a
pair in the zero level set of the moment map: a pair $(A,u)$ such that
$\overline{\partial}_{A}u=0,\quad
F_{A}+u^{*}(P\times_{G}\Phi)\operatorname{Vol}_{\Sigma}=0.$
Thus the moduli space $M(\Sigma,X)$ of symplectic vortices is the symplectic
quotient of the space of gauged maps by the group of gauge transformations. In
certain cases where the moduli spaces are compact Cieliebak, Gaio, Mundet, and
Salamon [4] and Mundet [16] constructed invariants that we will call gauged
Gromov-Witten invariants by integration over these moduli spaces. In general
$M(\Sigma,X)$ admits a compactification $\overline{M}(\Sigma,X)$ consisting of
polystable symplectic vortices given by allowing $u$ to develop holomorphic
sphere bubbles in the fibers of $P\times_{G}X$. A polystable vortex is
strongly stable if the principal component has finite automorphism group, and
regular if a certain linearized operator is surjective, that is, the moduli
space is formally smooth. Our main result is the following:
###### Theorem 1.0.2.
Let $\Sigma,X$ be as above. A regular strongly stable symplectic vortex from
$\Sigma$ to $X$ admits a universal stratified-smooth deformation.
Using the deformations constructed in Theorem 1.0.2 we prove that the moduli
space $\overline{M}^{{\operatorname{reg}}}(\Sigma,X)$ of regular strongly
stable symplectic vortices admits the structure of an oriented stratified-
smooth topological orbifold, and (non-canonically) the structure of a
$C^{1}$-orbifold. The first statement implies that if
$\overline{M}^{\operatorname{reg}}(\Sigma,X)$ is compact then it carries a
rational fundamental class. The second statement implies for example, that if
the target carries a group action then the usual equivariant localization
theorems hold for the induced group action on the moduli space. In the case
that $X$ is a smooth projective variety, algebraic methods explain in [11]
give similar results and provide virtual fundamental classes on the moduli
space. However, the symplectic gluing construction is interesting in its own
right, not in the least because it potentially extends to the case of
Lagrangian boundary conditions. We understand that a forthcoming paper of
Mundet i Riera and Tian gives a gluing construction for two symplectic
vortices, when the structure group is the circle group.
Acknowledgments: We thank Ignasi Mundet i Riera, Melissa Liu, and Robert
Lipshitz for helpful comments and discussions.
## 2\. Deformations of holomorphic curves
The following section is essentially a review of the material that can be
found at the beginning of Siebert [22], with a few additional comments
incorporating terminology of Hofer, Wysocki, and Zehnder [13, Appendix]. In
the first part we review the holomorphic construction of universal
deformations of stable curves. In the second part, we study smooth
deformations of curves.
### 2.1. Holomorphic families of stable curves
A compact, complex nodal curve ${\Sigma}$ is obtained from a collection
$(\Sigma_{1},\ldots,\Sigma_{k})$ of smooth, compact, complex curves by
identifying a collection of distinct nodal points
${w}=\\{\\{w_{1}^{-},w_{1}^{+}\\},\ldots,\\{w_{m}^{-},w_{m}^{+}\\}\\}.$
For $l=1,\ldots,m$, we denote by $\Sigma_{i^{\pm}(l)}$ the components such
that $w_{l}^{\pm}\in\Sigma_{i^{\pm}(l)}$. A point $z\in{\Sigma}$ is smooth if
it is not equal to any of the nodal points. A marked nodal curve is a nodal
curve together with a collection ${z}=(z_{1},\ldots,z_{n})$ of distinct,
smooth points. An isomorphism of marked nodal curves $({\Sigma}_{0},{z}_{0})$
to $({\Sigma}_{1},{z}_{1})$ is an isomorphism
$\phi:{\Sigma}_{0}\to{\Sigma}_{1}$ of nodal curves such that
$\phi(z_{0,i})=z_{1,i}$ for $i=1,\ldots,n$. A marked nodal curve is stable if
it has finite automorphism group, that is, each component contains at least
three marked or nodal points if genus zero, or one special point if genus one.
The combinatorial type $\Gamma({\Sigma})$ of ${\Sigma}$ is the graph whose
vertices are the components and edges are the nodes and markings of
${\Sigma}$. The map ${\Sigma}\mapsto\Gamma({\Sigma})$ extends to a functor
from the category of marked nodal curves to the category of graphs. In
particular, there is a canonical homomorphism
$\operatorname{Aut}({\Sigma})\to\operatorname{Aut}(\Gamma({\Sigma}))$, whose
kernel is the product of the automorphism groups of the components of
${\Sigma}$.
Let $S$ be a complex variety (or scheme). A family of nodal curves over $S$ is
a complex variety ${\Sigma}_{S}$ equipped with a proper flat morphism
$\pi:{\Sigma}_{S}\to S$, such that each fiber ${\Sigma}_{s},s\in S$ is a nodal
curve. A deformation of a marked nodal curve ${\Sigma}$ is a germ of a family
of marked nodal curves ${\Sigma}_{S}$ over a pointed space $(S,0)$ together
with an isomorphism $\varphi:{\Sigma}_{0}\to{\Sigma}$ of the central fiber
${\Sigma}_{0}$ with ${\Sigma}$. A deformation $({\Sigma}_{S},\varphi)$ of
${\Sigma}$ is versal iff any other deformation $({\Sigma}^{\prime}_{S}\to
S^{\prime},\varphi^{\prime})$ is induced from a map $\psi:S^{\prime}\to S$ in
the sense that there exists an isomorphism $\phi$ of ${\Sigma}^{\prime}$ with
the fiber product ${\Sigma_{S}}\times_{S}S^{\prime}$ in a neighborhood of the
central fiber ${\Sigma}_{0}$. A versal deformation is universal if the map
$\phi$ is the unique such map inducing the identity on ${\Sigma}_{0}$. A
deformation has fixed type if the combinatorial type of the fiber is constant.
A universal deformation of fixed type is a deformation of fixed type, which is
universal in the above sense for deformations of fixed type. The space
$\operatorname{Def}({\Sigma})$ of infinitesimal deformations of ${\Sigma}$ is
the tangent space $T_{0}S$ of the base $S$ of a universal deformation, well-
defined up to isomorphism. We write $\operatorname{Def}_{\Gamma}({\Sigma})$
for the space of infinitesimal deformations of fixed type. Let
$\tilde{\Sigma}$ be the normalization of ${\Sigma}$, so that
$\operatorname{Def}_{\Gamma}({\Sigma})$ is isomorphic to the space of
deformations of $\tilde{\Sigma}$ equipped with the additional markings
$w_{1}^{\pm},\ldots,w_{m}^{\pm}$ obtained by lifting the nodes. The general
theory of deformations, see for example [7] in the analytic setting, shows
that any marked nodal curve ${\Sigma}$ admits a versal deformation with smooth
parameter space $S$. ${\Sigma}$ admits a universal deformation
${\Sigma}_{S}\to S$ if and only if ${\Sigma}$ is stable. Furthermore, the
space $\operatorname{Def}({\Sigma})$ of the space of infinitesimal
deformations admits a canonical isomorphism with
$H^{0,1}({\Sigma},T{\Sigma}[-z_{1}-\ldots-z_{n}]),$ where
$T{\Sigma}[-z_{1}-\dots,-z_{n}]$ is the sheaf of vector fields vanishing at
$z_{1},\dots,z_{n}$.
The relationship between the various deformation spaces (in the case with
markings, fixed type, etc.) is given as follows. The space of infinitesimal
automorphisms $\operatorname{aut}({\Sigma},{z})$ of $({\Sigma},{z})$ is the
space
$\operatorname{Vect}({\Sigma},{z})=H^{0}({\Sigma},T{\Sigma}[-z_{1}-\ldots-
z_{n}])$ of holomorphic vector fields vanishing at the marked points. The
short exact sequence of sheaves
$0\to\oplus_{i=1}^{n}T_{z_{i}}{\Sigma}\to T{\Sigma}\to
T{\Sigma}[-z_{1}-\ldots-z_{n}]\to 0$
gives a long exact sequence in cohomology [12, p. 94]
$0\to\operatorname{Vect}({\Sigma},{z})\to\operatorname{Vect}({\Sigma})\to\bigoplus_{i=1}^{n}T_{z_{i}}{\Sigma}\to\operatorname{Def}({\Sigma},{z})\to\operatorname{Def}({\Sigma})\to
0.$
From now on, we omit the markings from the notation, and study deformations of
a nodal marked curve ${\Sigma}=({\Sigma},{z})$. By $T_{w_{i}^{\pm}}{\Sigma}$,
we mean the tangent space in the component of ${\Sigma}$ containing
$w_{i}^{\pm}$. A gluing parameter for the $j$-th node is an element
$\delta_{i}\in T_{w_{i}^{+}}{\Sigma}\otimes T_{w_{i}^{-}}{\Sigma}.$ The
canonical conormal sequence [12, p. 100] gives rise to an exact sequence
(1)
$0\to\operatorname{Def}_{\Gamma}({\Sigma})\to\operatorname{Def}({\Sigma})\to\bigoplus_{i=1}^{m}T_{w_{i}^{+}}{\Sigma}\otimes
T_{w_{i}^{-}}{\Sigma}\to 0.$
After trivialization of the tangent spaces the gluing parameters are
identified with complex numbers.
Universal deformations of a smooth marked curve can be constructed for example
using Teichmüller theory [8] or by Hilbert scheme methods [12, p. 102]. Later
we will need an explicit gluing construction of a universal deformation of a
stable marked curve. This construction seems to be well-known, but the only
proof we could find in the literature is Siebert [22]. The idea is to remove
small neighborhoods of the nodes, and glue the remaining components together.
A local coordinate near a smooth point $z\in{\Sigma}$ is a neighborhood $U$ of
$z$ and a holomorphic isomorphism $\kappa$ of $U$ with a neighborhood of $0$
in the tangent line $T_{z}{\Sigma}$, whose differential $T_{0}U\to
T_{z}\Sigma$ is the identity.
###### Remark 2.1.1.
The space of local coordinates near $z$ is convex, since if
$\kappa_{0},\kappa_{1}$ are local coordinates then any combination
$t\kappa_{0}+(1-t)\kappa_{1}$ is still holomorphic and has the same
differential at $z$, and so by the inverse function theorem is a holomorphic
isomorphism in a neighborhood of $z$.
Any gluing parameter $\delta_{i}$ induces an identification
$T_{w_{i}^{+}}{\Sigma}-\\{0\\}\to T_{w_{i}^{-}}{\Sigma}-\\{0\\},\ \
\lambda_{i}^{+}\mapsto\delta_{i}/\lambda_{i}^{-}.$
Given local coordinates for the nodes of ${\Sigma}$ and a set of gluing
parameters ${\delta}=(\delta_{1},\ldots,\delta_{m})$, define a (possibly
nodal) curve ${\Sigma}^{{\delta}}$ by gluing together small disks around the
node $w_{i}$ by $z\mapsto\delta_{i}/z$, for every gluing parameter
$\delta_{i}$ that is non-zero, where $z$ is the local coordinate given by
$\kappa_{i}$. That is,
(2)
${\Sigma}^{{\delta}}=\bigcup_{i=1}^{k}\Sigma_{i}-\\{w_{1}^{\pm},\ldots,w_{m}^{\pm}\\}/(z\sim(\kappa_{i}^{+})^{-1}(\delta_{i}/\kappa_{i}^{-}(z)),i=1,\ldots,m)$
for pairs of points in the two components such that both coordinates are
defined. In particular, the choice of local coordinates near the nodes defines
a splitting of the sequence (1).
The gluing construction works in families as follows. Let
$I^{i,\pm}_{\Gamma}\to S_{\Gamma}$ resp. ${I}_{\Gamma}\to S_{\Gamma}$ denote
the vector bundle whose fiber at $s\in S_{\Gamma}$ is the tangent line at the
$j$-node resp. tensor product of tangent lines at the nodes,
(3)
$I^{i,\pm}_{\Gamma,s}=T_{w_{i,s}^{\pm}}{\Sigma}_{s},\quad{I}_{\Gamma,s}=\bigoplus_{j=1}^{m}T_{w_{i,s}^{-}}{\Sigma}_{s}\otimes
T_{w_{i,s}^{+}}{\Sigma}_{s}.$
Let ${\Sigma}_{S_{\Gamma}}\to S_{\Gamma}$ be a family of nodal curves of the
same combinatorial type $\Gamma$, with nodal points
$(w_{S_{\Gamma},j}^{\pm})_{i=1}^{m}$. A holomorphic system of local
coordinates for the $i$-th node is a holomorphic map $\kappa_{i}$ from a
neighborhood $U_{i,\pm}$ of the zero section in $I^{i,\pm}_{S}$ to
${\Sigma}_{S}$ which is an isomorphism onto its image and induces the identity
at any point in the zero section. Given a holomorphic system of coordinates
for each node
${\kappa}=(\kappa_{1}^{+},\kappa_{1}^{-},\ldots,\kappa_{m}^{+},\kappa_{m}^{-})$
the gluing construction (2) produces a family ${\Sigma}_{S}\to S$ over an open
neighborhood $S$ of the zero section in the bundle $I\to S_{\Gamma}$ of gluing
parameters.
###### Theorem 2.1.2.
[22, Proposition 2.4] If ${\Sigma}_{\Gamma,S}$ is a family giving a universal
deformation of fixed type, then ${\Sigma}_{S}$ is a universal deformation of
any of its fibers, and in particular is independent up to isomorphism of
deformations of the choice of local coordinates $\kappa$.
The following properties of universal deformations of stable curves will be
used later:
###### Theorem 2.1.3.
[22, Lemma 2.7] For any universal deformation ${\Sigma}_{S}$, the action of
automorphisms $\operatorname{Aut}({\Sigma})$ of ${\Sigma}$ extends to an
action of $\operatorname{Aut}({\Sigma})$ on ${\Sigma}_{S}$, possibly after
shrinking $S$. For any universal deformation, there exists a neighborhood of
the central fiber such that any two fibers ${\Sigma}_{S}$ contained in the
neighborhood are isomorphic, if and only if they are related by an
automorphism of ${\Sigma}$.
If ${\Sigma}$ is not stable, then the above construction produces a minimal
versal deformation of ${\Sigma}$. That is, ${\Sigma}_{S}\to S$ is versal, and
any other versal deformation given by a family
${\Sigma}^{\prime}_{S^{\prime}}\to S^{\prime}$ is obtained by pull-back by a
map $S^{\prime}\to S$.
Algebraic families of connected stable nodal curves with genus $g$ and $n$
markings form the objects of a smooth Deligne-Mumford stack
$\overline{M}_{g,n}$ [6] which admits a coarse moduli space with the structure
of a normal projective variety. The maps
$\operatorname{Def}({\Sigma})\to\overline{M}_{g,n},\ s\mapsto[{\Sigma}_{s}]$
(restricted to a neighborhood of $0$) provide $\overline{M}_{g,n}$ with an
atlas of holomorphic orbifold charts.
### 2.2. Stratified-smooth families of stable curves
We extend the definition of families and deformations to smooth and
stratified-smooth settings. Given a family ${\Sigma}_{S}\to S$ of compact
complex nodal curves, let
$S=\bigcup S_{\Gamma},\quad S_{\Gamma}=\\{s\in S,\ \
\Gamma({\Sigma}_{s})=\Gamma\\}$
denote the stratification by combinatorial type of the fiber. It follows from
the gluing construction of the previous section that if ${\Sigma}_{S}\to S$ is
a family giving a universal deformation, then each $S_{\Gamma}$ is a smooth
manifold, and the restriction ${\Sigma}_{\Gamma,S_{\Gamma}}$ of
${\Sigma}_{S_{\Gamma}}$ to $S_{\Gamma}$ gives a universal deformation of fixed
type $\Gamma$. By a smooth family of curves of fixed type $\Gamma$ we mean a
fiber bundle ${\Sigma}_{\Gamma,S_{\Gamma}}\to S_{\Gamma}$ with fibers of type
$\Gamma$ and smoothly varying complex structure. In the nodal case, it is
obtained from a smooth family of smooth holomorphic curves, identified using a
collection of pairs of smooth sections (nodes).
###### Lemma 2.2.1.
Holomorphic universal deformations of fixed type are also universal in the
category of smooth deformations of ${\Sigma}$. That is, let ${\Sigma}_{S}\to
S,\varphi$ be a universal holomorphic deformation of fixed type of a nodal
curve ${\Sigma}$. Any smooth deformation ${\Sigma}_{S^{\prime}}^{\prime}\to
S^{\prime},\varphi^{\prime}$ of nodal curves of fixed type is obtained by
pull-back ${\Sigma}_{S}\to S$ by a smooth map $S^{\prime}\to S$.
###### Proof.
By the construction of local slices for the action of diffeomorphisms in [8],
[20, Chapter 9]. ∎
Similarly we can define continuous families of holomorphic curves, which
correspond to continuous maps $S^{\prime}\to S$ to the parameter space $S$ for
a universal holomorphic deformation. The following spells out the definition
without reference to the universal holomorphic deformation.
###### Definition 2.2.2.
Let $\Gamma_{0},\Gamma_{1}$ be graphs. A simple contraction $\tau$ is a pair
of maps
$\operatorname{Vert}(\tau):\operatorname{Vert}(\Gamma_{0})\to\operatorname{Vert}(\Gamma_{1})$
and a bijection
$\operatorname{Edge}(\tau):\operatorname{Edge}(\Gamma_{0})\to\operatorname{Edge}(\Gamma_{1})\cup\\{\emptyset\\}$
such that $\Gamma_{1}$ is obtained from $\Gamma_{0}$ identifying the head and
tail of the contracting edge $e$ such that
$\operatorname{Edge}(\tau)(e)=\emptyset$. A contraction is a sequence of
simple contractions.
###### Definition 2.2.3.
A continuous family of nodal holomorphic curves consists of topological spaces
${\Sigma}_{S}$, a surjection ${\Sigma}_{S}\to S$, and a collection of
(possibly nodal) holomorphic structures $j_{{{\Sigma}}_{s}}$ on the fibers
${\Sigma}_{s},s\in S$, which vary continuously in $s$ in the following sense:
for every $s_{0}\in S$ there exists for $s$ in a neighborhood of $s_{0}$ of
some combinatorial type $\Gamma$,
1. (a)
contractions $\tau_{s}:\Gamma({\Sigma}_{s_{0}})\to\Gamma$, constant in $s\in
S_{\Gamma}$;
2. (b)
for every node $\\{w_{i}^{\pm}\\}$ collapsed under $\tau_{s}$, a pair of local
coordinates $\kappa^{\pm}_{i}:W^{\pm}_{i}\to\mathbb{C}$
3. (c)
for every component $\Sigma_{s_{0},i}$ of ${\Sigma}_{s_{0}}$, a real number
$\epsilon_{s}>0$ and maps
$\phi_{i,s}:\Sigma_{s_{0},i}-\cup_{w_{k}^{\pm}\in\Sigma_{s_{0},i},\tau_{s}(w_{k}^{\pm})=\emptyset}(\kappa_{k}^{\pm})^{-1}(B_{\epsilon_{s}})\to{\Sigma}_{s,\tau_{s}(i)}$
such that
1. (a)
for any $s$, the images of the maps $\phi_{i,s}$ cover ${\Sigma}_{s}$;
2. (b)
for any nodal point $w_{i}^{\pm}$ of ${\Sigma}_{s}$ joining components
$\Sigma_{s,i^{\pm}(k)}$, there exists a constant
$\lambda_{s}\in\mathbb{C}^{*}$ such that
$(\kappa_{k}^{+}\circ\phi_{s,i^{+}(k)}^{-1}\circ\phi_{s,i^{-}(k)}\circ(\kappa_{k}^{-})^{-1})(z)=\lambda_{s}z$,
if the former is defined, and $\lambda_{s}\to 0$ as $s\to s_{0}$.
3. (c)
for any $z\in\Sigma_{s_{0},i}$ in the complement of the $W_{k,s}^{\pm}$,
$\lim_{s\to s_{0}}(\phi_{i,s}(z))=z$;
4. (d)
$\phi_{i,s}^{*}j_{\Sigma_{s,\tau_{s}(i)}}$ converges to $j_{\Sigma_{s_{0},i}}$
uniformly in all derivatives on compact sets;
5. (e)
if $z_{i}$ is contained in $\Sigma_{s_{0},k}$, then $z_{i}=\lim_{s\to
s_{0}}\phi_{s,k}^{-1}(z_{i,s})$.
A stratified-smooth family of curves is a continuous family ${\Sigma}_{S}\to
S$ over a stratified base $S=\bigcup_{\Gamma}S_{\Gamma}$ such that the
restriction ${\Sigma}_{S_{\Gamma}}$ of ${\Sigma}_{S}$ to $S_{\Gamma}$ is a
smooth family of fixed type $\Gamma$. A stratified-smooth deformation of a
nodal curve ${\Sigma}$ is a germ of a stratified-smooth family of nodal curves
${\Sigma}_{S}$ equipped with an isomorphism of the central fiber
${\Sigma}_{0}$ with ${\Sigma}$. A universal stratified-smooth deformation of
${\Sigma}$ is a deformation with the property that any other stratified-smooth
deformation ${\Sigma}_{S^{\prime}}^{\prime}\to S^{\prime}$ is obtained by
pull-back by maps $\psi:S^{\prime}\to S$,
$\phi:{\Sigma}\times_{S}S^{\prime}\to{\Sigma}^{\prime}$, and any two
isomorphisms $\phi,\phi^{\prime}$ inducing the identity on ${\Sigma}$ are
equal.
Any universal holomorphic deformation is also a universal stratified-smooth
deformation, essentially by Lemma 2.2.1. In the stratified-smooth setting, the
analog of Theorem 2.1.3 fails and we need an additional definition:
###### Definition 2.2.4.
A universal stratified-smooth deformation $(\pi:{\Sigma}_{S}\to S,\phi)$ is
strongly universal if $\pi$ is a universal deformation of any of its fibers,
and two fibers of $\pi$ are isomorphic, if and only if they are related by the
action of $\operatorname{Aut}({\Sigma})$.
The construction of universal deformations extends to the smooth setting as
follows. Let ${\Sigma}_{S_{\Gamma}}\to S_{\Gamma}$ be a smooth family of
curves of fixed type $\Gamma$. A smooth system of local coordinates for the
$i$-th node of ${\Sigma}_{S_{\Gamma}}$ is a smooth map $\kappa_{i}$ from a
neighborhood $U_{i,\pm}$ of the zero section in $I^{i,\pm}$ to
${\Sigma}_{S_{\Gamma}}$ which is an isomorphism onto its image and induces the
identity at zero. Given a universal deformation $({\Sigma}_{S_{\Gamma}}\to
S_{\Gamma},\varphi)$ of fixed type $\Gamma$ and a smooth system of local
coordinates, applying the gluing construction (2) gives a smooth family
${\Sigma}_{S}\to S$ over an open neighborhood $S$ of $0$ in
$\operatorname{Def}({\Sigma})$. We may identify $S$ with
$\operatorname{Def}({\Sigma})$, for simplicity of notation.
###### Theorem 2.2.5.
Let ${\Sigma}$ be a stable curve. The family ${\Sigma}_{S}\to
S\subset\operatorname{Def}({\Sigma})$ constructed by gluing from a family
${\Sigma}_{\Gamma,S}\to S\subset\operatorname{Def}_{\Gamma}({\Sigma})$ of
fixed type, using any smooth family of local coordinates ${\kappa}$ near the
nodes, gives a strongly universal stratified-smooth deformation of ${\Sigma}$.
###### Proof.
Let ${\Sigma}_{S^{{\kappa}}}^{{\kappa}}\to S^{{\kappa}}$ be a family
constructed via gluing using a smooth family of local coordinates ${\kappa}$
as in (2), and ${\Sigma}_{S}\to S$ a universal deformation using a holomorphic
family of local coordinates by the same construction (2). By universality,
there exists a map $\psi:S^{{\kappa}}\to S$ so that
${\Sigma}_{\psi(s)}\cong{\Sigma}_{s}^{{\kappa}}$. It suffices to show that
$\psi$ is a diffeomorphism on each stratum. Consider the canonical map from
$T_{{\delta}}S^{{\kappa}}$ to $\operatorname{Def}({\Sigma}^{{\delta}})$, which
maps an infinitesimal change in the parameter space $S^{{\kappa}}$ to the
corresponding infinitesimal deformation of ${\Sigma}^{{\delta}}$, which we
identify with an element of
$\Omega^{0}({\Sigma},\operatorname{End}(T{\Sigma}^{{\delta}}))$. Let
$U\subset{\Sigma}^{{\delta}}$ denote the gluing region, that is, the image of
the union of domains of the local coordinates. The deformations generated by
the gluing parameters are supported in the gluing region $U$. On the other
hand, linearly independent deformations of fixed type
$\operatorname{Def}_{\Gamma}({\Sigma})$ generate deformations of the glued
curve that are linearly independent on${{\delta}}-U$, for sufficiently small
$U$. (The generated deformations will not vanish on $U$, because of the
varying local coordinates.) Thus the map
$\operatorname{Def}_{\Gamma}({\Sigma})\to\Omega^{0}({\Sigma}-U)$ is injective;
it follows that $TS^{{\kappa}}\to\operatorname{Def}({\Sigma}^{{\delta}})$ is
injective, hence an isomorphism by a dimension count. This shows that the map
$S^{{\kappa}}\to S$ is a covering. Let ${\kappa}_{t}$ be a family of local
coordinates interpolating between ${\kappa}$ and a holomorphic family. The
corresponding family $\psi_{t}$ interpolates between the identity and $\psi$.
Since each $\psi_{t}$ is a covering and $\psi_{0}$ is the identity, each
$\psi_{t}$ is a diffeomorphism. ∎
The strongly universal deformations above defined using smooth families of
local coordinates provide smooth orbifold charts on $\overline{M}_{g,n}$.
Since the space of local coordinates is convex, one can construct the local
coordinates for each stratum compatibly. Namely, let $\Gamma^{\prime}$ be a
combinatorial type degenerating to $\Gamma$. Local coordinates for the nodes
of $M_{g,n,\Gamma}$ induce local coordinates for $M_{g,n,\Gamma^{\prime}}$, in
a neighborhood of $M_{g,n,\Gamma}$, via the gluing construction (2).
###### Definition 2.2.6.
A compatible system of local coordinates for $\overline{M}_{g,n}$ is a system
of local coordinates for the nodes of each stratum $M_{g,n,\Gamma}$, so that
the local coordinates on any stratum $M_{g,n,\Gamma^{\prime}}$ are induced
from those on $M_{g,n,\Gamma}$, in a neighborhood of $M_{g,n,\Gamma}$.
Compatible systems of local coordinates can be constructed by induction on the
dimension of $M_{g,n,\Gamma}$, using convexity on the space of local
coordinates in Remark 2.1.1.
One can modify the gluing construction above by choosing a different smooth
structure on the space of gluing parameters. In the language of Hofer, Wysocki
and Zehnder [13, Appendix],
###### Definition 2.2.7.
A gluing profile is a diffeomorphism $\varphi:(0,1]\to[0,\infty)$. The
diffeomorphism given by $\varphi(\delta)=-1+1/\delta$ will be called the
standard gluing profile; $\varphi(\delta)=e^{1/\delta}-e$ will be called the
exponential gluing profile, and $\varphi(\delta)=-\ln(\delta)$ the logarithmic
gluing profile.
The set of gluing profiles naturally forms a partially ordered set: Write
$\varphi_{1}\geq\varphi_{0}$ and say $\varphi_{1}$ is softer than
$\varphi_{0}$ if $\varphi_{1}^{-1}\varphi_{0}$ extends to a diffeomorphism of
$[0,1]$. Write $\varphi_{1}>\varphi_{0}$ and say that $\varphi_{1}$ is
strictly softer than $\varphi_{0}$ if the derivatives of
$\varphi_{1}^{-1}\varphi_{0}:[0,1]\to[0,1]$ vanish at $0$. The exponential
gluing profile, standard gluing profile, and logarithmic gluing profile form a
decreasingly soft sequence in this partial order.
Fix a gluing profile $\varphi$, and consider once again the gluing
construction.
###### Definition 2.2.8.
Given a nodal curve ${\Sigma}$ with local coordinates ${\kappa}$ near the
nodes, and a collection of gluing parameters
$\delta=(\delta_{1},\ldots,\delta_{m})$, the glued curve
${\Sigma}({\delta},\varphi)$ is defined by gluing together small disks:
(4)
${\Sigma}^{{\delta},\varphi,\kappa}:=\left(\bigcup_{i=1}^{m}\Sigma_{i}-\\{w_{1}^{\pm},\ldots,w_{m}^{\pm}\\}\right)/\sim$
where the equivalence relation $\sim$ is given by
$z\sim(\kappa_{i}^{+})^{-1}(\exp(-\varphi(|\delta_{i}|)-i\arg(\delta_{i}))/\kappa_{i}^{-}(z),\quad,z\in
U_{i}^{-},\quad i=1,\ldots,m.$
More generally, given a family ${\Sigma}_{S_{\Gamma}}\to S_{\Gamma}$ of curves
of constant combinatorial type $\Gamma$ and a system of local coordinates near
the nodes ${\kappa}$, the construction (4) produces a family of curves
${\Sigma}_{S^{{\kappa},\varphi}}^{\varphi,{\kappa}}\to S^{{\kappa},\varphi}$
where $S^{{\kappa},\varphi}$ is the product of $S$ with the space of gluing
parameters.
Let ${\Sigma}$ be a compact, complex nodal curve. For any gluing profile
$\varphi$ and any collection ${\kappa}$ of local coordinates near the nodes,
the family ${\Sigma}_{S^{{\kappa},\varphi}}^{{\kappa},\varphi}\to
S^{{\kappa},\varphi}$ is a stratified-smooth strongly universal deformation,
since it is so for the standard gluing profile. Let
${\kappa}=({\kappa}_{\Gamma})$ be a compatible system of local coordinates
near the nodes, for each combinatorial type $\Gamma$. Each stratified-smooth
universal deformation above defines a classifying map
(5)
$S^{{\kappa},\varphi}/\operatorname{Aut}({\Sigma})\to\overline{M}_{g,n},\quad
s\mapsto[{\Sigma}_{s}]$
which is a homeomorphism onto its image, possibly after shrinking the
parameter space $S^{{\kappa},\varphi}$. (To obtain a precise meaning for
“classifying map” it is necessary to pass to the stacks-theoretic viewpoint,
which we do not discuss here.) The maps (5) provide $\overline{M}_{g,n}$ with
a compatible set of stratified-smooth orbifold charts, since the transition
maps are the identity on the space of gluing parameters by construction, and
smooth on each stratum. We denote by $\overline{M}_{g,n}^{{\kappa},\varphi}$
the smooth structure on $\overline{M}_{g,n}$ defined by the system of local
coordinates ${\kappa}$ near the nodes and the gluing profile $\varphi$; the
use of this smooth structure seems to have been suggested by Hofer. It seems
that these smooth structures might depend on the choice of ${\kappa}$, except
in the case of the logarithmic gluing profile, in which case one has a
canonical smooth structure.
The forgetful maps with respect to these non-standard smooth structures have
regularity properties that are worse than those with respect to the standard
smooth structure. For $2g+n>3$ we have forgetful morphisms
$f_{i}:\overline{M}_{g,n}\to\overline{M}_{g,n-1}$ by forgetting the $i$-th
marking and collapsing unstable components. There are two possibilities: a
genus zero component with one marking and two nodes is replaced by a point; a
genus zero component with two markings and one node is replaced by a single
marking. For any gluing profile, the maps $f_{i}$ are smooth away from the
locus where collapsing occurs. We say a local coordinate on a genus zero curve
is standard if it extends to an isomorphism with the projective line. The
forgetful morphism $f_{i}$ is smooth near the locus of one node, two marking
components if the local coordinates are standard and
$\delta\mapsto\exp(\varphi(\delta))^{-1}$ is smooth, that is, $\varphi$ is at
least as hard as the logarithmic gluing profile. The forgetful morphism
$f_{i}$ is smooth near the locus of curves containing components with two
nodes and one marking if the map
$\delta_{1},\delta_{2}\mapsto\varphi^{-1}(\varphi(\delta_{1})+\varphi(\delta_{2}))$
is smooth. For example, in the logarithmic gluing profile we have
$(\delta_{1},\delta_{2})\mapsto\delta_{1}\delta_{2}$, which is smooth, while
for the standard gluing profile collapsing a component gives the map
$(\delta_{1},\delta_{2})\mapsto\delta_{1}\delta_{2}/(\delta_{1}+\delta_{2})$
in the local gluing parameters, which is not smooth.
## 3\. Deformations of holomorphic maps from curves
This section reviews the construction of a stratified-smooth universal
deformations for stable (pseudo)holomorphic maps. The proof relies on a gluing
theorem, of the sort given by Ruan-Tian [21]; our approach follows that of
McDuff-Salamon [15] who treat the genus zero case. A different set-up for
gluing is described in Fukaya-Oh-Ohta-Ono [9], and explained in more detail in
Abouzaid [1]. The gluing construction gives rise to charts for the moduli
space of regular stable maps.
### 3.1. Stable maps
Let $(X,\omega)$ be a compact symplectic manifold and $\mathcal{J}(X)$ the
space of compatible almost complex structures on $X$. Let
$J\in\mathcal{J}(X)$.
###### Definition 3.1.1.
A marked nodal $J$-holomorphic map to $X$ consists of a nodal curve
${\Sigma}$, a collection ${z}=(z_{1},\ldots,z_{n})$ of distinct, smooth points
on ${\Sigma}$, and a $J$-holomorphic map ${u}:{\Sigma}\to X$. An isomorphism
of marked nodal maps from $({\Sigma}_{0},{z}_{0},{u}_{0})$ to
$({\Sigma}_{1},{z}_{1},{u}_{1})$ is an isomorphism of nodal curves
${\psi}:{\Sigma}_{0}\to{\Sigma}_{1}$ such that ${\psi}(z_{0,i})=z_{1,i}$ for
$i=1,\ldots,n$ and ${u}_{1}\circ{\psi}={u}_{0}$. A marked nodal map
$({\Sigma},{u},{z})$ is stable if it has finite automorphism group or
equivalently each component $\Sigma_{i}$ of genus zero resp. one for which
$u_{i}$ is constant has at least three resp. one special (nodal or marked)
point. The homology class of stable map ${u}:{\Sigma}\to X$ is
${u}_{*}[{\Sigma}]\in H_{2}(X,\mathbb{Z})$.
A _continuous family_ of $J$-holomorphic maps over a topological space $S$ is
a continuous family of nodal curves ${\Sigma}_{S}\to S$ (see Definition 2.2.3)
and a continuous map ${u}:{\Sigma}_{S}\to X$ which is fiberwise holomorphic.
That is, for each $s_{0}\in S$ and each nearby combinatorial type $\Gamma$ we
have
1. (a)
a sequence of contractions $\tau_{s}:\Gamma({\Sigma}_{s_{0}})\to\Gamma$,
constant in $s\in S_{\Gamma}$;
2. (b)
for every node $\\{w_{i}^{\pm}\\}$ collapsed under $\tau_{s}$, a pair of local
coordinates $\kappa^{\pm}_{i}:W^{\pm}_{i}\to\mathbb{C}$
3. (c)
for every component $\Sigma_{s_{0},i}$ of ${\Sigma}_{s_{0}}$, a real number
$\epsilon_{s}>0$ converging to $0$ as $s\to s_{0}$ and maps
$\phi_{i,s}:\Sigma_{s_{0},i}-\cup_{w_{k}^{\pm}\in\Sigma_{s_{0},i},\tau_{s}(w_{k}^{\pm})=\emptyset}(\kappa_{k}^{\pm})^{-1}(B_{\epsilon_{s}})\to{\Sigma}_{s,\tau_{s}(i)}$
such that
1. (a)
for any $s$, the images of the maps $\phi_{i,s}$ cover ${\Sigma}_{s}$;
2. (b)
for any nodal point $w_{i}^{\pm}$ of ${\Sigma}_{s}$ joining components
$\Sigma_{s,i^{\pm}(k)}$, there exists a constant
$\lambda_{s}\in\mathbb{C}^{*}$ such that
$(\kappa_{k}^{+}\circ\phi_{s,i^{+}(k)}^{-1}\circ\phi_{s,i^{-}(k)}\circ(\kappa_{k}^{-})^{-1})(z)=\lambda_{s}z$
where defined, and $\lambda_{s}\to 0$ as $s\to s_{0}$.
3. (c)
for any $z\in\Sigma_{s_{0},i}$ in the complement of the $W_{k,s}^{\pm}$,
$\lim_{s\to s_{0}}(\phi_{i,s}(z))=z$;
4. (d)
$\phi_{i,s}^{*}j_{\Sigma_{s,\tau_{s}(i)}}$ converges to $j_{\Sigma_{s_{0},i}}$
uniformly in all derivatives on compact sets;
5. (e)
if $z_{i}$ is contained in $\Sigma_{s_{0},k}$, then $z_{i}=\lim_{s\to
s_{0}}\phi_{s,k}^{-1}(z_{i,s})$.
6. (f)
$\phi_{i,s}^{*}u_{s}$ converges to $u_{s_{0}}$ uniformly in all derivatives on
compact sets.
###### Remark 3.1.2.
It follows from the assumption that $u_{S}:\Sigma_{S}\to X$ is continuous that
the homology class $u_{s,*}[\Sigma_{s}]$ is locally constant in $s\in S$.
Indeed continuity implies that for $s$ sufficiently close to $s_{0}$, $u_{S}$
is homotopic to a map of the form $v_{S}\circ\gamma_{S}$ where
$\gamma_{s}:\Sigma_{S}\to\Sigma_{s_{0}}$ is a map to the central fiber
$\Sigma_{s_{0}}$ which collapses the gluing regions to the node. Since each
$\gamma_{s}=\gamma_{S}|\Sigma_{s}$ maps $[\Sigma_{s}]$ to $[\Sigma_{s_{0}}]$,
the claim follows.
In particular, taking $S$ to be the topological space given as the closure of
the set $S^{*}$ of rational numbers of the form $1/i,i\in\mathbb{Z}_{>0}$, we
say that a sequence of holomorphic maps $u_{i}:\Sigma_{i}\to X$ Gromov
converges if it extends to a continuous family over $S$. To state the Gromov
compactness theorem, recall that the energy of a map $u:\Sigma\to X$ is
$E(u)=\frac{1}{2}\int|du|^{2}.$
###### Theorem 3.1.3 (Gromov compactness).
Let $X,\omega,J$ be as above. Any sequence $u_{i}:\Sigma_{i}\to X$ of stable
holomorphic maps with bounded energy has a Gromov convergent subsequence.
Furthermore, the limit is unique.
For references and discussion, see for example [14, Theorem 1.8]. The
definition of Gromov convergence passes naturally to equivalence classes of
stable maps. A subset $C$ of $\overline{M}_{g,n}(X,d)$ is Gromov closed if any
sequence in $C$ has a limit point in $C$, and Gromov open if its complement is
closed. The Gromov open sets form a topology for which any convergent sequence
is Gromov convergent, by an argument using [15, Lemma 5.6.5]. Furthermore, any
convergent sequence has a unique limit. Gromov compactness implies that for
any $E>0$, the union of $\overline{M}_{g,n}(X,d)$ over $d\in
H_{2}(X,\mathbb{Z})$ with $(d,[\omega])<E$ is a compact, Hausdorff space.
###### Definition 3.1.4.
Let $X,\omega,J$ be as above. A stratified-smooth family of nodal
$J$-holomorphic maps over a space $S$ is a pair $({\Sigma}_{S},{u}_{S})$ of a
stratified-smooth family of nodal curves ${\Sigma}_{S}\to S$ together with a
continuous map ${u}_{S}:{\Sigma}_{S}\to X$ such that the restriction ${u}_{s}$
of ${u}$ to any fiber ${\Sigma}_{s}$ is holomorphic, and the restriction of
${u}_{S}$ to any stratum ${\Sigma}_{\Gamma}$ is smooth. A stratified-smooth
deformation of a stable $J$-holomorphic map $({\Sigma},{u})$ is a germ of a
stratified-smooth family $({\Sigma}_{S},{u}_{S})$ together with an isomorphism
of nodal maps $\iota:{\Sigma}_{0}\to{\Sigma}$ such that
$\iota^{*}{u}={u}_{0}$. A deformation $({\Sigma}_{S},{u}_{S},\iota)$ of
$({\Sigma},{u})$ is versal if any other (germ of) family of marked, nodal
curves $({\Sigma}^{\prime},{\Sigma}_{0})\to(S^{\prime},0)$ is induced from a
map $\psi:S^{\prime}\to S$ in the sense that there exists an isomorphism
$\phi:{\Sigma}^{\prime}\to{\Sigma}\times_{S}S^{\prime}$ in a neighborhood of
the central fiber ${\Sigma}_{0}$, and ${u}^{\prime}$ is obtained by composing
projection on the first factor with ${u}$. A versal deformation is universal
if the map $\phi$ above is the unique map inducing the identity on
${\Sigma}_{0}$.
### 3.2. Smooth universal deformations of regular stable maps of fixed
combinatorial type.
Let ${u}:{\Sigma}\to X$ be a stable map. For $p>2$ define a fiber bundle
$\mathcal{E}\to\mathcal{B}$ by
$\mathcal{B}=\mathcal{J}({\Sigma})\times\operatorname{Map}({\Sigma},X)_{1,p},\quad\mathcal{E}_{\zeta,{u}}=\Omega^{0,1}({\Sigma},{u}^{*}TX)_{0,p},$
where the latter is the space of $(0,1)$-forms with respect to the pair
$({j}(\zeta),J)$. Consider the Cauchy-Riemann section,
$\overline{\partial}:\mathcal{B}\to\mathcal{E},\ \ \
(j,{u})\mapsto\overline{\partial}_{j}{u},\ \
\overline{\partial}_{j}{u}={\frac{1}{2}}({\operatorname{d}}{u}\circ{j}(\zeta)-J_{u}\circ{\operatorname{d}}{u}).$
Let
$\operatorname{ev}:\mathcal{B}\to X^{2m},\ \ \
{u}\mapsto({u}(w_{1}^{-}),{u}(w_{1}^{+}),\ldots,{u}(w_{m}^{-}),{u}(w_{m}^{+}))$
denote the map evaluating at the nodal points. The space of stable maps of
type $\Gamma$ is given as $(\overline{\partial},\operatorname{ev})^{-1}(0)$.
To obtain a Fredholm map, we quotient by diffeomorphisms of $\Sigma$, or
equivalently, restrict to a minimal versal deformation ${\Sigma}_{S}\to S$ of
${\Sigma}$ of fixed type. This means that for each
$\zeta\in\operatorname{Def}_{\Gamma}({\Sigma})$ near $0$ we have a complex
structure ${j}(\zeta)$ on ${\Sigma}$, which we may assume agrees with
${j}={j}(0)$ near the nodes. Then the Cauchy-Riemann section induces a map
$\operatorname{Def}_{\Gamma}(\Sigma)\times\Omega^{0}(\Sigma,u^{*}TX)\to\mathcal{E}.$
Linearizing the Cauchy-Riemann section, together with the differences at the
nodes, gives rise to a Fredholm operator
$\tilde{D}_{{u}}:\operatorname{Def}_{\Gamma}({\Sigma})\times\Omega^{0}({\Sigma},{u}^{*}TX)\to\Omega^{0,1}({\Sigma},{u}^{*}TX)\oplus\bigoplus_{i=1}^{m}{u}(w_{i}^{\pm})^{*}TX$
(6)
$\tilde{D}_{{u}}({\zeta},{\xi}):=\left(\pi^{0,1}_{{\Sigma}}(\nabla{\xi}-{\frac{1}{2}}J({u}){\operatorname{d}}{u}Dj({\zeta})-{\frac{1}{2}}J_{{u}}(\nabla_{{\xi}}J)_{{u}}\partial{u})),({\xi}(w_{i}^{+})-{\xi}(w_{i}^{-}))_{i=1}^{m}\right)$
given by the linearized Cauchy-Riemann operator on each component, and the
difference of the values of the section at the nodes
$w^{\pm}_{1},\ldots,w^{\pm}_{m}$. The map ${u}=({\Sigma},{u},{z})$ is regular
if $\tilde{D}_{{u}}$ is surjective. This is independent of the choice of
representatives $j(\zeta)$: any two such choices
${j}^{\prime}(\zeta),{j}(\zeta)$ are related by a diffeomorphism of
${\Sigma}$. The space of infinitesimal deformations of ${u}$ of fixed type is
$\operatorname{Def}_{\Gamma}({u})=\operatorname{ker}(\tilde{D}_{{u}})/\operatorname{aut}({\Sigma}).$
The space of infinitesimal deformations of ${u}$ is
$\operatorname{Def}({u})=\operatorname{Def}_{\Gamma}({u})\oplus\bigoplus_{i=1}^{m}T_{w_{j}^{+}}{\Sigma}\otimes
T_{w_{j}^{-}}{\Sigma}$
where $\Gamma$ is the type of ${u}$.
###### Theorem 3.2.1.
Let $X,\omega,J$ be as above. A regular marked nodal $J$-holomorphic map
${u}=({\Sigma},{u},{z})$ admits a strongly universal deformation
$({\Sigma}_{S},{u}_{S},{z}_{S})$ with parameter space
$S\subset\operatorname{Def}_{\Gamma}({u})$ of fixed type if and only if ${u}$
is stable.
###### Proof.
Let $({\Sigma},{u})$ be a stable map to $X$ and ${\Sigma}_{S}\to
S\subset\operatorname{Def}_{\Gamma}({\Sigma})$ a minimal versal deformation of
${\Sigma}$ of fixed type constructed in (2). We may write any map
$C^{0}$-close to ${u}$ as $\exp_{{u}}({\xi})$ for some
${\xi}\in\Omega^{0}({\Sigma},{u}^{*}TX)$. Let
$\Psi_{{u}}({\xi}):{u}^{*}TX\to\exp_{{u}}({\xi})^{*}TX$ denote parallel
transport along geodesics with respect to the Hermitian connection
$\tilde{\nabla}=\nabla-{\frac{1}{2}}J(\nabla J);$ here $\nabla$ is the Levi-
Civita connection, see [15, Chapter 2]. This defines an isomorphism
(7)
$\Psi_{{u}}({\xi})^{-1}:\Omega^{0,1}_{{j}}({\Sigma},\exp_{{u}}({\xi})^{*}TX)\to\Omega^{0,1}_{{j}}({\Sigma},{u}^{*}TX).$
where subscript $j$ denotes the space of $0,1$-forms taken with respect to the
complex structure $j$ on ${\Sigma}$. There is an isomorphism of
$\Omega^{0,1}_{{j}(\zeta)}({\Sigma},{u}^{*}TX)$ with
$\Omega^{0,1}_{{j}}({\Sigma},{u}^{*}TX)$ given by composing the inclusion
$\Omega^{0,1}_{{j}(\zeta)}({\Sigma},{u}^{*}TX)\to\Omega^{1}({\Sigma},{u}^{*}TX)_{\mathbb{C}}=\Omega^{1}({\Sigma};{u}^{*}TX)\otimes_{\mathbb{R}}\mathbb{C}$
with the projection
$\Omega^{1}({\Sigma};{u}^{*}TX)\otimes_{\mathbb{R}}\mathbb{C}\to\Omega^{0,1}_{{j}}({\Sigma},{u}^{*}TX).$
We denote by
$\Psi_{j}(\zeta):\Omega^{0,1}_{{j}}({\Sigma},{u}^{*}TX)\to\Omega^{0,1}_{{j}(\zeta)}({\Sigma},{u}^{*}TX)$
the resulting map; one can think of this as a connection over the space of
complex structures on ${\Sigma}$ on the bundle whose fiber is the space of
$0,1$-forms with respect to $j(\zeta)$. By composing $\Psi_{{u}}({\xi})^{-1}$
and $\Psi_{j}(\zeta)^{-1}$ we obtain an identification
(8)
$\Psi_{j,{u}}(\zeta,\xi)^{-1}:\Omega^{0,1}_{{j}(\zeta)}({\Sigma},\exp_{{u}}({\xi})^{*}TX)\to\Omega^{0,1}_{{j}}({\Sigma},{u}^{*}TX).$
Define
$\mathcal{F}_{{u}}:\operatorname{Def}_{\Gamma}({\Sigma})\times\Omega^{0}({\Sigma},{u}^{*}TX)\to\Omega^{0,1}_{{j}}({\Sigma},{u}^{*}TX)$
$(\zeta,{\xi})\mapsto\Psi_{j,{u}}(\zeta,{\xi})^{-1}(\overline{\partial}_{{j}(\zeta)}(\exp_{{u}}({\xi}))).$
The operator $\tilde{D}_{{u}}$ is the linearization of $\mathcal{F}_{{u}}$.
The implicit function theorem implies that if ${u}$ is regular then the zero
set of $\mathcal{F}_{{u}}$ is modelled locally on a neighborhood of $0$ in
$\operatorname{ker}(\tilde{D}_{{u}})$. Furthermore, by elliptic regularity the
zero set consists entirely of smooth $J$-holomorphic maps [15, Section B.4].
Thus we obtain a smooth family of stable maps in a neighborhood of $0$ in
$\operatorname{ker}(\tilde{D}_{{u}})$. The action of $\operatorname{Aut}({u})$
on the space of stable maps with domain ${\Sigma}$ induces an inclusion of the
Lie algebra $\operatorname{aut}({u})$ into
$\operatorname{ker}(\tilde{D}_{{u}})$. Restricting to
$\operatorname{Def}_{\Gamma}({u})$, identified with a complement of
$\operatorname{aut}({\Sigma})$ (that is, a slice for the
$\operatorname{Aut}({u})$ action) gives a family $({\Sigma}_{S},{u}_{S})\to
S\subset\operatorname{Def}_{\Gamma}({u})$ of fixed type. The family
$({\Sigma}_{S},{u}_{S})$, together with the canonical identification $\iota$
of the central fiber with ${\Sigma}$, is a universal smooth deformation of
fixed type. Indeed, another smooth family
$({\Sigma}_{S^{\prime}},{u}_{S^{\prime}})$ over a base $S^{\prime}$ is in
particular a deformation of the underlying curve. After shrinking
$S^{\prime}$, each fiber of
$({\Sigma}_{s^{\prime}},{u}_{s^{\prime}}^{\prime})$ corresponds to a zero of
$\mathcal{F}_{{u}}$, and so lies in the image of the map given by the implicit
function theorem. The uniqueness part of the implicit function theorem gives a
smooth map $\psi:S^{\prime}\to\operatorname{Def}_{\Gamma}({u})$ and an
identification ${\Sigma}_{S^{\prime}}\to\psi^{*}{\Sigma}_{S}$. Any two such
maps inducing the same map on the central fiber are close in a neighborhood of
the central fiber. Since the automorphism groups of the central fibre are
discrete, any automorphism group is discrete. Thus any two such automorphisms
defined in a neighborhood of the central fiber, and equal on the central fiber
must be equal in a neighborhood of the central fiber. This shows that the
identification is unique, so that the deformation given by the gluing
construction is universal.
If ${u}$ is not stable, then it has no universal deformation since the
identification with the central fiber is unique only up to a continuous family
of automorphisms. ∎
Let $M_{g,n,\Gamma}^{{\operatorname{reg}}}(X,d)$ denote the moduli space of
regular stable maps of combinatorial type $\Gamma$. A family ${u}_{S}$ over
$S\subset\operatorname{Def}_{\Gamma}({u})$ induces a map
(9) $S\to M_{g,n,\Gamma}^{{\operatorname{reg}}}(X,d),\ s\mapsto[{u}_{s}]$
where $[{u}_{s}]$ denotes the isomorphism class of ${u}_{s}:{\Sigma}_{s}\to
X$.
###### Theorem 3.2.2.
For any $g,n,d$ and combinatorial type $\Gamma$ with $m$ nodes,
$M_{g,n,\Gamma}^{{\operatorname{reg}}}(X,d)$ has the structure of a smooth
orbifold of dimension $(1-g)(\dim(X)-6)+2(c_{1}(TX),d)-2m+2n$, with tangent
space at $[{u}]$ isomorphic to $\operatorname{Def}_{\Gamma}({u})$.
###### Proof.
By Theorem 3.2.1, the maps (9) for families giving universal deformations are
homeomorphisms onto their image and provide compatible charts. The dimension
formula follows from Riemann-Roch: The index of $\tilde{D}_{{u}}$ may be
deformed to a complex linear operator by homotoping the zero-th order terms
(which define a compact operator) to zero. ∎
### 3.3. Constructing stratified-smooth deformations of varying type
The main result of this section is Theorem 1.0.1, which is probably well-
known, cf. [19], [21], but for which we could not find an explicit reference.
The theorem itself will not be used, but the estimates involved in the proof
will be needed later for the corresponding result for vortices. The proof uses
a gluing construction for holomorphic maps, which produces from a smooth
family of holomorphic maps of fixed type, a stratified-smooth family of maps
of varying type.
Step 1: Approximate Solution
###### Definition 3.3.1.
Let ${\Sigma}$ be a compact, complex nodal curve. A gluing datum for
${\Sigma}$ consists of
1. (a)
a collection of gluing parameters ${\delta}=(\delta_{1},\ldots,\delta_{m})$ in
the bundle ${I}$ of (3);
2. (b)
local coordinates $\kappa_{j}^{\pm}$ near the nodes $w_{j}^{\pm}$ for
$j=1,\ldots,m$;
3. (c)
a parameter $\rho$ which describes the width of the annulus on which the
gluing of maps is performed;
4. (d)
a gluing profile $\varphi$, see Definition 2.2.7;
5. (e)
a smooth cutoff function
(10) $\alpha:\mathbb{C}\to[0,1],\ \ \
\alpha(z)=\left\\{\begin{array}[]{ll}0&|z|\leq 1\\\ 1&|z|\geq
2\end{array}\right\\}.$
We first treat the case that $\varphi$ is the standard gluing profile. Let a
gluing datum be given, and ${\Sigma}^{{\delta}}$ denote the glued curve from
(2). Let ${u}:{\Sigma}\to X$ be a holomorphic map. Near each node $w_{k}$ let
$i^{\pm}(k)$ denote the components on either side of $w_{k}$. In the
neighborhoods $U_{k}^{\pm}$ (assuming they have been chosen sufficiently
small) define maps
$\xi_{k}^{\pm}:U_{k}^{\pm}\to T_{x_{k}}X,\ \ \
u_{i^{\pm}(k)}(z)=\exp_{x_{k}}(\xi_{k}^{\pm}(z))$
where $x_{k}=u(w_{k})$ and $\exp_{x_{k}}:T_{x_{k}}X\to X$ denotes geodesic
exponentiation. Given a holomorphic map ${u}:{\Sigma}\to X$, and a gluing
datum $({\delta},{\kappa},\rho,\varphi,\alpha)$ define the pre-glued map by
interpolating between the maps on the various components using the given
cutoff function and local coordinates: ${u}^{{\delta}}={u}(z)$ for
$z\notin\cup_{k}U_{k}^{\pm}$ and otherwise
(11)
${u}^{{\delta}}(z)=\exp_{x_{k}}(\alpha(\kappa_{k}^{\pm}(z)/\rho|\delta_{k}|^{1/2})\xi_{k}^{\pm}(z))\quad
z\in U_{k}^{\pm}.$
###### Remark 3.3.2.
The same formula but with domain ${\Sigma}$ (not the glued curve) defines an
intermediate map ${u}_{0}^{{\delta}}:{\Sigma}\to X$ which is constant near the
nodes. The right inverse of $\tilde{D}_{{u}_{0}^{{\delta}}}$ will be used in
the gluing construction.
First we estimate the failure of ${u}^{{\delta}}$ to satisfy the Cauchy-
Riemann equation. Define on ${\Sigma}^{{\delta}}$ the $C^{0}$-metric $g$ by
the identification
(12)
${\Sigma}^{{\delta}}={\Sigma}-\bigcup_{k,\pm}\kappa_{k}^{\pm}(B_{|\delta_{k}|^{1/2}}(0))/\left(\kappa^{+}_{k}(\partial
B_{|\delta_{k}|^{1/2}}(0))\sim\kappa^{-}_{k}(\partial
B_{|\delta_{k}|^{1/2}}(0))\right)$
using a Kähler metric on $\Sigma$, see Figure 1.
Figure 1. Continuous metric on a glued curve
The generalized Sobolev spaces $W^{l,p}$ with respect to this metric are
defined for $p\geq 1$ and integers $l\in\\{0,1\\}$, see [2] or [3]. For any
vector bundle $E$ we denote by $\Omega({\Sigma}^{{\delta}},E)_{l,p,{\delta}}$
the space of $W^{l,p}$ forms with values in $E$. If $p=\infty$ the norm is
independent of ${\delta}$ and we drop it from the notation. Let
$\|\cdot\|_{k,p,{\delta}}$ denote the Sobolev $W^{k,p}$-norm on
$\Omega^{0}({u}^{*}TX)$ defined using the ${\delta}$-dependent metric (12).
###### Proposition 3.3.3.
Suppose that ${u}:{\Sigma}\to X$ is a stable map, and
${u}^{{\delta}}:{\Sigma}^{{\delta}}\to X$ is the pre-glued map defined in
(11), defined for ${\delta}$ sufficiently small. There is a constant $c$ and
an $\epsilon>0$ such that if $\|{\delta}\|<\epsilon,\rho>1/\epsilon,$ and
$|\delta_{k}|^{2}\rho<\epsilon,\ \ \ k=1,\ldots,m$ then
$\|\overline{\partial}{u}^{{\delta}}\|_{0,p,{\delta}}^{p}\leq
c\sum_{k=1}^{m}(|\delta_{k}|^{1/2}\rho)^{2}.$
###### Proof.
Compare with McDuff-Salamon [15, Chapter 10]. The error term
$\overline{\partial}{u}({\delta})$ can be estimated by terms of two types;
those involving derivatives of the cutoff functions and those involving
derivatives of the map $\xi_{k}$. The derivative of $\exp_{x_{k}}$ is
approximately the identity near the node. The derivatives of $\alpha$ grow
like $1/\rho|\delta_{k}|^{1/2}$, while the norm of $\xi_{k}^{\pm}$ is bounded
by a constant times $\rho|\delta_{k}|^{1/2}$ on the gluing region. The term
involving the derivatives of $\alpha$ is bounded and supported on a region of
area less than $\pi\rho^{2}|\delta_{k}|$ for each node. The derivatives of
$\xi_{k}^{\pm}$ are also uniformly bounded, and the area bound gives the
required estimate. ∎
Let ${\Sigma}_{S}\to S$ with $S\subset\operatorname{Def}_{\Gamma}({\Sigma})$
be a family giving a minimal versal deformation of ${\Sigma}$ of fixed type,
and ${\Sigma}_{S_{{\delta}}({\delta})}\to
S_{{\delta}}\subset\operatorname{Def}({\Sigma}^{{\delta}})$ a family giving a
minimal versal deformation of ${\Sigma}^{{\delta}}$. The gluing construction
(2) applied to the family ${\Sigma}_{S}$ produces a map
(13)
$\operatorname{Def}_{\Gamma}({\Sigma})\to\mathcal{J}({\Sigma}^{{\delta}}),\quad\zeta\mapsto
j^{{\delta}}(\zeta)$
which maps any deformation of the original curve to the corresponding
deformation of the glued curve. In other words, any variation of complex
structure on ${\Sigma}$ of fixed type induces a variation of complex structure
on ${\Sigma}^{{\delta}}$. Similarly, for any
$\xi\in\Omega^{0}({\Sigma},{u}^{*}TX)$ we obtain an element
$\xi^{{\delta}}\in\Omega^{0}({\Sigma}^{{\delta}},{u}^{*}TX)$.
###### Proposition 3.3.4.
Suppose that ${u},{u}^{{\delta}}$ are as above, and
$(\zeta,\xi)\in\operatorname{Def}_{\Gamma}(u)$. There is a constant $c$ and an
$\epsilon>0$ such that if $\|{\delta}\|<\epsilon,\rho>1/\epsilon$,
$\|\zeta\|+\|{\xi}\|_{1,p}\leq\epsilon$, and $|\delta_{k}|^{2}\rho<\epsilon$
for $k=1,\ldots,m$ then
$\|\overline{\partial}_{{j}^{{\delta}}(\zeta)}\exp_{{u}^{{\delta}}}({\xi}^{{\delta}})\|_{0,p,{\delta}}^{p}\leq
c\sum_{k=1}^{m}(|\delta_{k}|^{1/2}\rho)^{2}.$
Step 2: Uniformly bounded right inverse
We wish to show that the map in Proposition 3.3.4 can be corrected to obtain a
holomorphic map. Define
(14)
$\mathcal{F}_{{u}}^{{\delta}}:\operatorname{Def}_{\Gamma}({\Sigma})\times\Omega^{0}({\Sigma}^{{\delta}},{u}^{{\delta},*}TX)\to\Omega^{0,1}({\Sigma}^{{\delta}},{u}^{{\delta},*}TX)$
$(\zeta,\xi)\mapsto\Psi_{j,{u}^{{\delta}}}(\zeta,\xi)^{-1}(\overline{\partial}_{{j}^{{\delta}}(\zeta)}(\exp_{{u}^{{\delta}}}(\xi))).$
Here the operator $\Psi_{j,{u}^{\delta}}$ is as in (8). Let
$\tilde{D}_{{u}}^{{\delta}}(\xi)$ be the associated linear operator, that is,
the linearization of (14) at $\xi$. This operator naturally extends to a map
from Sobolev $1,p$-completion of the second factor of the domain to the
$0,p$-completion of the codomain. We denote by
$\tilde{D}_{{u}}^{{\delta}}:=\tilde{D}_{{u}}^{{\delta}}(0)$. We will construct
an approximate inverse
(15)
$T_{{\delta}}:\Omega^{0,1}({\Sigma}^{{\delta}},{u}^{{\delta},*}TX)\to\operatorname{Def}_{\Gamma}({\Sigma})\oplus\Omega^{0}({\Sigma}^{{\delta}},{u}^{{\delta},*}TX)$
to $\tilde{D}_{{u}}^{{\delta}}$. The construction depends on a carefully
chosen cutoff function:
###### Lemma 3.3.5.
[15, Section 10.3] For any $\delta>0,\rho>1$ there exists a function
$\beta_{\rho,\delta}:\mathbb{R}^{2}\to[0,1]$ that satisfies
$\beta_{\rho,\delta}(z)=\begin{cases}0&|z|\leq\sqrt{\delta/\rho}\\\
1&|z|\geq\sqrt{\delta\rho}\end{cases}$
and for all $\xi\in W^{1,p}(B_{\rho|\delta_{k}|})$ satisfying $\xi(0)=0$
(16) $\|(\nabla\beta_{2;\rho,\delta})\xi\|_{0,p}\leq
c\log(\rho^{2})^{-1+1/p}\|\xi\|_{1,p},\quad\|\beta_{2;\rho,\delta}\|_{1,2}\leq
C\log(\rho^{2})^{-1/2}.$
Recall the map ${u}_{0}$ from Remark 3.3.2.
###### Lemma 3.3.6.
For sufficiently small ${\delta}$ there exists a right inverse
$Q_{{u}_{0}^{{\delta}}}$ of $\tilde{D}_{{u}_{0}^{{\delta}}}$ with image the
$L^{2}$-perpendicular of the kernel of $\tilde{D}_{{u}_{0}^{{\delta}}}$.
###### Proof.
Consider the maps defined by parallel transport using the modified Levi-Civita
connection,
$\Pi_{{u}_{0}^{{\delta}}}^{{u}}:\Omega^{0}({\Sigma},({u}_{0}^{{\delta}})^{*}TX)\to\Omega^{0}({\Sigma},({u}^{{\delta}})^{*}TX)$
$\Psi_{{u}_{0}^{{\delta}}}^{{u}}:\Omega^{0,1}({\Sigma},({u}_{0}^{{\delta}})^{*}TX)\to\Omega^{0,1}({\Sigma},({u}^{{\delta}})^{*}TX).$
The operator
$\Psi_{{u}_{0}^{{\delta}}}^{{u}}\tilde{D}_{{\Sigma},{u}_{0}^{{\delta}}}(\Pi_{{u}_{0}^{{\delta}}}^{{u}})^{-1}$
approaches the operator $\tilde{D}_{{u}}$ as ${\delta}\to 0$, c.f. [15, Remark
10.2.2]. The statement of the lemma follows. ∎
Define the approximate right inverse for $\tilde{D}_{{u}}^{{{\delta}}}$ by
composing the right inverse $Q_{{u}_{0}^{{\delta}}}$ with a cutoff and
extension operator:
$T_{{\delta}}:=P_{{\delta}}Q_{{u}_{0}^{{\delta}}}K_{{\delta}},$ defined as
follows. The cutoff operator
$K_{{\delta}}:\Omega^{0,1}({u}^{{\delta}}*TX)_{0,p,{\delta}}\to\Omega^{0,1}({u}_{0}^{{\delta},*}TX)_{0,p}$
is defined by
$(K_{{\delta}}(\eta))(z)=\begin{cases}\eta(z)&z\notin\bigcup_{k,\pm}B_{|\delta_{k}(z)|^{1/2}}(w_{k}^{\pm})\\\
0&\text{otherwise}\end{cases}.$
We have $\|K_{{\delta}}\eta\|_{0,p}\leq\|\eta\|_{0,p,{\delta}}$ by definition
of the $0,p,{\delta}$ norm. The extension operator
$P_{{\delta}}:\operatorname{Def}_{\Gamma}({\Sigma})\oplus\Omega^{0}({\Sigma},{u}_{0}^{{\delta}}*TX)_{1,p,{\delta}}\to\operatorname{Def}({\Sigma}^{{\delta}})\oplus\Omega^{0}({\Sigma}^{{\delta}},{u}^{{\delta},*}TX)_{1,p,{\delta}}$
is defined as follows. For each component $\Sigma_{i}$ let $\Sigma_{i}^{*}$
denote the complements of small balls around the nodes
$\Sigma_{i}^{*}=\Sigma_{i}-\bigcup_{l,w_{l}^{\pm}\in\Sigma_{i}}B_{|\delta_{l}(z)|^{1/2}/\rho}(w_{l}^{\pm})$
and the inclusion $\pi_{i}:\Sigma_{i}^{*}\to{\Sigma}^{{\delta}}$ induces a map
$\pi_{i,*}:\Omega^{0}(\Sigma_{i}^{*},u_{i}^{*}TX)\to\Omega^{0}({\Sigma}^{{\delta}},{u}^{{\delta},*}TX)_{0,p}.$
Define
$P_{{\delta}}(\zeta,\xi)=(\zeta^{{\delta}},\xi^{{\delta}})$
where $\zeta^{{\delta}}$ is the image of $\zeta$ under the gluing map (13) and
$\xi^{{\delta}}$ is obtained by patching together the sections $\xi$; on the
gluing region arising from gluing the $k$-th node $w_{k}$ the section
$\xi^{{\delta}}$ is given by the sum
$\pi_{i^{+}(k),*}\beta_{\rho,\delta_{k}}(\xi_{i^{+}(k)}-\xi(w_{k}))\\\
+\pi_{i^{-}(k),*}\beta_{\rho,\delta_{k}}(\xi_{i^{-}(k)}-\xi(w_{k}))+\xi(w_{k}).$
Fix a metric $\|\cdot\|$ on the finite-dimensional space
$\operatorname{Def}_{\Gamma}({\Sigma})$ and define
$\|(\zeta,\xi)\|_{1,p,{\delta}}=\left(\|\zeta\|^{p}+\|\xi\|_{1,p,{\delta}}^{p}\right)^{1/p}.$
###### Proposition 3.3.7.
Let ${u}:{\Sigma}\to X$ be a stable map. There exist constants $c,C>0$ such
that if $\|{\delta}\|<c$ then the approximate inverse $T_{{\delta}}$ of (15)
satisfies
$\|(\tilde{D}_{{u}}^{{\delta}}T_{{\delta}}-I)\eta\|_{0,p,{\delta}}\leq{\frac{1}{2}}\|\eta\|_{0,p,{\delta}},\quad\|T_{{\delta}}\|<C.$
###### Proof.
By construction $T_{{\delta}}$ is an exact right inverse for
$\tilde{D}_{{u}}^{{\delta}}$ away from gluing region. In the gluing region the
variation of complex structure on the curve vanishes and
$D_{{u}}^{{\delta}}=D_{x_{k}}$, the standard Cauchy-Riemann operator with
values in $T_{x_{k}}X$. So
$\displaystyle\tilde{D}_{{u}}^{{\delta}}T_{{\delta}}\eta-\eta$
$\displaystyle=$ $\displaystyle\sum
D_{x_{k}}\beta_{2;\rho,\delta}(z)(\xi_{i^{\pm}(k)}(z)-\xi_{i^{\pm}(k)}(w_{k}))$
$\displaystyle=$
$\displaystyle\sum(D_{x_{k}}\beta_{2;\rho,\delta}(z))\xi_{i^{\pm}(k)}(z)$
since $K_{{\delta}}\eta=0$ on $B_{|\delta_{k}|^{1/2}}(0)$ in the components
adjacent to the node. Since $p>2$, the $0,p,{\delta}$-norm of the right hand
side is controlled by the ordinary $L^{p}$ norm. By (16) we have
$\|\tilde{D}_{{u}}^{{\delta}}T_{{\delta}}\eta-\eta\|_{0,p,{\delta}}\leq\sum_{k}c|\log(\rho)|^{2/p-2}\|\xi_{i^{\pm}(k)}-\xi(w_{k})\|_{1,p}.$
The last factor is bounded by $\|K_{{\delta}}\eta\|_{0,p}$, by the uniform
bound on $Q_{{\delta}}$, and hence $\|\eta\|_{0,p,{\delta}}$, by the uniform
bound on $K_{{\delta}}$. ∎
Define a right inverse $Q_{{\delta}}$ to $\tilde{D}_{{u}}^{{\delta}}$ by the
formula
$Q_{{\delta}}=T_{{\delta}}(\tilde{D}_{{u}}^{{\delta}}T_{{\delta}})^{-1}=\sum_{k\geq
0}T_{{\delta}}(\tilde{D}_{{u}}^{{\delta}}T_{{\delta}}-I)^{k}.$
The uniform bound on $T_{{\delta}}$ from Lemma 3.3.7 implies a uniform bound
on $Q_{{\delta}}$.
Step 3: Uniform quadratic estimate
###### Proposition 3.3.8.
Let ${u}:{\Sigma}\to X$ be a stable map, ${\delta}$ a collection of gluing
parameters, and ${u}^{{\delta}}:{\Sigma}^{{\delta}}\to X$ the approximate
solution defined above. For every constant $c>0$ there exist constants
$c_{0},\delta_{0}>0$ such that if
${u}\in\operatorname{Map}({\Sigma},X)_{1,p},\xi\in\Omega^{0}({\Sigma}^{{\delta}},{u}^{{\delta},*}TX)_{1,p}$
$\|{\operatorname{d}}{u}\|_{0,p}\leq c_{0},\ \ \ \|\xi\|_{L_{\infty}}\leq
c_{0}\ \ \ \ \|\zeta\|\leq c_{0},\ \ \ \ \|{\delta}\|<\delta_{0}$
then
$\|(D\mathcal{F}_{{u}}^{{\delta}}(\zeta,\xi,\zeta_{1},\xi_{1})-\tilde{D}_{{u}}^{{{\delta}}})(\zeta_{1},\xi_{1})\|_{0,p,{\delta}}\leq
c\|\zeta,\xi\|_{1,p}(\|\zeta_{1},\xi_{1}\|_{1,p}.$
Here $D\mathcal{F}_{{u}}^{{\delta}}(\zeta,\xi,\zeta_{1},\xi_{1})$ denotes the
derivative evaluated at $\zeta,\xi$, applied to $\zeta_{1},\xi_{1}$. We use
similar notation throughout the discussion. The proof uses uniform estimates
on Sobolev embedding:
###### Lemma 3.3.9.
There exists a constant $c>0$ independent of $\delta$ such that the embedding
$\Omega^{0}({\Sigma}^{{\delta}},{u}^{{\delta},*}TX)_{{1,p},{\delta}}\to\Omega^{0}({\Sigma}^{{\delta}},{u}^{{\delta},*}TX)_{0,\infty}$
has norm less than $c$.
###### Proof.
One writes the Sobolev norms as a contribution from each component of the
curve ${\Sigma}$. Then on each piece, the metric near the boundary is
uniformly comparable with the flat metric. The claim then follows from [2,
Chapter 4] which shows that the constants in the Sobolev embeddings depend
only on the dimensions of the cone in the cone condition. ∎
###### Proof of Proposition.
For simplicity we assume a single gluing parameter $\delta$. Let
$\Psi_{{u}}^{{\delta},x}(\zeta,\xi):\Lambda^{0,1}T_{z}^{*}{\Sigma}^{\delta}\otimes
T_{x}X\to\Lambda^{0,1}_{{j}^{{\delta}}(\zeta)}T_{z}^{*}{\Sigma}\otimes
T_{\exp_{x}(\xi)}X$ denote pointwise parallel transport as in (7) using the
parallel transport using the modified Levi-Civita connection, and projecting
onto the $0,1$-part of the form defined using the complex structure
${j}^{{\delta}}(\zeta)$ obtained from gluing ${j}(\zeta)$, see (13). Let
$\Theta_{{u}}^{{\delta},x}(\zeta,\xi,\zeta_{1},\xi_{1};\eta)=\tilde{\nabla}_{t}\Psi_{{u}}^{{\delta},x}(\zeta+t\zeta_{1},\xi+t\xi_{1})\eta.$
For $\xi,\eta$ sufficiently small there exists a constant $c$ such that
(17) $|\Theta_{{u}}^{\delta,x}(\zeta,\xi,\zeta_{1},\xi_{1};\eta)|\leq
c\|\xi,\zeta\|\|\xi_{1},\zeta_{1}\|\|\eta\|$
where the norms on the right-hand side are any norms on the finite dimensional
vector spaces $T_{{\Sigma}}M_{g,n,\Gamma}$ and $T_{x}X$. This estimate is
uniform in $\delta$, for the variation in complex structure vanishes in a
neighborhood of the nodes. Differentiate the equation
$\Psi_{{u}}^{\delta}(\zeta,\xi)\mathcal{F}_{{u}}^{\delta}(\zeta,\xi)=\overline{\partial}_{{j}^{\delta}(\zeta)}(\exp_{{u}^{\delta}}(\xi)))$
with respect to $(\zeta_{1},\xi_{1})$ to obtain
(18)
$\Theta_{{u}^{\delta}}(\zeta,\xi,\zeta_{1},\xi_{1},\mathcal{F}_{{u}}^{\delta}(\zeta,\xi))+\Psi_{{u}}^{\delta}(\zeta,\xi)(D\mathcal{F}_{{u}}^{\delta}(\zeta,\xi,\zeta_{1},\xi_{1}))=\\\
\tilde{D}_{{u}}^{\delta}(\xi,Dj^{\delta}(\zeta,\zeta_{1}),D\exp_{{u}^{\delta}}(\xi,\xi_{1})).$
Using the pointwise inequality
$|\mathcal{F}_{{u}}^{\delta}(\zeta,\xi)|<c|{\operatorname{d}}{\exp_{{u}^{\delta}}(\xi)}|<c(|{\operatorname{d}}{u}^{\delta}|+|\nabla\xi|)$
for $\zeta,\xi$ sufficiently small, the estimate (17) on $\Phi$ produces a
pointwise estimate
$|(\Psi_{{u}}^{\delta})^{-1}(\xi)\Theta_{{u}}^{\delta}(\zeta,\xi,\zeta_{1},\xi_{1},\mathcal{F}_{{u}}^{\delta}(\zeta,\xi))|\leq
c(|{\operatorname{d}}{u}^{\delta}|+|\nabla\xi|)\,|(\xi,\zeta)|\,|(\xi_{1},\zeta_{1})|.$
Hence
(19)
$\|\Psi_{{u}^{\delta}}^{-1}(\xi)\Theta_{{u}}^{\delta}(\zeta,\xi,\zeta_{1},\xi_{1},\mathcal{F}^{\delta}_{{u}}(\zeta,\xi))\|_{0,p}\\\
\leq
c(1+\|{\operatorname{d}}{u}^{\delta}\|_{0,p}+\|\nabla\xi\|_{0,p})\|(\xi,\zeta)\|_{0,\infty}\|(\xi_{1},\zeta_{1})\|_{0,\infty}.$
It follows that
(20)
$\|\Psi_{{u}}^{\delta}(\xi)^{-1}\Theta_{{u}}^{\delta}(\zeta,\xi,\zeta_{1},\xi_{1},\mathcal{F}_{{u}}^{\delta}(\zeta,\xi))\|_{0,p}\leq
c\|(\xi,\zeta)\|_{1,p}\|(\xi_{1},\zeta_{1})\|_{1,p}$
since the $W^{1,p}$ norm controls the $L^{\infty}$ norm by Lemma 3.3.9.
We next show that there exists a constant $c>0$ such that uniformly in
$\delta$,
(21)
$\|\Psi_{{u}^{\delta}}(\xi)^{-1}\tilde{D}_{{u}}^{\delta}(\xi,D{j}^{{\delta}}(\zeta,\zeta_{1}),D\exp_{{{u}^{\delta}}}(\xi,\xi_{1}))-\tilde{D}_{{u}}^{\delta}(\zeta_{1},\xi_{1})\|_{0,p}\leq
c\|\zeta,\xi\|_{1,p}\|\zeta_{1},\xi_{1}\|_{1,p}.$
Indeed differentiate (14) to obtain
(22)
$\tilde{D}_{{u}}^{\delta}(\xi,D{j}^{\delta}(\zeta,\zeta_{1}),D\exp_{{{u}^{\delta}}}(\xi,\xi_{1}))=\nabla^{0,1}_{j^{\delta}(\zeta)}D\exp_{{{u}^{\delta}}}(\xi,\xi_{1})-\\\
{\frac{1}{2}}\pi^{0,1}_{j^{\delta}(\zeta)}J_{\exp_{{u}^{\delta}}(\xi)}{\operatorname{d}}\exp_{{u}^{\delta}}(\xi)D{j}^{\delta}(\zeta,\zeta_{1})$$-J_{\exp_{{u}^{\delta}}(\xi)}(\nabla_{D\exp_{{{u}}}(\xi,\xi_{1})}J_{\exp_{{u}^{\delta}}(\xi)})\partial\exp_{{u}^{\delta}}(\xi).$
Hence
$\tilde{D}_{{u}}^{\delta}(\xi,D{j}^{\delta}(\zeta,\zeta_{1}),D\exp_{{{u}^{\delta}}}(\xi,\xi_{1}))-\Psi_{{u}^{\delta}}(\xi)\tilde{D}_{{u}}^{\delta}(\zeta_{1},\xi_{1})=\Pi_{1}+\Pi_{2}+\Pi_{3}$
where the three terms $\Pi_{1},\Pi_{2},\Pi_{3}$ are
$\Pi_{1}=\nabla^{0,1}_{{j}^{\delta}(\zeta)}D\exp_{{{u}^{\delta}}}(\xi,\xi_{1})-\Psi_{{u}^{\delta}}(\xi)\nabla^{0,1}_{{j}^{\delta}(0)}\xi_{1}$
$\Pi_{2}=-{\frac{1}{2}}\pi^{0,1}_{{j}^{\delta}(\zeta)}J_{\exp_{{u}}(\xi)}{\operatorname{d}}\exp_{{u}^{\delta}}(\xi)D{j}^{\delta}(\zeta,\zeta_{1})+\Psi_{{u}^{\delta}}(\xi)\pi^{0,1}_{{j}^{\delta}(0)}{\frac{1}{2}}J_{{u}^{\delta}}{\operatorname{d}}{u}^{\delta}D{j}^{\delta}(0,\zeta_{1})$
$\Pi_{3}=-{\frac{1}{2}}J_{\exp_{{u}}(\xi)}(\nabla^{0,1}_{D\exp_{{{u}}^{\delta}}(\xi,\xi_{1})}J_{\exp_{{u}^{\delta}}(\xi)})\partial_{{j}^{\delta}(\zeta)}\exp_{{u}^{\delta}}(\xi)+{\frac{1}{2}}\Psi_{{u}^{\delta}}(\xi)J_{{u}^{\delta}}(\nabla_{\xi_{1}}J_{{u}})\partial_{{j}^{\delta}(0)}{u}^{\delta}.$
The first difference has norm given by
(23)
$|\pi_{{j}^{\delta}(\zeta)}^{0,1}(\nabla(D\exp_{{{u}^{\delta}}}(\xi,\xi_{1}))-\Psi_{{u}^{\delta}}(\xi)\nabla\xi_{1})|\\\
\leq|\pi_{{j}^{\delta}(\zeta)}^{0,1}(\nabla(D\exp_{{{u}^{\delta}}}(\xi,\xi_{1}))-D\exp_{{{u}^{\delta}}}(\xi,\nabla\xi_{1}))|+|\pi^{0,1}_{{j}^{\delta}(\zeta)}(D\exp_{{{u}^{\delta}}}(\xi,\nabla\xi_{1})-\Psi_{{u}^{\delta}}(\xi)\nabla\xi_{1})|\\\
\leq
c|\nabla\xi||\xi_{1}|+c(|\zeta|+|\xi|)|\nabla\xi_{1}|+c|{\operatorname{d}}{u}^{\delta}||\xi||\xi_{1}|.$
We write for the second difference
(24)
$|\pi^{0,1}_{{j}^{\delta}(\zeta)}(J_{\exp_{{u}^{\delta}}(\xi)}{\operatorname{d}}\exp_{{u}^{\delta}}(\xi)D{j}^{\delta}(\zeta,\zeta_{1})-\Psi_{{u}^{\delta}}(\xi)J_{{u}^{\delta}}{\operatorname{d}}{u}^{\delta}D{j}^{\delta}(0,\zeta_{1}))|\\\
\leq|\pi^{0,1}_{{j}^{\delta}(\zeta)}(J_{\exp_{{u}^{\delta}}(\xi)}{\operatorname{d}}\exp_{{u}^{\delta}}(\xi)D{j}^{\delta}(\zeta,\zeta_{1})-J_{\exp_{{u}^{\delta}}(\xi)}\Psi_{{u}^{\delta}}(\xi){\operatorname{d}}{u}^{\delta}D{j}^{\delta}(0,\zeta_{1}))|+\\\
|\pi^{0,1}_{{j}^{\delta}(\zeta)}(J_{\exp_{{u}^{\delta}}(\xi)}\Psi_{{u}^{\delta}}(\xi){\operatorname{d}}{u}^{\delta}D{j}^{\delta}(0,\zeta_{1})-\Psi_{{u}^{\delta}}(\xi)J_{{u}}{\operatorname{d}}{u}^{\delta}D{j}^{\delta}(0,\zeta_{1}))|\\\
\leq
c(|\zeta|+|\xi|+|{\operatorname{d}}{u}^{\delta}|+|\nabla\xi|)||\zeta_{1}|.$
The third term can be estimated pointwise by
$|J_{\exp_{{u}^{\delta}}(\xi)}(\nabla_{D\exp_{{u}^{\delta}}(\xi,\xi_{1})}J_{\exp_{{u}^{\delta}}(\xi)})\partial_{{j}^{\delta}(\zeta)}\exp_{{u}^{\delta}}(\xi)-\Psi_{{u}^{\delta}}(\xi)J_{{u}^{\delta}}(\nabla_{\xi_{1}}J_{{u}^{\delta}})\partial_{{j}^{\delta}(0)}{u}^{\delta}|$
$\leq|J_{\exp_{{u}^{\delta}}(\xi)}(\nabla_{D\exp_{{{u}}}(\xi,\xi_{1})}J_{\exp_{{u}^{\delta}}(\xi)})\partial_{{j}^{\delta}(\zeta)}\exp_{{u}^{\delta}}(\xi)-J_{\exp_{{u}^{\delta}}(\xi)}(\nabla_{D\exp_{{u}^{\delta}}(\xi,\xi_{1})}J_{\exp_{{u}^{\delta}}(\xi)})\partial_{{j}^{\delta}(0)}\exp_{{u}^{\delta}}(\xi)|$
$+|J_{\exp_{{u}^{\delta}}(\xi)}(\nabla_{D\exp_{{{u}}}(\xi,\xi_{1})}J_{\exp_{{u}^{\delta}}(\xi)})\partial_{{j}^{\delta}(0)}\exp_{{u}^{\delta}}(\xi)-\Psi_{{u}^{\delta}}(\xi)J_{{u}^{\delta}}(\nabla_{\xi_{1}}J_{{u}^{\delta}})\partial_{{j}^{\delta}(0)}{u}^{\delta}|$
$\leq
c|\zeta|(|{\operatorname{d}}{u}^{\delta}|+|\nabla\xi|)|\xi_{1}|+c(|{\operatorname{d}}{u}^{\delta}|+|\nabla\xi|)(|\xi_{1}|)$
for $\xi$ sufficiently small. Combining these estimates and integrating, using
the $0,p,\delta$-norms on ${\operatorname{d}}{u}$, $\nabla\xi,\nabla\xi_{1}$
and the $L^{\infty}$ norms on the other factors and Lemma 3.3.9, completes the
proof. ∎
Step 4: Implicit Function Theorem
For any $(\zeta_{0},\xi_{0})\in\operatorname{Def}_{\Gamma}({u})$ we denote by
$\zeta_{0}^{{\delta}}$ the deformation of ${\Sigma}^{{\delta}}$ defined in
(13) and by $\xi_{0}^{{\delta}}$ the section of ${u}^{{\delta},*}TX$ defined
as in (11).
###### Theorem 3.3.10.
Let ${u}:{\Sigma}\to X$ be a stable map. There exist constants
$\epsilon_{0},\epsilon_{1}>0$ such that for any
$(\zeta_{0},\xi_{0},{\delta})\in\operatorname{ker}\tilde{D}_{{u}}\times\mathbb{R}^{m}$
of norm at most $\epsilon_{0}$, there is a unique
$(\zeta_{1},\xi_{1})=(\tilde{D}_{{u}}^{{\delta}})^{*}\eta_{1}$ of norm at most
$\epsilon_{1}$ such that the map
$\exp_{{u}^{{\delta}}}(\xi_{0}^{{\delta}}+\xi)$ is
${j}^{\delta}(\zeta_{0}+\zeta_{1})$-holomorphic, and depends smoothly on
$\zeta_{0},\xi_{0}$.
###### Proof.
The first claim is an application of the quantitative version of the implicit
function theorem (see for example [15, Appendix A.3]) using the uniform error
bound from Proposition 3.3.3, uniformly bounded right inverse from Proposition
3.3.7, and uniform quadratic estimate from Proposition 3.3.8. ∎
Step 5: Rigidification
In the previous step we have constructed a family of stable maps which we will
show eventually gives rise to a parametrization of all nearby stable maps. A
more natural way of parametrizing nearby stable maps involves examining the
intersections with a family of codimension two submanifolds. For example, this
construction of charts is that given in the algebraic geometry approach of
Fulton-Pandharipande [10]. In order to carry this out in the symplectic
approach, we study the differentiability of the evaluation maps. Let
$u_{S}:\Sigma_{S}\to X$ over a parameter space
$S\subset\operatorname{Def}({u})$ be the family of maps defined in the
previous step. The following is similar to [9, Lemma A1.59].
###### Theorem 3.3.11.
If ${u}_{S}$ is constructed using the exponential gluing profile $\varphi$ and
$U\subset{\Sigma}$ is an open neighborhood of the nodes then the map
$(z,s)\mapsto{u}_{s}(z)$ is differentiable on a neighborhood of
$({\Sigma}-U)\times\\{0\\}$.
###### Proof.
For simplicity, we assume that there is a single gluing parameter $\delta$.
Differentiability for $\delta$ is studied in McDuff-Salamon [15, Section
10.6]. The discussion in our case is somewhat easier, because we use a fixed
right inverse in the gluing construction. Given
$(\zeta,\xi)\in\operatorname{Def}_{\Gamma}({\Sigma})\times\Omega^{0}({\Sigma}^{\delta},({u}^{\delta})^{*}TX)$,
we constructed a unique correction $(\zeta_{1},\xi_{1})$ in the image of the
right inverse such that
$\overline{\partial}_{{j}^{\delta}(\zeta_{0}+\zeta_{1})}\exp_{{u}^{\delta}}(\xi_{0}+\xi_{1})=0.$
For $\delta$ fixed, $(\zeta_{1},\xi_{1})$ depends smoothly on
$(\zeta_{0},\xi_{0})$, by the implicit function theorem. Hence the evaluation
at $z\in{\Sigma}-U$ also depends smoothly on $(\zeta_{0},\xi_{0})$.
The computation of the derivative with respect to the gluing parameter is
complicated by the fact that for each $\delta$ a different implicit function
theorem is applied to obtain the correction. Let
$\tilde{D}_{\delta}=D\mathcal{F}_{{u}^{\delta}}$. Differentiating the equation
$\mathcal{F}_{{u}^{\delta}}(\zeta_{0}^{\delta}+\zeta_{1},\xi_{0}^{\delta}+\xi_{1})=0$
with respect to $\delta$ gives
$\tilde{D}_{\delta}\left(\frac{d}{d\delta}\zeta_{1},D\exp_{{u}^{\delta}}(\xi_{0}^{\delta}+\xi_{1},0,\frac{d}{d\delta}\xi_{1})\right)=-\tilde{D}_{\delta}\left(0,D\exp_{{u}^{\delta}}(\xi_{0}^{\delta}+\xi_{1},\frac{d}{d\delta}{u}^{\delta},0)\right).$
From (11) we have in the gluing region,
$\displaystyle\frac{d}{d\delta}\overline{\partial}{u}^{\delta}$
$\displaystyle=$
$\displaystyle\frac{d}{d\delta}\overline{\partial}\exp_{x}(\left(\alpha(\rho\varphi^{-1/2}|z|)\xi(z))\right)$
$\displaystyle=$ $\displaystyle
D\exp_{x}\left(\alpha(\rho\varphi^{-1/2}|z|)\xi(z),\alpha^{\prime}(\rho\varphi^{-1/2}|z|)|z|\xi(z)\frac{d}{d\delta}\varphi^{-1/2}\rho\right)^{0,1}.$
Hence there exists a constant $C$ depending on $\rho,\alpha$ but not on
$\delta$ such that
(25) $\left|\frac{d}{d\delta}\overline{\partial}{u}^{\delta}\right|\leq
C\left|\frac{d}{d\delta}\varphi^{-1/2}\right|.$
Now ${\operatorname{d}}\varphi$ is given by
${\operatorname{d}}(e^{1/\delta}-e)^{-1/2}=(1/2)(e^{1/\delta}-e)^{-3/2}e^{1/\delta}\delta^{-2}{\operatorname{d}}\delta=(1/2)(e^{1/3\delta}-e^{-2/3\delta+1})^{-3/2}\delta^{-2}{\operatorname{d}}\delta.$
For $\delta$ small, this is less than
${\frac{1}{2}}e^{-1/2\delta}\delta^{-2}$. Integrating and using the pointwise
estimate (25) we obtain for some constant $C>0$,
$\left\|\frac{d}{d\delta}\overline{\partial}{u}^{\delta}_{s}\right\|_{0,p}\leq
Ce^{-1/2\delta}\delta^{-2}\leq Ce^{-1/\delta}$
for sufficiently small $\delta$. Now the uniform quadratic estimates imply
that $\tilde{D}_{\delta}=D\mathcal{F}_{{u}^{\delta}}(\zeta,\xi)$ is uniformly
bounded from below on the right inverse of
$\tilde{D}_{{u}}^{\delta}=D\mathcal{F}_{{u}}^{\delta}(0,0)$, for
$(\zeta_{0},\xi_{0})$ sufficiently small. It follows that
$\left\|(\frac{d}{d\delta}\zeta_{1},\frac{d}{d\delta}\xi_{1})\right\|_{1,p}\leq
Ce^{-1/\delta}$
for $\zeta_{0},\xi_{0},\delta$ sufficiently small as well. Hence the same is
true for the evaluation $\frac{d}{d\delta}\xi_{1}(z)$ for $z\in{\Sigma}-U$. In
particular, $\lim_{\delta\to
0}(\partial_{\delta}\exp_{{u}^{\delta}}(\xi_{0}^{\delta}+\xi_{1}))(z)=0.$ It
follows that the differential of the evaluation map has a continuous limit as
${\delta}\to 0$, which completes the proof of the Theorem. ∎
Using the evaluation maps in the previous step, we construct embeddings of the
families constructed above into suitable moduli spaces of stable marked
curves, given by adding additional marked points which map to fixed
submanifolds in $X$. A codimension two submanifold $Y\subset X$ is transverse
to ${u}:{\Sigma}\to X$ if ${u}$ meets $Y$ transversally in a single point
${u}(z)$.
###### Definition 3.3.12.
Let ${u}:{\Sigma}\to X$ be a stable map. Given any family
${Y}=(Y_{1},\ldots,Y_{\ell})$ of codimension two submanifolds transverse to
${u}$ and a family ${\Sigma}_{S},{u}_{S},{z}_{S}$ with parameter space $S$ of
an $n$-marked stable map $({\Sigma},{u},{z})$, the rigidified family of
$n+\ell$-marked nodal surfaces is defined by
(26) ${\Sigma}_{S}^{{Y},{u}}:=({\Sigma}_{S},(z_{1,S},\ldots,z_{n+\ell,S}))\to
S,\ \ \ \ {u}_{s}(z_{n+i,s})\in Y_{i}.$
###### Proposition 3.3.13.
Let ${u}_{S}$ be a family of stable maps over a parameter space
$S\subset\operatorname{Def}({u})$ given by the gluing construction using a
gluing profile $\varphi$ and system of coordinates ${\kappa}$. Suppose that
the evaluation map $\operatorname{ev}:({\Sigma}-U)\times S\to X$ is $C^{1}$,
and that the rigidified family has stable underlying curves. Then the
rigidified family of curves ${\Sigma}_{S}^{{Y},{u}}$ is $C^{1}$ with respect
to the gluing profile and local coordinates, that is, the map
$S\mapsto\overline{M}_{g,n+l},\quad s\mapsto{\Sigma}_{s}^{{Y},{u}}$ is $C^{1}$
with respect to the smooth structure defined by $\varphi,{\kappa}$.
###### Proof.
By the implicit function theorem for $C^{1}$ maps and differentiability of
evaluation maps from the previous subsection. ∎
###### Definition 3.3.14.
Let ${Y},{u}$ be as above. The pair $({Y},{u})$ is compatible if
1. (a)
each $Y_{j}$ intersects ${u}$ transversally in a single point
$z_{j}\in{\Sigma}$;
2. (b)
if $\xi\in\operatorname{ker}(\tilde{D}_{{u}})$ satisfies $\xi(z_{n+j})\in
T_{{u}(z_{n+j})}Y_{j}$ for $j=1,\ldots,l$ then $\xi=0$;
3. (c)
the curve ${\Sigma}$ marked with the additional points
$z_{n+1},\ldots,z_{n+\ell}$ is stable;
4. (d)
if some automorphism of $({\Sigma},{u})$ maps $z_{i}$ to $z_{j}$ then $Y_{i}$
is equal to $Y_{j}$.
The second condition says that there are no infinitesimal deformations which
do not change the positions of the extra markings.
###### Proposition 3.3.15.
Let ${u}$ be a parametrized regular stable map, and ${u}_{S}$ the stratified-
smooth universal deformation constructed in Theorem 3.3.10 with base
$S\subset\operatorname{Def}({u})$. There exists a collection ${Y}$ compatible
with ${u}$. Furthermore if the evaluation map is $C^{1}$ as in Proposition
3.3.13 then ${\Sigma}^{{Y},{u}}_{S}$ defines an $C^{1}$-immersion of $S$ into
$\operatorname{Def}({\Sigma}^{{Y},{u}})$.
###### Proof.
First we show the existence of a compatible collection. Given a regular stable
map $({\Sigma},{z}=(z_{1},\ldots,z_{n}),{u}:{\Sigma}\to X)$, choose
$Y_{1},\ldots,Y_{k}$ transverse ${u}$ on the unstable components of
${\Sigma}$, so that ${\Sigma}_{1}=({\Sigma},(z_{1},\ldots,z_{n+k}))$ is a
stable curve. Let ${\Sigma}_{S_{1},1}\to S_{1}$ denote a universal deformation
of ${\Sigma}_{1}$. By universality, the family ${\Sigma}_{S}^{{Y},{u}}$ is
induced by a map $\psi:S\to S_{1}$. We successively add marked points until
$\psi$ is an immersion: Suppose that $\psi$ is not an immersion. Then we may
choose an additional marked point $z_{n+k+1}\in{\Sigma}$ such that
${\operatorname{d}}\operatorname{ev}_{n+k+1}$ is non-trivial on
$\operatorname{ker}D\psi$. Since ${u}$ is holomorphic,
${\operatorname{d}}{u}(z_{n+k+1})$ is rank two at $z_{n+k+1}$. Let
$Y_{n+k+1}\subset X$ be a codimension two submanifold containing
${u}(z_{n+k+1})$ such that ${u}$ is transverse to $Y_{n+k+1}$ at $z_{n+k+1}$,
and $Y_{n+k+1}$ is transversal to $\operatorname{ev}_{n+k+1}$ at
${\Sigma},{u}$. Suppose $z_{n+n^{\prime}+1}$ has orbit
$z_{n+k+1},z_{n+k+2},\ldots,z_{n+l}$ under the group
$\operatorname{Aut}({u})$. Repeating the same submanifold for each marking
related by automorphisms gives a collection invariant under the action of
automorphisms. The map $\psi_{1}$ for the new family has property that the
dimension of $\operatorname{ker}(D\psi_{1})$ has dimension at least two less
than that of $\operatorname{ker}(D\psi)$. It follows that the procedure
terminates after adding a finite number of markings. The last claim follows
from the second condition in Definition 3.3.14. ∎
Step 6: Surjectivity
In this section, we show that the family constructed above contains a Gromov
neighborhood of the central fiber. First we show:
###### Proposition 3.3.16.
There exists a constant $\epsilon>0$ such that any stable map
$({\Sigma}_{1},{u}_{1})$ with complex structure on ${\Sigma}_{1}$ given by
$j^{\delta}(\zeta)$ for some
$\zeta\in\operatorname{Def}({\Sigma}^{{\delta}})$, and
${u}_{1}:=\exp_{{u}^{{\delta}}}({\xi})$ with
$\|\zeta\|^{2}+\|\xi\|^{2}_{1,p,{\delta}}<\epsilon$ is of the form in Theorem
3.3.10 for some
$(\xi_{1},\zeta_{1})\in\operatorname{Im}(\tilde{D}_{{u}}^{{{\delta}}})^{*}$.
###### Proof.
Compare with [15, Section 10.7.3]. Let $(\zeta,{\xi})$ be as in the statement
of the Proposition. We claim that
$(\zeta,{\xi})=(\zeta_{0}^{{\delta}},{\xi}_{0}^{{\delta}})+(\zeta_{1},{\xi}_{1})$
for some $(\zeta_{0},{\xi}_{0})\in\operatorname{ker}(\tilde{D}_{{u}})$ and
$(\zeta_{1},{\xi}_{1})\in\operatorname{Im}((D_{{u}}^{{{\delta}}})^{*})$ with
small norm. It then follows by the implicit function theorem that
$(\zeta_{1},{\xi}_{1})$ is the solution given in Theorem 3.3.10. Now
$\zeta=\zeta_{0}^{{\delta}}$ for some
$\zeta_{0}\in\operatorname{Def}_{\Gamma}({\Sigma})$ and gluing parameters
${\delta}$, because $\operatorname{Def}({\Sigma}^{{\delta}})$ is the direct
sum of the image of $\operatorname{Def}_{\Gamma}({\Sigma})$ and
$\mathbb{C}^{m}$. By the gluing theorem for indices (see e.g. [23, Theorem
2.4.1]), the image of $\operatorname{Def}_{\Gamma}({u})$ under the gluing map
projects isomorphically onto $\operatorname{ker}(\tilde{D}_{{u}}^{{\delta}})$
for ${\delta}$ sufficiently small, and so $\operatorname{Def}_{\Gamma}({u})$
is transverse to $\operatorname{Im}(\tilde{D}_{{u}}^{{\delta}})^{*}$, for
${\delta}$ sufficiently small. The claim then follows from the inverse
function theorem. ∎
Given a regular stable ${u}$ with stable domain, consider the family of
$J$-holomorphic maps ${u}_{S}$ produced by Theorem 3.3.10 with parameter space
a neighborhood $S$ of $0$ in $\operatorname{Def}({u}),$ equipped with a
canonical identification $\iota$ of the central fiber with the original map
${u}$. In the case that the domain ${\Sigma}$ is not a stable (marked) curve,
we choose codimension two submanifolds ${Y}=(Y_{1},\ldots,Y_{l})$ meeting
${u}$ transversally so that ${\Sigma}$ with the additional marked points is
stable. Applying this to the family ${u}_{S}$ gives a family of marked stable
maps ${u}_{S}^{{Y}}$ with $n+l$ marked points over a parameter space
$S\subset\operatorname{Def}({u}^{{Y}})$ in the deformation space of the map
with the additional marked points. Now
$\operatorname{Def}({u}^{{Y}})\cong\operatorname{Def}({u})\oplus\bigoplus_{i=1}^{l}T_{z_{i}}{\Sigma}$
includes the deformations of the markings, but these are fixed by requiring
that the additional marked points map to the given collection ${Y}$.
Forgetting the additional marked points gives a family ${u}_{S}$ of stable
maps with $n$ marked points over a neighborhood of $0$ in
$\operatorname{Def}({u})$.
###### Proposition 3.3.17.
$({u}_{S},\iota)$ is a versal stratified-smooth deformation of ${u}$, and in
fact ${u}_{S}$ gives a versal stratified-smooth deformation of any of its
fibers.
###### Proof.
First suppose that ${\Sigma}$ is stable. Let $({u}_{S^{1}}^{1},\iota^{1})$ be
another stratified-smooth deformation of ${u}$ with parameter space $S^{1}$.
Let ${\Sigma}_{S}\to S\subset\operatorname{Def}({\Sigma})$ be a minimal versal
deformation of ${\Sigma}$. The family ${\Sigma}_{S^{1}}^{1}$ is obtained by
pull-back of ${\Sigma}_{S}$ by a stratified-smooth map $\psi:S^{1}\to S$. By
definition the map ${u}_{s}^{1}$ converges to the central fiber in the Gromov
topology as $s$ converges to the base point $0\in S^{1}$. The exponential
decay estimate of [15, Lemma 4.7.3] for holomorphic cylinders of small energy
imply that for $s$ sufficiently close to $0$, ${\Sigma}_{s}^{1},{u}_{s}^{1}$
is given by exponentiation, ${u}_{s}^{1}=\exp_{{u}^{{\delta}}}({\xi})$ for
some ${\xi}\in\Omega^{0}({u}^{{\delta},*}TX)$ with
$\|{\xi}\|_{1,p}<\epsilon_{1}$, for $s$ sufficiently close to $0$. Proposition
3.3.16 produces a stratified-smooth map $\psi:S^{1}\to\operatorname{Def}({u})$
such that ${u}_{S^{1}}^{1}$ is the pull-back of $\psi$. To show that the
deformation $({u}_{S},\iota)$ is universal, let
$\phi_{j}:{\Sigma}_{S^{1}}^{1}\to\psi_{j}^{*}{\Sigma}_{S},j=0,1$ be
isomorphisms of families inducing the identity on the central fiber. The
difference between the two automorphisms is an automorphism of the family
${\Sigma}_{S^{1}}^{1}$ inducing the identity on the central fiber; since the
automorphism group of the central fiber is discrete, the automorphism must be
the identity. In the case that ${\Sigma}$ is not stable, after adding marked
points passing through $Y_{1},\ldots,Y_{l}$, we obtain a family
${u}_{S^{1}}^{1,{Y}}$ of stable maps with $n+l$ marked points. By the case
with stable domain, this family is obtained by pull-back of ${u}_{S}^{{Y}}$ by
some map $S^{1}\to S$. Hence ${u}_{S^{1}}^{1}$ is obtained by pull-back by the
same map. The argument for an arbitrary fiber is similar and left to the
reader. ∎
###### Remark 3.3.18.
In the case that ${\Sigma}$ is unstable, it seems likely that restricting the
family of Theorem 3.3.10 to $\operatorname{Def}({u})$ (that is, the
perpendicular of $\operatorname{aut}({\Sigma})$) also gives a universal
deformation, but we do not know how to prove this. The problem is that in this
case, several different gluing parameters give the same curve, and we do not
have an implicit function theorem for varying gluing parameter.
Step 7: Injectivity
By injectivity, we mean that the family constructed above contains each nearby
stable map exactly once, up to the action of $\operatorname{Aut}(u)$. This is
part of what we called “strongly universal” in Definition 2.2.4.
###### Theorem 3.3.19.
The versal deformations constructed in Step 6 above are strongly universal.
###### Proof.
Let ${u}_{S}$ be a deformation constructed as in Step 6, using the exponential
gluing profile. Let ${\Sigma}_{1,S^{1}}\to S^{1}$ be a family giving a
universal deformation of the curve ${\Sigma}^{{Y},{u}}$ obtained by adding the
additional markings mapping to the given submanifolds. By Definition 3.3.14,
the family ${\Sigma}_{S}^{{Y},{u}}$ induces a map $\phi:S\to S_{1}$ whose
differential is injective in a neighborhood of $0$. By the inverse function
theorem for $C^{1}$ maps, $\phi$ induces a homeomorphism onto its image. In
particular, any two distinct fibers of ${\Sigma}_{S}^{{Y},{u}}$ are non-
isomorphic, and so two fibers of ${\Sigma}_{S}$ are isomorphic if and only if
they are related by a permutation of the markings. After shrinking $S$, this
happens only if the permutation is induced by an automorphism of ${u}$. Given
another family ${u}_{S^{\prime}}^{\prime}:{\Sigma}_{S^{\prime}}\to S^{\prime}$
corresponding to a deformation of a fiber of ${u}_{S}\to S$, by the uniqueness
part of the implicit function theorem, a map $\phi^{\prime}:S^{\prime}\to
S_{1}$ so that ${u}_{S^{\prime}}^{\prime}$ is obtained by pull-back from
${u}_{S}$, and this map is unique by the injectivity just proved. This shows
that ${u}_{S}$ gives a stratified-smooth universal deformation of any of its
fibers, and so is strongly universal. ∎
The Theorem implies that the families in the universal deformations
constructed above define stratified-smooth-compatible charts for the moduli
space $\overline{M}_{g,n}(X,d)$. That is, for any stratum
$M_{g,n,\Gamma}(X,d)$, the restriction of the charts given by the universal
deformation of some map of type $\Gamma$ to $M_{g,n,\Gamma}(X,d)$ are smoothly
compatible.
###### Corollary 3.3.20.
Let $X,J$ be as above. For any $g\geq 0,n\geq 0$, the strongly universal
stratified-smooth deformations of parametrized regular stable maps provide
$\overline{M}_{g,n}^{{\operatorname{reg}}}(X)$ with the structure of a
stratified-smooth topological orbifold.
In order to apply localization one needs to know that the fixed point sets
admit tubular neighborhoods. For this it is helpful to know that
$\overline{M}_{g,n}^{\operatorname{reg}}(X,d)$ admits a $C^{1}$ structure. In
order to obtain compatible charts, we construct the local coordinates
inductively as in Definition 2.2.6, starting with the strata of highest
codimension.
###### Proposition 3.3.21.
Let $X,J$ be as above. For any compatible system of local coordinates near the
nodes, the strongly universal deformations constructed using the exponential
gluing profile equip $\overline{M}_{g,n}^{\operatorname{reg}}(X)$ with the
structure of a $C^{1}$-orbifold.
###### Proof.
We claim that the charts induced by the universal deformations are
$C^{1}$-compatible, assuming they are constructed from the same system of
local coordinates near the nodes. Given two sets of submanifolds
${Y}_{1},{Y}_{2}$, define ${Y}={Y}_{1}\cup{Y}_{2}$. The family
${\Sigma}^{{Y},{u}}$ admits proper étale forgetful maps
${\Sigma}^{{Y},{u}}_{S}\to{\Sigma}_{S}^{{Y}_{j},{u}},\quad j=1,2.$ The fiber
consists of reorderings of the additional marked points induced by the action
of $\operatorname{Aut}({\Sigma},{u})$, and the diagram provided by
${\Sigma}^{{Y},{u}}$ expresses the composition as a smooth $C^{1}$-morphism of
orbifolds. ∎
###### Remark 3.3.22.
Any compact $C^{1}$ orbifold admits a compatible $C^{\infty}$ structure, in
analogy with the situation with manifolds. Indeed, as is well known any
orbifold admits a presentation as the quotient of a manifold (namely its
orthogonal frame bundle) by a locally free group action, and so the orbifold
case follows from the equivariant case proved in Palais [18]. Hence
$\overline{M}^{{\operatorname{reg}}}_{g,n}(X,d)$ if compact admits a (non-
canonical) smooth structure. Presumably the compactness assumption may be
removed but we have not proved that this is so. See however the construction
of smoothly compatible Kuranishi charts in [9, Appendix].
## 4\. Deformations of symplectic vortices
We begin by reviewing the theory of symplectic vortices introduced by Mundet i
Riera [16] and Salamon and collaborators [5]. Let $\Sigma$ be a compact
complex curve, $G$ a compact Lie group, and $\pi:P\to\Sigma$ a smooth
principal $G$-bundle. Given any left $G$-manifold $F$ we have a left action of
$G$ on $P\times F$ given by $g(p,f)=(pg^{-1},gf)$ and we denote by
$P(F)=(P\times F)/G$ the quotient, that is, the associated fiber bundle with
fiber $F$. Let $X$ be a compact Hamiltonian $G$-manifold with symplectic form
$\omega$ and moment map $\Phi:X\to\mathfrak{g}^{*}$. The action of $G$ on $X$
induces an action on $\mathcal{J}(X)$; and we denote by $\mathcal{J}(X)^{G}$
the invariant subspace. Let $\psi:\Sigma\to BG$ be a classifying map for $P$,
so that $P\cong\psi^{*}EG$ and
$P(X)\cong\psi^{*}EG\times_{G}X\cong\psi^{*}X_{G}$ where
$X_{G}=EG\times_{G}X$. Continuous sections $u:\Sigma\to P(X)$ are in one-to-
one correspondence with lifts of $\psi$ to $X_{G}$. The homology class
$\deg(u)$ of the section $u$ is defined to be the homology class $\deg(u)\in
H_{2}^{G}(X,\mathbb{Z})$ of the corresponding lift. Let $\mathcal{A}(P)$ be
the space of smooth connections on $P$, and $P(\mathfrak{g})$ the adjoint
bundle. For any $A\in\mathcal{A}(P)$, let
$F_{A}\in\Omega^{2}(\Sigma,P(\mathfrak{g}))$ the curvature of $A$. Any
connection $A\in\mathcal{A}(P)$ induces a map of spaces of almost complex
structures
$\mathcal{J}(X)^{G}\to\mathcal{J}(P(X)),\ \ J\mapsto J_{A}$
by combining the almost complex structures on $X$ and $\Sigma$ using the
splitting defined by the connection. Let $\Gamma(\Sigma,P(X))$ denote the
space of smooth sections of $P(X)$. Consider the vector bundle
(27)
$\bigcup_{u\in\Gamma(\Sigma,P(X))}\Omega^{0,1}(\Sigma,u^{*}T^{\operatorname{vert}}P(X))\to\Gamma(\Sigma,P(X)).$
We denote by $\overline{\partial}_{A}$ the section given by the Cauchy-Riemann
operator defined by $J_{A}$. A gauged map from $\Sigma$ to $X$ is a datum
$(P,A,u)$ where $A\in\mathcal{A}(P)$ and $u:\Sigma\to P(X)$ is a section. A
gauged holomorphic map is a gauged map $(P,A,u)$ such that
$\overline{\partial}_{A}u=0$. Let $\mathcal{H}(P,X)$ be the space of gauged
holomorphic maps with underlying bundle $P$. Let $\mathcal{G}(P)$ denote the
group of gauge transformations
$\mathcal{G}(P)=\\{a:P\to P,a(pg)=a(p)g,\ \ \ \pi\circ a=\pi\\}.$
The Lie algebra of $\mathcal{G}(P)$ is the space of sections
$\Omega^{0}(\Sigma,P(\mathfrak{g}))$ of the adjoint bundle
$P(\mathfrak{g})=P\times_{G}\mathfrak{g}$. We identify
$\mathfrak{g}\to\mathfrak{g}^{*}$, and hence $P(\mathfrak{g})\to
P(\mathfrak{g}^{*})$, using an invariant metric on $\mathfrak{g}$. Let
$P(\Phi):P(X)\to P(\mathfrak{g})$ denote the map induced by the equivariant
map $\Phi:X\to\mathfrak{g}$.
###### Definition 4.0.1.
A gauged holomorphic map $(A,u)\in\mathcal{H}(P,X)$ is a symplectic vortex (or
vortex for short) if it satisfies
$F_{A}+\operatorname{Vol}_{\Sigma}u^{*}P(\Phi)=0.$
An $n$-marked symplectic vortex is a vortex $(A,u)$ together with $n$-tuple
${z}=(z_{1},\ldots,z_{n})$ of distinct points on $\Sigma$. A marked vortex
$(A,u,{z})$ is stable if it has finite automorphism group.
The equation in the definition can be interpreted as the zero level set
condition for a formal moment map for the action of the group of gauge
transformations, see [16], [5]. The energy of a gauged holomorphic map $(A,u)$
is given by
$E(A,u)={\frac{1}{2}}\int_{\Sigma}\left(|{\operatorname{d}}_{A}u|^{2}+|F_{A}|^{2}+|u^{*}P(\Phi)|^{2}\right)\operatorname{Vol}_{\Sigma}.$
The _equivariant symplectic area_ of a pair $(A,u)$ is the pairing of the
homology class $\deg(u)$ with the class $[\omega_{G}=\omega+\Phi]\in
H^{2}(X_{G})$,
$D(A,u)=(\deg(u),[\omega_{G}])=([\Sigma],u^{*}[\omega_{G}]).$
###### Lemma 4.0.2.
Suppose $\operatorname{Vol}_{\Sigma}$ is the Kähler form for the metric on
$\Sigma$. The energy and equivariant area are related by
(28)
$E(A,u)=D(A,u)+\int_{\Sigma}\left(|\overline{\partial}_{A}u|^{2}+{\frac{1}{2}}|F_{A}+\operatorname{Vol}_{\Sigma}u^{*}P(\Phi)|^{2}\right)\operatorname{Vol}_{\Sigma}.$
###### Proof.
See [4, Proposition 2.2]. ∎
In particular, for any symplectic vortex the energy and action are equal. Let
$M(P,X)$ denote the moduli space of vortices
$M(P,X):=\mathcal{H}(P,X)/\kern-3.01385pt/\mathcal{G}(P)=\\{F_{A}+\operatorname{Vol}_{\Sigma}u^{*}P(\Phi)=0\\}/\mathcal{G}(P).$
Let $M_{n}(P,X)$ denote the moduli space of $n$-marked vortices, up to gauge
transformation, and $M_{n}(\Sigma,X)=\bigcup_{P\to\Sigma}M_{n}(P,X)$ the union
over types of bundles $P$. Clearly, $M_{n}(\Sigma,X)$ is homeomorphic to the
product $M(\Sigma,X)\times M_{n}(\Sigma)$ where $M(\Sigma,X):=M_{0}(\Sigma)$
and $M_{n}(\Sigma)$ denotes the configuration space of $n$-tuples of distinct
points on $\Sigma$.
We wish to study families and deformations of symplectic vortices. For
families with smooth domain, the definitions are straightforward:
###### Definition 4.0.3.
A smooth family of vortices on a principal $G$-bundle $P$ on $\Sigma$ over a
parameter space $S$ consists of a family of connections depending smoothly on
$s\in S$, that is, a smooth map $A_{S}:S\times P\to T^{*}P\otimes\mathfrak{g}$
on $P$ such that the restriction $A_{s}$ of $A_{S}$ to any $\\{s\\}\times P$
is a connection, together with a smooth family of (pseudo)holomorphic sections
$u_{S}=(u_{s})_{s\in S}$, such that each pair $(A_{s},u_{s}),s\in S$ is a
symplectic vortex. A deformation of $(A,u)$ is a germ of a smooth family
$(A_{S},u_{S})$ together with an isomorphism (gauge transformation) relating
$(A_{0},u_{0})$ with $(A,u)$. A deformation is universal if it satisfies the
condition as in Definition 3.1.4, and strongly universal if it satisfies the
conditions in Definition 2.2.4.
We define a linearized operator associated to a vortex as follows. Define
(29)
${\operatorname{d}}_{A,u}:\Omega^{1}(\Sigma,P(\mathfrak{g}))\oplus\Omega^{0}(\Sigma,u^{*}TP(X))\to\Omega^{2}(\Sigma,P(\mathfrak{g}))$
${\operatorname{d}}_{A,u}(a,\xi):={\operatorname{d}}_{A}a+\operatorname{Vol}_{\Sigma}u^{*}L_{\xi}P(\Phi).$
Here $L_{\xi}P(\Phi)$ denotes the derivative of $P(\Phi)$ with respect to the
vector field generated by $\xi$, and $u^{*}L_{\xi}P(\Phi)$ its evaluation at
$u$. Define an operator
(30)
${\operatorname{d}}_{A,u}^{*}:\Omega^{1}(\Sigma,P(\mathfrak{g}))\oplus\Omega^{0}(\Sigma,u^{*}TP(X))\to\Omega^{0}(\Sigma,P(\mathfrak{g}))$
${\operatorname{d}}_{A,u}^{*}(a,\xi)={\operatorname{d}}_{A}^{*}a+u^{*}L_{J\xi}P(\Phi).$
(This is not the adjoint of operator in (32), but rather defined by analogy
with the case $X$ trivial.) It is shown in [5, Section 4] that if $(A,u)$ is
stable then the set
$W_{A,u}=\\{(A+a,\exp_{u}(\xi)),(a,\xi)\in\operatorname{ker}{\operatorname{d}}_{A,u}^{*}\\}$
is a slice for the gauge group action near $(A,u)$. Define
(31)
$\mathcal{F}_{A,u}:\Omega^{1}(\Sigma,P(\mathfrak{g}))\oplus\Omega^{0}(\Sigma,u^{*}T^{\operatorname{vert}}P(X))\\\
\to(\Omega^{0}\oplus\Omega^{2})(\Sigma,P(\mathfrak{g}))\oplus\Omega^{0,1}(\Sigma,u^{*}T^{\operatorname{vert}}P(X))\\\
(a,\xi)\mapsto\left(F_{A+a}+\operatorname{Vol}_{\Sigma}\exp_{u}(\xi)^{*}P(\Phi),{\operatorname{d}}_{A,u}^{*}(a,\xi),\Psi_{u}(\xi)^{-1}\overline{\partial}_{A+a}\exp_{u}(\xi)\right).$
Let
$\Omega^{1}(\Sigma,P(\mathfrak{g}))\to\Omega^{1}(\Sigma,u^{*}T^{\operatorname{vert}}P(X)),\
\ \ a\mapsto a_{X}$
denote the map induced by the infinitesimal action. The linearization of the
last component (31) is
$D_{A,u}(a,\xi)=(\nabla_{A}\xi)^{0,1}+{\frac{1}{2}}J_{u}(\nabla_{\xi}J)_{u}\partial_{A}u+a_{X}^{0,1}.$
Here $0,1$ denotes projection on the $0,1$-component. The linearized operator
for a vortex $(A,u)$ is the operator
(32)
$\tilde{D}_{A,u}=({\operatorname{d}}_{A,u},{\operatorname{d}}_{A,u}^{*},D_{A,u}):\Omega^{1}(\Sigma,P(\mathfrak{g}))\oplus\Omega^{0}(\Sigma,u^{*}T^{\operatorname{vert}}P(X))\\\
\to(\Omega^{0}\oplus\Omega^{2})(\Sigma,P(\mathfrak{g}))\oplus\Omega^{0,1}(\Sigma,u^{*}T^{\operatorname{vert}}P(X))$
A vortex $(A,u)$ is regular if the operator $\tilde{D}_{A,u}$ is surjective. A
marked vortex $(A,u,{z})$ is regular if the underlying unmarked vortex is
regular. The space of infinitesimal deformations of $(A,u)$ is
$\operatorname{Def}(A,u):=\operatorname{ker}(\tilde{D}_{A,u}).$
###### Theorem 4.0.4.
Any regular vortex with smooth domain $(A,u)$ has a strongly universal smooth
deformation if and only if it is stable.
###### Proof.
Give the spaces of connections and sections the structure of Banach manifolds
by taking completions with respect to Sobolev norms $1,p$ for $1$-forms, and
$0,p$ for $0$ and $2$-forms. For $p>2$, the map $\mathcal{F}_{A,u}$ is a
smooth map of Banach spaces.
(33)
$\mathcal{F}_{A,u}:\Omega^{1}(\Sigma,P(\mathfrak{g}))_{1,p}\oplus\Omega^{0}(\Sigma,u^{*}T^{\operatorname{vert}}P(X))_{1,p}\\\
\to(\Omega^{0}\oplus\Omega^{2})(\Sigma,P(\mathfrak{g}))_{0,p}\oplus\Omega^{0,1}(\Sigma,u^{*}T^{\operatorname{vert}}P(X))_{0,p}$
equivariant for the action of the group $\mathcal{G}(P)_{2,p}$ of gauge
transformations of class $2,p$. Suppose that $(A,u)$ is regular and stable. By
the implicit function theorem, there is a local homeomorphism
$\operatorname{ker}(\tilde{D}_{A,u})\to\left\\{\begin{array}[]{c}F_{A+a}+\operatorname{Vol}_{\Sigma}(\exp_{u}(\xi))^{*}P(\Phi)=0\\\
\overline{\partial}_{A+a}(\exp_{u}(\xi))=0\\\ \
{\operatorname{d}}_{A,u}^{*}(a,\xi)=0\end{array}\right\\}.$
This gives rise to a family $(A_{S},u_{S})\to S$ over a neighborhood $S$ of
$0$ in $\operatorname{ker}(\tilde{D}_{A,u})$. By [5, Theorem 3.1],
$(A_{S},u_{S})$ is a smooth family, assuming $(A,u)$ is smooth. Given any
other family $(A_{S^{\prime}}^{\prime},u_{S^{\prime}}^{\prime})\to S^{\prime}$
of stable vortices with $(A^{\prime}_{0},u_{0}^{\prime})=(A,u)$, the implicit
function theorem provides a smooth map $S^{\prime}\to S$ so that
$(A^{\prime}_{S^{\prime}},u^{\prime}_{S^{\prime}})$ is obtained from $(A,u)$
by pull-back. The first property of the universal deformation is a consequence
of the slice condition; the second property follows from the fact that the
projection
$\operatorname{ker}(\tilde{D}_{A,u})\to\operatorname{ker}(\tilde{D}_{A_{s},u_{s}})$
is an isomorphism for sufficiently small $s$. ∎
Let $M^{\operatorname{reg}}_{n}(\Sigma,X)$ denote the moduli space of regular,
stable $n$-marked symplectic vortices from $\Sigma$ to $X$. We denote by
$(c_{1}^{G}(TX),d)$ the pairing of $d$ with the first Chern class
$c_{1}^{G}(P(TX)\to P(X))$
###### Theorem 4.0.5.
Let $\Sigma,X,J$ be as above. $M^{\operatorname{reg}}_{n}(\Sigma,X)$ has the
structure of a smooth orbifold with tangent space at $[A,u]$ isomorphic to
$\operatorname{Def}(A,u)$, and dimension of the component of homology class
$d\in H_{2}^{G}(X)$ is given by
$\dim(M^{\operatorname{reg}}_{n}(\Sigma,X,d))=(1-g)(\dim(X)-2\dim(G))+2((c_{1}^{G}(TX),d)+n).$
###### Proof.
Charts for $M^{{\operatorname{reg}}}_{n}(\Sigma,X)$ are provided by the
strongly universal deformations. The dimension of the tangent space at $[A,u]$
is given by the index of the linearized operator $\tilde{D}_{A,u}$, which
deforms via Fredholm operators to the sum of the operator
${\operatorname{d}}_{A}\oplus{\operatorname{d}}_{A}^{*}$ for the connection,
which has index $2\dim(G)(g-1)$, and the linearized Cauchy-Riemann operator on
the nodal curve, which has index $(1-g)\dim(X)+2n+2(c_{1}^{G}(TX),d)$ by
Riemann-Roch, if $(A,u)$ has equivariant homology class $d$ (which determines
the first Chern class of $P$ by projection.) ∎
### 4.1. Polystable vortices
The moduli space of symplectic vortices admits a compactification which allows
bubbling of the section in the fibers.
###### Definition 4.1.1.
A nodal gauged marked holomorphic map from $\Sigma$ to $X$ consists of a datum
$(\hat{\Sigma},P,A,{u},{z})$ where $P\to\Sigma$ is a principal $G$-bundle,
$A\in A(P)$ is a connection, $\hat{\Sigma}$ is a marked nodal curve,
$v:\hat{\Sigma}\to\Sigma$ is a holomorphic map of degree $[\Sigma]$, and
${u}:\hat{\Sigma}\to P(X)$ is a $J_{A}$-holomorphic map from a nodal curve
$\hat{\Sigma}$ such that $\pi\circ{u}$ has class $[\Sigma]$. In other words,
1. (a)
$\hat{\Sigma}$ is a connected nodal complex curve consisting of a principal
component $\Sigma_{0}$ equipped with an isomorphism with $\Sigma$ together
with a number of projective lines $\Sigma_{1},\ldots,\Sigma_{k}$. We denote by
$w_{1}^{\pm},\ldots,w_{m}^{\pm}$ the nodes. For each $i=1,\ldots,k$, we denote
by $w_{i}^{0}\in\Sigma_{0}$ the attaching point to the principal component.
2. (b)
$(A,u)\in\mathcal{H}(P,X)$ is a gauged holomorphic map from $\Sigma$ to $X$;
3. (c)
for each non-principal component $\Sigma_{i}$, a holomorphic map
$u_{i}:\Sigma_{i}\to P(X)_{w_{i}^{0}}$;
4. (d)
${z}=(z_{1},\ldots,z_{n})\in\hat{\Sigma}$ are distinct, smooth points of
$\hat{\Sigma}$.
Let $\mathcal{H}(\hat{\Sigma},P(X))$ denote the space of nodal gauged
holomorphic sections with domain $\hat{\Sigma}$ and bundle $P$. The group of
gauge transformations $\mathcal{G}(P)$ acts on
$\mathcal{H}(\hat{\Sigma},P(X))$ by $g(A,{u})=(g^{*}A,g\circ{u}).$ The
generating vector field for $\zeta\in\Omega^{0}(\Sigma,P(\mathfrak{g}))$
acting on $\mathcal{H}(\hat{\Sigma},P(X))$ at $(\hat{\Sigma},A,{u})$ is the
tuple given by
(34)
$\zeta_{\mathcal{H}(\hat{\Sigma},P(X))}(\hat{\Sigma},A,{u})=({\operatorname{d}}_{A}\zeta,u_{0}^{*}P(\zeta_{X}),(u_{i}^{*}P(\zeta_{X}(w_{i}^{0})))_{i=1}^{k})$
in
$\Omega^{1}(\Sigma,P(\mathfrak{g}))\oplus\Omega^{0}(\hat{\Sigma},{u}^{*}T^{\operatorname{vert}}P(X))$.
Here $P(\zeta_{X})\in\Omega^{0}(\Sigma,P(\operatorname{Vect}(X)))$ is the
fiber-wise vector field generated by $\zeta$ and $u_{0}^{*}P(\zeta)\in
T^{\operatorname{vert}}P(X)$ is the evaluation at $u_{0}$. Similarly for the
bubble components $u_{1},\ldots,u_{k}$ in the fibers
$P(X)_{w^{0}_{1}},\ldots,P(X)_{w^{0}_{k}}$. A slice is given by taking the
perpendicular to the tangent spaces to the $\mathcal{G}(P)$-orbits. We will
assume for simplicity that the stabilizer of the $\mathcal{G}(P)$ action on
the principal component is finite, so that a slice is given locally by the
kernel of ${\operatorname{d}}_{A,u_{0}}^{*}$, that is, the Coulomb gauge
condition on the principal component. The implicit function theorem shows that
any nearby pair $(A_{1},{u}_{1})$ is complex gauge equivalent to a pair of the
form $(A+a,\exp_{{u}}({v}))$ with
$(a,{v})\in\operatorname{ker}{\operatorname{d}}_{A,u_{0}}^{*}$.
###### Definition 4.1.2.
A nodal vortex is a stable nodal gauged holomorphic map such that the
principal component is an vortex. A nodal vortex $(\hat{\Sigma},A,{u},{z})$ is
polystable if each sphere bubble $\Sigma_{i}$ on which $u_{i}$ is constant has
at least three marked or singular points, and stable if it has finite
automorphism group. An isomorphism of nodal vortices
$(\hat{\Sigma},A,{u},{z}),(\hat{\Sigma}^{\prime},A^{\prime},{u}^{\prime},{z}^{\prime})$
consists of an automorphism of the domain, acting trivially on the principal
component, and a corresponding automorphism of the principal bundle mapping
$(A,{u})$ to $(A^{\prime},{u}^{\prime})$ and mapping the markings ${z}$ to
${z}^{\prime}$. For any nodal section ${u}:\hat{\Sigma}\to P(X)$, the homology
class of ${u}$ is defined as the sum of the homology class $d_{0}\in
H_{2}^{G}(X,\mathbb{Z})$ of the principal component $u_{0}$ and the homology
classes $d_{i}\in H_{2}(X,\mathbb{Z}),i=1,\ldots,k$ of the sphere bubbles,
using the inclusion $H_{2}(X,\mathbb{Z})\to H_{2}^{G}(X,\mathbb{Z})$ given by
equivariant formality. The combinatorial type $\Gamma(\hat{\Sigma},A,{u},{z})$
of a gauged nodal map is a rooted graph whose vertices represent the
components of $\hat{\Sigma}$, whose finite edges represent the nodes, semi-
infinite edges represent the markings, and whose root vertex represents the
principal component.
Note that there is no condition for points on the principal component. In
particular, nodal gauged holomorphic maps with no markings can be polystable.
The term polystable is borrowed from the vector bundle case. In that
situation, a bundle is stable if it is flat and has only central automorphisms
and polystable if it is a direct sum of stable bundles of the same slope. Any
flat bundle is automatically polystable; a bundle is semistable if it is grade
equivalent to a polystable bundle. In particular, the moduli space of stable
bundles is definitely not compact, and we feel that the vortex terminology
should include this fact as a special case.
From now on, we fix the bundle $P$.
###### Definition 4.1.3.
Let $X$ as above. A smooth family of fixed type of nodal vortices to $X$
consists of a smooth family $\hat{Sigma}_{S}\to S$ of nodal curves of fixed
type, a smooth family of holomorphic maps $v_{S}:\hat{\Sigma}_{S}\to\Sigma$ of
class $[\Sigma]$, a smooth family ${u}_{S}:\hat{\Sigma}_{S}\to P(X)$ of maps,
and a smooth family $A_{S}:S\times P\to T^{*}P$ of connections over $S$. A
smooth deformation of a nodal vortex $(A,\hat{\Sigma},{u},{z})$ of fixed type
consists of a germ of a smooth family
$(A_{S},\hat{\Sigma}_{S},{u}_{S},{z}_{S})$ of nodal vortices of fixed type
together with an identification $\iota$ of of the central fiber with
$(A,\hat{\Sigma},{u},{z})$. A stratified-smooth family of marked nodal
symplectic vortices is a datum $(\hat{\Sigma}_{S},A_{S},{u}_{S},{z}_{S})$
consisting of a stratified-smooth family $\hat{\Sigma}_{S}\to S$ of nodal
curves, a stratified-smooth family of holomorphic maps
$v:\hat{\Sigma}_{S}\to\Sigma$ of class $[\Sigma]$, a stratified-smooth family
$A_{S}$ of connections on $P$, a stratified-smooth family of maps
${u}_{S}:\hat{\Sigma}_{S}\to P(X)$; such that each triple
$(\hat{\Sigma}_{s},A_{s},{u}_{s},{z}_{s})$ is a marked nodal symplectic
vortex. A family of polystable symplectic vortices is a family of marked nodal
symplectic vortices such that any fiber is polystable.
A smooth vector bundle ${E}\to\hat{\Sigma}$ is a collection of smooth vector
bundles $E_{i}$ over the components $\Sigma_{i}$ of $\hat{\Sigma}$, equipped
with identifications of the fibers at nodal points $E_{i^{+}(w^{+}_{k})}\to
E_{i^{-}(w^{-}_{k})},k=1,\ldots,m$. We denote by $\Omega(\hat{\Sigma},{E})$
the sum over components,
$\Omega(\hat{\Sigma},{E})=\bigoplus_{i=1}^{k}\Omega(\Sigma_{i},E_{i})$ where
$E_{i}={E}|\Sigma_{i}.$
###### Definition 4.1.4.
For a polystable vortex $(\hat{\Sigma},A,{u})$, let $\tilde{D}_{A,{u}}$ denote
the linearized operator
(35)
$\Omega^{1}({\Sigma},P(\mathfrak{g}))\oplus\Omega^{0}(\hat{\Sigma},{u}^{*}T^{\operatorname{vert}}P(X))\\\
\to(\Omega^{0}\oplus\Omega^{2})({\Sigma},P(\mathfrak{g}))\oplus\Omega^{0,1}(\hat{\Sigma},{u}^{*}T^{\operatorname{vert}}P(X))\oplus\bigoplus_{k=1}^{m}T_{{u}(w_{k}^{\pm})}^{\operatorname{vert}}P(X)$
given by the linearized vortex operator
$({\operatorname{d}}_{A,u_{0}},D_{A,u_{0}})$ on the principal component, the
linearized Cauchy-Riemann operator $\tilde{D}_{u_{i}}$ on the bubbles, the
slice operator ${\operatorname{d}}_{A,u_{0}}^{*}$, and the difference operator
on the fibers over the nodes
$\Omega^{0}(\hat{\Sigma},{u}^{*}T^{\operatorname{vert}}P(X))\to\bigoplus_{i=1}^{m}T_{{u}(w_{i}^{\pm})}^{\operatorname{vert}}P(X),\quad{\xi}\mapsto({\xi}(w_{i}^{+})-{\xi}(w_{i}^{-}))_{i=1}^{m}.$
$(A,{u})$ is regular if $\tilde{D}_{A,{u}}$ is surjective. The space of
infinitesimal deformations of $A,{u}$ of fixed type is
$\operatorname{Def}_{\Gamma}(A,{u}):=\operatorname{ker}\tilde{D}_{A,{u}}/\operatorname{aut}(\hat{\Sigma})$
where $\operatorname{aut}(\hat{\Sigma})$ denotes the group of infinitesimal
automorphisms acting trivially on the principal component. The space of
infinitesimal deformations of $A,{u}$ is
$\operatorname{Def}(A,{u}):=\operatorname{Def}_{\Gamma}(A,{u})\oplus\bigoplus_{i=1}^{m}T_{w_{j}^{+}}\hat{\Sigma}\otimes
T_{w_{j}^{-}}\hat{\Sigma}$
consisting of a deformation of fixed type together with a collection of gluing
parameters.
### 4.2. Constructing deformations of symplectic vortices
First we construct deformations of fixed type.
###### Theorem 4.2.1.
A regular polystable vortex has a strongly universal smooth deformation of
fixed type if and only if it is stable.
The proof is by the implicit function theorem applied to the map
(36)
$\mathcal{F}_{A,{u}}(a,{\xi})=(F_{A+a}+\operatorname{Vol}_{\Sigma}(\exp_{u_{0}}(\xi_{0}))^{*}P(\Phi),{\operatorname{d}}_{A,u_{0}}^{*}(a,{\xi}),\\\
\Psi_{u_{0}}(\xi_{0})^{-1}\overline{\partial}_{A}u_{0},(\Psi_{u_{i}}(\xi_{i})^{-1}\overline{\partial}u_{i})_{i=1}^{k},({\xi}(w_{i}^{+})-{\xi}(w_{i}^{-}))_{i=1}^{m})$
whose linearization is $\tilde{D}_{A,{u}}$. The proof is left to the reader.
We denote by $M_{n,\Gamma}(\Sigma,X,d)$ of the moduli space of isomorphism
classes of polystable vortices of combinatorial type $\Gamma$ of homology
class $d\in H_{2}^{G}(X,\mathbb{Z})$, and
$M^{{\operatorname{reg}}}_{n,\Gamma}(\Sigma,X,d)$ the regular locus.
###### Corollary 4.2.2.
$M_{n,\Gamma}^{\operatorname{reg}}(\Sigma,X,d)$ has the structure of a smooth
orbifold of dimension given, if $\Sigma$ is connected, by
$(1-g)(\dim(X)-2\dim(G))+2(n+(c_{1}^{G}(TX),d)-m)$ where $m$ is the number of
nodes.
We now prove that a regular stable symplectic vortex from $\Sigma$ to $X$
admits a strongly universal stratified-smooth deformation if it is strongly
stable, that is, Theorem 1.0.2. We explain the construction for a single
bubble only, so that $\hat{\Sigma}$ is the union of a principal component
$\Sigma_{+}=\Sigma$ and a holomorphic sphere $\Sigma_{-}$, attached by a
single pair $w_{\pm}$ of nodes. We denote by $(A,u_{+})$ the restriction to
the principal component and by $u_{-}$ the bubble, so that
$x:=u_{+}(w_{+})=u_{-}(w_{-})$ and ${u}=(u_{+},u_{-})$. We choose a local
coordinate near $w$, equivariant for the action of the automorphism group
$\operatorname{Aut}(A,{u})$ in the sense that $\operatorname{Aut}(A,{u})$ acts
on the local coordinate by multiplication by roots of unity. The construction
depends on the following choices:
###### Definition 4.2.3.
A gluing datum for $(\hat{\Sigma},A,{u})$ consists of
1. (a)
neighborhoods $U_{\pm}$ of the nodes $w_{\pm}$;
2. (b)
local coordinates ${\kappa}=(\kappa_{+},\kappa_{-})$ on $U_{\pm}$;
3. (c)
a trivialization $P|_{U_{0}}\to G\times U_{0}$ on $U_{0}$;
4. (d)
a gluing parameter $\delta$;
5. (e)
an annulus parameter $\rho$ ;
6. (f)
a cutoff function $\alpha$ as in (10).
Step 1: Approximate Solution
Given a nodal vortex $(A,u)$ as above and a gluing datum we wish to define an
approximate solution to the vortex equations $(A,u^{\delta})$. Let
$\exp_{x}:T_{x}X\to X$ denote the exponential map defined by the metric on
$X$. Define sections
$\xi_{\pm}:U_{\pm}\to T_{x}X,\quad u(z)=\exp_{x}(\xi_{\pm}(z)).$
Let $\hat{\Sigma}^{\delta}$ denote the surface obtained by gluing; since the
bubble is genus zero, this surface is isomorphic to $\Sigma$ but not
canonically. Define the pre-glued section
${u}^{\delta}:\hat{\Sigma}^{\delta}\to P(X)$,
(37)
${u}^{\delta}(z)=\exp_{x}\left(\alpha(|\kappa_{+}(z)|/\rho|\delta|^{1/2})(\xi_{+}(z)-\xi_{\pm}(w^{\pm}))\right.+\\\
\left.\alpha(|\kappa_{-}(z)|/\rho|\delta|^{1/2})(\xi_{-}(z)-\xi_{\pm}(w^{\pm}))+\xi_{\pm}(w^{\pm})\right)$
for $|\kappa_{\pm}(z)|\leq 2|\delta|^{1/2}\rho^{2}$; elsewhere let
${u}^{\delta}(z)={u}(z)$, using the identification of $\Sigma$ with
$\hat{\Sigma}$ away from the gluing region. We do not modify $A$ in the bubble
region; this is because after re-scaling the connection on the bubble is
already close to the trivial connection.
The pair $(A,{u}^{\delta})$ is an approximate solution to the vortex equations
in the following Sobolev norms. Let
$\Omega(\Sigma^{\delta},P^{\delta}(\mathfrak{g}))_{k,p}$ denote the
$W^{k,p}$-completion using the standard metric. Let $g^{\delta}$ denote the
$C^{0}$ metric on the glued surface in (12). Let
$\Omega^{0,1}(\Sigma^{\delta},u^{\delta,*}T^{\operatorname{vert}}P(X))_{k,p,\delta}$.
Let
$\mathcal{H}_{\delta}:=\Omega^{2}(\Sigma,P(\mathfrak{g}))_{{0,3}}\oplus\Omega^{0}(\Sigma,P(\mathfrak{g}))_{0,3}\oplus\Omega^{0,1}(\Sigma^{\delta},u^{\delta,*}T^{\operatorname{vert}}P(X))_{{0,3},\delta}$
with norm
(38)
$\|(\phi,\psi,\eta)\|_{{\delta}}^{2}=\|\phi\|_{{0,3}}^{2}+\|\psi\|_{0,3}^{2}+\|\eta\|_{0,3,\delta}^{2}.$
Let
$\mathcal{I}_{\delta}:=\Omega^{1}(\Sigma,P(\mathfrak{g}))_{{1,3}}\oplus\Omega^{0}(\Sigma^{\delta},u^{\delta,*}T^{\operatorname{vert}}P(X))_{{1,3},\delta}$
with norm
$\|(a,\xi)\|_{{\delta}}^{2}=\|a\|_{{1,3}}^{2}+\|\eta\|_{{1,3},\delta}^{2}.$
Locally the moduli space of polystable vortices is in bijection with the zero
set of the map
(39)
$\mathcal{F}_{A,{u}}^{{\delta}}:\mathcal{I}_{{\delta}}\to\mathcal{H}_{{\delta}}\\\
(a,\xi)\mapsto\left(F_{A+a}+\operatorname{Vol}_{\Sigma}\exp_{{u}^{{\delta}}}(\xi)^{*}P(\Phi),{\operatorname{d}}_{A,u_{+}}^{*}(a,\xi),\Psi_{{u}^{{\delta}}}(\xi)^{-1}\overline{\partial}_{A+a}\exp_{{u}^{{\delta}}}(\xi)\right).$
Here the second map enforces a slice condition. That
$\mathcal{F}_{A,{u}}^{{\delta}}$ is well-defined follows from Sobolev
embedding: In particular, there is a $\delta$-uniform embedding
(40)
$\Omega^{0}(\Sigma^{\delta},u^{\delta,*}T^{\operatorname{vert}}P(X))_{{1,3},\delta}\to\Omega^{0}(\Sigma^{\delta},u^{\delta,*}T^{\operatorname{vert}}P(X))_{0,\infty},$
since the dimensions of the cone in the cone condition [2, Chapter 4] are
uniformly bounded in $\delta$, and the metric uniformly comparable to the flat
metric.
###### Lemma 4.2.4.
Let $(A,{u})$ be a symplectic vortex on a nodal curve with a single node
$w=(w^{+},w^{-})$. There exist constants $c_{0},c_{1}>0$ such that if
$|\delta|<c_{1},\rho>1/c_{1}$ and$|\delta|\rho^{4}<c_{1}$ then the pair
$(A,{u}^{\delta})\in\mathcal{A}(P)\times\Gamma(P(X))$ satisfies
(41) $\|\mathcal{F}_{A,{u}}^{{\delta}}(0,0)\|\leq
c_{0}|\delta|^{1/3}\rho^{2/3}.$
###### Proof.
The expression $\overline{\partial}_{A}{u}^{\delta}$ can be expressed as a sum
of terms involving derivatives of the cutoff function $\alpha$, terms
involving derivatives of $\xi_{j}$, and terms involving the connection $A$ on
the bubble region. The derivative of $\alpha$ is bounded by
$C/\rho|\delta|^{1/2}$, while the norm of $\xi_{j}$ is bounded by
$C\rho|\delta|^{1/2}$ on the gluing region. Hence the term involving the
derivative of $\alpha$ is bounded and supported on a region of area less than
$C|\delta|\rho^{2}$. In the given trivialization we have
$\overline{\partial}_{A}u=\overline{\partial}u+A_{X}^{0,1}(u)$
where $A^{0,1}$ is the $0,1$-form defined by
$A\in\Omega^{1}(B_{R},\mathfrak{g})$ is the connection $1$-form in the local
trivialization and $A_{X}^{0,1}(u)$ is the corresponding form with values in
$T^{\operatorname{vert}}P(X)\otimes_{\mathbb{R}}\mathbb{C}$. We have
$\|A_{X}^{0,1}(u)\|_{0,3,\delta}\leq\|A_{X}^{0,1}(u)\|_{0,3}$
since $p\geq 2$; for $p=2$ the $W^{0,3,\delta}$ and $W^{0,3}$ norms are the
same, by conformal invariance; for $p>2$ the $0,3$-norm is strictly greater.
Hence
$\|\overline{\partial}_{A}u^{\delta}\|_{0,3,\delta}\leq
C\max(|\delta|^{1/3}\rho^{2/3},|\delta|).$
The moment map term
$F_{A}+({u}^{\delta})^{*}P(\Phi)\operatorname{Vol}_{\Sigma}$ vanishes except
on $|\kappa_{+}|\leq\rho|\delta^{1/2}|$, where it is uniformly bounded. Hence
for $\delta$ small
$\|F_{A}+({u}^{\delta})^{*}P(\Phi)\operatorname{Vol}_{\Sigma}\|_{0,3}\leq
C\rho^{2/3}|\delta|^{1/3}.$
The statement of the lemma follows. ∎
We also wish to perform the gluing construction in families, that is, for each
nearby vortex and gluing parameter we wish to find a solution to the vortex
equations on $\Sigma^{\delta}$. Define
(42)
$\mathcal{F}_{A,{u}}^{D,\delta}:\operatorname{Def}_{\Gamma}(A,{u})\times\mathcal{I}_{{\delta}}\to\mathcal{H}_{{\delta}},\quad(a,{\xi},a_{1},\xi_{1})\mapsto\\\
\left(F_{A}+a_{0}+a_{1}+\operatorname{Vol}_{\Sigma}\exp_{{u}^{{\delta}}}({\xi}_{0}^{\delta}+\xi_{1})^{*}P(\Phi),{\operatorname{d}}_{A,{u}^{\delta}}^{*}(a_{0}+a_{1},{\xi}_{0}^{\delta}+\xi),\right.\\\
\left.\Psi_{{u}^{{\delta}}}({\xi}_{0}^{\delta}+\xi_{1})^{-1}\overline{\partial}_{A+a_{0}+a_{1}}\exp_{{u}^{{\delta}}}({\xi}_{0}^{\delta}+\xi_{1})\right)$
The following is proved in the same way as Lemma 4.2.4 and left to the reader:
###### Lemma 4.2.5.
Let $(A,{u})$ be as above. There exist constants $c_{0},c_{1}>0$ such that if
$|\delta|<c_{1},\rho>1/c_{1}$, $\|(a,\xi)\|\leq c_{1}$ and
$|\delta|\rho^{4}<c_{1}$ then
(43) $\|\mathcal{F}_{A,{u}}^{D,{\delta}}(a,\xi,0,0)\|\leq
c_{0}|\delta|^{1/3}\rho^{2/3}.$
Step 2: Uniformly bounded right inverse
In preparation for the construction of the uniformly bounded right inverse of
$\tilde{D}_{\delta}$ we define the intermediate family $(A,{u}_{0}^{\delta})$
of gauged holomorphic maps on the nodal curve $\hat{\Sigma}$ is the family
defined by the equations (4.2.4), using the identification of $\hat{\Sigma}$
and $\hat{\Sigma}^{\delta}$ away from the gluing region. Thus $u_{0}^{\delta}$
is constant in a neighborhood of the node $w^{\pm}$. We identify
$({u}_{0}^{\delta})^{*}T^{\operatorname{vert}}P(X)$ with
${u}^{*}T^{\operatorname{vert}}P(X)$ by geodesic parallel transport.
###### Lemma 4.2.6.
The operator $\tilde{D}_{A,{u}_{0}^{\delta}}$ converges in the operator norm
to $\tilde{D}_{A,{u}}$ as $\delta\to 0,\rho\to\infty,\delta\rho^{4}\to 0$.
###### Proof.
The section ${u}_{0}^{\delta}$ converges in the $W^{1,3}$ norm to $u$ as
$\rho^{2}|\delta|^{1/2}\to 0$. It follows that the operator
$\xi\mapsto\operatorname{Vol}_{\Sigma}(u^{\delta}_{0})^{*}L_{\xi}P(\Phi)$
converges to $\xi\mapsto\operatorname{Vol}_{\Sigma}u^{*}L_{\xi}P(\Phi)$. Hence
${\operatorname{d}}_{A,u_{0}^{\delta},\epsilon}$ converges to
${\operatorname{d}}_{A,u,\epsilon}$, and similarly for
${\operatorname{d}}_{A,u_{0},\epsilon}^{*}$. The operator
$D_{A,{u}_{0}^{\delta}}$ converges to $D_{A,{u}}$, as in Lemma 3.3.6. ∎
###### Proposition 4.2.7.
Let $(A,{u})$ be a nodal vortex, and $(A,{u}^{\delta})$ the approximate
solution constructed above. There exist constants $c,C>0$ such that if
$|\delta|<c$ then there exists an approximate right inverse $T_{\delta}$ of
the parametrized linear operator
$\tilde{D}_{\delta}:=\tilde{D}_{A,{u}^{\delta}}$ that is, a map
$T_{\delta}:\mathcal{H}_{\delta}\to\mathcal{I}_{\delta}$ such that
$\|(\tilde{D}_{\delta}T_{\delta}-I)\eta\|_{\delta}\leq{\frac{1}{2}}\|\eta\|_{\delta},\quad\|T_{{\delta}}\eta\|_{\delta}\leq
C\|\eta\|_{\delta}.$
Given such an approximate inverse, we obtain a uniformly bounded right inverse
$Q_{\delta}$ to $\tilde{D}_{\delta}$ by the formula
$Q_{\delta}=T_{\delta}(\tilde{D}_{\delta}T_{\delta})^{-1}=\sum_{k\geq
0}T_{\delta}(\tilde{D}_{\delta}T_{\delta}-I)^{k}.$
###### Proof of 4.2.7.
By the regularity assumption, $\tilde{D}_{A,{u}}$ is surjective when
restricted to the space of vectors $(a,{\xi})$ such that
$\xi_{0}(0)=\xi_{1}(\infty)$. By Lemma 4.2.6,
$\tilde{D}^{0}_{\delta}:=\tilde{D}_{A,{u}_{0}^{\delta}}$ is surjective for
sufficiently small $\rho,\delta$, when restricted to the same space. The
approximate right inverse is constructed by composing a cutoff operator
$K_{\delta}$, right inverse $Q_{\delta}$, and gluing operator $R_{\delta}$, as
follows. (In other words, the right inverse is defined by truncating the given
functions, applying the right inverse for the linearized operator on the nodal
curve, and then gluing back together using cutoff functions again.) For
simplicity we assume that there is a single node. Define the cutoff operator
(44)
$K_{\delta}:\Omega^{0,1}(\hat{\Sigma}^{\delta},{u}^{\delta,*}T^{\operatorname{vert}}P(X))_{0,3,\delta}\to\Omega^{0,1}(\hat{\Sigma},({u}^{\delta}_{0})^{*}T^{\operatorname{vert}}P(X))_{0,3}\\\
K_{\delta}(\eta)=\begin{cases}0&\kappa_{\pm}(z)\in
B_{|\delta|^{{\frac{1}{2}}}}(0)\\\ \eta&\text{otherwise}\end{cases}.$
Then $\|K_{\delta}(\eta)\|_{0,3}\leq\|\eta\|_{0,3,\delta}.$ Define the gluing
operator
$R_{\delta}:\Omega^{0}(\hat{\Sigma},({u}^{\delta}_{0})^{*}TX)_{{1,3}}\to\Omega^{0}(\hat{\Sigma}^{\delta},u^{\delta,*}T^{\operatorname{vert}}P(X))_{{1,3},\delta}$
as follows. Let $\hat{\Sigma}_{\pm}^{*}$ denote the complements of small balls
around the nodes
$\hat{\Sigma}_{\pm}^{*}=\Sigma_{\pm}-B_{\rho^{2}|\delta|^{1/2}}(w^{\pm})$. Let
$\pi_{\pm}:\hat{\Sigma}_{\pm}^{*}\to\hat{\Sigma}^{\delta}$ denote the
inclusions. These induce maps of sections with compact support in
$\hat{\Sigma}_{\pm}^{*}$,
$\pi_{\pm,*}:\,\Omega^{0}_{c}(\hat{\Sigma}_{\pm}^{*},u_{\pm}^{*}TX)\to\Omega^{0}(\hat{\Sigma}^{\delta},u^{\delta,*}TX).$
Define $R_{\delta}(\xi)=\xi^{\delta}$ where
$\xi^{\delta}=\pi_{+,*}\beta_{\rho,\delta}(\xi_{+}-\xi_{+}(w_{+}))+\pi_{-,*}\beta_{\rho,\delta}(\xi_{-}-\xi_{-}(w_{-}))+\xi(w)$
for $\kappa_{\pm}(z)\in B_{|\delta|^{1/2}\rho^{2}}(0)$ and $\xi^{\delta}=\xi$
otherwise. Here $x=\xi^{\pm}(w^{\pm})$ is the value of $\xi$ at the node.
Define $T_{\delta}:=(I\times R_{\delta})Q_{\delta}(I\times K_{\delta}).$ That
is, if $(a,\xi)=Q_{\delta}K_{\delta}(\phi,\psi,\eta)$ then
$T_{\delta}(\phi,\psi,\eta)=(a,\xi^{\delta}).$ The map $T_{\delta}$ is the
required approximate right inverse. The difference
$(\tilde{D}_{\delta}T_{\delta}-I)(\phi,\psi,\eta)$ is the sum of terms
${\operatorname{d}}_{A,u^{\delta}}(a,\xi^{\delta})-\phi,\quad{\operatorname{d}}_{A,u_{0}}^{*}(a,\xi^{\delta})-\psi,\quad
D_{A,u^{\delta}}(a,\xi^{\delta})-\eta.$
By definition ${\operatorname{d}}_{A,u_{0}^{\delta}}(a,\xi)=\phi$, so the
first difference has contributions involving the difference between
$u_{0}^{\delta}$ and $u^{\delta}$, and between $\xi^{\delta}$ and $\xi$. But
since
${\operatorname{d}}_{A,u}(a,\xi)={\operatorname{d}}_{A}a+u^{*}L_{\xi_{X}}P(\Phi)$,
these terms are zeroth order in $u,\xi$,
$\|(u^{\delta})^{*}L_{\xi^{\delta}_{X}}P(\Phi)-u_{0}^{\delta}L_{\xi_{X}}P(\Phi)\|_{0,3}\leq
C|\delta|\|\xi\|_{C^{0}};$
that is, a constant times the area of $\pi_{-,*}(\hat{\Sigma}_{-}^{*})$, which
goes to zero as $\delta$ does. A similar discussion holds for the second
difference. The third difference has terms arising from the cutoff function
and the term $a_{X}^{0,1}(u)$ on the bubble region
$|\kappa_{+}(z)|\leq|\delta|^{1/2}$. We have
$\|a_{X}^{0,1}(u)\|_{0,3,\delta}\leq\|a_{X}^{0,1}(u)\|_{0,3}\leq C|\delta|.$
since $p\geq 2$; for $p=2$ the $W^{0,3,\delta}$ and $W^{0,3}$ norms are the
same, by conformal invariance; for $p>2$ the $0,3$-norm is strictly greater.
The term involving the derivative of the cutoff function satisfies
$\|{\operatorname{d}}\beta_{\rho,\delta}(\xi-\xi(w))\|_{0,3,\delta}\leq
c\log(\rho^{2})^{-2/3}\|(\xi-\xi(w))\|_{1,3}$
by Lemma 3.3.5, and so vanishes in the limit $\rho\to\infty$. Using the
uniform bound on $Q_{\delta}$, the total difference is bounded by
$C(\log(\rho^{2})^{-2/3}+|\delta|)\|(\phi,\psi,\eta)\|$, and so vanishes in
the limit $\delta\to 0,\rho\to\infty,|\delta|\rho^{2}\to 0$. The uniform bound
on $T_{\delta}$ follows from the uniform bound on $Q_{\delta}$ and the cutoff
and extension operators. ∎
Step 3: Uniform quadratic estimate
###### Proposition 4.2.8.
There exist constants $c,C>0$ such that if $\|\xi\|_{1,3,\delta}<c$,
$|\delta|<c,\rho>1/c$ and $|\delta|\rho^{4}<c$ then the map
$\mathcal{F}^{\delta}_{A,{u}}$ satisfies a quadratic bound
$\|D\mathcal{F}_{A,{u}}^{\delta}(a_{1},\xi_{1})-\tilde{D}_{A,{u}^{\delta}}(a_{1},\xi_{1})\|_{\delta}\leq
C\|a,\xi\|_{\delta}\|a_{1},\xi_{1}\|_{\delta}.$
###### Proof.
The norm of the non-linear part of the curvature $\|[a,a_{1}]\|_{{0,3}}$ is
bounded by Sobolev multiplication. The other term appearing in the first
vortex equation satisfies
$\|\exp_{{u}^{\delta}}(\xi_{0})^{*}L_{\xi_{1,X}}P(\Phi)-({u}^{\delta})^{*}L_{\xi_{1,X}}P(\Phi)\|_{{0,3}}\leq
C\|\xi_{0}\|_{1,3,\delta}\|\xi_{1}\|_{1,3,\delta}$
for some constant $C$ independent of $\delta$, using that $W^{1,3,\delta}$
norm controls the $W^{0,3}$ norm uniformly. The non-linear terms in the
Cauchy-Riemann equation are estimated as in Theorem 4.2.8 and [15, Section
3.5, Lemma 10.3.1]; note that we are fixing the complex structure on $\Sigma$,
which avoids the more complicated analysis we gave in the previous section.
The second vortex equation also involves a term of mixed type
$\Psi_{u}(\xi+\xi_{1})^{-1}(a_{1})_{X}^{0,1}(\exp_{u}(\xi+\xi_{1}))-\Psi_{u}(\xi)^{-1}(a_{1})_{X}^{0,1}(\exp_{u}(\xi)).$
It follows from uniform Sobolev embedding that this difference has
$0,3,\delta$-norm bounded by $C\|a_{1}\|_{{1,3}}\|\xi_{1}\|_{1,3,\delta}$ for
some constant $C$ independent of $\delta$. ∎
Step 4: Implicit Function Theorem
###### Theorem 4.2.9.
Let $(A,{u})$ be a regular stable nodal vortex of combinatorial type $\Gamma$.
There exist constants $\epsilon_{0},\epsilon_{1}>0$ such that for every
$(a,{\xi},\delta)\in\operatorname{Def}_{\Gamma}(A,{u})$ with norm less than
$\epsilon_{0}$, there exists a unique $(\phi,\psi,\eta)$ of norm less than
$\epsilon_{1}$ such that if $(a_{1},\xi_{1})=Q_{\delta}(\phi,\psi,\eta)$ then
$(A+a_{0}+a_{1},\exp_{{u}^{\delta}}({\xi}_{0}^{\delta}+\xi_{1}))$ is a
symplectic vortex in Coulomb gauge with respect to $(A,{u}^{\delta})$. The
solution depends smoothly on $a_{0},{\xi}_{0}$, and transforms equivariantly
the action of $\mathcal{G}(P)_{A,{u}}$ on
$\operatorname{Def}_{\Gamma}(A,{u})$.
###### Proof.
Uniform error and quadratic estimates are those for
$\mathcal{F}_{A,{u}}^{\delta}$ in Lemmas 4.2.4, 4.2.7, and 4.2.8, in a
uniformly bounded neighborhood of $0$ in $\operatorname{Def}_{\Gamma}(A,{u})$.
Then the first claim is an application of the quantitative version of the
implicit function theorem (see for example [15, Appendix A.3]). Equivariance
follows from uniqueness of the solution given by the implicit function
theorem, since the map $\mathcal{F}^{D,\delta}_{A,{u}}$ is equivariant for the
action of $\mathcal{G}(P)_{A,{u}}$. ∎
Step 5: Rigidification
As in the case of holomorphic maps in the previous section, there is a more
natural way of parametrizing nearby symplectic vortices which involves
examining the intersections of the sections with submanifolds of $P(X)$, and
framings induced by parallel transport. First we study the differentiability
of the evaluation maps. The gluing construction of the previous step gives
rise to a deformation $(A_{S},{u}_{S})$ of $(A,{u})$ with parameter space a
neighborhood $S$ of $0$ in $\operatorname{Def}(A,{u})$, and so a map
$S\to\overline{M}_{n}(\Sigma,X),\ s\mapsto(\hat{\Sigma}_{s},A_{s},{u}_{s})$
Consider the map
(45) $\operatorname{ev}:(\hat{\Sigma}-U)\times S\to
P(X),\quad(z,s)\mapsto{u}_{s}(z).$
###### Proposition 4.2.10.
The map $\operatorname{ev}$ of (45) is $C^{1}$ for the family constructed by
gluing in Theorem 4.2.9 using the exponential gluing profile.
###### Proof.
We denote by ${u}_{S}^{\operatorname{pre}}:\hat{\Sigma}_{S}\to X$ the family
obtained by pre-gluing only, that is, omitting the step which solves for an
exact solution. We denote by $\operatorname{ev}^{\operatorname{pre}}$ the map
$\operatorname{ev}^{\operatorname{pre}}:(\hat{\Sigma}-U)\times S\to
P(X),\quad(z,s)\mapsto{u}_{s}^{\operatorname{pre}}(z).$
This map is independent of the gluing parameters, and is therefore $C^{1}$. We
write $s=(a_{0},\xi_{0})$ and
$A_{s}=A+a_{0}+a_{1},{u}_{s}=\exp_{{u}^{{\operatorname{pre}}}_{s}}(\xi_{0}^{\delta}+\xi_{1})$.
The corrections $a_{1},\xi_{1}$ depend smoothly on $a_{0},\xi_{0}$, by the
implicit function theorem, and so $\xi_{1}(z)$ depends smoothly on
$a_{0},\xi_{0}$. Next we take the derivative with respect to the gluing
parameter. Let $(A,{u})$ be a nodal symplectic vortex, $(A,{u}^{\delta})$ the
pre-glued pair (we omit the parameter $\rho$ controlling the diameter of the
gluing region from the notation) and consider the equation
$\mathcal{F}_{A,{u}^{\delta}}(a_{0}+a_{1},\xi_{0}^{\delta}+\xi_{1})=0.$ Let
$\tilde{D}_{\delta}$ denote the derivative of $\mathcal{F}_{A,{u}^{\delta}}$.
Differentiating with respect to $\delta$ gives
$\tilde{D}_{\delta}\left(\frac{d}{d\delta}a_{1},D\exp_{{u}^{\delta}}(\xi_{0}^{\delta};0,\frac{d}{d\delta}\xi_{1})\right)=-\tilde{D}_{\delta}\left(0,D\exp_{{u}^{\delta}}(\xi_{0}^{\delta};\frac{d}{d\delta}{u}^{\delta},\frac{d}{d\delta}\xi_{0}^{\delta})\right).$
The same arguments as in the proof of Theorem 3.3.11 show that there exists a
constant $C>0$ such that the right hand side is bounded in norm by
$Ce^{-1/\delta}$. On the other hand, the norm of the left-hand side
$\tilde{D}_{\delta}$ is uniformly bounded from below in terms of the norm of
$\frac{d}{d\delta}a_{1},\frac{d}{d\delta}\xi_{1}$, by the quadratic estimates.
It follows that $(\frac{d}{d\delta}a_{1},\frac{d}{d\delta}\xi_{1})$ is also
bounded in norm by $Ce^{-1/\delta}$. Hence $\lim_{\delta\to
0}\partial_{\delta}\operatorname{ev}=0$. It follows that $D\operatorname{ev}$
has a continuous limit as ${\delta}\to 0$. ∎
Choose a path $\gamma:[0,1]\to\Sigma$ in the principal component and an
element $\phi_{0}\in P_{\gamma(0)}$. Let $\tau_{\gamma}(A):P_{\gamma(0)}\to
P_{\gamma(1)}$ denote parallel transport. By an $m$-framed family of marked
curves, we mean a family of curves together with an $m$-tuple of points in
$P$. Given a family $(\hat{\Sigma}_{S},A_{S},{u}_{S})$ of gauged holomorphic
maps over a parameter space $S$, a collection of codimension two submanifolds
${Y}=(Y_{1},\ldots,Y_{k})$ in $P(X)$, and a collection of paths
${\gamma}=(\gamma_{1},\ldots,\gamma_{l})$ with the same initial point $y_{0}$
to $y_{j},j=1,\ldots,l$, define a family of marked, framed curves
$\hat{\Sigma}_{S}^{{Y},{u},{\gamma},A}\to S$ by requiring that the additional
marked points $z_{n+i}$ map to $Y_{i}$, and the framings are given by parallel
transport along the paths $\gamma_{i}$.
###### Definition 4.2.11.
The data ${Y},{\gamma},A,{u}$ are compatible if
1. (a)
each $Y_{j}$ intersects $u_{j}$ transversally in a single point
$z_{j}\in\hat{\Sigma}$;
2. (b)
if $(a,\xi)\in\operatorname{ker}\tilde{D}_{A,{u}}$ satisfies $\xi(z_{n+j})\in
T_{{u}(z_{n+j})}P(X)$ for $j=1,\ldots,k$ and $D_{A}\tau_{\gamma_{i}}(a)=0$ for
$i=1,\ldots,l$ then $(a,\xi)=0$.
3. (c)
the curve $\hat{\Sigma}$ marked with the additional points
$z_{n+1},\ldots,z_{n+k}$ is stable.
4. (d)
if some automorphism of $(\hat{\Sigma},{u})$ maps $z_{i}$ to $z_{j}$ then
$Y_{i}$ is equal to $Y_{j}$.
The second condition says that there are no infinitesimal deformations which
do not change the positions of the extra markings or framings.
###### Proposition 4.2.12.
Let $(A,{u})$ be a parametrized regular stable nodal vortex, and
$(A_{S},{u}_{S})\to S$ the stratified-smooth universal deformation constructed
by the gluing construction. There exists a collection $({Y},{\gamma})$
compatible with $({u},A)$. Furthermore, if $({Y},{\gamma})$ is compatible with
$(A,{u})$, then the rigidified family
$\hat{\Sigma}_{S}^{{Y},{u},{\gamma},A}\to S$ of (26) is a stratified-smooth
deformation of the marked-curve-with-framings
$\hat{\Sigma}^{{Y},{u},{\gamma},A}$ which defines an immersion of $S$ into the
parameter space for the universal deformation of the central fiber.
###### Proof.
First we show the existence of a compatible collection. Suppose that the
second condition is not satisfied for some $(a,\xi)$. Suppose first that
$\xi\neq 0$. Let $z_{n+1}$ be a point with $\xi(z_{n+1})\neq 0$, and choose a
codimension two submanifold $Y_{n+1}$ transverse to $u$ near $u(z_{n+1})$, and
such that $TY_{n+1}$ does not contain $\xi(z_{n+1})$. Adding $Y_{n+1}$ to the
list of submanifolds decreases the dimension of the space of $(a,\xi)$
satisfying the condition in (b) by at least one. Repeating this process, we
may assume that the only elements satisfying the condition in (b) have
$\xi=0$. Suppose that $\xi$ is zero, so that $a$ is necessarily non-zero.
Choose an additional marked point $y_{l+1}$ and a path $\gamma_{l+1}$ from the
base point $y_{0}$ to $y_{l+1}$ such that the derivative of the parallel
transport over $\gamma$ with respect to $a$ over is non-zero. Appending
$\gamma_{l+1}$ to the list of path decreases the dimension of $(a,\xi)$
satisfying the condition in (b) by at least one. Hence the process stops after
finitely many steps, after which the kernel is trivial. The proof of the
second claim is similar to Proposition 3.3.15 and will be omitted. ∎
Step 6: Surjectivity
We show that any nearby vortex appears in the family constructed above. First,
we show:
###### Proposition 4.2.13.
Let $(A,{u})$ be a regular strongly stable symplectic vortex. There exists a
constant $\epsilon>0$ such that if
$(A_{1},u_{1})=(A+a,\exp_{{u}^{\delta}}(\xi))$ with
$\|a\|_{1,3}+\|\xi\|_{1,3,\delta}\leq\epsilon$ then after gauge transformation
we have
$(A_{1},u_{1})=(A+a_{0}+a_{1},\exp_{{u}^{\delta}}(\xi_{0}^{\delta}+\xi_{1}))$
for some $(a_{0},\xi_{0})\in\operatorname{ker}(\tilde{D}_{A,{u}})$ and
$(a_{1},\xi_{1})$ in the image of $Q_{\delta}$.
###### Proof.
We claim that for some constant $C>0$, we have
$(a,\xi)=(a_{0},\xi_{0}^{{\delta}})+(a_{1},\xi_{1})$ for some
$(a_{0},\xi_{0})\in\operatorname{ker}\tilde{D}_{A,{u}}$ and
$(a_{1},\xi_{1})\in\operatorname{Im}\tilde{D}_{A,{u}^{{\delta}}}^{*}$ with
norm $\|(a_{1},\xi_{1})\|\leq C\|(a_{1},\xi_{1})\|.$ Given the claim, the
proposition follows by the uniqueness statement of the implicit function
theorem. For any $c>0$ there exists $\delta_{0}$ such that for
$\delta<\delta_{0}$,
$\|\tilde{D}_{A,{u}^{{\delta}}}(a_{0}^{{\delta}},{\xi}_{0}^{{\delta}})\|\leq
c\|(a_{0}^{{\delta}},{\xi}_{0}^{{\delta}})\|$
by estimates similar to those of Lemma 4.2.4. Thus the image of
$\operatorname{ker}\tilde{D}_{A,{u}}$ is transverse to
$\operatorname{Im}\tilde{D}_{A,{u}^{\delta}}^{*}$, for $\delta$ sufficiently
small, since it meets $\operatorname{Im}\tilde{D}_{A,{u}^{\delta}}^{*}$
trivially and projects isomorphically onto
$\operatorname{ker}\tilde{D}_{A,{u}^{\delta}}$, by gluing for indices, as in
[23, Theorem 2.4.1]. The claim then follows from the inverse function theorem.
∎
Given a strongly stable symplectic vortex $(A,{u})$ with stable domain
$\hat{\Sigma}$, let $(A_{S},{u}_{S})$ be the family given by the gluing
construction above. Otherwise, if ${Y}$ is not stable, let
${Y}=(Y_{1},\ldots,Y_{l})$ be a collection of codimension two submanifolds of
$P(X)$, and consider the family $(A,{u}^{{Y}})$ with additional marked points
given by requiring that the additional marked points $z_{n+i}$ map to $Y_{i}$.
Let $(A_{S},{u}_{S})$ denote the family obtained by applying the gluing
construction for $(A_{S},{u}_{S}^{{Y}})$, and then forgetting the additional
marked points.
###### Lemma 4.2.14.
Suppose that $(A_{i},u_{i})$ Gromov converges to $(A,u)$. After a sequence of
gauge transformations, for any $\epsilon$, there exists $i_{0}$ such that if
$i>i_{0}$ then there exists $\delta,(a_{i},\xi_{i})$ satisfying
$(A_{i},u_{i})=(A+a_{i},\exp_{{u}^{\delta}}(\xi_{i}))$ with
$\|a_{i}\|_{1,3}+\|\xi_{i}\|_{1,3,\delta}\leq\epsilon$.
###### Proof.
By definition of Gromov convergence, after gauge transformation $A_{i}$
$C^{0}$-converges to $A$ and converges uniformly in all derivatives on the
complement of the bubbling set [17]. The exponential decay estimate [17, Lemma
A.2.2] show that $u_{i}$ converges to $u$ on the complement of the nodes,
uniformly in all derivatives on compact sets, and whose derivative on the
gluing region is uniformly bounded in the $\delta$-dependent metric. It
follows that $u_{i}=\exp_{u^{\delta}}(\xi_{i})$ for some $\delta$ and
$\xi_{i}\in\Omega^{0}(\Sigma^{\delta},(u^{\delta})T^{\operatorname{vert}}P(X))$
with $\|\xi_{i}\|_{1,3,\delta}<\epsilon$. To obtain the improved convergence
for the connection, note that $F_{A_{i}}+(u_{i})^{*}P(\Phi)=0$ and the
corresponding equations for the limit $(A,u)$ imply that
$F_{A_{i}}-F_{A}={\operatorname{d}}_{A}(A_{i}-A)-(A-A_{i})\wedge(A-A_{i})=(u_{i})^{*}P(\Phi)-u^{*}P(\Phi).$
Since $u_{i}^{*}P(\Phi)$ is bounded and converges to $u^{*}P(\Phi)$ on the
complement of the bubbling set, and $A_{i}$ converges to $A$ in $C^{0}$ hence
$W^{0,3}$, the right hand side converges to $0$ in $W^{0,3}$ as $i\to\infty.$
After gauge transformation we may assume that
${\operatorname{d}}_{A}^{*}(A-A_{i})=0$. Then the elliptic estimate for the
operator ${\operatorname{d}}_{A}\oplus{\operatorname{d}}_{A}^{*}$ implies that
$A-A_{i}$ converges to zero in $W^{1,3}$. ∎
###### Corollary 4.2.15.
$(A_{S},{u}_{S})$ is a stratified-smooth versal deformation of
$(\hat{\Sigma},A,{u})$.
###### Proof.
Proposition 4.2.13 implies that any family $(A_{S^{1}}^{1},{u}_{S^{1}}^{1})$
is obtained by pull-back from $(A_{S},{u}_{S})$, in case $\hat{\Sigma}$ is
stable, or obtained from the family obtained by adding the marked points
mapping to submanifolds, in general. ∎
Step 7: Injectivity
We show that any nearby vortex appears once in our family, up to the action of
$\operatorname{Aut}(A,u)$; this is part of the following:
###### Theorem 4.2.16.
Any family $(A_{S},{u}_{S})$ constructed by gluing using the exponential
gluing profile is a strongly universal stratified-smooth deformation of
$(A,{u})$.
###### Proof.
Let $\overline{Z}_{n}(P,X)$ denote the moduli space of marked symplectic
vortices up to equivalences that involve only the identity gauge
transformation, so that
$\overline{M}_{n}(P,X)=\overline{Z}_{n}(P,X)/\mathcal{G}(P).$ Let $(A,{u})$ be
a stable marked vortex, and $W_{A,{u}}$ a slice for the gauge group action on
$\overline{Z}(P,X)$, so that
$W_{A,{u}}/\mathcal{G}(P)_{A,{u}}\to\overline{M}_{n}(P,X)$
is a homeomorphism onto its image. Let $\operatorname{Aut}_{0}(A,{u})$ denote
the subgroup of $\operatorname{Aut}(A,u)$ acting trivially on $P$, so that
$\mathcal{G}(P)_{A,{u}}=\operatorname{Aut}(A,u)/\operatorname{Aut}_{0}(A,{u})$
is the stabilizer of $(A,{u})$ under the gauge action. Let $(A_{S},{u}_{S})$
denote a universal deformation of $(A,{u})$ constructed by gluing using the
exponential gluing profile. We claim that the map
(46) $S/\operatorname{Aut}_{0}(A,{u})\to
W_{A,{u}},\quad[s]\mapsto[A_{s},{u}_{s}]$
is an injection. Indeed, rigidification produces an injection
(47)
$S/\operatorname{Aut}_{0}(A,{u})\to\overline{M}_{n+k,l}(\Sigma)/\operatorname{Aut}_{0}(A,{u}),\quad[s]\mapsto[\Sigma^{A_{s},{u}_{s},{Y},{\gamma}}]$
where $\operatorname{Aut}_{0}(A,{u})$ acts by re-ordering the marked points.
Since this map factors through (46), the claim follows. If
$(A_{S^{1}},{u}_{S^{1}}^{1})$ is a family of symplectic vortices giving a
deformation of any fiber of $(A_{S},{u}_{S})$¡ then Corollary 4.2.15 together
with injectivity shows that this family is obtained by pull-back by some map
$S^{1}\to S$. Hence $(A_{S},{u}_{S})$ is a stratified-smooth strongly
universal deformation of $(A,{u})$. ∎
###### Theorem 4.2.17.
Let $X$ be a Hamiltonian $G$-manifold equipped with a compatible invariant
almost complex structure $J\in\mathcal{J}(X)^{G}$. The maps
(48) $S\to\overline{M}_{n}(\Sigma,X),\quad s\mapsto[A_{s},{u}_{s}]$
associated to the universal deformations constructed above equip the locus
$\overline{M}^{\operatorname{reg}}_{n}(\Sigma,X)$ of regular stable symplectic
vortices with the structure of a stratified-smooth orbifold. If the local
coordinates near the nodes are chosen compatibly and the gluing profile is the
exponential gluing profile, then the deformations provide
$\overline{M}^{\operatorname{reg}}_{n}(\Sigma,X)$ with the structure of a
$C^{1}$-orbifold.
###### Proof.
It suffices to show that the charts given by two sets ${Y}_{j},{\gamma}_{j}$
are compatible. Define ${Y}={Y}_{1}\cup{Y}_{2}$ and $m=m_{1}+m_{2}$ the total
number of extra points. Similarly let ${\gamma}$ be the union of
${\gamma}_{1}$ and ${\gamma_{2}}$ of total number $l=l_{1}+l_{2}$. The family
$\hat{\Sigma}^{{Y},{u},{\gamma},A}_{S}$ admits a proper étale forgetful map
$\hat{\Sigma}_{S}^{{Y},{u},{\gamma},A}\to\hat{\Sigma}_{S}^{{Y}_{j},{u},{\gamma}_{j},A},\quad
j=1,2$ whose fiber consists of the re-orderings of the points for ${Y}$
induced by automorphisms of $\operatorname{Aut}(A,{u})$ that fix the ordering
for ${Y}_{j}$. It follows that the corresponding charts are
$C^{1}$-compatible. ∎
###### Remark 4.2.18.
As discussed in Remark 3.3.22, the Theorem implies that if
$\overline{M}^{{\operatorname{reg}}}_{n}(\Sigma,X)$ is compact then it admits
a (non-canonical) smooth structure.
Let $\overline{M}_{n}(\Sigma)$ denote the moduli space of stable maps to
$\Sigma$ with homology class $[\Sigma]$, $n$ markings and genus that of
$\Sigma$, or in other words, parametrized stable curves with principal
component isomorphic to $\Sigma$. Forgetting the pair $(A,u)$ gives a
forgetful morphism
$\overline{M}^{{\operatorname{reg}}}_{n}(\Sigma,X)\to\overline{M}_{n}(\Sigma).$
Using the differentiable structure defined above, the evaluation maps are
differentiable but unfortunately the forgetful morphisms are not, unless one
uses a different gluing profile for the moduli space of vortices with one less
marking. More precisely, the forgetful morphism
$\overline{M}^{\operatorname{reg}}_{n}(\Sigma,X)\to\overline{M}_{n}(\Sigma)$
is continuous and $C^{1}$ near any pair $(A,{u})$ whose domain is stable as an
element of $\overline{M}_{n}(\Sigma)$, and a submersion near the boundary of
$\overline{M}_{n}(\Sigma)$. For the standard smooth structure on
$\overline{M}_{n}(\Sigma)$, the forgetful morphism
$\overline{M}^{\operatorname{reg}}_{n}(\Sigma,X)\to\overline{M}_{n}(\Sigma)$
is smooth.
The gluing construction has various parametrized versions. For example, in
[11] we consider a moduli space of polystable polarized vortices, which
consist of a vortex together with a lift of the connection to the Chern-Simons
line bundle. In each of these cases one applies the implicit function theorem
using the linearized operator for the parametrized problem to prove that any
parametrized regular polystable vortex has a strongly universal deformation in
the parametrized sense. In particular, any regular polystable polarized vortex
has a strongly universal deformation etc.
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* [13] H. Hofer, K. Wysocki, and E. Zehnder. $sc$-smoothness, retractions and new models for smooth spaces. Discrete and Continuous Dyn. Systems, 28:665–788, 2010.
* [14] Eleny-Nicoleta Ionel and Thomas H. Parker. Relative Gromov-Witten invariants. Ann. of Math. (2), 157(1):45–96, 2003.
* [15] Dusa McDuff and Dietmar Salamon. $J$-holomorphic curves and symplectic topology, volume 52 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2004.
* [16] Ignasi Mundet i Riera. Hamiltonian Gromov-Witten invariants. Topology, 42(3):525–553, 2003.
* [17] A. Ott. Non-local vortex equations and gauged Gromov-Witten invariants. PhD thesis, ETH Zurich, 2010.
* [18] Richard S. Palais. $C^{1}$ actions of compact Lie groups on compact manifolds are $C^{1}$-equivalent to $C^{\infty}$ actions. Amer. J. Math., 92:748–760, 1970.
* [19] Joel W. Robbin, Yongbin Ruan, and Dietmar A. Salamon. The moduli space of regular stable maps. Math. Z., 259(3):525–574, 2008.
* [20] Joel W. Robbin and Dietmar A. Salamon. A construction of the Deligne-Mumford orbifold. J. Eur. Math. Soc. (JEMS), 8(4):611–699, 2006.
* [21] Yongbin Ruan and Gang Tian. Higher genus symplectic invariants and sigma models coupled with gravity. Invent. Math., 130(3):455–516, 1997.
* [22] Bernd Siebert. Gromov-Witten invariants of general symplectic manifolds. arXiv:dg-ga/9608005.
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|
arxiv-papers
| 2008-11-22T20:01:20 |
2024-09-04T02:48:58.911635
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Eduardo Gonzalez and Chris Woodward",
"submitter": "Chris T. Woodward",
"url": "https://arxiv.org/abs/0811.3711"
}
|
0811.3722
|
THE TORSION OF HOMOLOGY GROUPS OF $M(E,I)$-SETS
Lopatkin V.E
## Introduction
In this paper, following [1, 2], we consider the homology groups of right
pointed sets over a partially commutative monoid $M(E,I)$.
Definition. Let $E$ be a set and $I\subseteq E\times E$ an irreflexive and
summetric relation. A monoid guven by a set of generators $E$ and relations
$ab=ba$ for all $(a,b)\in I$ is called free partially commutative [3] and
denoted by $M(E,I)$. If $(a,b)\in I$ then the members $a,b\in E$ are said to
be commuting generators.
The motivation for the research of homology groups of the $M(E,I)$-sets came
from a desire to find topology’s invariants for asynchronous transition
systems. M. Bednarczyk [4] has introduced _asynchronous transition systems_ to
the modeling the concurrent processes. In [5] it was proved that the category
of asynchronous transition systems admits an inclusion into the category of
pointed sets over free partially commutative monoids. Thus asynchronous
transition systems may be considered as $M(E,I)$-sets.
Follow [2, Example 3.1] homology groups of $M(E,I)$-sets may have torsion
subgroups. The aim of this paper is to study torsion subgroups of homology
groups of followings $M(E,I)$-sets; denote by $X_{n}^{\bullet}$ a right
pointed $M(E,I)$-set, where a set $X_{n}^{\bullet}$ has $n+2$ elements
$X_{n}^{\bullet}=\\{x_{0},x_{1},\ldots,x_{n},*\\}$ here $n\geqslant 1$. Denote
by $X_{-1}^{\bullet}=*$ the base point. An action $X_{n}^{\bullet}\times
M(E,I)\to X_{n}^{\bullet}$ generated by formuls; $x_{0}\cdot e=x_{1}$,
$x_{1}\cdot e=x_{2}$,$\dots$, $x_{n-1}\cdot e=x_{n}$, $x_{n}\cdot e=*$,
$*\cdot e=*$, for all $e\in E$. This $M(E,I)$-set, we’ll denote by
$X_{n}^{\bullet}$.
For any $M(E,I)$-set $X_{n}^{\bullet}$ let us denote by
$(M(E,I)/X_{n}^{\bullet})^{op}$, or shortly $\mathscr{K}_{*}(X_{m}^{\bullet})$
a category which objects are elements of pointed set $X_{n}^{\bullet}$ and
morphisms are triples $(x,\mu,x^{\prime})$ with $x,x^{\prime}\in
X_{n}^{\bullet}$ and $\mu\in M(E,I)$ satisfying to $x\cdot\mu=x^{\prime}$.
Sometimes we denote this triples by $x\xrightarrow{\mu}x^{\prime}$.
Let $\mathscr{C}$ be a small category. Denote by
$\Delta_{\mathscr{C}}\mathbb{Z}:\mathscr{C}\to\mathrm{Ab}$, or shortly
$\Delta\mathbb{Z}$, the functor which has constant values
$\Delta\mathbb{Z}(c)=\mathbb{Z}$ at $c\in\mathrm{Ob}\,\mathscr{C}$ and
$\Delta\mathbb{Z}(\alpha)=1_{\mathbb{Z}}$ at
$\alpha\in\mathrm{Mor}\,\mathscr{C}$.
For any functor $F:\mathscr{C}\to\mathrm{Ab}$, here $\mathscr{C}$ is a small
category, denote by $\varinjlim_{k}^{\mathscr{C}}F$ values of the left
satellites of the colimit
$\varinjlim^{\mathscr{C}}:\mathrm{Ab}^{\mathscr{C}}\to\mathrm{Ab}$. It is well
[6, Prop. 3.3, Aplication 2] that thete exists an isomorphim of left
satellites of the colimit
$\varinjlim^{\mathscr{C}}:\mathrm{Ab}^{\mathscr{C}}\to\mathrm{Ab}$ and the
functors
$H_{n}(C_{*}(\mathscr{C},-)):\mathrm{Ab}^{\mathscr{C}}\to\mathrm{Ab}$. Since
the category $\mathrm{Ab}^{\mathscr{C}}$ has enough projectives, these
satellites are natural isomorphic to the left derived functor of
$\varinjlim^{\mathscr{C}}:\mathrm{Ab}^{\mathscr{C}}\to\mathrm{Ab}$. Denote the
values $H_{n}(C_{*}(\mathscr{C},-))$ of satellites at
$F\in\mathrm{Ab}^{\mathscr{C}}$ by $\varinjlim_{n}^{\mathscr{C}}F$.
We’ll be consider any monoid as the small category with the one object. This
exert influence on our terminology. In particular a right $M$-set $X$ will be
considered and denoted as a functor $X:M^{op}\to\mathrm{Set}$ (the value of
$X$ at the unique object will be denoted by $X(M)$ or shortly $X$.) Morphisms
of right $M$-sets are natural transformations.
## 1 General theorems
Let us show that there is following
###### Theorem 1.1
Let $X_{m}^{\bullet}$ and $X_{m+1}^{\bullet}$ are $M(E,I)$-sets, here
$m\geqslant-1$, then there exist an isomorphism
$H_{k}(X_{m+1}^{\bullet})\cong H_{k}(X_{m}^{\bullet})\oplus
H_{k-1}(E,\mathfrak{M}),$
here $k\geqslant 1$ and $H_{*}(E,\mathfrak{)}$ is homology groups of
simplicial schema $(E,\mathfrak{M})$.
Proof. Let us consider categories $\mathscr{K}_{*}(X_{m}^{\bullet})$ and
$\mathscr{K}_{*}(X_{m+1}^{\bullet})$. By
$\mathfrak{In}:\mathscr{K}_{*}(X^{\bullet}_{m})\to\mathscr{K}_{*}(X^{\bullet}_{m+1})$
denote the embedding functor. The functor $\mathfrak{In}$ is defined by
formulas $\mathfrak{In}(x_{i})=x_{i+1}$ for all $i\in\\{0,1,\ldots\,m\\}$ on
objects, and
$\mathfrak{In}(x_{j}\xrightarrow{e}x_{j+1})=x_{j+1}\xrightarrow{e}x_{j+2}$,
$\mathfrak{In}(*\xrightarrow{e}*)=*\xrightarrow{e}*$ for all
$j\in\\{0,\dots,m-1\\}$ at morphisms. Let us construct a left inverse functor
$\mathfrak{Re}:\mathscr{K}_{*}(X_{m+1}^{\bullet})\to\mathscr{K}_{*}(X_{m}^{\bullet})$
to functor $\mathfrak{In}$. We define the functor $\mathfrak{Re}$ by formulas
$\mathfrak{Re}(x_{\xi})=x_{{\xi}-1}$, $\mathfrak{Re}(x_{0})=x_{0}$,
$\mathfrak{Re}(*)=*$ for all ${\xi}\in\\{1,\dots,m+1\\}$ on objects and
$\mathfrak{Re}(x_{\xi}\xrightarrow{e}x_{{\xi}+1})=x_{{\xi}-1}\xrightarrow{e}x_{{\xi}}$,
$\mathfrak{Re}(x_{0}\xrightarrow{e}x_{1})=x_{0}\xrightarrow{1_{M(E,I)}}x_{0}$
and $\mathfrak{Re}(*\xrightarrow{e}*)=*\xrightarrow{e}*$ at morphisms. The
construction of functors $\mathfrak{In}$ and $\mathfrak{Re}$ is shown in
figure 1.
---
$\textstyle{x_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{In}}$$\scriptstyle{e_{1}}$$\scriptstyle{\vdots}$$\scriptstyle{e_{s}}$$\textstyle{x_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{In}}$$\scriptstyle{e_{1}}$$\scriptstyle{\vdots}$$\scriptstyle{e_{s}}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{1}}$$\scriptstyle{\vdots}$$\scriptstyle{e_{s}}$$\textstyle{x_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{In}}$$\scriptstyle{e_{1}}$$\scriptstyle{\vdots}$$\scriptstyle{e_{s}}$$\textstyle{{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vdots}$$\scriptstyle{e_{1}}$$\scriptstyle{e_{s}}$$\scriptstyle{\mathfrak{In}}$$\textstyle{x_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{Re}}$$\scriptstyle{e_{1}}$$\scriptstyle{\vdots}$$\scriptstyle{e_{s}}$$\textstyle{x_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{Re}}$$\scriptstyle{e_{1}}$$\scriptstyle{\vdots}$$\scriptstyle{e_{s}}$$\textstyle{x_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{Re}}$$\scriptstyle{e_{1}}$$\scriptstyle{\vdots}$$\scriptstyle{e_{s}}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{1}}$$\scriptstyle{\vdots}$$\scriptstyle{e_{s}}$$\textstyle{x_{m+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{Re}}$$\scriptstyle{e_{1}}$$\scriptstyle{\vdots}$$\scriptstyle{e_{s}}$$\textstyle{{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vdots}$$\scriptstyle{e_{1}}$$\scriptstyle{e_{s}}$$\scriptstyle{\mathfrak{Re}}$
Figure 1: Constructions of functor
$\mathfrak{In}:\mathscr{K}_{*}(X^{\bullet}_{m})\to\mathscr{K}_{*}(X^{\bullet}_{m+1})$
and functor
$\mathfrak{Re}:\mathscr{K}_{*}(X^{\bullet}_{m+1})\to\mathscr{K}_{*}(X^{\bullet}_{m})$,
here the cardinality of $E$ is $s$.
Denote by $\Delta_{m}\mathbb{Z}$ the functor
$\Delta_{\mathscr{K}_{*}(X^{\bullet}_{m})}\mathbb{Z}:\mathscr{K}_{*}(X^{\bullet}_{m})\to\mathrm{Ab}$
for any $m\geqslant-1$. From [2, Theorem 3.1] it follows that homology groups
$\varinjlim_{n}^{\mathscr{K}_{*}(X^{\bullet}_{m})}\Delta_{m}\mathbb{Z}$
isomorphic to homology groups of differential object
$\left({\mathstrut}_{m}\mathscr{C}_{*}(\mathscr{K}_{*}(X^{\bullet}_{m}),\Delta_{m}\mathbb{Z}),{\mathstrut}_{m}d_{*}\right)$.
Since $\varinjlim_{k}^{(-)}\Delta\mathbb{Z}$ is exact with respect to first
argument, we have homomorphisms;
$\mathfrak{In}_{k}:\varinjlim_{k}^{\mathscr{K}_{*}(X^{\bullet}_{m})}\Delta_{m}\mathbb{Z}\to\varinjlim_{k}^{\mathscr{K}_{*}(X^{\bullet}_{m+1})}\Delta_{m+1}\mathbb{Z}$
and
$\mathfrak{Re}_{k}:\varinjlim_{k}^{\mathscr{K}_{*}(X^{\bullet}_{m+1})}\Delta_{m}\mathbb{Z}\to\varinjlim_{k}^{\mathscr{K}_{*}(X^{\bullet}_{m})}\Delta_{m}\mathbb{Z}$,
here $k\geqslant 0$. We have exact sequence of functors
$0\to\widetilde{\Delta_{m}\mathbb{Z}}\to\Delta_{m+1}\mathbb{Z}\to{\Delta_{m+1}\mathbb{Z}}/\widetilde{{\Delta_{m}\mathbb{Z}}}\to
0,$
here we’ve denote by $\widetilde{\Delta_{m}\mathbb{Z}}$ the functor
$\Delta_{m}\mathbb{Z}$ with zeros. This exact sequence we can show by
following diagram
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vdots}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{\mathbb{Z}}}$$\scriptstyle{\vdots}$$\scriptstyle{1_{\mathbb{Z}}}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{\mathbb{Z}}}$$\scriptstyle{\vdots}$$\scriptstyle{1_{\mathbb{Z}}}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{\mathbb{Z}}}$$\scriptstyle{\vdots}$$\scriptstyle{1_{\mathbb{Z}}}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{\mathbb{Z}}}$$\scriptstyle{\vdots}$$\scriptstyle{1_{\mathbb{Z}}}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vdots}$$\scriptstyle{1_{\mathbb{Z}}}$$\scriptstyle{1_{\mathbb{Z}}}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{\mathbb{Z}}}$$\scriptstyle{\vdots}$$\scriptstyle{1_{\mathbb{Z}}}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{\mathbb{Z}}}$$\scriptstyle{\vdots}$$\scriptstyle{1_{\mathbb{Z}}}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{\mathbb{Z}}}$$\scriptstyle{\vdots}$$\scriptstyle{1_{\mathbb{Z}}}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{\mathbb{Z}}}$$\scriptstyle{\vdots}$$\scriptstyle{1_{\mathbb{Z}}}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{\mathbb{Z}}}$$\scriptstyle{\vdots}$$\scriptstyle{1_{\mathbb{Z}}}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vdots}$$\scriptstyle{1_{\mathbb{Z}}}$$\scriptstyle{1_{\mathbb{Z}}}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vdots}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vdots}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vdots}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vdots}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vdots}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vdots}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{\ldots}$$\textstyle{0}$$\textstyle{0}$
In this diagram all vertical sequences are exact. Since the functor
${\mathstrut}_{m}\mathscr{C}_{*}(\mathscr{K}_{*}(X^{\bullet}_{m}),\Delta_{m}\mathbb{Z})$
is exact with respect to $\Delta_{m}\mathbb{Z}$, we obtain the long exact
sequence
$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\partial}_{k}}$$\textstyle{\varinjlim_{k}^{\mathscr{K}_{*}(X^{\bullet}_{m})}\Delta_{m}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{In}_{k}}$$\textstyle{\varinjlim_{k}^{\mathscr{K}_{*}(X^{\bullet}_{m+1})}\Delta_{m+1}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{Re}_{k}}$$\scriptstyle{\mathfrak{P}_{k}}$$\textstyle{\varinjlim_{k}^{\mathscr{K}_{*}(X^{\bullet}_{0})}\mathbb{Z}[x_{0}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\partial}_{k-1}}$$\textstyle{\ldots}$
From this long exact sequence we can make up the following shot exact sequence
$0\to\mathrm{Im}\,\mathfrak{In}_{k}\to\underrightarrow{\mathrm{lim}}_{k}^{\mathscr{K}_{*}(X_{m+1}^{\bullet})}\Delta_{m+1}\mathbb{Z}\to\mathrm{Im}\,\mathfrak{P}_{k}\to
0$
Since
$\mathrm{Im}\,\mathfrak{In}_{k}\cong\varinjlim_{k}^{\mathscr{K}_{*}(X^{\bullet}_{m})}\Delta_{m}\mathbb{Z}$,
we claim that
$\mathrm{Im}\,\mathfrak{P}_{k}\cong\varinjlim_{k}^{\mathscr{K}_{*}(X^{\bullet}_{0})}\mathbb{Z}[x_{0}]$.
Indeed, this follows from long exact sequence and homomorphism’s theorem;
$\mathrm{Im}\mathfrak{P}_{k}=\mathrm{Ker}\,\partial_{k-1},\quad\mathrm{Im}\,\partial_{k-1}=\mathrm{Ker}\,\mathfrak{In}_{k-1}=0$
And using homomorphism’s theorem, we get
$\mathrm{Ker}\,\partial_{k-1}\cong\varinjlim_{k}^{\mathscr{K}_{*}(X^{\bullet}_{0})}\mathbb{Z}[x_{0}]$.
Since there exist the homomorphism
$\mathfrak{Re}_{k}:\varinjlim_{k}^{\mathscr{K}_{*}(X^{\bullet}_{m+1})}\Delta_{m+1}\mathbb{Z}\to\varinjlim_{k}^{\mathscr{K}_{*}(X^{\bullet}_{m})}\Delta_{m}\mathbb{Z}$,
we see that shot exact sequence is split, thus we get
$\varinjlim_{k}^{\mathscr{K}_{*}(X^{\bullet}_{m+1})}\Delta_{m+1}\mathbb{Z}\cong\varinjlim_{k}^{\mathscr{K}_{*}(X^{\bullet}_{m})}\Delta_{m}\mathbb{Z}\oplus\varinjlim_{k}^{\mathscr{K}_{*}(X^{\bullet}_{0})}\mathbb{Z}[x_{0}]$
using [2, Theorem 3.1, Example 3.2], we’ll complete the proof of theorem.
From this thoerem, we get the following
###### Corollary 1.2
Let $X_{m}^{\bullet}$ and $X_{m+1}^{\bullet}$ are $M(E,I)$-sets, then there
exist a isomorphism
${\rm Tor\,}(H_{k}(X_{m+1}^{\bullet}))\cong{\rm
Tor\,}(H_{k}(X_{m}^{\bullet}))\oplus{\rm Tor\,}(H_{k}(X_{0}^{\bullet})),$
here $m\geqslant-1$ and $k\geqslant 1$.
For one-dimensional homology groups, we get the following
###### Theorem 1.3
One-dimensional homology groups of $M(E,I)$-sets form $X_{m}^{\bullet}$ are
free, here $m\geqslant-1$.
Proof. Indeed, using the theorem 1.1, where $m=0$, we get the isomorphism
$H_{1}(X_{0}^{\bullet})\cong
H_{1}(X_{-1}^{\bullet})\oplus\varinjlim_{0}^{\mathscr{K}_{*}(X^{\bullet}_{0})}\mathbb{Z}[x_{0}]$.
From [2, Example 3.1], it follows that the group $H_{1}(X_{-1}^{\bullet})$ is
free. Further, using induction on $m$ and theorem 1.1, we’ll complete the
proof of this theorem.
###### Theorem 1.4
Let $X^{\bullet}=\\{x_{0},x_{1},\dots,x_{n},*\\}$ is poset. Suppose that we
have a free partially commutative monoid
$M(E,I)=M\left(\coprod\limits_{i=1}^{n}E_{i},\coprod\limits_{i=1}^{n}I_{i}\right)$.
Let there is an action $X^{\bullet}\times M(E,I)\to X^{\bullet}$ on the
pointed set $X^{\bullet}$, thus we get the $M(E,I)$-set $X^{\bullet}$. Suppose
that this action generated by formulas, for all $i\in\\{1,\dots,n\\}$;
$x_{0}\cdot e^{E_{i}}_{j_{i}}=x_{1},\,\,x_{1}\cdot
e^{E_{i}}_{j_{i}}=x_{2},\dots,x_{n}\cdot e^{E_{i}}_{j_{i}}=*,\,\,*\cdot
e^{E_{i}}_{j_{i}}=*,$
here $j_{i}\in\\{j_{1},\dots,j_{\mathrm{card}E_{i}}\\}$, and
$\mathrm{card}E_{i}$ is cardinality of $E_{i}$. Then there exist a isomorphism
$H_{m}\left((M(E,I)/X^{\bullet})^{op}\right)\cong\bigoplus\limits_{i=1}^{n}H_{m}\left((M(E_{i},I_{i})/X^{\bullet})^{op}\right),\qquad
m\geqslant 1.$
Proof. Indeed, since
$M(E,I)=M\left(\coprod\limits_{i=1}^{n}E_{i},\coprod\limits_{i=1}^{n}I_{i}\right)$,
we see that there exist a isomorphism for differentional objects
$(\mathscr{C}_{*}((M(E,I)/X^{\bullet})^{op}),d_{*})\cong\bigoplus\limits_{i=1}^{n}({\mathstrut}_{i}\mathscr{C}_{*}((M(E_{i},I_{i})/X^{\bullet})^{op}),{\mathstrut}_{i}d_{*}),$
but, it knows that the functor $H_{*}(-,\mathbb{Z})$ is permutable with
respect to direct sums. Therefor, we get the isomorphism
$H_{m}\left((M(E,I)/X^{\bullet})^{op}\right)\cong\bigoplus\limits_{i=1}^{n}H_{m}\left((M(E_{i},I_{i})/X^{\bullet})^{op}\right).$
## References
* [1] Husainov A. On the homology of small categories and asynchronous systems. Homology Homotopy Appl., 2004. V. 6, N 1. P. 439-471. htpp//www.rmi.acnet.ge/hha
* [2] Husainov A. On the Cubical Homology Groups of Free Partially Commutative Monoids // New York: Cornell Univ, Preprint, 2006. 47 pp. http://arxiv.org/abs/math.CT/0611011
* [3] Diekert V., Métivier Y. Partial Commutation and Traces. // Handbook of formal languages. V. 3. Springer-Verlag, 1997. P. 457–533.
* [4] Bednarczyk M. A. Categoris of Asynchronous Systems Ph. D. Thesis, Unicersity of Syssex, report 1/88, 1988, 222p. http://www.ipipan.gda.pl/ matek
* [5] Husainov A. A., Tkachenko V. V. Homology groups of asynchronous transition systems. Mathematical modeling and the near questions of mathematics. Collection of the scientifics works. Khabarovsk: KhHPU, 2003. P. 23-33. (Russian) http://www.knastu.ru/husainov site/index.html
* [6] Gabriel P., Zisman M. Calculus of fractions and homotopy theory. Berlin-Heidelberg-New York: Springer-Verland, 1967.
|
arxiv-papers
| 2008-11-23T02:30:36 |
2024-09-04T02:48:58.929077
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lopatkin Viktor",
"submitter": "Viktor Lopatkin",
"url": "https://arxiv.org/abs/0811.3722"
}
|
0811.3813
|
# Anomalous isotope effects of fulleride superconductors
Wei Fan
Key Laboratory of Materials Physics,
Institute of Solid State Physics, Chinese Academy of Sciences,
230031-Hefei, People’s Republic of China
###### Abstract
The numerical calculations of the standard Eliashberg-Nambu strong coupling
theory and a formula of isotope effect derived in this paper provide direct
evidences that an-harmonic vibrations of lattice enhance isotope effect with
anomalous coefficient $\alpha>1/2$. The results in this paper explain very
well the wide distributed $\alpha$ values for the samples with different
ratios of substitutions of 12C by 13C of fulleride superconductor Rb3C60. The
calculations of isotope effects indicate that the intra-molecule radial modes
have more important contributions to superconductivity than the intra-molecule
tangential modes with higher phonon frequencies
PACS: 74.20.Fg, 74.70.Wz, 71.20.Tx, 81.05.Tp
Keywords: Fulleride superconductor, Anomalous isotope effect, An-harmonic
effect, Eliashberg-Nambu Theory
## 1 Introduction
The discoveries of isotope effects in metallic superconductors confirm the
electron-phonon mechanism as the origin of attractive interaction between
electrons. Anomalous isotope effects such as the negative isotope effects with
$\alpha<0$ and the enhanced isotope effects with $\alpha>1/2$ are found in
many new types of superconductors. Understanding the origin of these anomalous
isotope effects is very important to understand the microscopic mechanism of
superconductivity.
Coefficient of isotope effect $0<\alpha<1/2$ is correct for most conventional
superconductors. Negative isotope effect $\alpha<0$, where Tc increases with
mass M, had been found experimentally in conventional superconductors such as
PdH [1, 2] ($\alpha=-0.25$), organic superconductors [3, 4] and
$\alpha$-Uranium ($\alpha=-2.0$) [5]. The negative isotope effect had also
been found in Sr2RuO4 [6] when 16O atoms were substituted by 18O atoms.
Large positive Oxygen isotope effects $\alpha>1/2$ beyond the BCS value $1/2$
had been found in HTSC material La2-xSrxCuO4 ($\alpha$=0.75) at doping level
x=0.12 [7] and in fullerides Rb3C60 ($\alpha>1.0$) with different ratios of
substitutions of 12C with 13C [8, 9, 10]. For La2-xSrxCuO4, near x=0.12 the
large coefficient of isotope effect is probably due to the strong an-harmonic
vibrations companying with structural transitions. Similarly, the transition
from partial substitution to completed substitution of 12C by 13C for Rb3C60
leads to the decrease of $\alpha$ from 1.275 or 1.189 [8, 9] to 0.30 [11]
because the an-harmonic effect is relatively weak at completed substitution.
The Eliashberg theory combined with different models of an-harmonic lattice
vibrations had been used to qualitatively explain these anomalous isotope
effects [12, 13, 14]. Beyond Migdal’s theorem, the coefficient of isotope
effect $\alpha>1/2$ had also been obtained by including non-adiabatic effect
[15].
It’s very convenient to find a formula for isotope effect which is able to
explain anomalous isotope effects such as the enhanced isotope effects with
$\alpha>1/2$ and large negative isotope effect. Most of previous formulas of
isotope effects don’t include the frequency shift due to the isotope
substitution. A formula for isotope effect is derived in this paper using
McMillan’s Tc formula, which explicitly includes the frequency shift of phonon
and is qualitatively consistent with the more accurate numerical results of
Eliashberg-Nambu theory. The formula is similar to a previous formula derived
with similar idea [16]. In this work, we not only compare the results of the
formula with the strong-coupling theory, but also we study in details the
meaning of the an-harmonic coefficient $A^{*}$ and its relation with the
potential $V(r)$ of the vibrations of atoms around their equilibrium positions
in a crystal. We concentrate on the anomalous isotope effects of fulleride
superconductors [17] in the present paper.
An important relation is the key to understand isotope effect and written as
$M\langle\omega^{2}\rangle\lambda=\eta=const$ (1)
where the McMillan-Hopfield parameter $\eta$ characterizes the chemical
environment of atoms in a material [18, 19] and is expressed as
$\eta=N(0)\langle J^{2}\rangle$, where $N(0)$ is the density of state at Fermi
energy, $J$ the matrix element of electron-phonon interaction. Certainly, we
assume $\eta$ is constant for isotope substitution or others small structural
changes. We will see in section (4) that, under the constraint, the an-
harmonic effect is equivalent to the fact that the parameter $\lambda$ is
dependent on $M$ the mass of atom. In a previous work, using electron-phonon
mechanism and the constraint Eq.(1), we have successfully explained the
spatial anti-correlation of the energy gap and the phonon energy of Bi2212
superconductors observed in STM experiments [20]. In this work, using the
constraint combining with McMillan’s Tc formula, we derive a formula of
isotope effect that can explain almost all anomalous isotope effects,
especially for fulleride superconductors.
## 2 Theory
The energy-gap equation of Eliasgberg-Nambu theory in the Matsubara’s
imaginary energy form is standard [18]. When temperature is very close to
transition temperature Tc, the Eliashberg equation can be simplified as
$\displaystyle\sum_{n=0}^{N}(K_{mn}-\rho\delta_{mn})\bar{\Delta}_{n}=0~{}~{}(m\geq
0)$ (2) $\displaystyle K_{mn}$ $\displaystyle=$
$\displaystyle\lambda(m-n)+\lambda(m+n+1)-2\mu^{*}(N)$ $\displaystyle-$
$\displaystyle\delta_{mn}(2m+1+\lambda(0)+2\sum_{l=1}^{m}\lambda(l)),$
where $\bar{\Delta}_{n}=\Delta_{n}/|\omega_{n}|$ is the energy-gap parameter
and
$\lambda(n)=2\int_{0}^{\infty}d\omega\alpha^{2}F(\omega)\omega/(\omega^{2}+(2\pi
nT)^{2})$. The pair-breaking parameter $\rho$ is introduced to form a
eigenvalue problem with $\rho$=0 corresponding the physical energy-gap
equation. The transition temperature Tc is defined as the temperature when the
maximum of eigenvalue of kernel matrix $K_{mn}$ crosses to zero and changes
its sign. We use about N=200 Matsubara energies to solve above equation.
The Eliashberg function is expressed as
$\alpha^{2}F(\omega)=\left\\{\begin{tabular}[]{cc}$\frac{c}{(\omega-\Omega_{P})^{2}+(\omega_{2})^{2}}-\frac{c}{(\omega_{3})^{2}+(\omega_{2})^{2}}$,&$|\omega-\Omega_{P}|<\omega_{3}$\\\
0&others,\end{tabular}\right.$ (3)
where $\Omega_{P}$ is the energy (or frequency) of phonon mode, $\omega_{2}$
the half-width of peak of phonon mode and $\omega_{3}=2\omega_{2}$. We can
write the parameter of electron-phonon interaction
$\lambda=\lambda(0)=2\int_{0}^{\infty}d\omega\alpha^{2}F(\omega)/\omega$. The
moments $\langle\omega^{n}\rangle$ of the distribution function
$(2/\lambda)\alpha^{2}F(\omega)/\omega$ are defined as
$\langle\omega^{n}\rangle=2/\lambda\int^{\infty}_{0}d\omega\alpha^{2}F(\omega)\omega^{n-1}$.
The parameter $\lambda$ characterizes the strength of electron-phonon
interaction by $\lambda\propto N(0)\langle
J^{2}\rangle/M\langle\omega^{2}\rangle$ in terms of Eq.(1). The Coulomb
pseudo-potential is defined as $\mu_{0}=N(0)U$ and its renormalized value as
$\mu^{*}=\mu_{0}/(1+\mu_{0}\ln(E_{C}/\omega_{0}))$, where $U$ is the Coulomb
parameter, $E_{C}$ the characteristic energy for electrons such as the Fermi
energy or band width, and $\omega_{0}$ the characteristic phonon energy such
as the energy cutoff of phonon energy or Debye energy. If
$\omega_{2}\ll\Omega_{P}$, Eq.(1) can be simplified as
$M\Omega_{P}^{2}\lambda=\eta=const,$ (4)
which is obviously correct for Einstein model
$\alpha^{2}F(\omega)=(\lambda/2)\Omega_{P}\delta(\omega-\Omega_{P})$.
## 3 A formula of coefficient of isotope effect
We will derive a formula of isotope effect by including an-harmonic effect.
The shift of phonon energy (or frequency) is explicitly included in the
formula. For single-frequency like mode, the $\alpha$ values in terms of the
formula are qualitatively consistent with the numerical solutions of
Eliashberg equation. Most importantly, the formula clearly shows that the an-
harmonic vibration of lattice leads to the enhanced isotope effect with
$\alpha>1/2$. It’s conveniently to define a parameter $A^{*}$ by
$\Omega_{P}\propto M^{-(1-A^{*})/2}$ to measure the an-harmonic effect. We can
easily see that $A^{*}\neq 0$ represents the an-harmonic effect. The an-
harmonic parameter $A^{*}$ can be expressed as
$A^{*}=1+2M/\Omega_{P}(\delta\Omega_{P}/\delta M).$ (5)
Under harmonic approximation $\Omega_{P}\propto M^{-1/2}$, the parameter
$\lambda$ is a constant ($\delta\lambda/\delta M=0$) because
$A^{*}=1+2M/\Omega_{P}(\delta\Omega_{P}/\delta
M)=-M/\lambda(\delta\lambda/\delta M)=0$
by using the relation $M\Omega_{P}^{2}\lambda=const$.
We start from the McMillan formula of transition temperature Tc of
superconductor
$T_{c}=\frac{\Theta_{D}}{1.45}\exp[-\frac{1.04(1+\lambda)}{\lambda-\mu^{*}(1+0.62\lambda)}].$
(6)
where $\lambda$ is the strength of electron-phonon interaction. The
coefficient $\alpha$ of isotope effect defined by T${}_{c}\propto M^{-\alpha}$
can be obtained from above McMillan formula by the direct mass-dependent from
$\Theta_{D}\propto M^{-1/2}$ and implicit mass-dependent from $\mu^{*}$ by
$\omega_{0}\propto M^{-1/2}$. If an-harmonic effect is included then
$\Theta_{D},\omega_{0}\propto M^{-(1-A^{*})/2}$. In this work we consider
additionally mass-dependent from $\lambda$ by the well known constraint
$M\Omega^{2}_{P}\lambda=\eta$, we obtain a formula of isotope effect which is
expressed as
$\alpha=\frac{1}{2}-\frac{1.04(1+\lambda)(1+0.62\lambda)(\mu^{*})^{2}}{2[\lambda-\mu^{*}(1+0.62\lambda)]^{2}}+A^{*}T(\lambda,\mu^{*})$
(7)
where $A^{*}$ is defined in Eq.(5) and the function $T(\lambda,\mu^{*})$ is
expressed as
$\displaystyle
T(\lambda,\mu^{*})=\frac{1.04\lambda(1+0.38\mu^{*})}{[\lambda-\mu^{*}(1+0.62\lambda)]^{2}}$
(8)
or
$\displaystyle T(\lambda,\mu^{*})$ $\displaystyle=$
$\displaystyle\frac{\lambda(2.08-\lambda)+\lambda(2.7904+1.24\lambda)\mu^{*}}{2[\lambda-\mu^{*}(1+0.62\lambda)]^{2}}$
$\displaystyle+$
$\displaystyle\frac{(0.04+0.42\lambda)(1+0.62\lambda)(\mu^{*})^{2}}{2[\lambda-\mu^{*}(1+0.62\lambda)]^{2}},$
dependent on whether $\Theta_{D},\omega_{0}\propto M^{-1/2}$ for Eq.(8) or
$\Theta_{D},\omega_{0}\propto M^{-(1-A^{*})/2}$ for Eq.(3). We notice that,
for the Eq.(8), the an-harmonic effect enters into the coefficient $\alpha$
only by the M-dependent $\lambda$, however for the Eq.(3) not only by
M-dependent $\lambda$ but also by the $\Theta_{D}$ and $\omega_{0}$. It’s very
important that $T(\lambda,\mu^{*})>0$ for reasonable values
$0<\lambda<2<2.08$, so the sign of the third term of Eq.(7) is determined by
the an-harmonic parameter $A^{*}$. It’s obviously that $A^{*}=0$ under
harmonic approximation $\Omega_{P}\propto M^{-1/2}$ and the third term in the
Eq.(7) is also zero. The derivative $\delta\Omega_{P}/\delta M$ used in the
calculation of an-harmonic parameter $A^{*}$ is approximately obtained from
the experimental energy (or frequency) shift of phonon. The first two terms
give $\alpha<1/2$. The properties of the third term to isotope effect is
determined by the sign of the an-harmonic parameter $A^{*}$. Generally,
$\Omega_{P}$ decreases with increasing mass M, thus
$2(M/\Omega_{P})\delta\Omega_{P}/\delta M<0$. If
$2|(M/\Omega_{P})\delta\Omega_{P}/\delta M|<1$, $A^{*}>0$. So we can get
$\alpha>1/2$ if the third term has larger absolute value than the second term.
## 4 The anomalous isotope effects with $\alpha>1/2$
In many literatures of AnC60, the average mass $m$ of C60 is used in the
definition of coefficient of isotope effect by
$\tilde{\alpha}=-(m/T_{c})\delta T_{c}/\delta m$. If the ratio of substitution
is $p$, the average mass of C60 molecule $m=60[M_{12}(1-p)+M_{13}p]$, so
$\delta m=m-60M_{12}=60p\delta M$ with $\delta M=M_{13}-M_{12}$. The relation
$\tilde{\alpha}=\alpha/p$ connects the parameter $\tilde{\alpha}$ with the
usual definition of $\alpha=-(M/T_{c})(\delta T_{c}/\delta M)$. The usual
coefficient of isotope effect $\alpha=p\tilde{\alpha}=1.275$ is obtained with
$\tilde{\alpha}$=2.125 and $p$=0.60 [8], $\alpha=1.189$ with
$\tilde{\alpha}$=1.45 and $p$=0.82 [9], $\alpha=0.462$ with
$\tilde{\alpha}$=1.4 and $p$=0.33 [10]. For completed substitution with
$p$=1.0, $\alpha=\tilde{\alpha}$=0.30 [11].
The relation $M\langle\omega^{2}\rangle\lambda=\eta=const$ services as a
constraint to determine the parameters $\lambda$ and $\Omega_{P}$ in numerical
calculations. The Coulomb pseudo-potential $\mu^{*}$ has to change in isotope
substitution because $\mu^{*}$ is dependent on the cutoff of phonon energy
$\omega_{0}\propto M^{-1/2}$ or $\propto M^{-(1-A^{*})/2}$ in an-harmonic
approximation. In harmonic approximation, $\delta\mu^{*}=-(\mu^{*})^{2}\delta
M/2M$, the an-harmonic effect is included in calculations only by the
M-dependent $\lambda$. The an-harmonic effect can be realized by shifting
phonon energy $\delta\Omega_{P}$ to make
$A^{*}=1+2M/\Omega_{P}(\delta\Omega_{P}/\delta M)\neq 0$.
The intra-molecule radial mode, which is about $\Omega_{P}$=65 meV or 525 cm-1
from infrared spectrum [10, 8] and the intra-molecule tangential modes around
1400-1 or 174 meV [17] have strong intensity. However we concentrate
attentions on the intra-molecule radial mode with energy $\Omega_{P}$=65 meV
and half-width $\omega_{2}$=8 meV. The Coulomb parameter $\mu_{0}=UN(0)$ and
the corresponding renormalized Coulomb parameter
$\mu^{*}=\mu_{0}/[1+\mu_{0}\ln(E_{C}/\omega_{0})]$ can be estimated from ab-
initio density functional theory (DFT) based on pseudo-potential method using
atomic orbital basis functions[21]. In the DFT calculation, the super-cell
includes one C60 molecule and three Rb atoms. The effect of orientation of
different C60 molecules is ignored. The parameter $U$ is the charge energy
defined as $\delta^{2}E_{tot}/\delta n^{2}=E(n+1)+E(n-1)-2E(n)$, $E(n)$ the
total energy of electric-neutral system of n valence electrons, $E(n+1)$ and
$E(n-1)$ the total energies with one negative and one positive charge
respectively. We choose the possible valence-electron configuration
$4p^{6}5s^{1}$ for Rb atoms and $2s^{2}2p^{2}$for Carbon atoms. The core
electrons are presented by Troulier-Martins pseudo-potentials. The electrons
in semicore state 4p of Rb atoms having already treated as valence electrons
have the single-$\zeta$ basis set and all others valence electrons for all
atoms have the split valence double-$\zeta$ plus polarized basis sets. We use
$\Gamma$ point sampling the first Brillouin zone. The exchange-correlation
potential is GGA Perdew-Burke-Ernzerhof type[22] and the spin-polarization
effects are included in the self-consistent calculations. The results in
table(1) show that the Coulomb parameter $\mu_{0}$ is equal to 4.374 and the
renormalized Coulomb parameters $\mu^{*}$ is 0.127 if the maximum of phonon
energy $\omega_{0}$ is 150meV and $E_{C}$=15 eV when all valence electrons are
included. The common value $\mu^{*}$=0.10 is close 0.127 obtained in this
work. We have known the Coulomb parameter $\mu^{*}$ and the phonon energy
$\Omega_{P}$, the parameter $\lambda$ of electron-phonon interaction can be
defined by the experimental transition temperature. From table(1), we can see
that the parameters $\mu^{*}$ have small influence on isotope effects. Below
we present the calculations of coefficient $\alpha$ of isotope effects with
$\mu^{*}$=0.1.
Table 1: The Coulomb parameter $\mu^{*}$ is calculated when we choose valence-electron configuration (a) $4p^{6}5s^{1}$ for Rb atom and $2s^{2}2p^{2}$for Carbon atom in DFT calculations. The case (b) is corresponding to the general value $\mu^{*}$=0.10. The coefficients $\alpha$ of isotope effects are calculated using the phonon-energy shifts from 65 meV to 62.3 meV after the isotope substitution (A∗=0.003). | U(eV) | N(0)(1/eV) | UN(0) | $\ln(E_{F}/\omega_{0})$ | $\mu^{*}$ | $\lambda$ | Tc(K) | $\alpha$
---|---|---|---|---|---|---|---|---
(a) | 1.991 | 2.197 | 4.374 | 6.9078 | 0.127 | 0.700 | 29.544 | 0.342
(b) | | | | | 0.100 | 0.670 | 29.820 | 0.358
Table 2: The simulation parameters and the results with different an-harmonic parameters $A^{*}$. The coefficients $\alpha$ are calculated based on the numerical solutions of Eliashberg equation and $\alpha^{\prime}$ from the formula Eq.(7) and Eq.(8) derived in this paper. 13C | M (u) | $\Omega_{P}$ (meV) | $\lambda$ | $T_{c}$ (K) | $A^{*}$ | $\alpha$ | $\alpha^{\prime}$
---|---|---|---|---|---|---|---
(1) | 13 | 64.3 | 0.632 | 25.5 | 0.742 | 1.929$\pm$0.10 | 2.377
(2) | 13 | 63.4 | 0.650 | 27.2 | 0.409 | 1.207$\pm$0.09 | 1.516
(3) | 13 | 62.3 | 0.674 | 29.0 | 0.003 | 0.358$\pm$0.09 | 0.464
(4) | 13 | 62.2 | 0.676 | 29.2 | -0.034 | 0.282$\pm$0.09 | 0.368
(5) | 13 | 62.0 | 0.680 | 29.5 | -0.108 | 0.132$\pm$0.08 | 0.177
(6) | 13 | 61.8 | 0.685 | 29.8 | -0.182 | -0.02$\pm$0.08 | -0.014
(7) | 13 | 61.6 | 0.689 | 30.3 | -0.255 | -0.165$\pm$0.08 | -0.205
(8) | 13 | 61.2 | 0.698 | 30.9 | -0.403 | -0.457$\pm$0.08 | -0.588
(9) | 13 | 60.0 | 0.727 | 33.2 | -0.846 | -1.305$\pm$0.08 | -1.736
12C | 12 | 65.0 | 0.670 | 29.82 | | |
The parameter of electron-phonon interaction $\lambda$=0.67 is defined by the
experimental transition temperature Tc about 29.5(K) using parameters
$\mu^{*}$=0.1 and $\Omega_{P}$=65 meV. We calculate the $\eta=\eta_{12}$ for
12C. The phonon energies after 13C substitutions are unknown because the
phonon energies are dependent on the ratios of 13C substitutions. We choose
the nine possible energies for phonons in table (2), which are all smaller
than $\Omega_{P}$=65 meV because the energies (or frequency) decrease with
increasing mass of atoms. We also assume that the parameters $\lambda$ of
electron-phonon interaction alter after isotope substitutions. The new
Eliashberg function $\alpha^{2}F(\omega)$ after isotope substitution by 13C is
obtained from the old one before isotope substitution by simply shifting
energy of phonon and scaling it to satisfy the constraint
$M\langle\omega^{2}\rangle\lambda=\eta_{13}=\eta_{12}$. Based on the new
Eliashberg function $\alpha^{2}F(\omega)$, we can calculate the Tc and
$\lambda$ after the isotope substitution. The new Coulomb pseudo-potential
after isotope substitution $\mu^{*}$=0.099583 is obtained from the formula
$\delta\mu^{*}=-(\mu^{*})^{2}\delta M/2M$. The nine possible values of
$\lambda$ corresponding to nine possible energies of phonon are collected in
table (2). The transition temperatures Tc are obtained by solving Eliashberg
equation Eq.(2). The coefficients $\alpha$ are easily calculated in terms of
the transition temperatures Tc after and before isotope substitutions by
$\alpha=-\ln(^{13}T_{c}/^{12}T_{c})/\ln(M_{13}/M_{12})$. The an-harmonic
parameters $A^{*}$ are calculated in terms of the shifts of phonon energies.
For the case (2) in table (2), we get $\alpha=1.207$ which is very close to
the already known maximum value 1.275 in experiments with uncompleted
substitution ($\tilde{\alpha}$=2-2.25, $p$=0.60) [8]. For sample with
completed substitution with 13C, the value of $\alpha$ decreases to $\sim$0.30
[11]. To explain the interesting results, we assume the increase of ratio of
substitution makes the distribution of 13C more homogenous and the an-harmonic
effect become weak with small $A^{*}$. If the phonon energy shifts to 62.3 meV
with smaller an-harmonic parameter $A^{*}$=0.003, the value of $\alpha$=0.358
is close to the experimental values from 0.30 to 0.37 for the samples with
completed substitution.
Figure 1: The comparison of coefficients of isotope effect having obtained
from numerical calculations of Eliashberg equation and from Eq.(7) for radial
intra-molecule mode $\Omega_{P}$=65 meV (a) and the tangential intra-molecule
mode $\Omega_{P}$=150 meV (b)
For negative an-harmonic parameter$A^{*}$=-0.182, we find $\alpha$=0.02 close
to zero. After having made $A^{*}$ more negative further, at $A^{*}$=-0.403,
we get the negative coefficient $\alpha$=-0.457. However, the negative isotope
effect wasn’t found in fulleride superconductors. From above calculations we
can see that the sign of $A^{*}$ determines the sign of $\alpha$ at relative
larger absolute values of $A^{*}$. If the absolute values of an-harmonic
parameters are not too large $0.07>A^{*}>-0.21$ or an-harmonic effect is weak,
the coefficient $\alpha$ are from 0 to 0.5 within the range of general values.
The above calculations explain the wide distributed values of coefficients for
Rb3C60 in experiments. This is because the an-harmonic parameters are
dependent on the ratios of substitutions.
From table (2), we can see that the values obtained from numerical
calculations are very close to the values from Eq.(7) with Eq.(8). The values
of $\alpha$ and $\alpha$’ have the same sign on the same rows. It’s very
important that if $\alpha>1/2$, $A^{*}>0$ must be satisfied. Thus, the
enhanced coefficients $\alpha$ larger than $1/2$ for Rb3C60 in uncompleted
substitution samples mean that the strong an-harmonic effects. If the an-
harmonic effect becomes weak as approaching the completed substitution, the
value of $\alpha$ will decrease and reach to smaller value about 0.358 at
$A^{*}=0.003$.
Fig. 1(a) shows the results of the numerical solutions of Eliashberg equation
and the formula Eq.(7). We can see that Eq.(7) with Eq.(8) is more close to
the numerical solution than using the Eq.(3) although for Eq.(3) more
completed an-harmonic effects are included. We have preformed the same
calculations for intra-molecule tangential mode with energy $\Omega_{P}$=150
meV, $\mu^{*}$=0.285, $\lambda$=0.67 and Tc=29.5(K). From Fig.1(b), there is
larger slope of $\alpha$-$\delta\Omega_{P}/\Omega_{P}$ curve compared with the
intra-molecule radial mode with energy $\Omega_{P}$=65 meV. The coefficient of
isotope effect of the intra-molecular radial mode $\alpha$=0.358 at
$A^{*}\simeq$0.0 is more close to experiments at completed substitutions
$\alpha\sim 0.30-0.37$. So it is more correlated with superconductivity of
fulleride than the intra-molecule tangential mode.
## 5 An-harmonic parameter $A^{*}$ and models of lattice vibrations
Figure 2: An-harmonic coefficients $A^{*}$ for the models of interaction with
form $V(r)=hr^{d}$. The inserted figures (1),(2) and (3) illustrate the shapes
of interacting potentials and the quantized energy levels for $h$=0.01 and
$d$=0.5, 2.0 and 4.0 respectively. The maximum of quantum number of angular
momentum is up to $l_{max}$=8
In this paper, we introduce an important parameter, the an-harmonic parameter
$A^{*}$. To make parameter $A^{*}$ more realistic, we study the models of
lattice vibrations in a crystal. The atoms in crystal move around their
equilibrium positions. The change of potential energy of an atom away from its
equilibrium position is written as $V(r)=hr^{d}$ with radially local
displacement $r$. The harmonic vibration is corresponding $d=2$. We had
numerically solved the Schrödinger equation
$[-\hbar^{2}\nabla^{2}/2M+V(r)]\psi(\vec{r})=E\psi(\vec{r})$ under isotropic
approximation, especially, the one-dimensional radial part converts into
difference equation. We calculated the relation $E_{0}(M)$ between energy of
ground state and mass of atom for a series of discrete values of $M$. It’s
well known that for harmonic approximation $E_{0}(M)\propto 1/M^{0.5}$. The
effective an-harmonic parameters $A^{*}$ for interaction with the form
$V(r)=hr^{d}$ are obtained by fitting the discrete $E_{0}(M)$ functions with
function $\tilde{E}_{0}(M)\propto 1/M^{(1-A^{*})/2}$. From the Fig.2, if $d<2$
we can obtain $A^{*}>0$. So the enhanced isotope effects with $\alpha>0.5$ are
hopefully found based on the formula Eq.(7) obtained in this work. If $d>2$ so
$A^{*}<0$, we can get the normal isotope effects with $\alpha<0.5$ and the
negative isotope effects with $\alpha<0$. We can see that the simple model
$V(r)=hr^{d}$ is suitable for the isotope effects of fulleride Rb3C60 for
different ratios of substitutions because the an-harmonic parameters are
within the range to obtain experimental values of $\alpha$.
## 6 Discussion and Summary
The Figure 1(b) shows clearly that the formula Eq.(7) isn’t good approximation
to more accurate numerical solution for high-energy mode. The reason is
probably that McMillan’s formula isn’t correct in some regions of parameter
space. The McMillan’s formula is good approximation when the parameter
$\lambda$ of electron-phonon interaction is not too large ($\lambda<1$) and
the phonon energy $\Omega_{P}$ is not too high. The inter-molecule phonon
modes such as the vibration between $C_{60}$ molecules and the between alkali-
metal atoms and $C_{60}$ molecules are ignored in this work because their
energies generally smaller than 12 meV. To obtain $T_{c}$=30K, the parameters
$\lambda$ of electron-phonon interaction are at least 3.0 so the instability
of lattice will destroy superconductivity. However, these inter-molecule modes
still have significant influence on the properties of fulleride
superconductors such as the differences of isotope effects for different
substitution configurations $Rb_{3}[(^{13}C_{60})_{x}(^{12}C_{60})_{1-x}]$ and
$Rb_{3}(^{13}C_{x}^{12}C_{1-x})_{60}$ [11].
In summary, the coefficients $\alpha$ of isotope effects obtained in this
paper are very close to the values in experiments or within the range of
experimental values. The reductions of $\alpha$ with increasing the
substitution ratios of 13C are due to the reductions of an-harmonic effects of
lattice vibrations when the ratios of substitution tend to 100%. The enhanced
coefficients of isotope effects with $\alpha>1/2$ generally happen at the
intermediate stage of transition from one phase to another. Finally, the
formula Eq.(7) and the numerical methods used in this paper are also suitable
to study isotope effects of other superconductors if the electron-phonon
interaction is the pairing mechanism for electrons.
The author benefits from the discussions with members within CMS Group at
ISSP-CAS, especially from Dr. Xu Yong, Huang Ling-Feng, Li Long-Long and Li
Yan-Ling. This work is supported by Director Grants of Hefei Institutes of
Physical Sciences, Knowledge Innovation Program of Chinese Academy of Sciences
and National Science Foundation of China.
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|
arxiv-papers
| 2008-11-24T10:20:45 |
2024-09-04T02:48:58.935335
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "W.Fan",
"submitter": "Wei Fan",
"url": "https://arxiv.org/abs/0811.3813"
}
|
0811.3859
|
# On the Complexity of Matroid Isomorphism Problem
Raghavendra Rao B.V The Institute of Mathematical Sciences,C.I.T.
Campus,Chennai 600 113, India. bvrr@imsc.res.in Jayalal Sarma M.N Institute
for Theoretical Computer Science,Tsinghua University,Beijing 100 084, China.
jayalal@tsinghua.edu.cn The work was done when this author was also a graduate
student at the Institute of Mathematical Sciences, Chennai, India.
###### Abstract
We study the complexity of testing if two given matroids are isomorphic. The
problem is easily seen to be in $\Sigma_{2}^{p}$. In the case of linear
matroids, which are represented over polynomially growing fields, we note that
the problem is unlikely to be $\Sigma_{2}^{p}$-complete and is
${\mathsf{co}}{\mathsf{NP}}$-hard. We show that when the rank of the matroid
is bounded by a constant, linear matroid isomorphism, and matroid isomorphism
are both polynomial time many-one equivalent to graph isomorphism.
We give a polynomial time Turing reduction from graphic matroid isomorphism
problem to the graph isomorphism problem. Using this, we are able to show that
graphic matroid isomorphism testing for planar graphs can be done in
deterministic polynomial time. We then give a polynomial time many-one
reduction from bounded rank matroid isomorphism problem to graphic matroid
isomorphism, thus showing that all the above problems are polynomial time
equivalent.
Further, for linear and graphic matroids, we prove that the automorphism
problem is polynomial time equivalent to the corresponding isomorphism
problems. In addition, we give a polynomial time membership test algorithm for
the automorphism group of a graphic matroid.
## 1 Introduction
Isomorphism problems over various mathematical structures have been a source
of intriguing problems in complexity theory (see [AT05]). The most important
problem of this domain is the well-known graph isomorphism problem. Though the
complexity characterization of the general version of this problem is still
unknown, there have been various interesting special cases of the problem
which are known to have polynomial time algorithms [BGM82, Luk80]. In this
paper we talk about isomorphism problem associated with matroids.
A matroid $M$ is a combinatorial object defined over a finite set $S$ (of size
$m$) called the ground set, equipped with a non-empty family ${\mathcal{I}}$
of subsets of $S$ (containing the empty subset) which is closed under taking
of subsets and satisfies the exchange axiom : for any
$I_{1},I_{2}\in{\mathcal{I}}$ such that $|I_{1}|>|I_{2}|$, $\exists x\in
I_{1}\setminus I_{2}$, $I_{2}\cup\\{x\\}\in{\mathcal{I}}$. The sets in
${\mathcal{I}}$ are called independent sets. The rank of the matroid is the
size of the maximal independent set. This provides useful abstractions of many
concepts in combinatorics and linear algebra [Whi35]. The theory of matroids
is a well studied area of combinatorics [Oxl92]. We study the problem of
testing isomorphism between two given matroids.
Two matroids $M_{1}$ and $M_{2}$ are said to be isomorphic if there is a
bijection between the elements of the ground set which maps independent sets
to independent sets, (or equivalently circuits to circuits, or bases to bases,
see section 2). Quite naturally, the representation of the input matroids is
important in deciding the complexity of the algorithmic problem.
There are several equivalent representations of a matroid. For example,
enumerating the maximal independent sets (called bases) or the minimal
dependent sets (called circuits) also defines the matroid. These
representations, although can be exponential in the size of the ground set,
indeed exist for every matroid, by definition. With this enumerative
representation, Mayhew [May08] studied the matroid isomorphism problem, and
shows that the problem is equivalent to graph isomorphism problem. However, a
natural question is whether the problem is difficult when the representation
of the matroid is more implicit?. In a black-box setting, one can also
consider the input representation in the form of an oracle or a black-box,
where the oracle answers whether a given set is independent or not.
More implicit (and efficient) representation of matroids have been studied.
One natural way is to identify the given matroid with matroids defined over
combinatorial or algebraic objects which have implicit descriptions. A general
framework in this direction is the representation of a matroid over a field. A
matroid $M=(S,\mathcal{I})$ of rank $r$ is said to be representable over a
field $\mathbb{F}$ if there is a map, $\phi:S\to\mathbb{F}^{r}$ such that,
$\forall A\subseteq S$, $A\in\mathcal{I}\iff\phi(A)$ is linearly independent
over $\mathbb{F}^{r}$ as a vector space. However, there are matroids which do
not admit linear representations over any field. (For example, the Vamós
Matroid, See Proposition 6.1.10, [Oxl92].). In contrast, there are matroids
(called regular matroids) which admit linear representations over all fields.
Another natural representation for a matroid is over graphs. For any graph
$X$, we can associate a matroid $M(X)$ as follows: the set of edges of $X$ is
the ground set, and the acyclic subgraphs of the given graph form the
independent sets. A matroid $M$ is called a graphic matroid (also called
polygon matroid or cyclic matroid) if it is isomorphic to $M(X)$ for some
graph $X$. It is known that graphic matroids are linear. Indeed, the incidence
matrix of the graph will give a representation over $\mbox{$\mathbb{F}$}_{2}$.
There are linear matroids which are not graphic. (See [Oxl92] for more
details.)
The above definitions themselves highlight the importance of testing
isomorphism between two given matroids. We study the isomorphism problem for
the case of linear matroids (Linear Matroid Isomorphism problem ($\mbox{\sc
LMI})$ and graphic matroids (Graphic Matroid Isomorphism problem (GMI)).
From a complexity perspective, the general case of the problem is in
$\Sigma_{2}^{p}$. However, it is not even clear a priori if the problem is in
${\mathsf{NP}}$ even in the above restricted cases where there are implicit
representations. But we note that for the case of graphic matroids the problem
admits an ${\mathsf{NP}}$ algorithm. Hence an intriguing question is about the
comparison of this problem to the well studied graph isomorphism problem.
At an intuitive level, in the graph isomorphism problem we ask for a map
between the vertices that preserves the adjacency relations, whereas in the
case of graphic matroid isomorphism, we ask for maps between the edges such
that the set of cycles (or spanning trees) in the graph are preserved. As an
example, in the case of trees, any permutation gives a 2-isomorphism, where as
computing the isomorphism between trees is known to be
${\mathsf{L}}$-complete. This indicates that the reduction between the
problems cannot be obtained by a local replacement of edges with gadgets, and
has to consider the global structure.
An important result in this direction, due to Whitney (see [Whi32]), says that
in the case of 3-connected graphs, the graphs are isomorphic if and only if
the corresponding matroids are isomorphic (see section 5). Thus the problem of
testing isomorphism of graphs and the corresponding graphic matroids are
equivalent for the case of 3-connected graphs are equivalent. Despite this
similarity between the problems, to the best of our knowledge, there has not
been a systematic study of GMI and its relationships to graph isomorphism
problem (GI). This immediately gives a motivation to study the isomorphism
problem for 3-connected graphs. In particular, from the recent results on
graph isomorphism problem for these classes of graphs [DLN08, TW08], it
follows that graphic matroid isomorphism problem for 3-connected planar graphs
${\mathsf{L}}$-complete.
In this context we study the general, linear and graphic matroid isomorphism
problems. Our main contributions in the paper are as follows:
* •
Matroid isomorphism problem is easily seen to be in $\Sigma_{2}^{p}$. In the
case of linear matroids where the field is also a part of the input we observe
that the problem is ${\mathsf{co}}{\mathsf{NP}}$-hard (Proposition 3.4), and
is unlikely to be $\Sigma_{2}^{p}$-complete (Proposition 3.2). We also observe
that when the rank of the matroid is bounded, linear matroid isomorphism, and
matroid isomorphism are both equivalent to GI (Theorem 3.5)111We note that,
although not explicitly stated, the equivalence of bounded rank matroid
isomorphism and and graph isomorphism also follows from the results of Mayhew
[May08]. However, it is not immediately clear if the GI-hard instances are
linearly representable. Our proofs are different and extends this to linear
matroids.
* •
We develop tools to handle colouring of ground set elements in the context of
isomorphism problem. We show that coloured version of the linear matroid
isomorphism and graphic matroid isomorphisms are as hard as the general
version (Lemma 4.2, 4.1). As an immediate application of this, we show that
the automorphism problems for graphic matroids and linear matroids are
polynomial time Turing equivalent to the corresponding isomorphism problems.
In this context, we also give a polynomial time membership test algorithm for
the automorphism group of a graphic matroid (Theorem 6.5).
* •
We give a polynomial time Turing reduction from graphic matroid isomorphism
problem to the graph isomorphism problem by developing an edge colouring
scheme which algorithmically uses a decomposition given by [HT73] (and [CE80])
and reduce the graphic matroid (Theorem 5.3). Our reduction, in particular
implies that the graphic matroid isomorphism testing for planar graphs can be
done in deterministic polynomial time (Corollary 5.9).
* •
Finally, we give a reduction from bounded rank matroid isomorphism problem to
graphic matroid isomorphism (Theorem 5.11), thus showing that all the above
problems are poly-time equivalent.
Table 1 below summarizes the complexity of matroid isomorphism problem under
various input representations.
Repn. of $M_{1},M_{2}$ | Complexity Bounds for MI
---|---
List of Ind. sets | GI-complete [May08]
Linear | GI-hard, ${\mathsf{co}}{\mathsf{NP}}$-hard ([Hli07, OW02]).
Linear (bounded rank) | GI complete
Graphic | Turing equivalent to GI
Planar | ${\mathsf{P}}$
Planar 3-connected | ${\mathsf{L}}$-complete
Table 1: Complexity of MI under various input representations
## 2 Notations and Preliminaries
All the complexity classes used here are standard and we refer the reader to
any standard text book (for e.g. see [Gol08]). Now we collect some basic
definitions on matroids (see also [Oxl92]). Formally, a matroid $M$ is a pair
$(S,{\mathcal{I}})$, where $S$ is a finite set called the ground set of size
$m$ and ${\mathcal{I}}$ is a collection of subsets of $S$ such that: (1) the
empty set $\phi$, is in ${\mathcal{I}}$. (2) If $I_{1}\in I$ and $I_{2}\subset
I_{1}$, then $I_{2}\in{\mathcal{I}}$. (3) If $I_{1},I_{2}\in{\mathcal{I}}$
with $|I_{1}|<|I_{2}|$, then $\exists x\in I_{2}\setminus I_{1}$ such that
$I_{1}\cup\\{x\\}$ is in ${\mathcal{I}}$.
The Rank function of a matroid is a map rank: $2^{S}\to\mathbb{N}$, is defined
for a $T\subseteq S$, as the maximum size of any element of ${\mathcal{I}}$
that is contained in $T$. The rank of the matroid is the maximum value of this
function. A circuit is a minimal dependent set. Spanning sets are subsets of
$S$ which contains at least one basis as its subset. Notice that a set
$X\subseteq S$ is spanning if and only if $rank(X)=rank(S)$. Moreover, $X$ is
a basis set if and only if it is a minimal spanning set. For any $F\subseteq
S$, $cl(F)=\\{x\in S~{}:~{}rank(F\cup x)=~{}rank(F)\\}$. A set $F\subseteq S$
is a flat if $cl(F)=F$. Hyperplanes are flats which are of rank $r-1$, where
$r=\mbox{\sc Rank}(S)$. $X\subseteq S$ is a hyperplane if and only if it is a
maximal non-spanning set.
An isomorphism between two matroids $M_{1}$ and $M_{2}$ is a bijection
$\phi:S_{1}\to S_{2}$ such that $\forall C\subseteq
S_{1}:C\in{\mathcal{C}}_{1}~{}\iff~{}\phi(C)\in{\mathcal{C}}_{2}$, where
${\mathcal{C}}_{1}$ and ${\mathcal{C}}_{2}$ are the family of circuits of the
matroids $M_{1}$ and $M_{2}$ respectively. Now we state the computational
problems more precisely.
###### Problem 1 ($\mbox{\sc Matroid Isomorphism}(\mbox{\sc MI})$).
Given two matroids $M_{1}$ and $M_{2}$ as their independent set oracles, does
there exist an isomorphism between the two matroids?
Given a matrix $A$ over a field $\mathbb{F}$, we can define a matroid $M[A]$
with columns of $A$ as the ground set and linearly independent columns as the
independent sets of $M[A]$. A matroid $M=(E,{\mathcal{I}})$ with rank$=r$ is
said to be representable over $\mathbb{F}$, if there is amap
$\Phi:E\to\mbox{$\mathbb{F}$}^{r}$ such that $I\in{\mathcal{I}}\iff\Phi(I)$ is
linearly independent in $\mbox{$\mathbb{F}$}^{n}$. Linear matroids are
matroids representable over fields. Without loss of generality we can assume
that the representation is of the form of a matrix where the columns of the
matrix correspond to the ground set elements. We assume that the field on
which the matroid is represented is also a part of the input, also that the
field has at least $m$ elements and at most $poly(m)$ elements, where
$m=poly(n)$.
###### Problem 2 ($\mbox{\sc Linear Matroid Isomorphism}(\mbox{\sc LMI})$).
Given two matrices $A$ and $B$ over a given field $\mathbb{F}$ does there
exist an isomorphism between the two linear matroids represented by them?.
As mentioned in the introduction, given a graph $X=(V,E)$ ($|V|=n,|E|=m$), a
classical way to associate a matroid $M(X)$ with $X$ is to treat $E$ as ground
set elements, the bases of $M(X)$ are spanning forests of $X$. Equivalently
circuits of $M(X)$ are simple cycles in $X$. A matroid $M$ is called graphic
iff $\exists X$ such that $M=M(X)$.
Evidently, adding vertices to a graph $G$ with no incident edges will not
alter the matroid of the graph. Without loss of generality we can assume that
$G$ does not have self-loops.
###### Problem 3 ($\mbox{\sc Graphic Matroid Isomorphism}(\mbox{\sc GMI})$).
Given two graphs $X_{1}$ and $X_{2}$ does there exist an isomorphism between
$M(X_{1})$ and $M(X_{2})$?.
Another associated terminology in the literature is about 2-ismorphism. Two
graphs $X_{1}$ and $X_{2}$ are said to $2$-isomorphic (denoted by
$X_{1}\cong_{2}X_{2}$) if their corresponding graphic matroids are isomorphic.
Thus the above problem asks to test if two given graphs are 2-isomorphic.
In a rather surprising result, Whitney [Whi33] came up with a combinatorial
characterisation of 2-isomorpic graphs. We briefly describe it here. Whitney
defined the following operations.
* •
Vertex Identification: Let $v$ and $v^{\prime}$ be vertices of distinct
components of $X$. We modify $X$ by identifying $v$ and $v^{\prime}$ as a new
vertex $\bar{v}$.
* •
Vertex Cleaving: This is the reverse operation of vertex identification so
that a graph can only be cleft at the a cut-vertex or at a vertex incident
with a loop.
* •
Twisting: Suppose that the graph $X$ is obtained from the disjoint graphs
$X_{1}$ and $X_{2}$ by identifying vertices $u_{1}$ of $X_{1}$ and $u_{2}$ of
$X_{2}$ as the vertex $u$ of $X$, identifying vertices $v_{1}$ of $X_{1}$ and
$v_{2}$ of $X_{2}$ as the vertex $v$ of $X$. In a twisting of $X$ about
$\\{u,v\\}$, we identify, instead $u_{1}$ with $v_{2}$ and $u_{2}$ with
$v_{1}$ to get a new graph $X^{\prime}$.
###### Theorem 2.1 (Whitney’s 2-ismorphism theorem).
([Whi33], see also [Oxl92]) Let $X_{1}$ and $X_{2}$ be two graphs having no
isolated vertices. Then $M(X_{1})$ and $M(X_{2})$ are isomorphic if and only
if $X_{1}$ can be transformed to a graph isomorphic to $X_{2}$ by a sequence
of operations of vertex identification, cleaving and/or twisting.
The graphic matroids of planar graphs are called planar matroids. We now
define the corresponding isomorphism problem for graphic matroids,
###### Problem 4 ($\mbox{\sc Planar Matroid Isomorphism}(\mbox{\sc PMI})$).
Given two planar graphs $X_{1}$ and $X_{2}$ does there exist an isomorphism
between their graphic matroids ?.
As a basic complexity bound, it is easy to see that $\mbox{\sc
MI}\in\Sigma_{2}^{p}$. Indeed, the algorithm will existentially guess a
bijection $\sigma:S_{1}\to S_{2}$ and universally verify if for every subset
$C\subseteq S_{1}$, $C\in{\mathcal{C}}_{1}\iff\sigma(C)\in{\mathcal{C}}_{2}$
using the independent set oracle.
## 3 Linear Matroid Isomorphism
In this section we present some observations and results on Linear Matroid
Isomorphism. Some of these follow easily from the techniques in the
literature. We make them explicit in a form that is relevant to the problem
that we are considering.
We first observe that using the arguments similar to that of [KST93] one can
show $\overline{\mbox{\sc LMI}}\in{\mathsf{BP}}.\Sigma_{2}^{\mathsf{P}}$
(Notice that an obvious upper bound for this problem is $\Pi_{2}$). We include
some details of this here while we observe some points about the proof.
###### Proposition 3.1.
$\overline{\mbox{\sc LMI}}\in{\mathsf{BP}}.\Sigma_{2}^{\mathsf{P}}$
###### Proof.
Let $M_{1}$ and $M_{2}$ be the given linear matroids having $m$ columns each.
We proceed as in [KST93], for the case of GI. To give a
${\mathsf{BP}}.\Sigma_{2}^{\mathsf{P}}$ algorithm for $\overline{\mbox{\sc
LMI}}$, define the following set:
$N(M_{1},M_{2})=\left\\{(N,\phi):(N\cong M_{1})\lor(N\cong M_{2})\land\phi\in
Aut(N)\right\\}$
where $Aut(H)$ contains all the permutations (bijections) which are
isomorphisms of matroid $N$ to itself. The key property that is used in
[KST93] has the following easy counterpart in our context.
For any matroid $M$ on a ground set of size $m$, if $Aut(M)$ denote the
automorphism group of $M$, $\\#M$ denotes the number of different matroids
isomorphic to $M$, $|Aut(M)|*(\\#M)=|S_{m}|$.
$M_{1}\cong M_{2}\implies|N(M_{1},M_{2})|=m!$ $M_{1}\not\cong
M_{2}\implies|N(M_{1},M_{2})|=2.m!$
As in [KST93], we can amplify this gap and then using a good hash family and
utilise the gap to distinguish between the two cases. In the final protocol
(before amplifying) the verifier chooses a hash function and sends it to the
prover, the prover returns a tuple $(N,\phi)$ along with a proof that this
belongs to $N(M_{1},M_{2})$. (Notice that this will not work over very large
fields, especially over infinite fields.) Verifier checks this claim along
with the hash value of the tuple. This can be done in $\Sigma_{2}^{p}$. Hence
the entire algorithm gives an upper bound of
${\mathsf{BP}}.\exists.\Sigma_{2}^{p}={\mathsf{BP}}.\Sigma_{2}^{p}$, and thus
the result follows. ∎
Now, we know that [Sch99], if
$\Pi_{2}^{p}\subseteq{\mathsf{BP}}.\Sigma_{2}^{p}$ then
${\mathsf{PH}}={\mathsf{BP}}.\Sigma_{2}^{p}=\Sigma_{3}^{p}$. Thus we get the
following:
###### Theorem 3.2.
$\mbox{\sc LMI}\in\Sigma_{2}^{p}$. In addition, $\mbox{\sc LMI}\textrm{ \em is
}\Sigma_{2}^{\mathsf{P}}\textrm{\em-
hard}\implies{\mathsf{PH}}=\Sigma_{3}^{\mathsf{P}}$.
We notice that a special case of this is already known to be
${\mathsf{co}}{\mathsf{NP}}$-hard. A matroid of rank $k$ is said to be uniform
if all subsets of size at most $k$ are independent. Testing if a given linear
matroid of rank $k$ is uniform is known to be ${\mathsf{coNP}}$-complete
[OW02]. We denote them by $U_{k,m}$, the uniform matroid whose ground set is
of $m$ elements. Now notice that the above result is equivalent to checking if
the given linear matroid of rank $k$ is isomorphic to $U_{k,m}$. To complete
the argument, we use a folklore result that $U_{k,m}$ is representable over
any field $\mathbb{F}$ which has at least $m$ non-zero elements. We give some
details here since we have not seen an explicit description of this in the
literature.
###### Claim 3.3.
Let $|\mbox{$\mathbb{F}$}|>m$, $U_{k,m}$ has a representation over
$\mathbb{F}$.
###### Proof.
Let $\\{\alpha_{1},\ldots,\alpha_{m}\\}$ be distinct elements of $\mathbb{F}$,
and $\\{s_{1},\ldots,s_{m}\\}$ be elements of the ground set of $U_{k,m}$
Assign the vector
$(1,\alpha_{i},\alpha_{i}^{2},\ldots,\alpha_{i}^{k-1})\in\mbox{$\mathbb{F}$}^{k}$
to the element $s_{i}$. Any $k$ subset of these vectors forms a Vandermonde
matrix, and hence linearly independent. Any larger set is dependent since the
vectors are in $\mbox{$\mathbb{F}$}^{k}$. ∎
This gives us the following proposition.
###### Proposition 3.4.
LMI is ${\mathsf{co}}{\mathsf{NP}}$-hard.
The above proposition also holds when the representation is over infinite
fields. In this case, the proposition also more directly follows from a result
of Hlinený [Hli07], where it is shown that the problem of testing if a spike
(a special kind of matroids) represented by a matrix over $\mathbb{Q}$ is the
free spike is ${\mathsf{co}}{\mathsf{NP}}$ complete. He also derives a linear
representation for spikes.
Now we look at bounded rank variant of the problem. We denote by $\mbox{\sc
LMI}_{b}$ ($\mbox{\sc MI}_{b}$), the restriction of LMI (MI) for which the
input matrices have rank bounded by $b$. In the following we use the following
construction due to Babai [Bab78] to prove $\mbox{\sc
LMI}_{b}\equiv_{m}^{p}\mbox{\sc GI}$.
Given a graph $X=(V,E)$ ($3\leq k\leq d$, where, $d$ is the minimum vertex
degree of $X$), define a matroid $M=St_{k}(X)$ of rank $k$ with the ground set
as $E$ as follows: every subset of $k-1$ edges is independent in $M$ and every
subset of $E$ with $k$ edges is independent if and only if they do not share a
common vertex. Babai proved that $Aut(X)\cong Aut(St_{k}(X))$ and also gave a
linear representation for $St_{k}(X)$ (Lemma 2.1 in [Bab78]) for all $k$ in
the above range.
###### Theorem 3.5.
For any constant $b\geq 3$, $\mbox{\sc LMI}_{b}\equiv_{m}^{p}\mbox{\sc GI}$.
###### Proof.
$\mbox{\sc GI}\leq_{m}^{p}\mbox{\sc LMI}_{b}$: Let $X_{1}=(V_{1},E_{1})$ and
$X_{2}=(V_{2},E_{2})$ be the given GI instance. We can assume that the minimum
degree of the graph is at least $3$ since otherwise we can attach cliques of
size $n+1$ at every vertex. We note that from Babai’s proof we can derive the
following stronger conclusion.
###### Lemma 3.6.
$X_{1}\cong X_{2}\iff\forall~{}k\in[3,d],~{}St_{k}(X_{1})\cong St_{k}(X_{2})$
###### Proof.
Suppose $X_{1}\cong X_{2}$ via a bijection $\pi:V_{1}\to V_{2}$. (The
following proof works for any $k\in[3,d]$.) Let $\sigma:E_{1}\to E_{2}$ be the
map induced by $\pi$. That is $\sigma(\\{u,v\\})=\\{\pi(u),\pi(v)\\}$.
Consider an independent set $I\subseteq E_{1}$ in $St_{k}(X_{1})$. If $|I|\leq
k-1$ then $|\sigma(I)|\leq k-1$ and hence $\sigma(I)$ is independent in
$St_{k}(X_{2})$. If $|I|=k$, and let $\sigma(I)$ be dependent. This means that
the edges in $\sigma(I)$ share a common vertex $w$ in $X_{2}$. Since $\pi$ is
an isomorphism which induces $\sigma$, $\pi^{-1}(w)$ must be shared by all
edges in $I$. Thus $I$ is independent if and only if $\sigma(I)$ is
independent. Suppose $St_{k}(X_{1})\cong St_{k}(X_{2})$ via a bijection
$\sigma:E_{1}\to E_{2}$. By definition, any subset $H\subseteq E_{1}$ is a
hyperplane of $St_{k}(X_{1})$ if and only if $\sigma(H)$ is a hyperplane of
$St_{k}(X_{2})$. Now we use the following claim which follows from [Bab78].
###### Claim 3.7 ([Bab78]).
For any graph $X$, any dependent hyperplane in $St_{k}(X)$ is a maximal set of
edges which share a common vertex (forms a star) in $X$, and these are the
only dependent hyperplanes.
Now we define the graph isomorphism $\pi:V_{1}\to V_{2}$ as follows. For any
vertex $v$, look at the star $E_{1}(v)$ rooted at $v$, we know that
$\sigma(E_{1}(v))=E_{2}(v^{\prime})$ for some $v^{\prime}$. Now set
$\pi(v)=v^{\prime}$. From the above claim, $\pi$ is an isomorphism. ∎
It remains to show that representation for $St_{k}(X)$ ($X=(V,E)$) can be
computed in polynomial time. We choose $k=3$ (by the above proof, $\exists k$
and $\forall k$ in the Lemma 3.6 are equivalent). Now we show that the
representation of $St_{k}(X)$ given in [Bab78] is computable in polynomial
time. The representation of $St_{k}(X)$ is over a field $\mathbb{F}$ such that
$|\mbox{$\mathbb{F}$}|\geq|V|^{2k-1}$. For $e=\\{u,v\\}\in E$ assign a vector
$b_{e}=[1,(x_{u}+x_{v}),(x_{u}x_{v}),y_{e,1},\ldots,y_{e,k-3}]\in\mbox{$\mathbb{F}$}^{k}$,
where $x_{u},x_{v}$ and $y_{e,i}$ are distinct unknowns. To represent
$St_{k}(X)$ we need to ensure that the $k$-subsets of the columns
corresponding to a basis form a linearly independent set, and all the
remaining $k$-subsets form a dependent set. Babai [Bab78] showed that by the
above careful choice of $b_{e}$, it will be sufficient to ensure only the
independence condition. He also proved the existence of a choice of values for
the variables which achieves this if $|\mbox{$\mathbb{F}$}|\geq|V|^{2k-1}$.
We make this constructive. As $k$ is a constant, the number of bases is
bounded by ${\mathsf{poly}}(m)$. We can greedily choose the value for each
variable at every step, such that on assigning this value, the resulting set
of constant ($k\times k$) size matrices are non-singular. Since there exists a
solution, this algorithm will always find one. Thus we can compute a
representation for $St_{k}(X)$ in polynomial time.
$\mbox{\sc LMI}_{b}\leq_{m}^{p}\mbox{\sc GI}$: Let $A_{k\times m}$ and
$B_{k\times m}$ be two matrices of rank $b$ at the input. Now define the
following bipartite graph $X_{A}=(U_{A},V_{A},E_{A})$ (similarly for $X_{B}$),
where $U_{A}$ has a vertex for each column of $A$, and $V_{A}$ has a vertex
for each maximal independent set of $A$ (Notice that there are at most
${m\choose b}=O(m^{b})$ of them) and $\forall i\in U_{A},I\in V_{A},\
\\{i,I\\}\in E_{A}\iff i\in I$. Now we claim that $M(A)\cong M(B)\iff$
$X_{A}\cong X_{B}$ where the isomorphism maps $V_{A}$ to $V_{B}$, and which is
reducible to GI. It is easy to see that the matroid isomorphism can be
recovered from the map between the sets. ∎
Observe that the reduction $\mbox{\sc LMI}_{b}\leq_{m}^{p}\mbox{\sc GI}$ can
be done even if the input representation is an independent set oracle. This
gives the following corollary.
###### Corollary 3.8.
$\mbox{\sc LMI}_{b}\equiv_{m}^{p}\mbox{\sc MI}_{b}\equiv_{m}^{p}\mbox{\sc
GI}$.
## 4 Isomorphism Problem of Coloured Matroids
Vertex or edge colouring is a classical tool used extensively in proving
various results in graph isomorphism problem. We develop similar techniques
for matroid isomorphism problems too.
An edge-$k$-colouring of a graph $X=(V,E)$ is a function
$f:E\to\\{1,\ldots,k\\}$. Given two coloured graphs
$X_{1}=(V_{1},E_{1},f_{1})$ and $X_{2}=(V_{2},E_{2},f_{2})$, the Coloured-GMI
asks for an isomorphism which preserves the colours of the edges. Not
surprisingly, we can prove the following.
###### Lemma 4.1.
Coloured-GMI is ${\mathsf{AC}}^{0}$ many-one reducible to GMI.
###### Proof.
Let $X_{1}=(V_{1},E_{1},f_{1})$ and $X_{2}=(V_{2},E_{2},f_{2})$, be the two
$k$-coloured graphs at the input, with $n=|V_{1}|=|V_{2}|$. For every edge
$e=(u,v)\in E_{1}$ (respectively $E_{2}$), add a path
$P_{e}=\\{(u,v_{e,1}),(v_{e,1},v_{e,2}),\ldots,(v_{e,n+f_{1}(e)},v)\\}$ of
length $n+f_{1}(e)$ (respectively $n+f_{2}(e)$)Where $v_{e,1},\ldots
v_{e,n+f_{1}(e)}$ are new vertices. Let $X_{1}^{\prime}$ and $X_{2}^{\prime}$
be the two new graphs thus obtained. By definition, any 2-isomorphism between
$X_{1}^{\prime}$ and $X_{2}^{\prime}$ can only map cycles of equal length to
themselves. There are no simple cycles of length more than $n$ in the original
graphs. Thus, given any 2-isomorphism between $X_{1}^{\prime}$ and
$X_{2}^{\prime}$, we can recover a 2-isomorphism between $X_{1}$ and $X_{2}$
which preserves the colouring and vice versa. ∎
Now we generalize the above construction to the case of linear matroid
isomorphism. Coloured-LMI denotes the variant of LMI where the inputs are the
linear matroids $M_{1}$ and $M_{2}$ along with colour functions
$c_{i}:\\{1,\ldots,m\\}\to\mbox{$\mathbb{N}$},i\in\\{1,2\\}$. The problem is
to test if there is an isomorphism between $M_{1}$ and $M_{2}$ which preserves
the colours of the column indices. We have,
###### Lemma 4.2.
Coloured-LMI is ${\mathsf{AC}}^{0}$ many-one reducible to LMI.
###### Proof.
Let $M_{1}$ and $M_{2}$ be two coloured linear matroids represented over a
field $\mathbb{F}$. We illustrate the reduction where only one column index of
$M_{1}$ (resp. $M_{2}$) is coloured. Without loss of generality, we assume
that there are no two vectors in $M_{1}$ (resp.$M_{2}$) which are scalar
multiples of each other.
We transform $M_{1}$ and $M_{2}$ to get two matroids $M_{1}^{\prime}$ and
$M_{2}^{\prime}$. In the transformation, we add more columns to the matrix
(vectors to the ground set) and create dependency relations in such a way that
any isomorphism between the matroids must map these new vectors in $M_{1}$ to
the corresponding ones $M_{2}$.
We describe this transformation in a generic way for a matroid $M$. Let
$\\{e_{1},\ldots,e_{m}\\}$ be the column vectors of $M$, where
$e_{i}\in\mbox{$\mathbb{F}$}^{n}$. Let $e=e_{1}$ be the coloured vector in
$M$.
Choose $m^{\prime}>m$, we construct $\ell=m+m^{\prime}$ vectors $f_{1},\ldots
f_{\ell}\in\mbox{$\mathbb{F}$}^{n+m^{\prime}}$ as the columns of the following
$(n+m^{\prime})\times\ell$ matrix. The $i^{\mathrm{th}}$ column of the matrix
represents $f_{i}$.
$\left[\begin{array}[]{cccc|cccccccc}e_{11}&e_{21}&\ldots&e_{m1}&e_{11}&0&\ldots&0&0&\ldots&0\\\
e_{12}&e_{22}&\ldots&e_{m2}&0&e_{12}&\ldots&0&0&\ldots&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
e_{1m}&e_{2m}&\ldots&e_{mm}&0&0&\ldots&e_{1m}&0&\ldots&0\\\ \hline\cr
0&0&\ldots&0&1&-1&0&0&{2}&0\\\ \vdots&\vdots&\ldots&\vdots&0&1&-1&0&{2}&0\\\
\vdots&\vdots&\ldots&\vdots&\vdots&\vdots&\ddots&\ddots&{2}&\vdots\\\
0&0&\ldots&0&0&0&\ldots&&0&1&-1\\\ 0&0&\ldots&0&-1&0&\ldots&&0&0&1\\\
\end{array}\right]$
where $-1$ denotes the additive inverse of $1$ in $\mathbb{F}$. Denote the
above matrix as $M^{\prime}=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$. Let
$S=\\{f_{m+1},\ldots,f_{m+m^{\prime}}\\}$. We observe the following:
1. 1.
Columns of $B$ generate $e_{1}$. Since $C$ is a $0$-matrix $f_{1}\in Span(S)$.
2. 2.
Columns of $D$ are minimal dependent. Any proper subset of columns of $D$ will
split the $1$, $-1$ pair in at least a row and hence will be independent.
3. 3.
$S$ is linearly independent. Suppose not. Let
$\sum_{i=m}^{m+m^{\prime}}\alpha_{i}f_{i}=0$. Restricting this to the columns
of $B$ gives that $\alpha_{j}=0$ for first $j$ such that $e_{1j}\neq 0$. Thus
this gives a linearly dependent proper subset of columns of $B$, and
contradicts the above observation.
4. 4.
If for any $f\notin S$, $f=\sum_{f_{i}\in S}\alpha_{i}f_{i}$, then
$\alpha_{i}$’s must be the same.
Now we claim that the newly added columns respect the circuit structure
involving $e_{1}$. Let $\mathcal{C}$ and $\mathcal{C}^{\prime}$ denote the set
of circuits of $M$ and $M^{\prime}$ respectively.
###### Claim 4.3.
$\displaystyle\left\\{e_{1},e_{i_{2}},\ldots,e_{i_{k}}\right\\}\in\mathcal{C}$
$\displaystyle\iff$
$\displaystyle\left\\{f_{1},f_{i_{2}}\ldots,f_{i_{k}}\right\\}\in\mathcal{C}^{\prime}\textrm{
and }$
$\displaystyle\left\\{f_{i_{2}},\ldots,f_{i_{k}},f_{m+1},\ldots,f_{m+m^{\prime}}\right\\}\in\mathcal{C}^{\prime}$
###### Proof.
Suppose $c=\\{e_{1},e_{i_{2}},\ldots,e_{i_{k}}\\}$ is a circuit in $M$. Then
clearly $\\{f_{1},f_{i_{2}},\ldots,f_{i_{k}}\\}$ is a cycle, since they are
nothing but vectors in $c$ extended with $0$s. Since
$\\{f_{i_{2}},\ldots,f_{i_{k}}\\}$ and $\\{f_{m+1},\ldots,f_{m+m^{\prime}}\\}$
both generate $f_{1}$, the set
$F=\\{f_{i_{2}},\ldots,f_{i_{k}},f_{m+1},\ldots,f_{m+m^{\prime}}\\}$ is a
linearly dependent set. Now we argue that $F$ is a minimal dependent set, and
hence is a circuit. Denote by $G$ the set $\\{f_{i_{2}},\ldots,f_{i_{k}}\\}$.
Suppose not, let $F^{\prime}\subset F$ be linearly dependent. Since $S$ is
linearly independent (property 3 above), we note that
$F^{\prime}\not\subseteq\\{f_{m+1},\ldots,f_{m+m^{\prime}}\\}$. Therefore,
$f_{i_{j}}\in F^{\prime}$ for some $0\leq j\leq k$. Since $F^{\prime}$ is
dependent, express $f_{j}$ in terms of the other elements in $F^{\prime}$:
$f_{j}=\sum_{g\in G}\gamma_{g}g+\sum_{s\in S}\delta_{s}s$
Since $G$ is linearly independent, at least one of the $\delta_{s}$ should be
non-zero. Restrict this to the matrices $C$ and $D$. This gives a non-trivial
dependent proper subset of $D$ and hence a contradiction. ∎
From the above two observations and the fact that there is no other column in
$M$ which is a multiple of $e$, the set
$f(e)=\\{f_{1},f_{m+1},\ldots,f_{m+m^{\prime}}\\}$ is a unique circuit of
length $m^{\prime}+1$ in $M^{\prime}$, where $e$ is column which is coloured.
Now we argue about the isomorphism between $M_{1}^{\prime}$ and
$M_{2}^{\prime}$ obtained from the above operation, and there is a unique
circuit of length $m^{\prime}+1>m$ in both $M_{1}^{\prime}$ and
$M_{2}^{\prime}$ corresponding to two vectors $e\in M_{1}$ and $e^{\prime}\in
M_{2}$. Hence any matroid isomorphism should map these sets to each other.
From such an isomorphism, we can recover the a matroid isomorphism between
$M_{1}$ and $M_{2}$ that maps between $e$ and $e^{\prime}$, thus preserving
the colours. Indeed, if there is a matroid isomorphism between $M_{1}$ and
$M_{2}$, that can easily be extended to $M_{1}^{\prime}$ and $M_{2}^{\prime}$.
For the general case, let $k$ be the number of different colour classes and
$c_{i}$ denote the size of the $i$th colour class. Then for each vector $e$ in
the color class $i$, we add $l_{i}=m+m^{\prime}+i$ many new vectors, which
also increases the dimension of the space by $l_{i}$. Thus the total number of
vectors in the new matroid is $\sum c_{i}(l_{i})\leq m^{3}$. Similarly, the
dimension of the space is bounded by $m^{3}$. This completes the proof of
Lemma 4.2.
∎
## 5 Graphic Matroid Isomorphism
In this section we study GMI. Unlike in the case of the graph isomorphism
problem, an ${\mathsf{NP}}$ upper bound is not so obvious for GMI. We start
with the discussion of an ${\mathsf{NP}}$ upper bound for GMI.
As stated in Theorem 2.1, Whitney gave an exact characterization of when two
graphs are 2-isomorphic, in terms of three operations; twisting, cleaving and
identification. Note that it is sufficient to find 2-isomorphisms between
2-connected components of $X_{1}$ and $X_{2}$. In fact, any matching between
the sets of 2-connected components whose edges connect 2-isomorphic components
will serve the purpose. This is because, any 2-isomorphism preserves simple
cycles, and any simple cycle of a graph is always within a 2-connected
component. Hence we can assume that both the input graphs are 2-connected and
in the case of 2-connected graphs, twist is the only possible operation.
The set of separating pairs does not change under a twist operation. Despite
the fact that the twist operations need not commute, Truemper [Tru80] gave the
following bound.
###### Lemma 5.1 ([Tru80]).
Let $X$ be a 2-connected graph of $n$ vertices, and let $Y$ be a graph
$2$-isomorphic to $X$, then: $X$ can be transformed to graph $X^{\prime}$
isomorphic to $Y$ through a sequence at most $n-2$ twists.
Using this lemma we get an ${\mathsf{NP}}$ upper bound for GMI. Given two
graphs, $X_{1}$ and $X_{2}$, the ${\mathsf{NP}}$ machine just guesses the
sequence of $n-2$ separating pairs which corresponding to the 2-isomorphism.
For each pair, guess the cut w.r.t which the twist operation is to be done,
and apply each of them in sequence to the graph $X_{1}$ to obtain a graph
$X_{1}^{\prime}$. Now ask if $X_{1}^{\prime}\cong X_{2}^{\prime}$. This gives
an upper bound of $\exists.\mbox{\sc GI}\subseteq{\mathsf{NP}}$. Thus we have,
###### Proposition 5.2.
GMI is in ${\mathsf{NP}}$.
This can also be seen as an ${\mathsf{NP}}$-reduction from GMI to GI. Now we
will give a deterministic reduction from GMI to GI. Although, this does not
improve the ${\mathsf{NP}}$ upper bound, it implies that it is unlikely that
GMI is hard for ${\mathsf{NP}}$ (Using methods similar to that of Proposition
3.2, one can also directly prove that if GMI is ${\mathsf{NP}}$-hard, then
${\mathsf{PH}}$ collapses to the second level).
Now we state the main result of the paper:
###### Theorem 5.3.
$\mbox{\sc GMI}\leq_{T}^{p}\mbox{\sc GI}$
Let us first look into the case of 3-connected graphs. A separating pair is a
pair of vertices whose deletion leaves the graph disconnected. A 3-connected
graph is a connected graph which does not have any separating pairs. Whitney
([Whi32]) proved the following equivalence,
###### Theorem 5.4 (Whitney, [Whi32]).
$X_{1}$ and $X_{2}$ be 3-connected graphs, $X_{1}\cong_{2}X_{2}\iff X_{1}\cong
X_{2}$.
Before giving a formal proof of Theorem 5.3, we describe the idea roughly
here:
### Basic Idea:
Let $X_{1}$ and $X_{2}$ be the given graphs. From the above discussion, we can
assume that the given graph is 2-connected.
In [HT73], Hopcroft and Tarjan proved that every 2-connected graph can be
decomposed uniquely into a tree of 3-connected components, bonds or
polygons.222Cunningham et al. [CE80] shows that any graphic matroid $M(X)$ is
isomorphic to $M(X_{1})\oplus M(X_{2})\ldots\oplus
M(X_{k})/\\{e_{1},e_{2},\ldots,e_{k}\\}$, where $M(X_{1}),\ldots,M(X_{k})$ are
3-connected components, bonds or polygons of $M(X)$ and $e_{1},\ldots,e_{k}$
are the virtual edges. However, it is unclear if this can be turned into a
reduction from GMI to GI using edge/vertex colouring. Moreover, [HT73] showed
that this decomposition can be computed in polynomial time. The idea is to
then find the isomorphism classes of these 3-connected components using
queries to GI (see theorem 5.4), and then colour the tree nodes with the
corresponding isomorphism class, and then compute a coloured tree isomorphism
between the two trees produced from the two graphs.
A first mind block is that these isomorphisms between the 3-connected
components need not map separating pairs to separating pairs. We overcome this
by colouring the separating pairs (in fact the edge between them), with a
canonical label of the two sub trees which the corresponding edge connects. To
support this, we observe the following. There may be many isomorphisms between
two 3-connected components which preserves the colours of the separating
pairs. However, the order in which the vertices are mapped within a separating
pair is irrelevant, since any order will be canonical up to a twist operation
with respect to the separating pair.
So with the new colouring, the isomorphism between 3-connected components maps
a separating pair to a separating pair, if and only if the two pairs of sub
trees are isomorphic. However, even if this is the case, the coloured sub
trees need not be isomorphic. This creates a simultaneity problem of colouring
of the 3-connected components and the tree nodes and thus a second mind block.
We overcome this by colouring again using the code for coloured sub trees, and
then finding the new isomorphism classes between the 3-connected components.
This process is iterated till the colours stabilize on the tree as well as on
the individual separating pairs (since there are only linear number of
3-connected components). Once this is ensured, we can recover the
2-isomorphism of the original graph by weaving the isomorphism of the
3-connected components guided by the tree adjacency relationship. In addition,
if two 3-connected components are indeed isomorphic in the correctly aligned
way, the above colouring scheme, at any point, does not distinguish between
them.
Now we convert this idea into an algorithm and a formal proof.
### Breaking into Tree of 3-connected components:
We use the algorithm of Hopcroft and Tarjan [HT73] to compute the set of
3-connected components of a 2-connected graph in polynomial time. We will now
describe some details of the algorithm which we will exploit.
Let $X(V,E)$ be a $2$-connected graph. Let $Y$ be a connected component of
$X\setminus\\{a,b\\}$, where $a,b$ is a separating pair. $X$ is an excisable
component w.r.t $\\{a,b\\}$ if $X\setminus Y$ has at least $2$ edges and is
$2$-connected. The operation of excising $Y$ from $X$ results in two graphs:
$C_{1}=X\setminus Y$ plus a virtual edge joining $(a,b),$ and $C_{2}=$ the
induced subgraph on $X\cup\\{a,b\\}$ plus a virtual edge joining $(a,b)$. This
operation may introduce multiple edges.
The decomposition of $X$ into its 3-connected components is achieved by the
repeated application of the excising operation (we call the corresponding
separating pairs as excised pairs) until all the resulting graphs are free of
excisable components. This decomposition is represented by a graph $G_{X}$
with the 3-connected components of $X$ as its vertices and two components are
adjacent in $G_{X}$ if and only if they share a virtual edge. In the above
explanation, the graph $G_{X}$ need not be a tree as the components which
share a separating pair will form a clique.
To make it a tree, [HT73] introduces another component corresponding to the
virtual edges thus identifying all the virtual edges created in the same
excising operation with each other.
Instead, we do a surgery on the original graph $X$ and the graph $G_{X}$. We
add an edge between all the excised pairs (excised while obtaining $G_{X}$) to
get graph $X^{\prime}$. Notice that, following the same series of
decomposition gives a new graph $T_{X}$ which is the same as $G_{X}$ except
that the cliques are replaced by star centered at a newly introduced vertex
(component) corresponding to the newly introduced excised edges in
$X^{\prime}$. The newly introduced edges form a 3-connected component
themselves with one virtual edge corresponding to each edge of the clique they
replace.
We list down the properties of the tree $T_{X}$ for further reference. (1) For
every node in $t\in T_{X}$, there is exactly one 3-connected component in
$X^{\prime}$. We denote this by $c_{t}$. (2) For every edge $e=(u,v)\in
T_{X}$, there are exactly two virtual edges, one each in the 3-connected
components $c_{u}$ and $c_{v}$. We call these virtual edges as the twin edges
of each other. (3) For any given graph $X$, $T_{X}$ is unique up to
isomorphism (since $G_{X}$ is unique [HT73]). In addition, $T_{X}$ can be
obtained from $G_{X}$ in polynomial time.
In the following claim, we prove that this surgery in the graphs does not
affect the existence of 2-isomorphisms.
###### Claim 5.5.
$X_{1}\cong_{2}X_{2}\iff X_{1}^{\prime}\cong_{2}X_{2}^{\prime}$.
###### Proof.
Suppose $X_{1}\cong_{2}X_{2}$, via a bijection $\phi:E_{1}\to E_{2}$. This
induces a map $\psi$ between the sets of 3-connected components of $X_{1}$ and
$X_{2}$. By theorem 5.4, for every 3-connected component $c$ of $X_{1}$,
$c\cong\psi(c)$ (via say $\tau_{c}$; when $c$ is clear from the context we
refer to it as $\tau$).
We claim that $\psi$ is an isomorphism between $G_{1}$ and $G_{2}$. To see
this, consider an edge $e=(u,v)\in T_{1}$. This corresponds to two 3-connected
components $c_{u}$ and $c_{v}$ of $X_{1}$ which share a separating pair
$s_{1}$. The 3-connected components $\psi(c_{u})$ and $\psi(c_{v})$ must share
a separating pair say $s_{2}$; otherwise, the cycles spanning across $c_{u}$
and $c_{v}$ will not be preserved by $\phi$ which contradicts the fact that
$\phi$ is a 2-isomorphism. Hence $(\psi(c_{u}),\psi(c_{v}))$ correspond to an
edge in $G_{2}$. Therefore, $\psi$ is an isomorphism between $G_{1}$ and
$G_{2}$. In fact, this also gives an isomorphism between $T_{1}$ and $T_{2}$,
which in turn gives a map between the excised pairs of $X_{1}$ and $X_{2}$. To
define the 2-isomorphism between $X_{1}^{\prime}$ and $X_{2}^{\prime}$, we
extend the map $\psi$ to the excised edges.
To argue the reverse direction, let $X_{1}^{\prime}\cong_{2}X_{2}^{\prime}$
via $\psi$. In a very similar way, this gives an isomorphism between $T_{1}$
and $T_{2}$. The edge map of this isomorphism gives the map between the
excised pairs. Restricting $\psi$ to the edges of $X_{1}$ gives the required
2-isomorphism between $X_{1}$ and $X_{2}$. This is because, the cycles of
$X_{1}$($X_{2}$) are anyway contained in $X_{1}^{\prime}$ ($X_{2}^{\prime}$),
and the excised pairs does not interfere in the mapping. ∎
Thus it is sufficient to give an algorithm to test if
$X_{1}^{\prime}\cong_{2}X_{2}^{\prime}$, which we describe as follows.
Input: 2-connected graphs $X_{1}^{\prime}$ and $X_{2}^{\prime}$ and tree of
3-connected components $T_{1}$ and $T_{2}$.
Output:Yes if $X_{1}^{\prime}\cong_{2}X_{2}^{\prime}$, and No otherwise.
Algorithm:
Notation: $\mbox{\sc code}(T)$ denotes the canonical label333When $T$ is
coloured, $\mbox{\sc code}(T)$ is the code of the tree obtained after
attaching the necessary gadgets to the coloured nodes. Notice that even after
colouring, the graph is still a tree. In addition, for any $T$, $\mbox{\sc
code}(T)$ can be computed in ${\mathsf{L}}$ [Lin92]. for a tree $T$.
1. 1.
Initialize $T_{1}^{\prime}=T_{1}$, $T_{2}^{\prime}=T_{2}$.
2. 2.
Repeat
1. (a)
Set $T_{1}=T_{1}^{\prime}$, $T_{2}=T_{2}^{\prime}$.
2. (b)
For each edge $e=(u,v)\in T_{i}$, $i\in\\{1,2\\}$:
Let $T_{i}(e,u)$ and $T_{i}(e,v)$ be subtrees of $T_{i}$ obtained by deleting
the edge $e$, containing $u$ and $v$ respectively.
Colour virtual edges corresponding to the separating pairs in the components
$c_{u}$ and $c_{v}$ with the set $\\{\mbox{\sc code}(T_{i}(e,u)),\mbox{\sc
code}(T_{i}(e,v))\\}$. From now on, $c_{t}$ denotes the coloured 3-connected
component corresponding to node $t\in T_{1}\cup T_{2}$.
3. (c)
Let $S_{1}$ and $S_{2}$ be the set of coloured 3-connected components of
$X_{1}^{\prime}$ and $X_{2}^{\prime}$ and let $S=S_{1}\cup S_{2}$. Using
queries to GI (see observation 5.8) find out the isomorphism classes in $S$.
Let $C_{1},\ldots,C_{q}$ denote the isomorphism classes.
4. (d)
Colour each node $t\in T_{i}$, $i\in\\{1,2\\}$, with colour $\ell$ if
$c_{t}\in C_{\ell}$. (This gives two coloured trees $T_{1}^{\prime}$ and
$T_{2}^{\prime}$.)
Until ($\mbox{\sc code}(T_{i})\neq\mbox{\sc code}(T_{i}^{\prime})$, $\forall
i\in\\{1,2\\}$)
3. 3.
Check if $T_{1}^{\prime}\cong T_{2}^{\prime}$ preserving the colours. Answer
Yes if $T_{1}^{\prime}\cong T_{2}^{\prime}$, and No otherwise.
First we prove that the algorithm terminates in linear number of iterations of
the repeat-until loop. Let $q_{i}$ denote the number of isomorphism classes of
the set of the coloured 3-connected components after the $i^{th}$ iteration.
We claim that, if the termination condition is not satisfied, then
$|q_{i}|>|q_{i-1}|$. To see this, suppose the termination is not satisfied.
This means that the coloured tree $T_{1}^{\prime}$ is different from $T_{1}$.
This can happen only when the colour of a 3-connected component $c_{v}$, $v\in
T_{1}\cup T_{2}$ changes. In addition, this can only increase the isomorphism
classes. Thus $|q_{i}|>|q_{i-1}|$. Since $q$ can be at most $2n$, this shows
that the algorithm exits the loop after at most $2n$ steps.
Now we prove the correctness of the algorithm. We follow the notation
described in the algorithm.
###### Lemma 5.6.
$X_{1}^{\prime}\cong_{2}X_{2}^{\prime}$. $\iff$ $T_{1}^{\prime}\cong
T_{2}^{\prime}$.
###### Proof.
($\Rightarrow$) Suppose $X_{1}^{\prime}\cong_{2}X_{2}^{\prime}$, via a
bijection $\phi:E_{1}\to E_{2}$. This induces a map $\psi$ between the sets of
3-connected components of $X_{1}^{\prime}$ and $X_{2}^{\prime}$. By theorem
5.4, for every 3-connected component $c$ of $X_{1}^{\prime}$, $c\cong\psi(c)$
(via say $\tau_{c}$; when $c$ is clear from the context we refer to it as
$\tau$).
We claim that $\psi$ is an isomorphism between $T_{1}$ and $T_{2}$. To see
this, consider an edge $e=(u,v)\in T_{1}$. This corresponds to two 3-connected
components $c_{u}$ and $c_{v}$ of $X_{1}^{\prime}$ which share a separating
pair $s_{1}$. The 3-connected components $\psi(c_{u})$ and $\psi(c_{v})$ must
share a separating pair say $s_{2}$; otherwise, the cycles spanning across
$c_{u}$ and $c_{v}$ will not be preserved by $\phi$ which contradicts the fact
that $\phi$ is a 2-isomorphism. Hence $(\psi(c_{u}),\psi(c_{v}))$ correspond
to an edge in $T_{2}$. Therefore, $\psi$ is an isomorphism between $T_{1}$ and
$T_{2}$. So in what follows, we interchangeably use $\psi$ to be a map between
the set of 3-connected components as well as between the vertices of the tree.
Note that $\psi$ also induces (and hence denotes) a map between the edges of
$T_{1}$ and $T_{2}$.
Now we prove that $\psi$ preserves the colours attached to $T_{1}$ and $T_{2}$
after all iterations of the repeat-until loop in step 2. To simplify the
argument, we do it for the first iteration and the same can be carried forward
for any number of iterations. Let $T_{1}^{\prime}$ and $T_{2}^{\prime}$ be the
coloured trees obtained after the first iteration. We argue that $\psi$ itself
is an isomorphism between $T_{1}^{\prime}$ and $T_{2}^{\prime}$.
To this end, we prove that for any vertex $u$ in $T_{1}$,
$c_{u}\cong\psi(c_{u})$ even after colouring as in step 2b. That is, the map
preserves the colouring of the virtual edges in step 2b.
Consider any virtual edge $f_{1}$ in $c_{u}$, we know that $f_{2}=\tau(f_{1})$
is a virtual edge in $\psi(c_{u})$. Let $e_{1}=(u_{1},v_{1})$ and
$e_{2}=(u_{2},v_{2})$ be the tree edges in $T_{1}$ and $T_{2}$ corresponding
to $f_{1}$ and $f_{2}$ respectively. We know that, $e_{1}=\psi(e_{2})$. Since
$T_{1}\cong T_{2}$ via $\psi$, we have
$\left\\{\mbox{\sc code}(T_{1}(e_{1},u_{1})),\mbox{\sc
code}(T_{1}(e_{1},v_{1}))\right\\}=\left\\{\mbox{\sc
code}(T_{2}(e_{2},u_{2})),\mbox{\sc code}(T_{2}(e_{2},v_{2}))\right\\}.$
Thus, in Step 2b, the virtual edges $f_{1}$ and $f_{2}$ get the same colour.
Therefore, $c_{u}$ and $\psi(c_{u})$ belong to the same colour class after
step 2b. Hence $\psi$ is an isomorphism between $T_{1}^{\prime}$ and
$T_{2}^{\prime}$.
($\Leftarrow$) First, we recall some definitions needed in the proof. A center
of a tree $T$ is defined as a vertex $v$ such that $\max_{u\in T}d(u,v)$ is
minimized at $v$, where $d(u,v)$ is the number of edges in the unique path
from $u$ to $v$. It is known [Har69] that every tree $T$ has a center
consisting of a single vertex or a pair of adjacent vertices. The minimum
achieved at the center is called the height of the tree, denoted by $ht(T)$.
###### Claim 5.7.
Let $\psi$ be a colour preserving isomorphism between $T_{1}^{\prime}$ and
$T_{2}^{\prime}$, and $\chi_{t}$ is an isomorphism between the 3-connected
components $c_{t}$ and $c_{\psi(t)}$. Then,
$X_{1}^{\prime}\cong_{2}X_{2}^{\prime}$ via a map $\sigma$ such that $\forall
t\in T_{1}^{\prime}$, $\forall e\in c_{t}\cap E_{1}:\sigma(e)=\chi_{t}(e)$
where $E_{1}$ is the set of edges in $X_{1}^{\prime}$.
###### Proof.
The proof is by induction on height of the trees
$h=ht(T_{1}^{\prime})=ht(T_{2}^{\prime})$, where the height (and center) is
computed with respect to the underlying tree ignoring colours on the vertices.
Base case is when $h=0$; that is, $T_{1}^{\prime}$ and $T_{2}^{\prime}$ have
just one node (3-connected component) without any virtual edges. Simply define
$\sigma=\chi$. By Theorem 5.4, this gives the required 2-isomorphism.
Suppose that if $h=ht(T_{1}^{\prime})=ht(T_{2}^{\prime})<k$, the above claim
is true. For the induction step, suppose further that $T_{1}^{\prime}\cong
T_{2}^{\prime}$ via $\psi$, and $ht(T_{1}^{\prime})=ht(T_{2}^{\prime})=k$.
Notice that $\psi$ should map the center(s) of $T_{1}$ to that of $T_{2}$. We
consider two cases (we present one case here, and the other in the appendix).
In the first case, $T_{1}^{\prime}$ and $T_{2}^{\prime}$ have unique centers
$\alpha$ and $\beta$. It is clear that $\psi(\alpha)=\beta$. Let $c_{1}$ and
$c_{2}$ be the corresponding coloured (as in step 2b) 3-connected components.
Therefore, there is a colour preserving isomorphism $\chi=\chi_{\alpha}$
between $c_{\alpha}$ and $c_{\beta}$. Let $f_{1},\ldots f_{k}$ be the virtual
edges in $c_{\alpha}$ corresponding to the tree edges
$e_{1}=(\alpha,v_{1}),\ldots,e_{k}=(\alpha,v_{k})$ where $v_{1},\ldots,v_{k}$
are neighbors of $\alpha$ in $T_{1}^{\prime}$. Denote $\psi(e_{i})$ by
$e_{i}^{\prime}$, and $\psi(v_{i})$ by $v_{i}^{\prime}$.
Observe that only virtual edges are coloured in the 3-connected components in
step 2b while determining their isomorphism classes. Therefore, for each $i$,
$\chi(f_{i})$ will be a virtual edge in $c_{\beta}$, and in addition, with the
same colour as $f_{i}$. That is,
$\left\\{\mbox{\sc code}(T_{1}(e_{i},\alpha)),\mbox{\sc
code}(T_{1}(e_{i},v_{i}))\right\\}=\left\\{\mbox{\sc
code}(T_{2}(e_{i}^{\prime},\beta)),\mbox{\sc
code}(T_{2}(e_{i}^{\prime},v_{i}^{\prime})))\right\\}$.
Since $\alpha$ and $\beta$ are the centers of $T_{1}^{\prime}$ and
$T_{2}^{\prime}$, it must be the case that in the above set equality,
$\mbox{\sc code}(T_{1}(e_{i},v_{i}))$ $=$ $\mbox{\sc
code}(T_{2}(e_{i}^{\prime},v_{i}^{\prime}))$. From the termination condition
of the algorithm, this implies that $\mbox{\sc
code}(T_{1}^{\prime}(e_{i},v_{i}))=\mbox{\sc
code}(T_{2}^{\prime}(e_{i}^{\prime},v_{i}^{\prime}))$. Hence,
$T_{1}^{\prime}(e_{i},v_{i})\cong
T_{2}^{\prime}(e_{i}^{\prime},v_{i}^{\prime})$. In addition,
$ht(v_{i})=ht(v_{i}^{\prime})<k$. Let $X_{f_{i}}^{\prime}$ and
$X_{\chi(f_{i})}^{\prime}$ denote the subgraphs of $X_{1}^{\prime}$ and
$X_{2}^{\prime}$ corresponding to $T_{1}^{\prime}(e_{i},v_{i})$ and
$T_{2}^{\prime}(e_{i}^{\prime},v_{i}^{\prime})$ respectively. By induction
hypothesis, the graphs $X_{f_{i}}^{\prime}$ and $X_{\chi(f_{i})}^{\prime}$ are
2-isomorphic via $\sigma_{i}$ which agrees with the corresponding $\chi_{t}$
for $t\in T_{1}^{\prime}(e_{i},v_{i})$. Define $\pi_{i}$ as a map between the
set of all edges, such that it agrees with $\sigma_{i}$ on all edges of
$X_{f(i)}^{\prime}$ and with $\chi_{t}$ (for $t\in
T_{1}^{\prime}(e_{i},v_{i})$) on the coloured virtual edges.
We claim that $\pi_{i}$ must map the twin-edge of $f_{i}$ to twin-edge of
$\tau(f_{i})$. Suppose not. By the property of the colouring, this implies
that there is a subtree of $T_{1}^{\prime}(e_{i},v_{i})$ isomorphic to
$T_{1}^{\prime}\setminus T_{1}^{\prime}(e_{i},v_{i})$. This contradicts the
assumption that $c_{\alpha}$ is the center of $T_{1}^{\prime}$.
For each edge $e\in E_{1}$, define $\sigma(e)$ to be $\chi(e)$ when $e\in
c_{\alpha}$ and to be $\pi_{i}(e)$ when $e\in E_{f_{i}}~{}(\textrm{edges of
$X_{f_{i}}$})$. From the above argument, $\chi=\chi_{\alpha}$ and $\sigma_{i}$
indeed agrees on where it maps $f_{i}$ to. This ensures that every cycle
passing through the separating pairs of $c_{\alpha}$ gets preserved. Thus
$\sigma$ is a 2-isomorphism between $X_{1}^{\prime}$ and $X_{2}^{\prime}$.
For case 2, let $T_{1}^{\prime}$ and $T_{2}^{\prime}$ have two centers
$(\alpha_{1},\alpha_{2})$ and $(\beta_{1},\beta_{2})$ respectively. It is
clear that $\psi(\\{\alpha_{1},\alpha_{2}\\})=\\{\beta_{1},\beta_{2}\\}$.
Without loss of generality, we assume that $\psi(\alpha_{1})=\beta_{1}$,
$\psi(\alpha_{2})=\beta_{2}$. Therefore, there are colour preserving
isomorphisms $\chi_{1}$ from $c_{\alpha_{1}}$ to $c_{\beta_{1}}$ and
$\chi_{2}$ from $c_{\alpha_{2}}$ and $c_{\beta_{2}}$. Define $\chi(e)$ as
follows:
$\chi(e)=\left\\{\begin{array}[]{ll}\chi_{1}(e)&~{}~{}e\in c_{\alpha_{1}}\\\
\chi_{2}(e)&~{}~{}e\in c_{\alpha_{2}}\end{array}\right.$
$c_{\alpha}=\cup_{i}c_{\alpha_{i}},~{}~{}~{}c_{\beta}=\cup_{i}c_{\beta_{i}}$
With this notation, we can appeal to the proof in the case 1, and construct
the 2-isomorphism $\sigma$ between $X_{1}^{\prime}$ and $X_{2}^{\prime}$.
∎
This completes the proof of correctness of the algorithm (Lemma 5.6).
∎
To complete the proof of Theorem 5.3, we need the following observation,
###### Observation 5.8.
Coloured-GMI for 3-connected graphs reduces to GI.
Observing that the above construction does not use non-planar gadgets, we get
the following.
###### Corollary 5.9.
Given two planar matroids, $M(X_{1})$ and $M(X_{2})$, testing if
$M(X_{1})\cong M(X_{2})$ can be performed in ${\mathsf{P}}$.
Now we give a polynomial time many-one reduction from $\mbox{\sc MI}_{b}$ to
GMI. Let $M_{1}$ and $M_{2}$ be two matroids of rank $b$ over the ground set
$S_{1}$ and $S_{2}$. Let ${\mathcal{C}}_{1}$ and ${\mathcal{C}}_{2}$
respectively denote the set of cycles of $M_{1}$ and $M_{2}$. Note that
$|{\mathcal{C}}_{1}|,|{\mathcal{C}}_{2}|\leq m^{b+1}$.
Define graphs $X_{1}=(V_{1},E_{1})$ (respectively for $X_{2}=(V_{2},E_{2})$)
as follows. For each circuit $c=\\{s_{i_{1}},\ldots,s_{i_{\ell}}\\}\subseteq
S_{1}$ in $M_{1}$, $X_{1}$ contains a simple cycle
$\\{e(c,s_{i_{1}}),\ldots,e(c,s_{i_{\ell}})\\}$. Now pairwise interconnect all
the endpoints of the edges corresponding to each of the ground set elements
(these edges form a clique), and colour these edges as red and the remaining
edges as blue. Now we claim the following.
###### Lemma 5.10.
$M_{1}\cong M_{2}$ if and only if $X_{1}\cong_{2}X_{2}$.
###### Proof.
Suppose $M_{1}\cong M_{2}$, via a map $\phi:S_{1}\to S_{2}$. This gives a map
$\psi$ between the blue edges of the graphs $X_{1}$ and $X_{2}$ which
preserves blue cycles. Now we extend this to the red edges. Take a red edge
$r$, there are two blue edges $e_{1}$ and $e_{2}$ which share an endpoint with
$r$. We know that $e_{1}$ and $e_{2}$ are corresponding to the same ground set
element (say $s$). Thus $\psi(e_{1})$ and $\psi(e_{2})$ correspond to the same
ground set element $\phi(s)$, and hence shares a red edge in $X_{2}$. Thus
$\psi$ can be extended to preserve the red edges. Hence $X_{1}\cong_{2}X_{2}$.
Conversely, suppose $X_{1}\cong_{2}X_{2}$ via $\psi:E_{1}\to E_{2}$. Define
$\phi:S_{1}\to S_{2}$ as follows: For $s\in S_{1}$ let $R_{s}$ denote the
clique in $X_{1}$ corresponding to $s$. $R_{s}$ is either a single blue edge
or a clique on at least $4$ vertices (in the latter case it is 3-connected).
Thus $\psi$ should map $R_{s}$ to $R^{\prime}_{s^{\prime}}$ for some
$s^{\prime}$ in $S_{2}$.. Define $\phi(s)=s^{\prime}$. Now we argue that
$\psi$ is an isomorphism between $M_{1}$ and $M_{2}$. Let
$c=\\{s_{1},\ldots,s_{\ell}\\}\subseteq S_{1}$ be a cycle in $M_{1}$.
$\displaystyle c\in{\mathcal{C}}_{1}$ $\displaystyle\iff$
$\displaystyle\bigcap_{i}\psi(R_{s_{i}})\textrm{ is a {\sc blue} cycle in
$X_{1}$}$ $\displaystyle\iff$
$\displaystyle\bigcap_{i}\psi(R^{\prime}_{s^{\prime}_{i}})\textrm{ is a {\sc
blue} cycle in $X_{2}$}$ $\displaystyle\iff$
$\displaystyle\phi(c)\in{\mathcal{C}}_{2}$
∎
From the above construction, we have the following theorem.
###### Theorem 5.11.
$\mbox{\sc MI}_{b}\leq^{p}_{m}\mbox{\sc GMI}$.
Thus we have,
###### Theorem 5.12.
$\mbox{\sc GI}\equiv^{p}_{T}\mbox{\sc GMI}\equiv^{p}_{T}\mbox{\sc
MI}_{b}\equiv^{p}_{T}\mbox{\sc LMI}_{b}$
## 6 Matroid Automorphism Problem
With any isomorphism problem, there is an associated automorphism problem i.e,
to find a generating set for the automorphism group of the underlying object.
Relating the isomorphism problem to the corresponding automorphism problem
gives access to algebraic tools associated with the automorphism groups. In
the case of graphs, studying automorphism problem has been fruitful.(e.g. see
[Luk80, BGM82, AK02].) In this section we turn our attention to Matroid
automorphism problem.
An automorphism of a matroid $M=(S,{\mathcal{C}})$ (where $S$ is the ground
set and ${\mathcal{C}}$ is the set of circuits) is a permutation $\phi$ of
elements of $S$ such that $\forall C\subseteq
S,~{}C\in{\mathcal{C}}\iff\phi(C)\in{\mathcal{C}}$. $Aut(M)$ denotes the group
of automorphisms of the matroid $M$. When the matroid is graphic we denote by
$Aut(X)$ and $Aut(M_{X})$ the automorphism group of the graph and the graphic
matroid respectively.
To begin with, we note that given a graph $X$, and a permutation $\pi\in
S_{m}$, it is not clear apriori how to check if $\pi\in Aut(M_{X})$
efficiently. This is because we need to ensure that $\pi$ preserves all the
simple cycles, and there could be exponentially many of them. Note that such a
membership test (given a $\pi\in S_{n}$) for $Aut(X)$ can be done easily by
testing whether $\pi$ preserves all the edges. We provide an efficient test
for this problem.
We use the notion of a cycle bases of $X$. A cycle basis of a graph $X$ is a
minimal set of cycles $\mathcal{B}$ of $X$ such that every cycle in $X$ can be
written as a linear combination (viewing every cycle as a vector in
$\mbox{$\mathbb{F}$}_{2}^{m}$) of the cycles in ${\mathcal{B}}$. Let
$\mathscr{B}$ denote the set of all cycle basis of the graph $X$.
###### Lemma 6.1.
Let $\pi\in S_{n}$,
$\exists{\mathcal{B}}\in{\mathscr{B}}:\pi({\mathcal{B}})\in{\mathscr{B}}\implies\forall{\mathcal{B}}\in{\mathscr{B}}:\pi({\mathcal{B}})\in{\mathscr{B}}$
###### Proof.
Let ${\mathcal{B}}=\\{b_{1},\ldots b_{\ell}\\}\in{\mathscr{B}}$ such that
$\pi({\mathcal{B}})=\\{\pi(b_{1}),\ldots,\pi(b_{\ell})\\}$ is a cycle basis.
Now consider any other cycle basis
${\mathcal{B}}^{\prime}=\\{b_{1}^{\prime},\ldots,b_{k}^{\prime}\\}\in{\mathscr{B}}$.
Thus, $b_{i}=\sum_{j}\alpha_{j}b_{j}^{\prime}$. This implies,
$\pi(b_{i})=\sum_{j}\alpha_{j}\pi(b_{j}^{\prime}).$
Thus,
$\pi(B^{\prime})=\\{\pi(b_{1}^{\prime}),\ldots,\pi(b_{\ell}^{\prime})\\}$
forms a cycle basis. ∎
###### Lemma 6.2.
Let $\pi\in S_{m}$, and let ${\mathcal{B}}\in{\mathscr{B}}$, then $\pi\in
Aut(M_{X})\iff\pi({\mathcal{B}})\in{\mathscr{B}}$.
###### Proof.
Let ${\mathcal{B}}=\\{b_{1},\ldots,b_{\ell}\\}$ be the given cycle basis.
For the forward direction, suppose $\pi\in Aut(M_{X})$. That is, $C\subseteq
E$ is a cycle in $X$ if and only if $\pi(C)$ is also a cycle in $X$. Let $C$
be any cycle in $X$, and let $D=\pi^{-1}(C)$. Since
${\mathcal{B}}\in{\mathscr{B}}$, we can write, $D=\sum_{i}\alpha_{i}b_{i}$,
and hence $C=\sum_{i}\alpha_{i}\pi(b_{i})$. Hence $\pi({\mathcal{B}})$ forms a
cycle basis for $X$.
For the reverse direction, suppose $\pi({\mathcal{B}})$ is a cycle basis of
$X$. Let $C$ be any cycle in $X$. We can write $C=\sum_{i}\alpha_{i}b_{i}$.
Hence, $\pi(C)=\sum_{i}\alpha_{i}\pi(b_{i})$. As $\pi$ is a bijection, we have
$\pi(b_{i}\cap b_{j})=\pi(b_{i})\cap\pi(b_{j})$. Thus $\pi(C)$ is also a cycle
in $X$. Since $\pi$ extends to a permutation on the set of cycles, we get that
$C$ is a cycle if and only if $\pi(C)$ is a cycle. ∎
Using Lemmas 6.1 and 6.2 it follows that, given a permutation $\pi$, to test
if $\pi\in Aut(M_{X})$ it suffices to check if for a cycle basis
${\mathcal{B}}$ of $X$, $\pi({\mathcal{B}})$ is also a cycle basis. Given a
graph $X$ a cycle basis ${\mathcal{B}}$ can be computed in polynomial time
(see e.g, [Hor87]). Now it suffices to show:
###### Lemma 6.3.
Given a permutation $\pi\in S_{m}$, and a cycle basis
${\mathcal{B}}\in{\mathscr{B}}$, testing whether $\pi({\mathcal{B}})$ is a
cycle basis, can be done in polynomial time.
###### Proof.
To check if $\pi({\mathcal{B}})$ is a cycle basis, it is sufficient to verify
that every cycle in ${\mathcal{B}}=\\{b_{1},\ldots,b_{\ell}\\}$ can be written
as a $\mbox{$\mathbb{F}$}_{2}$-linear combination of the cycles in
${\mathcal{B}}^{\prime}=\\{b_{1}^{\prime},\ldots,b_{\ell}^{\prime}\\}=\pi({\mathcal{B}})$.
This can be done as follows.
For $b_{i}\in{\mathcal{B}}$, let $\pi(b_{i})=b_{i}^{\prime}$. View $b_{i}$ and
$b_{i}^{\prime}$ as vectors in $\mbox{$\mathbb{F}$}_{2}^{m}$. Let $b_{ij}$
(resp. $b_{ij}^{\prime}$) denote the $j^{\mathrm{th}}$ component of $b_{i}$
(resp. $b_{i}^{\prime}$). Construct the set of linear equations,
$b_{ij}^{\prime}=\sum_{b_{k}\in{\mathcal{B}}}x_{ik}b_{kj}$ where $x_{ik}$ are
unknowns. There are exactly $\ell$ $b_{i}^{\prime}$s and each of them gives
rise to exactly $m$ equations like this. This gives a system $I$ of $\ell m$
linear equations in $\ell^{2}$ unknowns such that, $\pi(B)$ is a cycle basis
if and only if $I$ has a non-trivial solution. This test can indeed be done in
polynomial time. ∎
This gives us the following:
###### Theorem 6.4.
Given any $\pi\in S_{m}$, the membership test for $\pi$ in $Aut(M_{X})$ is in
${\mathsf{P}}$.
Notice that similar arguments can also give another proof of Proposition 5.2.
As in the case of graphs, we can define automorphism problems for matroids.
$\mbox{\sc Matroid Automorphism}(\mbox{\sc MA})$: Given a matroid $M$ as
independent set oracle, compute a generating set for $Aut(M)$.
We define GMA and LMA as the corresponding automorphism problems for graphic
and linear matroids, when the input is a graph and matrix respectively. Using
the colouring techniques from Section 4, we prove the following equivalence.
###### Theorem 6.5.
$\mbox{\sc LMI}\equiv_{T}^{p}\mbox{\sc LMA}$, and $\mbox{\sc
GMI}\equiv_{T}^{p}\mbox{\sc GMA}$.
###### Proof.
This proof follows a standard idea due to Luks [Luk93]. We argue the forward
direction as follows. Given two matrices $M_{1}$ and $M_{2}$, form the new
matrix $M$ as,
$M=\begin{bmatrix}M_{1}&0\\\ 0&M_{2}\end{bmatrix}$
Now using queries to LMA construct the generating set of $Aut(M)$. Check if at
least one of the generators map the columns in $M$ corresponding to columns of
$M_{1}$ to those corresponding to the columns of $M_{2}$.
To see the other direction, we use the colouring idea, and the rest of the
details is standard. The idea is to find the orbits of each element of the
ground set as follows: For every element of $e\in S$, for each $f\in S$,
colour $e$ and $f$ by the same colour to obtain coloured matroids $M_{1}$ and
$M_{2}$. Now by querying to LMI we can check if $f$ is in the orbit of $e$.
Thus we can construct the orbit structure of $Aut(M)$ and hence compute a
generating set.
Using similar methods we can prove $\mbox{\sc GMI}\equiv_{T}^{p}\mbox{\sc
GMA}$. ∎
## 7 Conclusion and Open Problems
We studied the matroid isomorphism problem under various input representations
and restriction on the rank of the matroid. We proved that graph isomorphism,
graphic matroid isomorphism and bounded rank version of matroid isomorphism
are all polynomial time equivalent.
In addition, we find it interesting that in the bounded rank case, $\mbox{\sc
MI}_{b}$ and $\mbox{\sc LMI}_{b}$ are equivalent, though there exist matroids
of bounded rank which are not representable over any field. Some of the open
questions that we see are as follows:
* •
Our results provide new possibilities to attack the graph isomorphism problem.
For example, it will be interesting to prove a ${\mathsf{co}}{\mathsf{NP}}$
upper bound for $\mbox{\sc LMI}_{b}$. Note that this will imply that
$\mbox{\sc GI}\in{\mathsf{NP}}\cap{\mathsf{co}}{\mathsf{NP}}$. Similarly, are
there special cases of GMI (other than what is translated from the bounds for
GI) which can be solved in polynomial time?
* •
The representations of the matroid in the definition of LMI is over fields of
size at least $m$ and at most ${\mathsf{poly}}(m)$, where $m$ is the size of
the ground set of the matroid. This is critically needed for the observation
of ${\mathsf{co}}{\mathsf{NP}}$-hardness. One could ask if the problem is
easier over fixed finite fields independent on the input. However, we note
that, by our results, it follows that this problem over
$\mbox{$\mathbb{F}$}_{2}$ is already hard for GI. It will still be interesting
to give a better (than the trivial $\Sigma_{2}$) upper bound for linear
matroids represented over fixed finite fields (even for
$\mbox{$\mathbb{F}$}_{2}$).
* •
Can we use the colouring technique of linear matroid isomorphism to reduce the
general instances of linear matroid isomorphism to isomorphism testing of
“simpler components” of the matroid?
* •
Can we make the reduction from GMI to GI many-one? Can we improve the
complexity of this reduction in the general case?
## 8 Acknowledgements
We thank V. Arvind and Meena Mahajan for providing us with useful inputs and
many insightful discussions, James Oxley for sharing his thoughts while
responding to our queries about matroid isomorphism. We also thank the
anonymous referees for providing us many useful pointers to the literature.
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## Appendix A Proof of Claim 3.7
Claim 3.7: [Bab78] For any graph $X$, any hyperplane of $St_{k}(X)$ is a
maximal set edges which share a common vertex (forms a star) in $X$, and these
are the only hyperplanes.
###### Proof.
To see the first part, note that since $k$ is at most the min-degree of the
graph, any maximal set of edges which forms a star is a hyperplane. To argue
that these are the only hyperplanes, suppose $H$ is a hyperplane whose edges
do not share a common vertex in $X$. Since any independent set cannot be a
hyperplane, $H$ has size at least $k$, and the rank of $H$ by definition is
$k-1$. Take any $k$ sized subset $S$ of $H$. It has to be a star (say at
vertex $v$), since otherwise $H$ will have rank $k$. Since $H$ by itself is
not a star, there is an edge $e$ which does not have $v$ as its end point. But
then, $H$ contains a $k$ sized subset which is not a star, and hence
independent. This is a contradiction. ∎
## Appendix B 3-connected Coloured-GMI
By combining Lemma 4.1 with Theorem 5.4, we prove the following corollary
which we present although we do not need it explicitly in the paper.
###### Corollary B.1.
Let $X_{1}$ and $X_{2}$ be two 3-connected graphs with given edge colourings,
testing if there is a colour preserving 2-isomorphism between $X_{1}$ and
$X_{2}$, can be reduced to GI.
###### Proof.
We follow the notation in the proof of Lemma 4.1. Let $X_{1}^{\prime}$ and
$X_{2}^{\prime}$ be the two graphs obtained from $X_{1}$ and $X_{2}$ by
attaching the colouring gadgets. Now we claim that
$X_{1}^{\prime}\cong_{2}X_{2}^{\prime}\iff X_{1}^{\prime}\cong
X_{2}^{\prime}$.
One direction is easy to see. $X_{1}^{\prime}\cong X_{2}^{\prime}\Rightarrow
X_{1}^{\prime}\cong_{2}X_{2}^{\prime}$. To see the other direction, suppose
$X_{1}^{\prime}\cong_{2}X_{2}^{\prime}$ via $\sigma$. By proof of Lemma 4.1
this induces a 2-isomorphism between $X_{1}$ and $X_{2}$, which in turn
induces an isomorphism $\tau$ between $X_{1}$ and $X_{2}$. Observe that $\tau$
also preserves colours on the edges of $X_{1}$ and $X_{2}$.
Let $\pi$ be the map between the vertices of $X_{1}^{\prime}$ and
$X_{2}^{\prime}$ induced by $\tau$. Note that $\pi$ is not defined on the new
vertices introduced by the colouring gadgets. However, $\pi$ already preserves
all the edges between $X_{1}^{\prime}$ and $X_{2}^{\prime}$ except the newly
introduced edges.
Since $\tau$ preserves edge colours in $X_{1}$, any coloured edge $e\in
X_{1}$, the paths $P_{e}$ and $P_{\tau(e)}$, introduced while constructing
$X_{1}^{\prime}$ and $X_{2}^{\prime}$ are of the same length. Hence the vertex
map $\pi$ can be extended to the vertices in these paths to an edge preserving
vertex map between $X_{1}^{\prime}$ and $X_{2}^{\prime}$. ∎
## Appendix C A hardness result for GMI
We show that known hardness results of GI ([Tor04]) can be adapted to the case
of GMI. This is subsumed by the many-one reduction from GI to GMI. But we
state this observation here anyway.
###### Theorem C.1.
GMI is ${\mathsf{NL}}$-hard under ${\mathsf{AC}}^{0}$ many one reductions.
The hardness is proved using ideas similar to that in [Tor04]. We include the
details of this modified part of the proof.
Consider the graph $X(k)=(V,E)$ where,
$\displaystyle V$ $\displaystyle=$
$\displaystyle\\{x_{0},\ldots,x_{k-1},y_{0},\ldots,y_{k-1},z_{0},\ldots,z_{k-1}\\}$
$\displaystyle\cup\\{u_{i,j}~{}|~{}0\leq i,j\leq k-1\\}$ $\displaystyle E$
$\displaystyle=$ $\displaystyle\\{(x_{i},u_{i,j})~{}|~{}0\leq i,j\leq k-1\\}$
$\displaystyle\cup\\{(y_{j},u_{i,j})~{}|~{}0\leq i,j\leq k-1\\}$
$\displaystyle\cup\\{(u_{i,j},z_{i\oplus j})~{}|~{}0\leq i,j\leq k-1\\}$
The following is easy to verify for the above graph.
###### Observation C.2.
For $k\geq 3$, the graph $X(k)$ is 3-connected.
Toran [Tor04] argued that for any $a,b\in\\{0,\ldots,k-1\\}$ the graph $X(k)$
has a unique automorphism which maps vertices $x_{i}\to x_{a\oplus i}$ and
$y_{i}\to y_{b\oplus i}$, which also maps $z_{i}\to z_{i\oplus a\oplus b}$.
Combining this with Proposition 5.4, and the above observation, we get the
following lemma.
###### Lemma C.3.
For any $k\geq 3$, given any $a,b\in\\{0,\ldots,k-1\\}$, there exists a unique
automorphism for the matroid $M_{X(k)}$ which maps
$\displaystyle(x_{i},u_{i,j})$ $\displaystyle\to$ $\displaystyle(x_{i\oplus
a},u_{(i\oplus a,j)})$ $\displaystyle(y_{j},u_{i,j})$ $\displaystyle\to$
$\displaystyle(y_{i\oplus b},u_{(i,j\oplus a)})$
$\displaystyle(u_{i,j},z_{i\oplus j})$ $\displaystyle\to$
$\displaystyle(u_{i\oplus a,j\oplus b},z_{i\oplus j\oplus a\oplus b})$
###### Lemma C.4.
For $k\geq 3$, GMI is ${\mathsf{Mod}}_{k}{\mathsf{L}}$-hard under
${\mathsf{AC}}^{0}$ many-one reductions.
###### Proof.
Let $C$ be a circuit with ${\mathsf{Mod}}_{k}$ gates for $k\geq 3$. Without
loss of generality, we assume that each gate is of fan-in 2. We construct a
new graph $X$ as follows : For each gate $g$ in $C$, $X$ contains a copy of
the graph $X(k)$, denoted by $X_{g}$. Colour each edge of $X_{g}$ with colour
$g$ (see Lemma 4.1). If gate $g$ has $h_{1}$ and $h_{2}$ as its inputs
identify $x_{0},\ldots x_{k-1}$ and $y_{0},\ldots,y_{k-1}$ of $X_{g}$ with
$z_{0},\ldots,z_{k-1}$ of $X_{h_{1}}$ and $X_{h_{2}}$ respectively. Let $r$ be
the root gate of $C$, $g_{1},\ldots g_{m}$ be the gates which receive the
inputs directly, then the following claim can be verified by induction on the
circuit structure.
###### Claim C.5.
$C$ evaluates to $\ell\in\\{0,\ldots,k-1\\}$ on input $a_{1},\ldots,a_{n}$ if
and only if $M_{X}$ has an automorphism which maps, for the gate $g_{i}$
receiving the input $a_{j}$ as its ’x’ input, in $X_{g_{i}}$
$(x_{i},u_{i,t})\to(x_{i\oplus a_{j}},u_{(i\oplus a_{j},t)})$
for all $0\leq t\leq k-1$, and in $X_{r}$, for all $a\oplus b=\ell$, $j\oplus
t=i$,
$(u_{t,j},z_{i})\to(u_{t\oplus a,j\oplus b},z_{i\oplus\ell})$
∎
###### Proof.
(of theorem C.1) Using ideas from [Tor04], it is easy to see that we can write
an ${\mathsf{NL}}$ computation as a series ${\mathsf{Mod}}_{k}{\mathsf{L}}$
computations, for $3\leq k\leq 2n$. Thus, the theorem follows. ∎
The following corollary is immediate as in (Theorem 4.4, [Tor04]), using
Chinese remaindering, and the above reduction.
###### Corollary C.6.
Every ${\mathsf{Gap}}{\mathsf{L}}$-function is ${\mathsf{AC}}^{0}$ many-one
reducible to GMI.
## Appendix D Testing Uniformity of Matroids
A problem of testing if a given matroid is uniform, is clearly as a special
case of the matroid isomorphism problem, where one of the matroid is uniform.
The problem is known to be ${\mathsf{co}}{\mathsf{NP}}$-complete as shown
below.
###### Proposition D.1.
Given a representable matroid of rank $k$ (input is a $k\times n$ matrix $A$),
the problem of checking if the matroid is uniform is
${\mathsf{co}}{\mathsf{NP}}$-complete. In other words, testing if the matroid
$M$ represented by columns of $A$ is isomorphic to $U_{k,n}$ is
${\mathsf{co}}{\mathsf{NP}}$-complete.
###### Proof.
This proof is due to Oxley and Welsh [OW02]. We make it more explicit. A set
of $n$ points in $d$ dimensions is linearly degenerate if there is a set of
$d$ points which is linearly dependent. In other words, the set of $n$ points
is said to be in general position if all subsets of size $d$ are linearly
independent. Khanchiyan [Kha95] proved that given $n$ points testing if they
are in general position is ${\mathsf{NP}}$-hard. Now notice that this exactly
answers the question of testing if the matroid is uniform. ∎
|
arxiv-papers
| 2008-11-24T14:19:02 |
2024-09-04T02:48:58.942706
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Raghavendra Rao B.V. and Jayalal M.N. Sarma",
"submitter": "Raghavendra Rao",
"url": "https://arxiv.org/abs/0811.3859"
}
|
0811.4016
|
# Solutions of the (2+1)-dimensional KP, SK and KK equations generated by
gauge transformations from non-zero seeds
$\begin{array}[]{l}\text{\sl Jingsong He , Xiaodong Li}\\\ \text{\small
Department of Mathematics }\\\ \text{\small University of Science and
Technology of China}\\\ \text{\small Hefei, 230026 Anhui}\\\ \text{\small P.R.
China }\end{array}$
By using gauge transformations, we manage to obtain new solutions of
($2+1$)-dimensional Kadomtsev-Petviashvili(KP), Kaup-Kuperschmidt(KK) and
Sawada-Kotera(SK) equations from non-zero seeds. For each of the preceding
equations, a Galilean type transformation between these solutions $u_{2}$ and
the previously known solutions $u_{2}^{\prime}$ generated from zero seed is
given. We present several explicit formulas of the single-soliton solutions
for $u_{2}$ and $u_{2}^{\prime}$, and further point out the two main
differences of them under the same value of parameters, i.e., height and
location of peak line, which are demonstrated visibly in three figures.
## 1 Introduction
In the 1980s, Sato and his colleagues brought us the famous Sato theory [1,
2]. Since then, the pseudo-differential operator has been playing an important
role in the research of the Kadomtsev-Petviashvili(KP) hierarchy [3], which
can yield many important nonlinear partial differential equations, such as the
generalized nonlinear Schrödinger equation, the KdV equation, the Sine-Gordon
equation and the famous KP equation. To be self-consistent, we would like to
give a brief review of the KP hierarchy [1, 2, 3, 4].
Let
$L=\partial+u_{2}\partial^{-1}+u_{3}\partial^{-2}+\cdots,$ (1.1)
be a pseudo-differential operator($\Psi$DO), here
$\\{u_{i}\\},u_{i}=u_{i}(t_{1},t_{2},t_{3},\ldots)$ serve as generators of a
differential algebra $\mathcal{A}$. The corresponding generalized Lax
equations are defined as
$\frac{\partial L}{\partial t_{n}}=[B_{n},L],\quad n=1,2,3,\ldots,$ (1.2)
which give rise to infinite number of partial differential equations of the KP
hierarchy, $B_{n}$ is defined as $B_{n}=[L^{n}]_{+}$. It can be easily showed
that eq.(1.2) is equivalent to the so-called Zakharov-Shabat(ZS) equation [5]
$\frac{\partial B_{m}}{\partial t_{n}}-\frac{\partial B_{n}}{\partial
t_{m}}+[B_{m},B_{n}]=0,\quad(m,n=2,3,\ldots).$ (1.3)
The eigenfunction $\phi$ and conjugate eigenfunction $\psi$ corresponding to
$L$ are defined by
$\displaystyle\frac{\partial\phi}{\partial t_{n}}$ $\displaystyle=$
$\displaystyle B_{n}\phi,$ (1.4) $\displaystyle\frac{\partial\psi}{\partial
t_{n}}$ $\displaystyle=$ $\displaystyle-B_{n}^{*}\psi.$ (1.5)
The first non-trivial example is the KP equation given by the $t_{2}$-flow and
$t_{3}$-flow of the KP hierarchy
$(4u_{t}-12uu_{x}-u_{xxx})_{x}-3u_{yy}=0,$ (1.6)
in which $x=t_{1}$, $y=t_{2}$ and $t=t_{3}$.
Suppose $L$ given by eq.(1.1) and $L^{*}$ defined by
$L^{*}=-\partial+\sum_{i=1}^{\infty}(-1)^{i}\partial^{-i}u_{i+1}.$
If $L$ satisfies $L^{*}+L=0$ , then $L$ is called the Lax operator of the CKP
hierarchy [4, 6], and the corresponding flow equations of the CKP hierarchy
are described by
$\frac{\partial L}{\partial t_{n}}=[B_{n},L],\quad n=1,3,5,\cdots.$ (1.7)
The first non-trivial example is the CKP equation [4, 7]
$u_{t}=\frac{5}{9}\left(\partial_{x}^{-1}u_{yy}+3u_{x}\partial_{x}^{-1}u_{y}-\frac{1}{5}u_{xxxxx}-3uu_{xxx}-\frac{15}{2}u_{x}u_{xx}-9u^{2}u_{x}+u_{xxy}+3uu_{y}\right)$
(1.8)
which is generated by $t_{3}$-flow and $t_{5}$-flow and also called the
($2+1$)-dimensional Kaup-Kuperschmidt(KK) equation [8]. Here $x=t_{1}$,
$y=t_{3}$ and $t=t_{5}$. Moreover, $L$ is called the Lax operator of the BKP
hierarchy [2, 9] if it satisfies $L^{*}=-\partial L\partial^{-1}$, and the
flow equations of the BKP hierarchy associated with it are also described by
eq.(1.7). The first non-trivial example is the BKP equation [10, 11]
$u_{t}=\frac{5}{9}\left(\partial_{x}^{-1}u_{yy}+3u_{x}\partial_{x}^{-1}u_{y}-\frac{1}{5}u_{xxxxx}-3uu_{xxx}-3u_{x}u_{xx}-9u^{2}u_{x}+u_{xxy}+3uu_{y}\right),$
(1.9)
which is generated by $t_{3}$-flow and $t_{5}$-flow and also called the
($2+1$)-dimensional Sawada-Kotera(SK) equation [8]. Here $x=t_{1}$, $y=t_{3}$
and $t=t_{5}$.
If we find a set of functions ${u_{2},u_{3},\ldots}$ which makes the
corresponding pseudo-differential operator $L$ satisfies eq.(1.3), then we
have a solution of the KP hierarchy. It’s a well-known result that this set of
solutions can be generated from one single function $\tau(x)$ as the following
way
$\displaystyle u_{2}$ $\displaystyle=$
$\displaystyle\frac{\partial^{2}}{\partial t_{1}^{2}}\log{\tau},$ (1.10)
$\displaystyle u_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[\frac{\partial^{2}}{\partial t_{1}\partial
t_{2}}-\frac{\partial^{3}}{\partial t_{1}^{3}}]\log{\tau},$ (1.11)
$\vdots$
During the last two decades, in order to solve the KP hierarchy, the gauge
transformation was formally introduced in reference [12]. The basic idea
behind gauge transformation is to find a transformation for the initial Lax
operator $L^{(0)}$ of the KP hierarchy after which the new operator $L^{(1)}$
and $B_{n}^{(1)}$ still satisfies Lax equation eq.(1.2) and eq.(1.3)
respectively. Here
$L^{(1)}=T\circ L^{(0)}\circ T^{-1},\qquad B_{n}^{(1)}=(L^{(1)})_{+}^{n},$
(1.12)
$T$ is a suitable pseudo-differential operator. There exist two kinds of gauge
transformation operators [12]
$\displaystyle T_{D}(\phi^{(0)})$ $\displaystyle=$
$\displaystyle\phi^{(0)}\,\partial\,(\phi^{(0)})^{-1},$ (1.13) $\displaystyle
T_{I}(\psi^{(0)})$ $\displaystyle=$
$\displaystyle(\psi^{(0)})^{-1}\,\partial^{-1}\,\psi^{(0)},$ (1.14)
in which $\phi^{(0)}$, $\psi^{(0)}$ are eigenfunction and conjugate
eigenfunction of $L^{(0)}$ respectively and they are also called the
generating functions of the gauge transformation. $T_{D}$ is called
differential type of gauge transformation, $T_{I}$ is called integral type of
gauge transformation. After one gauge transformation $T_{D}$, the new
$\tau$-function
$\tau^{(1)}=\phi^{(0)}\tau^{(0)},$ (1.15)
is transformed from an initial $\tau$-function $\tau^{(0)}$ associated with
the initial Lax operator $L^{(0)}$. A similar result can be formulated for the
case of $T_{I}$
$\tau^{(1)}=\psi^{(0)}\tau^{(0)}.$ (1.16)
With the help of formulas eq.(1.10), eq.(1.11), eq.(1.15) and eq.(1.16), we
can obtain new solutions {$u_{i}^{(1)}$} from the known seed solutions
{$u_{i}^{(0)}$} in the $L^{(0)}$. For example,
$u_{2}^{(1)}=u_{2}^{(0)}+(\log\phi^{(0)})_{xx}$ by the gauge transformation in
eq.(1.15). By a successive application of gauge transformations, the
determinant representation of $\tau^{(n+k)}$ is given in [13] and further more
$u_{2}^{(n+k)}$ can be deduced by using eq.(1.10).
In the last decade, the method of gauge transformation has been developed by
several researchers. The original form of this transformation proposed in
reference [12] cannot be applied directly to the sub-hierarchies of the KP
hierarchy. So in [14, 15, 16], an improvement was made which makes it
applicable to the BKP and CKP hierarchies, and in [17, 18, 19, 20, 21, 22]
another improvement was made so that the gauge transformation can be used on
the constrained KP hierarchy. Besides gauge transformation, some other methods
have been used to solve the KP, BKP, CKP equations. In [23], Hirota method was
considered on the KP equation. Darboux transformation was applied on this
equation in Chapter 3 of [24]. N-soliton solutions of the BKP equation was
obtained through Hirota method in [25, 26], lump solutions was obtained
through this method in [27], the same method was applied to the
($2+1$)-dimensional KK equations in [28] and 3-soliton solutions were obtained
explicitly. Darboux transformation was applied to ($2+1$)-dimensional KK, SK
equations in [29]. In [30], $\bar{\partial}$\- dressing method was used on the
($2+1$)-dimensional KK, SK equations and line solitons and line rational lumps
were obtained. It is easy to recognize that all these known solutions are
corresponding to the solutions given by gauge transformation from zero seed.
However, solving the soliton equations starting from a ”non-zero seed” has not
attracted enough attention. There are very few works on the KPI and KP II
equations with a non-decay initial background[31, 32, 33] by dressing method
and classical inverse scattering method. On the other hand, gauge
transformation from non-zero seeds was not considered before to our knowledge.
One possible reason is that in the case of the KdV equation, solutions
obtained by gauge transformation from zero seed can be transformed to those
solutions from non-zero seeds by a Galilean transformation [34]. So far, we
have not seen any similar discussions on solutions of ($2+1$)-dimensional KP,
KK, SK equations. Therefore in this paper, we solve these equations by gauge
transformation from non-zero seeds and manage to find out the relations
between new solutions and those from zero seed.
The organization of this paper is as follows. In section two we consider the
KP equation. In section three and section four, we discuss (2+1)-dimensional
KK and SK equations respectively. Section five is devoted to the conclusions
and discussions. The notations we use in this paper is the same as in [18].
## 2 Successive Gauge transformation for KP equation
It’s a natural thought to consider successive application of gauge
transformation for KP hierarchy. In [12, 13], a very useful theorem was
introduced about the result after successive gauge transformations.
###### Lemma 1 ([12, 13]).
After $n$ times $T_{D}$ and $k$ times $T_{I}$ transformations $(n\geq k)$ , we
have :
$\displaystyle\tau^{(k+n)}$
$\displaystyle=\psi_{k}^{(k-1+n)}\cdot\psi_{k-1}^{(k-2+n)}\cdots\cdot\psi_{1}^{(n)}\cdot\tau^{(n)}$
$\displaystyle={\rm
IW}_{k,n}(\psi_{k}^{(0)},\psi_{k-1}^{(0)},\cdots,\psi_{1}^{(0)};\phi_{1}^{(0)},\phi_{2}^{(0)},\cdots,\phi_{n}^{(0)})\cdot\tau^{(0)},$
(2.1)
in which ${\rm
IW}_{k,n}(\psi_{k}^{(0)},\psi_{k-1}^{(0)},\cdots,\psi_{1}^{(0)};\phi_{1}^{(0)},\phi_{2}^{(0)},\cdots,\phi_{n}^{(0)})$
stands for
${\rm
IW}_{k,n}=\begin{vmatrix}\int\phi_{1}^{(0)}\cdot\psi_{k}^{(0)}&\int\phi_{2}^{(0)}\cdot\psi_{k}^{(0)}&\cdots&\int\phi_{n}^{(0)}\cdot\psi_{k}^{(0)}\\\
\int\phi_{1}^{(0)}\cdot\psi_{k-1}^{(0)}&\int\phi_{2}^{(0)}\cdot\psi_{k-1}^{(0)}&\cdots&\int\phi_{n}^{(0)}\cdot\psi_{k-1}^{(0)}\\\
\vdots&\vdots&\cdots&\vdots\\\
\int\phi_{1}^{(0)}\cdot\psi_{1}^{(0)}&\int\phi_{2}^{(0)}\cdot\psi_{1}^{(0)}&\cdots&\int\phi_{n}^{(0)}\cdot\psi_{1}^{(0)}\\\
\phi_{1}^{(0)}&\phi_{2}^{(0)}&\cdots&\phi_{n}^{(0)}\\\
\phi_{1,x}^{(0)}&\phi_{2,x}^{(0)}&\cdots&\phi_{n,x}^{(0)}\\\
\vdots&\vdots&\cdots&\vdots\\\
(\phi_{1}^{(0)})^{(n-k-1)}&(\phi_{2}^{(0)})^{(n-k-1)}&\cdots&(\phi_{n}^{(0)})^{(n-k-1)}\end{vmatrix}$
$\phi_{i}^{(0)}$ and $\psi_{i}^{(0)}$ are solutions of eq.(1.4) and eq.(1.5)
associated with the initial value $\tau^{(0)}$, further we have
$u_{2}^{(k+n)}=(\log{{\rm IW}_{k,n}})_{x,x}+u_{2}^{(0)}.$ (2.2)
By using the above theorem, we now start to construct the new solutions of the
KP equation in eq.(1.6) from non-zero seeds. To the end, we choose the initial
Lax operator of the KP hierarchy to be
$L^{(0)}=\partial+\partial^{-1}+\partial^{-2}+\partial^{-3}+\cdots,$
such that all $u_{i}^{(0)}=1$ and then the seed solution of the KP equation is
$u^{(0)}=u_{2}^{(0)}=1$. We know that the KP equation is generated by
$t_{2}$-flow and $t_{3}$-flow of the KP hierarchy, so the generating functions
$\phi_{i}^{(0)}$ and $\psi_{i}^{(0)}$ for the gauge transformation satisfy
$\displaystyle\begin{cases}\phi_{i,t_{2}}^{(0)}&=B^{(0)}_{2}\phi_{i}^{(0)}=(\partial^{2}+2)\phi_{i}^{(0)},\quad
B^{(0)}_{2}=(L^{(0)})^{2}_{+}\\\
\phi_{i,t_{3}}^{(0)}&=B^{(0)}_{3}\phi_{i}^{(0)}=(\partial^{3}+3\partial+3)\phi_{i}^{(0)},\quad
B^{(0)}_{3}=(L^{(0)})^{3}_{+}\end{cases}$ (2.3)
$\displaystyle\begin{cases}\psi_{i,t_{2}}^{(0)}&=-(B_{2}^{(0)})^{*}\psi_{i}^{(0)}=-(\partial^{2}+2)\psi_{i}^{(0)},\\\
\psi_{i,t_{3}}^{(0)}&=-(B_{3}^{(0)})^{*}\psi_{i}^{(0)}=(\partial^{3}+3\partial-3)\psi_{i}^{(0)}.\end{cases}$
(2.4)
###### Lemma 2.
The solutions of eq.(2.3), eq.(2.4) are in form of
$\displaystyle\phi_{i}^{(0)}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}k_{j}e^{\frac{\beta_{j}-3}{\alpha_{j}+1}x+\alpha_{j}y+\beta_{j}t},\quad\beta_{j}=\beta_{j}(\alpha_{j}),$
(2.5) $\displaystyle\psi_{i}^{(0)}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{m}\widetilde{k_{j}}e^{\frac{\widetilde{\beta_{j}}+3}{-\widetilde{\alpha_{j}}+1}x+\widetilde{\alpha_{j}}y+\widetilde{\beta_{j}}t},\quad\widetilde{\beta_{j}}=\widetilde{\beta_{j}}(\widetilde{\alpha_{j}}).$
(2.6)
Here $\alpha_{j}$ , $\beta_{j}$ ,$\widetilde{\alpha_{j}}$ ,
$\widetilde{\beta_{j}}$ should satisfy the following relations
$\displaystyle(\beta_{j}-3)^{2}$ $\displaystyle=$
$\displaystyle(\alpha_{j}+1)^{2}(\alpha_{j}-2),$ (2.7)
$\displaystyle(\widetilde{\beta_{j}}+3)^{2}$ $\displaystyle=$
$\displaystyle(-\widetilde{\alpha_{j}}+1)^{2}(-\widetilde{\alpha_{j}}-2).$
(2.8)
###### Proof.
We assume the solutions of eq.(2.3) have the form
$\widehat{\phi}=X(x)\,Y(y)\,T(t)$, then eq.(2.3) is equivalent to
$\displaystyle\begin{cases}\frac{Y_{y}}{Y}&=\frac{X_{xx}}{X}+2,\\\
\frac{T_{t}}{T}&=\frac{X_{xxx}}{X}+3\,\frac{X_{x}}{X}+3.\end{cases}$ (2.9)
Let
$\frac{Y_{y}}{Y}=\alpha,\qquad\frac{T_{t}}{T}=\beta,$ (2.10)
where $\alpha$ and $\beta$ are constants, we have
$\displaystyle\begin{cases}(\alpha-2)\,X&=X_{xx},\\\
(\beta-3)\,X&=X_{xxx}+3\,X_{x},\end{cases}$ (2.11)
which can be reduced to
$\displaystyle\begin{cases}X_{x}&=\frac{(\alpha+1)\,(\alpha-2)}{\beta-3}\,X,\\\
X_{x}&=\frac{\beta-3}{\alpha+1}\,X.\end{cases}$ (2.12)
Under the consistency condition $(\beta-3)^{2}=(\alpha+1)^{2}\,(\alpha-2)$ we
can obtain
$X(x)=c_{1}\,e^{\frac{\beta-3}{\alpha+1}\,x}.$ (2.13)
From eq.(2.10), we have
$Y(y)=c_{2}\,e^{\alpha y},\qquad T(t)=c_{3}\,e^{\beta t},$
which infer the solutions of eq.(2.3)
$\widehat{\phi}=k\,e^{\frac{\beta-3}{\alpha+1}\,x+\alpha y+\beta t},$ (2.14)
with the help of (2.13), where $k=c_{1}\,c_{2}\,c_{3}$. By linear
superposition, the linear combination of $\widehat{\phi}$ in eq.(2.14) with
respect to different $\alpha$ and $\beta$ is still a solution of eq.(2.3),
that is
$\phi_{i}^{(0)}=\sum_{j=1}^{n}k_{j}\,\widehat{\phi_{j}}=\sum_{j=1}^{n}k_{j}\,e^{\frac{\beta_{j}-3}{\alpha_{j}+1}x+\alpha_{j}y+\beta_{j}t}$
(2.15)
A similar procedure can be applied to $\psi_{i}^{(0)}$ which yields eq.(2.6).
∎
Having these results, it’s sufficient to perform gauge transformation on
$L^{(0)}$. But according to lemma 1, the transformed $\tau$-function may not
be satisfactory, since it may vanish on some point. To rule out this
situation, we need the following theorem.
###### Theorem 1.
Let the generating functions of n-steps $T_{D}$ be
$\phi_{m}^{(0)}\,(m=1,2,\cdots,n)$ in eq.(2.5) and rewritten as
$\phi_{m}^{(0)}=\sum_{i=1}^{p_{m}}k_{m,i}\exp^{a_{m,i}x+\alpha_{m,i}y+\beta_{m,i}t}$
for simplicity, then the new $\tau$-function
$\tau^{(n)}={\rm IW}_{0,n}\cdot\tau^{(0)}={\rm
W}_{n}(\phi_{1}^{(0)},\phi_{2}^{(0)},\cdots,\phi_{n}^{(0)})\cdot\tau^{(0)},$
(2.16)
and ${\rm W}_{n}(\phi_{1}^{(0)},\phi_{2}^{(0)},\cdots,\phi_{n}^{(0)})>0$ if
$k_{m,i}>0$ , $a_{m,i}<a_{m^{{}^{\prime}},j}$ for all $m<m^{{}^{\prime}}$ and
$\forall\,i,j$. The transformed solution $u_{2}^{(n)}$ of KP equation is
$u_{2}^{(n)}=1+\left(\log\left({\rm
W}_{n}(\phi_{1}^{(0)},\phi_{2}^{(0)},\cdots,\phi_{n}^{(0)})\right)\right)_{xx}$
(2.17)
###### Proof.
First, ${\rm W}_{n}$ takes the following form
${\rm
W}_{n}=\begin{vmatrix}\phi_{1}^{(0)}&\phi_{2}^{(0)}&\cdots&\phi_{n}^{(0)}\\\
\frac{\partial}{\partial x}\phi_{1}^{(0)}&\frac{\partial}{\partial
x}\phi_{2}^{(0)}&\cdots&\frac{\partial}{\partial x}\phi_{n}^{(0)}\\\
\vdots&\vdots&\cdots&\vdots\\\ \frac{\partial^{n-1}}{\partial
x^{n-1}}\phi_{1}^{(0)}&\frac{\partial^{n-1}}{\partial
x^{n-1}}\phi_{2}^{(0)}&\cdots&\frac{\partial^{n-1}}{\partial
x^{n-1}}\phi_{n}^{(0)}\end{vmatrix}_{n\times n}$
then we expand the determinant with respect to columns using the equation
$\phi_{m}^{(0)}=\sum_{i=1}^{p_{m}}k_{m,i}\,e^{a_{m,i}\,x+\alpha_{m,i}\,y+\beta_{m,i}\,t},\quad
m=1\ldots n.$
then we have:
${\rm W}_{n}=\sum_{1\leq i_{q}\leq p_{q},\,q=1\ldots
n}\Pi_{j=1}^{n}k_{j,i_{j}}\,e^{a_{j,i_{j}}x+\alpha_{j,i_{j}}y+\beta_{j,i_{j}}t}\,\begin{vmatrix}1&1&\cdots&1\\\
a_{1,i_{1}}&a_{2,i_{2}}&\cdots&a_{n,i_{n}}\\\ \vdots&\vdots&\cdots&\vdots\\\
a_{1,i_{1}}^{n-1}&a_{2,i_{2}}^{n-1}&\cdots&a_{n,i_{n}}^{n-1}\end{vmatrix}$
(2.18)
Notice the Vendermonde determinants in the above equation. Since $k_{m,i}>0$,
the coefficients of these Vendermonde determinants are positive. Using
$a_{m,i}<a_{m^{{}^{\prime}},j}$ for all $m<m^{{}^{\prime}}$ and $\forall i,j$,
it’s easy to prove that all Vendermonde determinants in the above equation are
positive, so ${\rm W}_{n}>0$. Using eq.(2.16), eq.(2.2) and $u_{2}^{(0)}=1$,
we can obtain eq.(2.17). ∎
Next we give single-soliton solutions of the KP equation from a zero seed and
a non-zero seed respectively. Notations with prime are corresponding to the
results of gauge transformation from a zero seed. The generating functions are
$\displaystyle\left(\phi_{1}^{(0)}\right)^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle
k^{{}^{\prime}}\,e^{\xi_{1}^{{}^{\prime}}}+k^{{}^{\prime}}\,e^{\xi_{2}^{{}^{\prime}}},$
(2.19) $\displaystyle\phi_{1}^{(0)}$ $\displaystyle=$ $\displaystyle
k\,e^{\xi_{1}}+k\,e^{\xi_{2}},$ (2.20)
where
$\displaystyle\xi_{1}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle{\frac{{\beta_{1}^{{}^{\prime}}}}{{\alpha_{1}^{{}^{\prime}}}}}\,x+{{\alpha_{1}^{{}^{\prime}}}}\,y+{{\beta_{1}^{{}^{\prime}}}}\,t,$
(2.21) $\displaystyle\xi_{2}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle{\frac{{\beta_{2}^{{}^{\prime}}}}{{\alpha_{2}^{{}^{\prime}}}}\,x}+{{\alpha_{2}^{{}^{\prime}}}}\,y+{{\beta_{2}^{{}^{\prime}}}}\,t,$
(2.22) $\displaystyle\xi_{1}$ $\displaystyle=$
$\displaystyle{\frac{({\beta_{1}}-3)}{{\alpha_{1}}+1}}\,x+{\alpha_{1}}\,y+{\beta_{1}}\,t,$
(2.23) $\displaystyle\xi_{2}$ $\displaystyle=$
$\displaystyle{\frac{({\beta_{2}}-3)}{{\alpha_{2}}+1}}\,x+{\alpha_{2}}\,y+{\beta_{2}}\,t,$
(2.24)
and $(\alpha_{i}^{{}^{\prime}})^{3}=(\beta_{i}^{{}^{\prime}})^{2}$,
$(\beta_{i}-3)^{2}=(\alpha_{i}+1)^{2}\,(\alpha_{i}-2)$, $i=1,2$. The two
single-solitons of the KP equation can be written as
$\displaystyle(u_{2}^{(1)})^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\,(\frac{\beta_{1}^{{}^{\prime}}}{\alpha_{1}^{{}^{\prime}}}-\frac{\beta_{2}^{{}^{\prime}}}{\alpha_{2}^{{}^{\prime}}})^{2}\,{\rm
sech}^{2}(\frac{\xi_{2}^{{}^{\prime}}-\xi_{1}^{{}^{\prime}}}{2}),$ (2.25)
$\displaystyle u_{2}^{(1)}$ $\displaystyle=$ $\displaystyle
1+\frac{1}{4}\,(\frac{\beta_{1}-3}{\alpha_{1}+1}-\frac{\beta_{2}-3}{\alpha_{2}+1})^{2}\,{\rm
sech}^{2}(\frac{\xi_{1}-\xi_{2}}{2}).$ (2.26)
There are two differences between $u_{2}$ and $u_{2}^{\prime}$ under the same
parameters $\alpha$: 1) the height of solitons, 2) the location of the peak
line of the solitons, which are demonstrated visibly in figure 1. In figure 2,
we demonstrate the solution obtained by a two-step gauge transformation by
using eq.(2.17) and
$\displaystyle\phi_{1}^{(0)}$ $\displaystyle=$ $\displaystyle
e^{2\,y+3\,t}+e^{x+3\,y+7\,t},$ (2.27) $\displaystyle\phi_{2}^{(0)}$
$\displaystyle=$ $\displaystyle
e^{\sqrt{2}\,x+4\,y+(3+5\sqrt{2})\,t}+e^{\sqrt{6}\,x+8\,y+(3+9\sqrt{6})\,t}.$
(2.28)
###### Corollary 1.
There exists a Galilean type transformation
$\displaystyle u_{2}^{\prime}$ $\displaystyle\longmapsto$ $\displaystyle
u_{2}(x,y,t)=1+u_{2}^{\prime}(x+3\,t,\ y,\ t).$ (2.29)
between $u_{2}^{{}^{\prime}}$ in eq.(2.25) and $u_{2}$ in eq.(2.26).
Obviously, this result is consistent with the Galilean transformation [34] of
the KdV equation by a dimensional reduction.
## 3 Gauge transformation for (2+1)-dimensional KK equation
Gauge transformation of the CKP hierarchy is somewhat different from that of
the KP hierarchy, because a transformed Lax operator $L^{(1)}$ by one-step
gauge transformation has to satisfy $(L^{(1)})^{*}+L^{(1)}=0$. To meet this
requirement, we introduce the following lemma.
###### Lemma 3 ([16]).
1. 1.
The appropriate gauge transformation $T_{n+k}$ is given by $n=k$ and
generating functions $\psi_{i}^{(0)}=\phi_{i}^{(0)}$ for $i=1,2,\cdots,n$.
2. 2.
The $\tau$-function $\tau_{{\rm CKP}}^{(n+n)}$ of the CKP hierarchy has the
form
$\displaystyle\tau_{\text{\rm CKP}}^{(n+n)}$ $\displaystyle=$
$\displaystyle{\rm
IW}_{n,n}(\phi_{n}^{(0)},\phi_{n-1}^{(0)},\cdots,\phi_{1}^{(0)};\phi_{1}^{(0)},\phi_{2}^{(0)},\cdots,\phi_{n}^{(0)})\cdot\tau_{\text{\rm
CKP}}^{(0)}$ (3.1) $\displaystyle=$
$\displaystyle\begin{vmatrix}\int\phi_{n}^{(0)}\cdot\phi_{1}^{(0)}&\cdots&\int\phi_{n}^{(0)}\cdot\phi_{n}^{(0)}\\\
\vdots&\cdots&\vdots\\\
\int\phi_{1}^{(0)}\cdot\phi_{1}^{(0)}&\cdots&\int\phi_{1}^{(0)}\cdot\phi_{n}^{(0)}\end{vmatrix}\cdot\tau_{\text{\rm
CKP}}^{(0)}.$
and further we have
$u_{2}^{(n+n)}=u_{2}^{(0)}+(\log{{\rm IW}_{n,n}})_{xx}.$ (3.2)
To solve the (2+1)-dimensional KK equation from non-zero seed solution, we
choose a initial Lax operator $L^{(0)}$ of the CKP hierarchy to be
$L^{(0)}=\partial+\partial^{-1}+\partial^{-3}+\partial^{-5}+\cdots.$
Since the (2+1)-dimensional KK equation is generated by $t_{3}$-flow and
$t_{5}$-flow of the CKP hierarchy, we solve
$\displaystyle\begin{cases}\phi_{i,t_{3}}^{(0)}&=B_{3}^{(0)}\phi_{i}^{(0)}=(\partial^{3}+3\partial)\,\phi_{i}^{(0)},\quad
B_{3}^{(0)}=(L^{(0)})^{3}_{+},\\\
\phi_{i,t_{5}}^{(0)}&=B_{5}^{(0)}\phi_{i}^{(0)}=(\partial^{5}+5\partial^{3}+15\partial)\,\phi_{i}^{(0)},\quad
B_{5}^{(0)}=(L^{(0)})^{5}_{+},\end{cases}$ (3.3)
in order to obtain the eigenfunctions.
###### Lemma 4.
The solutions of eq.(3.3) are
$\phi_{i}^{(0)}=\sum_{j=1}^{n}k_{j}\,e^{\frac{\alpha_{j}^{3}-18\alpha_{j}+9\beta_{j}}{\alpha_{j}^{2}+\alpha_{j}\beta_{j}+81}x+\alpha_{j}y+\beta_{j}t},\quad\beta_{j}=\beta_{j}(\alpha_{j}),$
(3.4)
here $\alpha_{j}$ , $\beta_{j}$ should satisfy the relation
$\alpha_{j}^{5}-25\alpha_{j}^{3}+30\beta_{j}\alpha_{j}^{2}+1215\alpha_{j}-\beta_{j}^{3}-243\beta_{j}=0.$
(3.5)
###### Proof.
First, we assume the solution of eq.(3.3) has the form
$\widehat{\phi}=X(x)\,Y(y)\,T(t)$ then we have
$\begin{cases}\frac{Y_{y}}{Y}&=\frac{X_{xxx}}{X}+3\,\frac{X_{x}}{X},\\\
\frac{T_{t}}{T}&=\frac{X_{xxxxx}}{X}+5\,\frac{X_{xxx}}{X}+15\,\frac{X_{x}}{X}.\end{cases}$
(3.6)
Let
$\frac{Y_{y}}{Y}=\alpha,\qquad\frac{T_{t}}{T}=\beta,$ (3.7)
where $\alpha$ and $\beta$ are constants, eq.(3.6) become
$\begin{cases}X_{xxx}&=\alpha\,X-3\,X_{x},\\\
X_{xxxxx}&=\beta\,X-15\,X_{x}-5\,X_{xxx},\end{cases}$ (3.8)
which can be further reduced to
$\begin{cases}9\,X_{xx}-(\alpha+\beta)\,X_{x}+\alpha^{2}\,X&=0,\\\
\alpha\,X_{xx}+9\,X_{x}+(2\,\alpha-\beta)\,X&=0.\end{cases}$ (3.9)
Combining the two equations in eq.(3.9) together, we have
$(\alpha^{2}+\alpha\,\beta+81)\,X_{x}=(\alpha^{3}-18\,\alpha+9\,\beta)\,X.$
(3.10)
The solution of eq.(3.10)
$X(x)=c_{1}\,e^{\frac{\alpha^{3}-18\alpha+9\beta}{\alpha^{2}+\alpha\beta+81}x}.$
(3.11)
By substituting eq.(3.11) back into eq.(3.8), we have
$\alpha^{5}-25\alpha^{3}+30\beta\alpha^{2}+1215\alpha-\beta^{3}-243\beta=0,$
(3.12)
that means if $\alpha$ and $\beta$ satisfy eq.(3.12), then eq.(3.11) is the
solution of eq.(3.8). From eq.(3.7), we have
$Y(y)=c_{2}\,e^{\alpha y},\qquad T(t)=c_{3}\,e^{\beta t},$
together with eq.(3.11) we have
$\widehat{\phi}=k\,e^{\frac{\alpha^{3}-18\alpha+9\beta}{\alpha^{2}+\alpha\beta+81}x+\alpha
y+\beta t},$ (3.13)
where $k=c_{1}\,c_{2}\,c_{3}$. Using the linear superposition as we did in
lemma 2, we can obtain
$\phi_{i}^{(0)}=\sum_{j=1}^{n}k_{j}\,\widehat{\phi_{j}}=\sum_{j=1}^{n}k_{j}\,e^{\frac{\alpha_{j}^{3}-18\alpha_{j}+9\beta_{j}}{\alpha_{j}^{2}+\alpha_{j}\beta_{j}+81}x+\alpha_{j}y+\beta_{j}t}.$
(3.14)
∎
Similar to the previous section about KP equation, we need the following
theorem to assure that the solutions we get are without singularities.
###### Theorem 2.
Let eigenfunctions $\phi_{m}^{(0)}$ take the form as in lemma 4
$\phi_{m}^{(0)}=\sum_{i=1}^{n}k_{m,i}\,e^{a_{m,i}x+\alpha_{m,i}y+\beta_{m,i}t},$
(3.15)
where $m=1,2$, if $k_{m,i}>0$, $a_{1,i}<a_{2,j}$, then ${\rm
IW}_{2,2}(\phi_{2}^{(0)},\phi_{1}^{(0)};\phi_{1}^{(0)},\phi_{2}^{(0)})<0$. The
solution of the (2+1)-dimensional KK equation can be written as
$u_{2}^{(2+2)}=1+\left(\log{\rm IW}_{2,2}\right)_{xx}$ (3.16)
###### Proof.
We rewrite $\phi_{1}^{(0)}$ and $\phi_{2}^{(0)}$ in eq.(3.15) as
$\begin{cases}\phi_{1}^{(0)}&=\sum_{i=1}^{n}R_{i}e^{a_{i}x},\\\
\phi_{2}^{(0)}&=\sum_{i=1}^{n}S_{i}e^{b_{i}x}.\end{cases}$
Here the values of $R_{i}$ and $S_{i}$ are greater than zero. Then we have
$\displaystyle\int(\phi_{1}^{(0)})^{2}$
$\displaystyle=\sum_{i,j=1}^{n}R_{i}R_{j}\frac{e^{(a_{i}+a_{j})x}}{a_{i}+a_{j}},$
(3.17) $\displaystyle\int(\phi_{2}^{(0)})^{2}$
$\displaystyle=\sum_{i,j=1}^{n}S_{i}S_{j}\frac{e^{(b_{i}+b_{j})x}}{b_{i}+b_{j}},$
(3.18) $\displaystyle\int\phi_{1}^{(0)}\phi_{2}^{(0)}$
$\displaystyle=\sum_{i,j=1}^{n}R_{i}S_{j}\frac{e^{(a_{i}+b_{j})x}}{a_{i}+b_{j}}.$
(3.19)
Since $a_{i}<b_{j}$ for $i,j=1\ldots n$, it’s easy prove the following
inequality
$R_{i}R_{j}\frac{e^{(a_{i}+a_{j})x}}{a_{i}+a_{j}}S_{k}S_{l}\frac{e^{(b_{k}+b_{l})x}}{b_{k}+b_{l}}>R_{i}S_{k}R_{j}S_{l}\frac{e^{(a_{i}+b_{k})x}}{a_{i}+b_{k}}\frac{e^{(a_{j}+b_{l})x}}{a_{j}+b_{l}},$
(3.20)
where $1\leq i,j,k,l\leq n$, then
$\begin{vmatrix}\int\phi_{1}^{(0)}\phi_{2}^{(0)}&\int(\phi_{1}^{(0)})^{2}\\\
\int(\phi_{2}^{(0)})^{2}&\int\phi_{1}^{(0)}\phi_{2}^{(0)}\end{vmatrix}=(\int\phi_{1}^{(0)}\phi_{2}^{(0)})^{2}-\int(\phi_{1}^{(0)})^{2}\int(\phi_{2}^{(0)})^{2}<0.$
(3.21)
can be directly verified by using eq.(3.17), eq.(3.18), eq.(3.19). Eq.(3.16)
can be obtained by eq.(3.2) and $u_{2}^{(0)}=1$. ∎
###### Remark 1.
For $T_{(1+1)}=T_{I}\,T_{D}$, with the generating function $\phi_{1}^{(0)}$ as
in eq.(3.15), it’s easy to show that
$\tau^{(1+1)}=(\int(\phi_{1}^{(0)})^{2})\,\tau^{(0)}$ (3.22)
is positive. The corresponding new solution of the (2+1)-dimensional KK
equation can be represented as
$u_{2}^{(1+1)}=1+(\log\int(\phi_{1}^{(0)})^{2})_{xx}$ (3.23)
Here we give the single-soliton solution of the (2+1)-dimensional KK equation
from the generating function
$\phi_{1}^{(0)}=e^{\xi_{1}}+e^{\xi_{2}},$ (3.24)
where
$\xi_{i}=\frac{\alpha_{i}^{3}-18\alpha_{i}+9\beta_{i}}{\alpha_{i}^{2}+\alpha_{i}\beta_{i}+81}\,x+\alpha_{i}\,y+\beta_{i}\,t$,
the solution is
$u_{2}^{(1+1)}=1+\frac{(a_{1}-a_{2})^{2}}{a_{1}+a_{2}}\frac{(\frac{e^{\frac{\xi_{1}-\xi_{2}}{2}}}{a_{1}}+\frac{e^{\frac{\xi_{2}-\xi_{1}}{2}}}{a_{2}})\,(e^{\frac{\xi_{1}-\xi_{2}}{2}}+e^{\frac{\xi_{2}-\xi_{1}}{2}})}{(\frac{e^{\xi_{1}-\xi_{2}}}{2a_{1}}+\frac{e^{\xi_{2}-\xi_{1}}}{2a_{2}}+\frac{2}{a_{1}+a_{2}})^{2}},$
(3.25)
where
$a_{i}=\frac{\alpha_{i}^{3}-18\alpha_{i}+9\beta_{i}}{\alpha_{i}^{2}+\alpha_{i}\beta_{i}+81}$.
The solution $(u_{2}^{(1+1)})^{\prime}$ generated from zero seed have the form
$(u_{2}^{(1+1)})^{{}^{\prime}}=\frac{(a_{1}^{{}^{\prime}}-a_{2}^{{}^{\prime}})^{2}}{a_{1}^{{}^{\prime}}+a_{2}^{{}^{\prime}}}\frac{(\frac{e^{\frac{\xi_{1}^{{}^{\prime}}-\xi_{2}^{{}^{\prime}}}{2}}}{a_{1}^{{}^{\prime}}}+\frac{e^{\frac{\xi_{2}^{{}^{\prime}}-\xi_{1}^{{}^{\prime}}}{2}}}{a_{2}^{{}^{\prime}}})\,(e^{\frac{\xi_{1}^{{}^{\prime}}-\xi_{2}^{{}^{\prime}}}{2}}+e^{\frac{\xi_{2}^{{}^{\prime}}-\xi_{1}^{{}^{\prime}}}{2}})}{(\frac{e^{\xi_{1}^{{}^{\prime}}-\xi_{2}^{{}^{\prime}}}}{2a_{1}^{{}^{\prime}}}+\frac{e^{\xi_{2}^{{}^{\prime}}-\xi_{1}^{{}^{\prime}}}}{2a_{2}^{{}^{\prime}}}+\frac{2}{a_{1}^{{}^{\prime}}+a_{2}^{{}^{\prime}}})^{2}},$
(3.26)
where
$\xi_{i}^{{}^{\prime}}=\frac{(\alpha_{i}^{{}^{\prime}})^{2}}{\beta_{i}^{{}^{\prime}}}\,x+\alpha_{i}^{{}^{\prime}}\,y+\beta_{i}^{{}^{\prime}}\,t$,
$a_{i}^{{}^{\prime}}=\frac{(\alpha_{i}^{{}^{\prime}})^{2}}{\beta_{i}^{{}^{\prime}}}$
and $(\alpha_{i}^{{}^{\prime}})^{5}=(\beta_{i}^{{}^{\prime}})^{3}$. The
differences between $u_{2}^{(1+1)}$ and $(u_{2}^{(1+1)})^{{}^{\prime}}$ under
the same value of parameters are showed in figure 3. By taking
$\displaystyle\phi_{1}^{(0)}$ $\displaystyle=$
$\displaystyle{e^{0.0001999999974\,x+0.0006\,y+0.003\,t}}+{e^{0.0006666665679\,x+0.002\,y+0.01\,t}}$
(3.27)
$\displaystyle+{e^{0.003333320988\,x+0.01\,y+0.05\,t}}+{e^{0.006666567904\,x+0.02\,y+0.1\,t}},$
$\displaystyle\phi_{2}^{(0)}$ $\displaystyle=$
$\displaystyle{e^{1.218304787\,x+5.463203409\,y+30\,t}}+{e^{0.4917724251\,x+1.594247576\,y+8\,t}}$
(3.28)
$\displaystyle+{e^{0.6835764081\,x+2.370148557\,y+12\,t}}+{e^{0.970831384\,x+3.827515914\,y+20\,t}}.$
in eq.(3.16), we can obtain solution of the ($2+1$)-dimensional KK equation
which is plotted in figure 4.
## 4 Gauge transformation for (2+1)-dimensional SK equation
The procedure of this section is mostly the same as the previous section
except that the transformed Lax operator $L^{(1)}$ by one-step gauge
transformation should satisfy $(L^{(1)})^{*}=-\partial L^{(1)}\partial^{-1}$,
so we need lemma 5 about gauge transformation for BKP hierarchy.
###### Lemma 5 ([16]).
1. 1.
The appropriate gauge transformation $T_{n+k}$ is given by $n=k$ and
generating functions $\psi_{i}^{(0)}=\phi_{i,x}^{(0)}$ for $i=1,2,\ldots,n$.
2. 2.
The $\tau$-function $\tau_{{\rm BKP}}^{(n+n)}$ of the BKP hierarchy has the
form
$\displaystyle\tau_{\text{\rm BKP}}^{(n+n)}$ $\displaystyle=$
$\displaystyle{\rm
IW}_{n,n}(\phi_{n,x}^{(0)},\phi_{n-1,x}^{(0)},\ldots,\phi_{1,x}^{(0)};\phi_{1}^{(0)},\phi_{2}^{(0)},\ldots,\phi_{n}^{(0)})\cdot\tau_{\text{\rm
BKP}}^{(0)}$ (4.1) $\displaystyle=$
$\displaystyle\begin{vmatrix}\int\phi_{n,x}^{(0)}\cdot\phi_{1}^{(0)}&\cdots&\int\phi_{n,x}^{(0)}\cdot\phi_{n}^{(0)}\\\
\vdots&\cdots&\vdots\\\
\int\phi_{1,x}^{(0)}\cdot\phi_{1}^{(0)}&\cdots&\int\phi_{1,x}^{(0)}\cdot\phi_{n}^{(0)}\end{vmatrix}\cdot\tau_{\text{\rm
BKP}}^{(0)}.$
and we have
$u_{2}^{(n+n)}=u_{2}^{(0)}+(\log{{\rm IW}_{n,n}})_{xx}.$ (4.2)
With this theorem, we can write down the solutions of the (2+1)-dimensional SK
equation explicitly after successive application of gauge transformations. We
take the initial Lax operator $L^{(0)}$ of the BKP hierarchy as
$L^{(0)}=\partial+\partial^{-1}+\partial^{-3}+\partial^{-5}+\cdots.$
The corresponding eigenfunction $\phi_{i}^{(0)}$ and conjugate eigenfunction
$\psi_{i}^{(0)}=\phi_{i,x}^{(0)}$ are given by lemma 4 and lemma 5, i.e.
$\displaystyle\phi_{i}^{(0)}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}k_{j}\,e^{\frac{\alpha_{j}^{3}-18\alpha_{j}+9\beta_{j}}{\alpha_{j}^{2}+\alpha_{j}\beta_{j}+81}x+\alpha_{j}y+\beta_{j}t},$
(4.3) $\displaystyle\psi_{i}^{(0)}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}k_{j}\,\frac{\alpha_{j}^{3}-18\alpha_{j}+9\beta_{j}}{\alpha_{j}^{2}+\alpha_{j}\beta_{j}+81}\,e^{\frac{\alpha_{j}^{3}-18\alpha_{j}+9\beta_{j}}{\alpha_{j}^{2}+\alpha_{j}\beta_{j}+81}x+\alpha_{j}y+\beta_{j}t},\qquad\beta_{j}=\beta_{j}(\alpha_{j}).$
(4.4)
Similar as section two and section three, we need the following theorem to
assure that the new $\tau$-function we get after gauge transformations will
not vanish at any point.
###### Theorem 3.
Let eigenfunction $\phi_{m}^{(0)}$ take the form as in eq.(4.3)
$\sum_{i=1}^{n}k_{m,i}\,e^{a_{m,i}x+\alpha_{m,i}y+\beta_{m,i}t}$, $m=1,2$, if
$\,0<3\cdot a_{1,i}<a_{2,j}$,
then we have ${\rm
IW}_{2,2}(\phi_{2,x}^{(0)},\phi_{1,x}^{(0)};\phi_{1}^{(0)},\phi_{2}^{(0)})<0$.
The solution can be written as
$u_{2}^{(2+2)}=1+\left(\log{\rm IW}_{2,2}\right)_{xx}.$ (4.5)
###### Proof.
$\phi_{1}^{(0)}$ and $\phi_{2}^{(0)}$ can be rewritten as
$\begin{cases}\phi_{1}^{(0)}&=\sum_{i=1}^{n}R_{i}e^{a_{i}x},\\\
\phi_{2}^{(0)}&=\sum_{i=1}^{n}S_{i}e^{b_{i}x},\end{cases}$
where the value of $R_{i}$ and $S_{i}$ are greater than zero, then we have
$\displaystyle\frac{(\phi_{1}^{(0)})^{2}}{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\sum_{i,j=1}^{n}R_{i}R_{j}e^{(a_{i}+a_{j})x},$
(4.6) $\displaystyle\frac{(\phi_{2}^{(0)})^{2}}{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\sum_{i,j=1}^{n}S_{i}S_{j}e^{(b_{i}+b_{j})x},$
(4.7) $\displaystyle\int\phi_{1,x}^{(0)}\phi_{2}^{(0)}$ $\displaystyle=$
$\displaystyle\sum_{i,j=1}^{n}R_{i}S_{j}\frac{a_{i}}{a_{i}+b_{j}}e^{(a_{i}+b_{j})x},$
(4.8) $\displaystyle\int\phi_{2,x}^{(0)}\phi_{1}^{(0)}$ $\displaystyle=$
$\displaystyle\sum_{i,j=1}^{n}R_{j}S_{i}\frac{b_{i}}{a_{j}+b_{i}}e^{(a_{j}+b_{i})x}.$
(4.9)
The following inequality
$(a_{i}+b_{k})\,(a_{j}+b_{l})>4\,a_{i}\,b_{l},$
is trivial if we use $0<3\cdot a_{1,i}<a_{2,j}$ which means $0<3\cdot
a_{i}<b_{k}$, together with eq.(4.6), eq.(4.7), eq.(4.8) and eq.(4.9), we can
prove
$\begin{vmatrix}\int\phi_{1}^{(0)}\phi_{2,x}^{(0)}&\frac{(\phi_{2}^{(0)})^{2}}{2}\\\
\frac{(\phi_{1}^{(0)})^{2}}{2}&\int\phi_{1,x}^{(0)}\phi_{2}^{(0)}\end{vmatrix}=(\int\phi_{1,x}^{(0)}\phi_{2}^{(0)})(\int\phi_{2,x}^{(0)}\phi_{1}^{(0)})-\frac{(\phi_{1}^{(0)})^{2}(\phi_{2}^{(0)})^{2}}{4}<0,$
(4.10)
by a direct calculation. Eq.(4.5) can be obtained by eq.(4.2) and
$u_{2}^{(0)}=1$. ∎
###### Remark 2.
For $T_{1+1}=T_{I}\,T_{D}$, with the generating function $\phi_{1}^{(0)}$ as
in eq.(4.3), it’s easy to show that
$\tau^{(1+1)}=\frac{(\phi_{1}^{(0)})^{2}}{2}\,\tau^{(0)}$ (4.11)
is positive. The corresponding new solution of the (2+1)-dimensional SK
equation can be represented as
$u_{2}^{(1+1)}=1+(\log(\frac{(\phi_{1}^{(0)})^{2}}{2}))_{xx}$ (4.12)
To obtain a single-soliton solution of the (2+1)-dimensional SK equation, we
start from a generating function
$\phi_{1}^{(0)}=e^{\xi}+e^{-\xi},$ (4.13)
and the solution is
$u_{2}^{(1+1)}=1+2\,a^{2}\,{\rm sech}^{2}(\xi),$ (4.14)
here
$\xi=\frac{\alpha^{3}-18\alpha+9\beta}{\alpha^{2}+\alpha\beta+81}\,x+\alpha\,y+\beta\,t$
and $a=\frac{\alpha^{3}-18\alpha+9\beta}{\alpha^{2}+\alpha\beta+81}$. A
solution generated from zero seed is
$(u_{2}^{(1+1)})^{{}^{\prime}}=2\,(a^{{}^{\prime}})^{2}\,{\rm
sech}^{2}(\xi^{{}^{\prime}}),$ (4.15)
in which
$\xi^{{}^{\prime}}=\frac{(\alpha^{{}^{\prime}})^{2}}{\beta^{{}^{\prime}}}\,x+\alpha^{{}^{\prime}}\,y+\beta^{{}^{\prime}}\,t$,
$(\alpha^{{}^{\prime}})^{5}=(\beta^{{}^{\prime}})^{3}$ and
$a^{{}^{\prime}}=\frac{(\alpha^{{}^{\prime}})^{2}}{\beta^{{}^{\prime}}}$. The
differences between $u_{2}^{(1+1)}$ and $(u_{2}^{(1+1)})^{{}^{\prime}}$ are
showed in figure 5\. In figure 6, we plot the solution of the
($2+1$)-dimensional SK equation by taking
$\displaystyle\phi_{1}^{(0)}$ $\displaystyle=$
$\displaystyle{e^{0.009999666694\,x+0.02999999998\,y+0.15\,t}}+{e^{0.01333254332\,x+0.03999999992\,y+0.2\,t}}$
(4.16) $\displaystyle+{e^{0.006666567904\,x+0.02\,y+0.1\,t}},$
$\displaystyle\phi_{2}^{(0)}$ $\displaystyle=$
$\displaystyle{e^{0.5924749002\,x+1.985399095\,y+10\,t}}+{e^{0.06656825084\,x+0.1999997386\,y+t}}$
(4.17) $\displaystyle+{e^{1.218304787\,x+5.463203409\,y+30\,t}},$
in eq.(4.5).
###### Corollary 2.
For the (2+1)-dimensional KK equation and (2+1)-dimensional SK equation, there
exist a common Galilean type transformation between $(u_{2}^{(1+1)})^{\prime}$
(generated from zero seed) and $u_{2}^{(1+1)}$ (generated from non-zero seed),
i.e.
$u^{\prime}_{2}(x,y,t)\longmapsto
u_{2}(x,y,t)=1+u_{2}^{{}^{\prime}}(x+3y+15t,y+5t,t).$ (4.18)
## 5 Conclusions and Discussions
By now we have obtained new solutions $u_{2}^{(n)}$ in theorem 1 for KP
equation, $u_{2}^{(2+2)}$ in theorem 2 for (2+1)-dimensional KK equation and
$u_{2}^{(2+2)}$ in theorem 3 for (2+1)-dimensional SK equation by using the
the gauge transformations of the KP hierarchy, CKP hierarchy and BKP hierarchy
respectively. The corresponding generating functions of the gauge
transformations previously mentioned are explicitly expressed in lemma 2 and
lemma 4. For these three equations, the single-soliton $u^{(1)}_{2}$(or
$u_{2}^{(1+1)}$) generated from non-zero seeds and
$(u_{2}^{(1)})^{{}^{\prime}}$(or $(u_{2}^{(1+1)})^{{}^{\prime}}$ ) generated
from zero seed are constructed. The main differences between the $u_{2}$ and
$(u_{2})^{{}^{\prime}}$ are height and locations of the peak line under the
same value of parameters, which are demonstrated visibly in figures 1, 2 and
3. We also found a Galilean type transformation in eq.(2.29) between
$(u_{2}^{(1)})^{{}^{\prime}}$ and $u_{2}^{(1)}$ for the KP equation, and
another one in eq.(4.18) between $(u_{2}^{(1+1)})^{{}^{\prime}}$ and
$u_{2}^{(1+1)}$ for the (2+1)-dimensional KK and SK equations. To guarantee
the new solutions $u_{2}$ generated by gauge transformations is smooth, in
other words, the transformed $\tau$-function doesn’t vanish at any point, we
only consider the ${\rm W}_{n}$ in theorem 1 and ${\rm IW}_{2,2}$ in theorem 2
and theorem 3.
The corollary 1 and corollary 2 show that we can establish a one-parameter
transformation group (specifically, Galilean type transformation) of the
solutions of these three equations by setting the seeds
$u_{2}^{(0)}=\epsilon$(arbitrary constant) instead of $u_{2}^{(0)}=1$. The
advantage of this new method to find one-parameter group is to avoid solving
the characteristic line equation, which is not easy to solve, as usual
approach of Lie point transformation. We will try to do this in the future. On
the other hand, if we can choose some more complicated initial Lax operator
$L^{(0)}$ in which {$u^{(0)}_{i}$} are not constants and we are able to solve
the corresponding generating functions, then we can get some other new
solutions. Of course, the calculation is much tedious although the idea is
straightforward. The present work is the first step to this difficult purpose.
Acknowledgement
This work is supported partly by the NSFC grant of China under No.10671187. We
thank Professor Li Yishen(USTC, China) for many valuable suggestions on this
paper.
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Figure 1: Single-soliton solutions at $t=1$ of the KP equation. The lower one
is $(u_{2}^{(1)})^{{}^{\prime}}$ with $k^{{}^{\prime}}=1$,
$\alpha_{1}^{{}^{\prime}}=2.7225$ and $\alpha_{2}^{{}^{\prime}}=3.24$; the
higher one is $(u_{2}^{(1)}-1)$ with parameters $k=1$, $\alpha_{1}=2.7225$ and
$\alpha_{2}=3.24$. Figure 2: Two-soliton solution at $t=0$ of the KP equation.
Figure 3: Single-soliton solutions at $t=1$ of the (2+1)-dimensional KK
equation. The higher one is $(u_{2}^{(1+1)})^{{}^{\prime}}$ with
$\alpha_{1}^{{}^{\prime}}=0.970299$ and $\alpha_{2}^{{}^{\prime}}=0.075$; the
lower one is $(u_{2}^{(1+1)}-1)$ with parameters $\alpha_{1}=0.970299$ and
$\alpha_{2}=0.075$. Figure 4: Solution at $t=0$ of the (2+1)-dimensional KK
equation.
Figure 5: Single-soliton solutions at $t=1$ of (2+1)-dimensional SK equation.
The higher one is $(u_{2}^{(1+1)})^{{}^{\prime}}$ with
$\alpha^{{}^{\prime}}=4.096$ ,the lower one is $(u_{2}^{(1+1)}-1)$ with
parameters $\alpha=4.096$
.
Figure 6: Solution at $t=0$ of (2+1)-dimensional SK equation.
|
arxiv-papers
| 2008-11-25T05:47:55 |
2024-09-04T02:48:58.957845
|
{
"license": "Public Domain",
"authors": "Jingsong He, Xiaodong Li",
"submitter": "Jingsong He",
"url": "https://arxiv.org/abs/0811.4016"
}
|
0811.4215
|
# Well-posedness in critical spaces for the compressible Navier-Stokes
equations with density dependent viscosities
Qionglei Chen † Changxing Miao † and Zhifei Zhang ‡
† Institute of Applied Physics and Computational Mathematics, Beijing 100088,
China
E-mail: chen_qionglei@iapcm.ac.cn and miao_changxing@iapcm.ac.cn
‡ School of Mathematical Science, Peking University, Beijing 100871, China
E-mail: zfzhang@math.pku.edu.cn
(25 November, 2008)
###### Abstract
In this paper, we prove the local well-posedness in critical Besov spaces for
the compressible Navier-Stokes equations with density dependent viscosities
under the assumption that the initial density is bounded away from zero.
Key Words: compressible Navier-Stokes equations, Besov spaces, Bony’s
paraproduct, Fourier localization.
AMS Classification: 35Q30,35D10.
## 1 Introduction
In this paper, we consider the compressible Navier-Stokes equations with
density dependent viscosities in $\mathop{\bf R\kern
0.0pt}\nolimits^{+}\times\mathop{\bf R\kern 0.0pt}\nolimits^{N}(N\geq 2)$:
$\left\\{\begin{array}[]{ll}\partial_{t}\rho+\textrm{div}(\rho u)=0,\\\
\partial_{t}(\rho u)+\textrm{div}(\rho u\otimes
u)-\textrm{div}(2\mu(\rho)D(u))-\nabla(\lambda(\rho)\textrm{div}u)+\nabla
P(\rho)=0,\\\ (\rho,u)|_{t=0}=(\rho_{0},u_{0}).\end{array}\right.$ (1.1)
Here $\rho(t,x)$ and $u(t,x)$ are the density and velocity of the fluid. The
pressure $P$ is a smooth function of $\rho$, $D(u)=\frac{1}{2}(\nabla u+\nabla
u^{t})$ is the strain tensor, the Lamé coefficients $\mu$ and $\lambda$ depend
smoothly on $\rho$ and satisfy
$\displaystyle\mu>0\quad\textrm{and}\quad\lambda+2\mu>0,$ (1.2)
which ensures that the operator
$-\textrm{div}(2\mu(\rho)D\cdot)-\nabla(\lambda(\rho)\textrm{div}\cdot)$ is
elliptic. An important example is included in the system (1.1): the viscous
shallow water equations($N=2,\mu(\rho)=\rho,\lambda(\rho)=0$ and
$P(\rho)=\rho^{2}$).
The local existence and uniqueness of smooth solutions for the system (1.1)
were proved by Nash [23] for smooth initial data without vacuum. Later on,
Matsumura and Nishida[20] proved the global well-posedness for smooth data
close to equilibrium, see also [18] for one dimension. Concerning the global
existence of weak solutions for the large initial data, we refer to [2, 3, 19,
21]. We may refer to [4, 10, 25] and references therein for the viscous
shallow water equations.
This paper is devoted to the study of the well-posedness of the system (1.1)
in the critical spaces. Recently, Danchin has obtained several important well-
posedness results in the critical spaces for the compressible Navier-Stokes
equations [11, 12, 14]. To explain the precise meaning of critical spaces, let
us consider the incompressible Navier-Stokes equations
$\displaystyle(NS)\quad\left\\{\begin{array}[]{ll}\partial_{t}u-\Delta
u+u\cdot\nabla u+\nabla p=0,\\\ \mbox{div}u=0.\end{array}\right.$
It is easy to find that if $(u,p)$ is a solution of (NS), then
$\displaystyle u_{\lambda}(t,x)\buildrel\hbox{\footnotesize
def}\over{=}\lambda u(\lambda^{2}t,\lambda x),\quad
p_{\lambda}(t,x)\buildrel\hbox{\footnotesize
def}\over{=}\lambda^{2}p(\lambda^{2}t,\lambda x)$ (1.4)
is also a solution of (NS). For the (NS) equations, a functional space $X$ is
critical if the corresponding norm is invariant under the scaling of (1.4).
Obviously, $\dot{H}^{\frac{N}{2}-1}$ is a critical space. Fujita and Kato[16]
proved the well-posedness of (NS) in $\dot{H}^{\frac{N}{2}-1}$, see also [5,
6, 22] and references therein for the well-posedness in the other critical
spaces. For the compressible Navier-Stokes equations, let us introduce the
following transformation
$\displaystyle\rho_{\lambda}(t,x)\buildrel\hbox{\footnotesize
def}\over{=}\rho(\lambda^{2}t,\lambda x),\quad
u_{\lambda}(t,x)\buildrel\hbox{\footnotesize def}\over{=}\lambda
u(\lambda^{2}t,\lambda x).$
Then if $(\rho,u)$ solves (1.1), so does $(\rho_{\lambda},u_{\lambda})$
provided the viscosity coefficients are constants and the pressure law has
been changed into $\lambda^{2}P$. This motivates the following definition:
###### Definition 1.1
We will say that a functional space is critical with respect to the scaling of
the equations if the associated norm is invariant under the transformation:
$\displaystyle(\rho,u)\longrightarrow(\rho_{\lambda},u_{\lambda})$
(up to a constant independent of $\lambda$).
A natural candidate is the homogenous Sobolev space
$\dot{H}^{N/2}\times\bigl{(}\dot{H}^{N/2-1}\bigr{)}^{N}$, but since
$\dot{H}^{N/2}$ is not included in $L^{\infty}$, we can not obtain a
$L^{\infty}$ control of the density when $\rho_{0}\in\dot{H}^{N/2}$. Instead,
we choose the initial data $(\rho_{0},u_{0})$ for some $\bar{\rho_{0}}$ in a
critical homogenous Besov spaces:
$\displaystyle(\rho_{0}-\bar{\rho}_{0},u_{0})\in\dot{B}^{\frac{N}{p}}_{p,1}\times\bigl{(}\dot{B}^{\frac{N}{p}-1}_{p,1}\bigr{)}^{N},$
since $\dot{B}^{\frac{N}{p}}_{p,1}$ is continuously embedded in $L^{\infty}$.
However, working in the critical spaces, if we deal with the elliptic
operators of the momentum equations as a constant coefficient second order
operator plus a perturbation induced by the density and viscosity
coefficients, the perturbation will be a trouble term. In the case when
$\rho-\bar{\rho}_{0}$ is small in $\dot{B}^{\frac{N}{p}}_{p,1}$ or has more
regularity, the perturbation can be treated as a harmless source term and the
corresponding local-well posedness can be obtained by following the argument
of Danchin [12], see [17].
The purpose of the present paper is to obtain a local well-posedness result in
the critical Besov spaces under the natural physical assumption that the
initial density is bounded away from zero. Our new observation is that if
$\rho-\bar{\rho}_{0}$ is small in the weighted Besov spaces
$\dot{B}^{\frac{N}{p}}_{p,1}(\omega)$(see Section 3 for the definition), the
perturbation can still be treated as a harmless source term. Similar idea has
been used by the authors of this paper to prove the local well-posedness in
$\dot{B}^{1}_{2,1}\times\bigl{(}\dot{B}^{0}_{2,1}\bigr{)}^{2}$ for the viscous
shallow water equations [10]. Very rencently, Danchin[15] proved a similar
result for the system (1.1) with constant coefficients. The key of his proof
is a new and interesting estimate for a class of parabolic systems with the
coefficients in $C([0,T];\dot{B}^{N/2}_{2,1})$. It seems to be possible to
adapt his method to the present model. Here we would like to present a general
functional framework to deal with the local well-posedness in the critical
spaces for the compressible fluids.
Our main result is as follows:
###### Theorem 1.2
Let $\bar{\rho}_{0}$ and $c_{0}$ be two positive constants. Assume that the
initial data satisfies
$\displaystyle(\rho_{0}-\bar{\rho}_{0},u_{0})\in\dot{B}^{\frac{N}{p}}_{p,1}\times\bigl{(}\dot{B}^{\frac{N}{p}-1}_{p,1}\bigr{)}^{N}\quad\textrm{and}\quad\rho_{0}\geq
c_{0}.$
Then there exists a positive time $T$ such that
(a) Existence: If $p\in(1,N]$, the system (1.1) has a solution
$(\rho-\bar{\rho}_{0},u)\in E^{p}_{T}$ with
$\displaystyle E^{p}_{T}\buildrel\hbox{\footnotesize
def}\over{=}C([0,T];\dot{B}^{\frac{N}{p}}_{p,1})\times\Bigl{(}C([0,T];\dot{B}^{\frac{N}{p}-1}_{p,1})\cap
L^{1}(0,T;\dot{B}^{\frac{N}{p}+1}_{p,1})\Bigr{)}^{N},\quad\rho\geq\frac{1}{2}c_{0};$
(b) Uniqueness: If $p\in(1,N]$, then the uniqueness holds in $E^{p}_{T}$.
###### Remark 1.3
If the Lamé coefficients $\mu$ and $\lambda$ are constants satisfying (1.2),
then the range of $p$ in the existence result of the system (1.1) can be
extended to $p\in(1,2N)$, since we can take $p\in(1,2N)$ in Proposition 5.1
for the case when $\overline{\lambda}$ and $\overline{\mu}$ are constants.
The structure of this paper is as follows:
In Section 2, we recall some basic facts about the Littlewood-Paley
decomposition and the functional spaces. In Section 3, we firstly introduce
the weighted Besov spaces, then present some nonlinear estimates. Section 4 is
devoted to the estimates in the weighted Besov spaces for the linear transport
equation. Section 5 is devoted to the estimates in the weighted Besov spaces
for the linearized momentum equation. In Section 6, we prove the existence of
the solution. In Section 7, we prove the uniqueness of the solution.
## 2 Littlewood-Paley theory and the functional spaces
Let us introduce the Littlewood-Paley decomposition. Choose a radial function
$\varphi\in{{\cal S}}(\mathop{\bf R\kern 0.0pt}\nolimits^{N})$ supported in
${{\cal C}}=\\{\xi\in\mathop{\bf R\kern
0.0pt}\nolimits^{N},\,\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\\}$ such that
$\displaystyle\sum_{j\in{\mathbf{Z}}}\varphi(2^{-j}\xi)=1\quad\textrm{for
all}\,\,\xi\neq 0.$
The frequency localization operator $\Delta_{j}$ and $S_{j}$ are defined by
$\displaystyle\Delta_{j}f=\varphi(2^{-j}D)f,\quad S_{j}f=\sum_{k\leq
j-1}\Delta_{k}f\quad\mbox{for}\quad j\in{\mathbf{Z}}.$
With our choice of $\varphi$, one can easily verify that
$\displaystyle\Delta_{j}\Delta_{k}f=0\quad\textrm{if}\quad|j-k|\geq
2\quad\textrm{and}\quad$ (2.1)
$\displaystyle\Delta_{j}(S_{k-1}f\Delta_{k}f)=0\quad\textrm{if}\quad|j-k|\geq
5.$
We denote the space ${{\cal Z}^{\prime}}(\mathop{\bf R\kern
0.0pt}\nolimits^{N})$ by the dual space of ${{\cal Z}}(\mathop{\bf R\kern
0.0pt}\nolimits^{N})=\\{f\in{{\cal S}}(\mathop{\bf R\kern
0.0pt}\nolimits^{N});\,D^{\alpha}\hat{f}(0)=0;\forall\alpha\in\mathop{\bf
N\kern 0.0pt}\nolimits^{d}\,\mbox{multi-index}\\}$, it also can be identified
by the quotient space of ${{\cal S}^{\prime}}(\mathop{\bf R\kern
0.0pt}\nolimits^{N})/{{\cal P}}$ with the polynomials space ${{\cal P}}$. The
formal equality
$\displaystyle f=\sum_{k\in{\mathbf{Z}}}\Delta_{k}f$
holds true for $f\in{{\cal Z}^{\prime}}(\mathop{\bf R\kern
0.0pt}\nolimits^{N})$ and is called the homogeneous Littlewood-Paley
decomposition.
The operators $\Delta_{j}$ help us recall the definition of the Besov
space(see also [24]).
###### Definition 2.1
Let $s\in\mathop{\bf R\kern 0.0pt}\nolimits$, $1\leq p,r\leq+\infty$. The
homogeneous Besov space $\dot{B}^{s}_{p,r}$ is defined by
$\dot{B}^{s}_{p,r}=\\{f\in{{\cal Z}^{\prime}}(\mathop{\bf R\kern
0.0pt}\nolimits^{N}):\,\|f\|_{\dot{B}^{s}_{p,r}}<+\infty\\},$
where
$\displaystyle\|f\|_{\dot{B}^{s}_{p,r}}\buildrel\hbox{\footnotesize
def}\over{=}\Bigl{\|}2^{ks}\|\Delta_{k}f(t)\|_{p}\Bigr{\|}_{\ell^{r}}.$
We next introduce the Besov-Chemin-Lerner space
$\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,r})$ which is initiated in [9].
###### Definition 2.2
Let $s\in\mathop{\bf R\kern 0.0pt}\nolimits$, $1\leq p,q,r\leq+\infty$,
$0<T\leq+\infty$. The space $\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,r})$ is
defined as the set of all the distributions $f$ satisfying
$\|f\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,r})}<+\infty,$
where
$\|f\|^{r}_{\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,r})}\buildrel\hbox{\footnotesize
def}\over{=}\Bigl{\|}2^{ks}\|\Delta_{k}f(t)\|_{L^{q}(0,T;L^{p})}\Bigr{\|}_{\ell^{r}}.$
Obviously,
$\widetilde{L}^{1}_{T}(\dot{B}^{s}_{p,1})=L^{1}_{T}(\dot{B}^{s}_{p,1}).$ In
the sequel, we will constantly use the Bony’s decomposition from [1] that
$uv=T_{u}v+T_{v}u+R(u,v),$ (2.2)
with
$T_{u}v=\sum_{j\in{\mathbf{Z}}}S_{j-1}u\Delta_{j}v,\quad
R(u,v)=\sum_{j\in{\mathbf{Z}}}\Delta_{j}u\widetilde{\Delta}_{j}v,\quad\widetilde{\Delta}_{j}v=\sum_{|j^{\prime}-j|\leq
1}\Delta_{j^{\prime}}v.$
Let us conclude this section by collecting some useful lemmas.
###### Lemma 2.3
[7] Let $1\leq p\leq q\leq+\infty$. Assume that $f\in L^{p}(\mathop{\bf R\kern
0.0pt}\nolimits^{N})$, then for any $\gamma\in(\mathop{\bf N\kern
0.0pt}\nolimits\cup\\{0\\})^{N}$, there exist constants $C_{1}$, $C_{2}$
independent of $f$, $j$ such that
$\displaystyle{\rm supp}\hat{f}\subseteq\\{|\xi|\leq
A_{0}2^{j}\\}\Rightarrow\|\partial^{\gamma}f\|_{q}\leq
C_{1}2^{j{|\gamma|}+jN(\frac{1}{p}-\frac{1}{q})}\|f\|_{p},$ $\displaystyle{\rm
supp}\hat{f}\subseteq\\{A_{1}2^{j}\leq|\xi|\leq
A_{2}2^{j}\\}\Rightarrow\|f\|_{p}\leq
C_{2}2^{-j|\gamma|}\sup_{|\beta|=|\gamma|}\|\partial^{\beta}f\|_{p}.$
###### Lemma 2.4
[12] Let $1<p<\infty$, and $a\geq\bar{a}>0$ be a bounded continuous function.
Assume that $u\in L^{p}(\mathop{\bf R\kern 0.0pt}\nolimits^{N})$ and
$\textrm{supp}\,\hat{u}\subset\\{\xi:R_{1}\leq|\xi|\leq R_{2}\\}$. Then there
exists a constant $c$ depending only on $N$ and $R_{2}/R_{1}$ such that
$\displaystyle c\bar{a}R_{1}^{2}\frac{(p-1)}{p^{2}}\int_{\mathop{\bf R\kern
0.0pt}\nolimits^{N}}|u|^{p}\operatorname{d}x\leq-\int_{\mathop{\bf R\kern
0.0pt}\nolimits^{N}}{\rm div}(a\nabla u)|u|^{p-2}u\operatorname{d}x.$
###### Lemma 2.5
Let $s>0$, and $1\leq p\leq\infty$. Assume that
$f,g\in\dot{B}^{s_{1}}_{p,1}\cap L^{\infty}$. Then there holds
$\displaystyle\|fg\|_{\dot{B}^{s}_{p,1}}\leq
C(\|f\|_{\dot{B}^{s}_{p,1}}\|g\|_{{L}^{\infty}}+\|f\|_{{L}^{\infty}}\|g\|_{\dot{B}^{s}_{p,1}}).$
###### Lemma 2.6
Let $s_{1},s_{2}\leq\frac{N}{p},\,s_{1}+s_{2}>N\max(0,\frac{2}{p}-1)$, and
$1\leq p,q,q_{1},q_{2}\leq\infty$ with
$\frac{1}{q_{1}}+\frac{1}{q_{2}}=\frac{1}{q}$. Assume that
$f\in\widetilde{L}^{q_{1}}_{T}(\dot{B}^{s_{1}}_{p,1})$ and
$g\in\widetilde{L}^{q_{2}}_{T}(\dot{B}^{s_{2}}_{p,1})$. Then there holds
$\displaystyle\|fg\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s_{1}+s_{2}-\frac{N}{p}}_{p,1})}\leq
C\|f\|_{\widetilde{L}^{q_{1}}_{T}(\dot{B}^{s_{1}}_{p,1})}\|g\|_{\widetilde{L}^{q_{2}}_{T}(\dot{B}^{s_{2}}_{p,1})}.$
###### Lemma 2.7
Let $s_{1}\leq\frac{N}{p},s_{2}<\frac{N}{p},\,s_{1}+s_{2}\geq
N\max(0,\frac{2}{p}-1)$, and $1\leq p,q,q_{1},q_{2}\leq\infty$ with
$\frac{1}{q_{1}}+\frac{1}{q_{2}}=\frac{1}{q}$. Assume that
$f\in\widetilde{L}^{q_{1}}_{T}(\dot{B}^{s_{1}}_{p,1})$ and
$g\in\widetilde{L}^{q_{2}}_{T}(\dot{B}^{s_{2}}_{p,\infty})$. Then there holds
$\displaystyle\|fg\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s_{1}+s_{2}-\frac{N}{p}}_{p,\infty})}\leq
C\|f\|_{\widetilde{L}^{q_{1}}_{T}(\dot{B}^{s_{1}}_{p,1})}\|g\|_{\widetilde{L}^{q_{2}}_{T}(\dot{B}^{s_{2}}_{p,\infty})}.$
###### Lemma 2.8
Let $s\in(-N\min\big{(}\frac{1}{p},\frac{1}{p^{\prime}}\big{)},\frac{N}{p}+1]$
and $1\leq p,q,q_{1},q_{2}\leq\infty$ with
$\frac{1}{q_{1}}+\frac{1}{q_{2}}=\frac{1}{q}$. Assume that
$f\in\widetilde{L}^{q_{1}}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})$ and
$g\in\widetilde{L}^{q_{2}}_{T}(\dot{B}^{s}_{p,1})$. Then there holds
$\displaystyle\sum_{j}2^{j(s-1)}\|{\rm div}[\Delta_{j},f]\nabla
g\|_{L^{q}_{T}(L^{p})}\leq
C\|f\|_{\widetilde{L}^{q_{1}}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})}\|g\|_{\widetilde{L}^{q_{2}}_{T}(\dot{B}^{s}_{p,1})}.$
###### Lemma 2.9
Let $s>0$ and $1\leq p,q\leq\infty$. Assume that $F\in
W^{[s]+3,\infty}_{loc}(\mathop{\bf R\kern 0.0pt}\nolimits)$ with $F(0)=0$.
Then for any $f\in
L^{\infty}_{T}(L^{\infty})\cap\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,1})$, we
have
$\displaystyle\|F(f)\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,1})}\leq
C\bigl{(}1+\|f\|_{L^{\infty}_{T}(L^{\infty})}\bigr{)}^{[s]+2}\|f\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,1})}.$
Lemma 2.6-Lemma 2.9 can be easily proved by using Bony’s decomposition and
Lemma 2.3, see also [8, 12] or Section 3 for similar results.
###### Remark 2.10
Lemma 2.6-Lemma 2.9 still remain true for the usual homogenous Besov spaces.
For example, the estimate in Lemma 2.6 becomes
$\displaystyle\|fg\|_{\dot{B}^{s_{1}+s_{2}-\frac{N}{p}}_{p,1}}\leq
C\|f\|_{\dot{B}^{s_{1}}_{p,1}}\|g\|_{\dot{B}^{s_{2}}_{p,1}},$
with $p,s_{1},s_{2}$ satisfying the conditions as in Lemma 2.6.
## 3 Nonlinear estimates in the weighted Besov spaces
Let us firstly introduce the weight function. Let
$\\{e_{k}(t)\\}_{k\in{\mathbf{Z}}}$ be a sequence defined in $[0,+\infty)$
satisfying the following conditions:
$\displaystyle e_{k}(t)\in[0,1],\quad e_{k}(t)\leq
e_{k^{\prime}}(t)\quad\textrm{if}\quad k\leq k^{\prime}\quad\textrm{and}\quad
e_{k}(t)\sim e_{k^{\prime}}(t)\quad\textrm{if}\quad k\sim k^{\prime}.$ (3.1)
Then the weight function $\\{\omega_{k}(t)\\}_{k\in{\mathbf{Z}}}$ is defined
by
$\omega_{k}(t)=\sum_{\ell\geq k}2^{k-\ell}e_{\ell}(t),\quad k\in{\mathbf{Z}}.$
It is easy to verify that for any $k\in{\mathbf{Z}}$,
$\begin{split}&\omega_{k}(t)\leq 2,\quad e_{k}(t)\leq\omega_{k}(t),\\\
&\omega_{k}(t)\leq
2^{k-k^{\prime}}\omega_{k^{\prime}}(t)\quad\textrm{if}\,\,k\geq
k^{\prime},\quad\omega_{k}(t)\leq
3\omega_{k^{\prime}}(t)\quad\textrm{if}\,\,k\leq k^{\prime},\\\
&\omega_{k}(t)\sim\omega_{k^{\prime}}(t)\quad\textrm{if}\,\,k\sim
k^{\prime}.\end{split}$ (3.2)
###### Definition 3.1
Let $s\in\mathop{\bf R\kern 0.0pt}\nolimits$, $1\leq p,r\leq+\infty$,
$0<T<+\infty$. The weighted Besov space $\dot{B}^{s}_{p,r}(\omega)$ is defined
by
$\dot{B}^{s}_{p,r}(\omega)=\\{f\in{{\cal Z}^{\prime}}(\mathop{\bf R\kern
0.0pt}\nolimits^{N}):\,\|f\|_{\dot{B}^{s}_{p,r}(\omega)}<+\infty\\},$
where
$\displaystyle\|f\|_{\dot{B}^{s}_{p,r}(\omega)}\buildrel\hbox{\footnotesize
def}\over{=}\bigl{\|}2^{ks}\omega_{k}(T)\|\Delta_{k}f\|_{p}\bigr{\|}_{\ell^{r}}.$
###### Definition 3.2
Let $s\in\mathop{\bf R\kern 0.0pt}\nolimits$, $1\leq p,q\leq+\infty$,
$0<T<+\infty$. The weighted function space
$\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,1}(\omega))$ is defined by
$\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,1}(\omega))=\\{f\in
L^{q}_{T}(\dot{B}^{s}_{p,1}(\omega)):\,\|f\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,1}(\omega))}<+\infty\\},$
where
$\displaystyle\|f\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,1}(\omega))}\buildrel\hbox{\footnotesize
def}\over{=}\sum_{k\in{\mathbf{Z}}}2^{ks}\omega_{k}(T)\bigg{(}\int_{0}^{T}\|\Delta_{k}f(t)\|^{q}_{p}dt\bigg{)}^{\frac{1}{q}}.$
###### Remark 3.3
If $e_{k}(t)$ is continuous on $[0,+\infty)$ and $e_{k}(0)=0$ for
$k\in{\mathbf{Z}}$, $f\in\widetilde{L}^{\infty}_{T}(\dot{B}^{s}_{p,1})$, then
for any $\varepsilon>0$, there exists a $\widetilde{T}\in(0,T]$ such that
$\displaystyle\|f\|_{\widetilde{L}^{\infty}_{\widetilde{T}}(\dot{B}^{s}_{p,1}(\omega))}\leq\varepsilon.$
Indeed, due to $f\in\widetilde{L}^{\infty}_{T}(\dot{B}^{s}_{p,1})$ and
$\omega_{k}(T)\leq 2$, there exists $N_{1}\in\mathop{\bf N\kern
0.0pt}\nolimits$ such that
$\displaystyle\sum_{|k|\geq
N_{1}+1}2^{ks}\omega_{k}(T)\|\Delta_{k}f\|_{L^{\infty}_{T}(L^{p})}\leq\varepsilon/3,\quad\;\sum_{|k|\leq
N_{1}}2^{ks}\sum_{\ell\geq
k+N_{1}+1}2^{k-\ell}e_{\ell}(T)\|\Delta_{k}f\|_{L^{\infty}_{T}(L^{p})}\leq\varepsilon/3.$
Thus, we have
$\displaystyle\|f\|_{\widetilde{L}^{\infty}_{\widetilde{T}}(\dot{B}^{s}_{p,1}(\omega))}$
$\displaystyle\leq$ $\displaystyle 2\varepsilon/3+\sum_{|k|\leq
N_{1}}2^{ks}\sum_{k\leq\ell\leq
k+N_{1}}2^{k-\ell}e_{\ell}(\widetilde{T})\|\Delta_{k}f\|_{L^{\infty}_{\widetilde{T}}(L^{p})}$
$\displaystyle\leq$ $\displaystyle
2\varepsilon/3+2e_{2N_{1}}(\widetilde{T})\sum_{|k|\leq
N_{1}}2^{ks}\|\Delta_{k}f\|_{L^{\infty}_{\widetilde{T}}(L^{p})}$
$\displaystyle\leq$ $\displaystyle\varepsilon,$
if $\widetilde{T}\in(0,T]$ is chosen such that
$\displaystyle 2e_{2N_{1}}(\widetilde{T})\sum_{|k|\leq
N_{1}}2^{ks}\|\Delta_{k}f\|_{L^{\infty}_{\widetilde{T}}(L^{p})}\leq\varepsilon/3.$
Next, we present some estimates in the weighted Besov spaces.
###### Lemma 3.4
Let $1\leq p\leq\infty$. Assume that
$f\in\dot{B}^{s_{1}}_{p,1}(\omega),g\in\dot{B}^{s_{2}}_{p,1}$. Then there hold
(a) if $s_{2}\leq\frac{N}{p}$, we have
$\displaystyle\|T_{g}f\|_{\dot{B}^{s_{1}+s_{2}-\frac{N}{p}}_{p,1}(\omega)}\leq
C\|f\|_{\dot{B}^{s_{1}}_{p,1}(\omega)}\|g\|_{\dot{B}^{s_{2}}_{p,1}};$
(b) if $s_{1}\leq\frac{N}{p}-1$, we have
$\displaystyle\|T_{f}g\|_{\dot{B}^{s_{1}+s_{2}-\frac{N}{p}}_{p,1}(\omega)}\leq
C\|f\|_{\dot{B}^{s_{1}}_{p,1}(\omega)}\|g\|_{\dot{B}^{s_{2}}_{p,1}};$
(c) if $s_{1}+s_{2}>N\max(0,\frac{2}{p}-1)$, we have
$\displaystyle\|R(f,g)\|_{\dot{B}^{s_{1}+s_{2}-\frac{N}{p}}_{p,1}(\omega)}\leq
C\|f\|_{\dot{B}^{s_{1}}_{p,1}(\omega)}\|g\|_{\dot{B}^{s_{2}}_{p,1}}.$
Proof. Due to (2.1), we have
$\displaystyle\Delta_{j}(T_{g}f)=\sum_{|j^{\prime}-j|\leq
4}\Delta_{j}(S_{j^{\prime}-1}g\Delta_{j^{\prime}}f),$
then we get by Lemma 2.3 and (3.2) that
$\displaystyle\|T_{g}f\|_{\dot{B}^{s_{1}+s_{2}-\frac{N}{p}}_{p,1}(\omega)}$
$\displaystyle=$
$\displaystyle\sum_{j}2^{j(s_{1}+s_{2}-\frac{N}{p})}\omega_{j}(T)\|\Delta_{j}(T_{g}f)\|_{p}$
$\displaystyle\leq$ $\displaystyle
C\sum_{j}2^{j(s_{1}+s_{2}-\frac{N}{p})}\omega_{j}(T)\|S_{j-1}g\|_{\infty}\|\Delta_{j}f\|_{p}$
$\displaystyle\leq$ $\displaystyle
C\|f\|_{\dot{B}^{s_{1}}_{p,1}(\omega)}\|g\|_{\dot{B}^{s_{2}}_{p,1}},$
where we used in the last inequality
$\displaystyle\|S_{j-1}g\|_{\infty}\leq C\sum_{\ell\leq
j-2}2^{\ell\frac{N}{p}}\|\Delta_{\ell}g\|_{p}\leq
C2^{j(-s_{2}+\frac{N}{p})}\|g\|_{\dot{B}^{s_{2}}_{p,1}}.$
This proves (a). We next prove (b). Similarly, we have
$\displaystyle\|T_{f}g\|_{\dot{B}^{s_{1}+s_{2}-\frac{N}{p}}_{p,1}(\omega)}$
$\displaystyle=$
$\displaystyle\sum_{j}2^{j(s_{1}+s_{2}-\frac{N}{p})}\omega_{j}(T)\|\Delta_{j}(T_{f}g)\|_{p}$
$\displaystyle\leq$ $\displaystyle
C\sum_{j}2^{j(s_{1}+s_{2}-\frac{N}{p})}\omega_{j}(T)\|S_{j-1}f\|_{\infty}\|\Delta_{j}g\|_{p},$
and by Lemma 2.3 and (3.2), we have
$\displaystyle\omega_{j}(T)\|S_{j-1}f\|_{\infty}$ $\displaystyle\leq$
$\displaystyle C2^{j}\sum_{\ell\leq
j-2}2^{\ell(\frac{N}{p}-1)}\omega_{\ell}(T)\|\Delta_{\ell}f\|_{p}$
$\displaystyle\leq$ $\displaystyle
C2^{j(\frac{N}{p}-s_{1})}\|f\|_{\dot{B}^{s_{1}}_{p,1}(\omega)},$
which lead to (b). Now we prove (c). Notice that
$\displaystyle\Delta_{j}(R(f,g))=\sum_{j^{\prime}\geq
j-3}\Delta_{j}(\Delta_{j^{\prime}}f\widetilde{\Delta}_{j^{\prime}}g),$
then we get by Lemma 2.3 that if $p\geq 2$
$\displaystyle\|R(f,g)\|_{\dot{B}^{s_{1}+s_{2}-\frac{N}{p}}_{p,1}(\omega)}$
$\displaystyle\leq C\sum_{j}\sum_{j^{\prime}\geq
j-3}2^{j(s_{1}+s_{2})}\omega_{j}(T)\|\Delta_{j^{\prime}}f\|_{p}\|\widetilde{\Delta}_{j^{\prime}}g\|_{p}$
$\displaystyle\leq C\sum_{j}\sum_{j^{\prime}\geq j-3}\sum_{\ell\geq
j}2^{j-\ell}e_{\ell}(T)2^{j(s_{1}+s_{2})}\|\Delta_{j^{\prime}}f\|_{p}\|\widetilde{\Delta}_{j^{\prime}}g\|_{p}$
$\displaystyle=\sum_{j}\sum_{j^{\prime}\geq j-3}\sum_{\ell\geq
j,j^{\prime}}\square+\sum_{j}\sum_{j^{\prime}\geq
j-3}\sum_{j^{\prime}\geq\ell\geq j}\square$ $\displaystyle\triangleq I+II.$
For $II$, using the fact that $e_{\ell}(T)\leq
e_{j^{\prime}}(T)\leq\omega_{j^{\prime}}(T)$ if $\ell\leq j^{\prime}$, we get
$\displaystyle II$ $\displaystyle\leq$ $\displaystyle
C\sum_{j}\sum_{j^{\prime}\geq
j-3}\omega_{j^{\prime}}(T)2^{j(s_{1}+s_{2})}\|\Delta_{j^{\prime}}f\|_{p}\|\widetilde{\Delta}_{j^{\prime}}g\|_{p}$
$\displaystyle\leq$ $\displaystyle C\sum_{j}\sum_{j^{\prime}\geq
j-3}\omega_{j^{\prime}}(T)2^{(j-j^{\prime})(s_{1}+s_{2})}2^{j^{\prime}s_{1}}\|\Delta_{j^{\prime}}f\|_{p}\|g\|_{\dot{B}^{s_{2}}_{p,1}}$
$\displaystyle\leq$ $\displaystyle
C\|f\|_{\dot{B}^{s_{1}}_{p,1}(\omega)}\|g\|_{\dot{B}^{s_{2}}_{p,1}},$
and for $I$, using the fact that
$\displaystyle\sum_{\ell\geq j,j^{\prime}}2^{j-\ell}e_{\ell}(T)\leq
2^{j-j^{\prime}}\sum_{\ell\geq
j^{\prime}}2^{j^{\prime}-\ell}e_{\ell}(T)=2^{j-j^{\prime}}w_{j^{\prime}}(T),$
we obtain
$\displaystyle I$ $\displaystyle\leq$ $\displaystyle
C\sum_{j}\sum_{j^{\prime}\geq
j-3}\omega_{j^{\prime}}(T)2^{j(s_{1}+s_{2})}2^{j-j^{\prime}}\|\Delta_{j^{\prime}}f\|_{p}\|\widetilde{\Delta}_{j^{\prime}}g\|_{p}$
$\displaystyle\leq$ $\displaystyle C\sum_{j}\sum_{j^{\prime}\geq
j-3}\omega_{j^{\prime}}(T)2^{(j-j^{\prime})(s_{1}+s_{2}+1)}2^{j^{\prime}s_{1}}\|\Delta_{j^{\prime}}f\|_{p}\|g\|_{\dot{B}^{s_{2}}_{p,1}}$
$\displaystyle\leq$ $\displaystyle
C\|f\|_{\dot{B}^{s_{1}}_{p,1}(\omega)}\|g\|_{\dot{B}^{s_{2}}_{p,1}}.$
If $p<2$, we get by Lemma 2.3 that
$\displaystyle\|R(f,g)\|_{\dot{B}^{s_{1}+s_{2}-\frac{N}{p}}_{p,1}(\omega)}$
$\displaystyle\leq C\sum_{j}\sum_{j^{\prime}\geq
j-3}2^{j(s_{1}+s_{2}-N(\frac{2}{p}-1))}\omega_{j}(T)\|\Delta_{j^{\prime}}f\|_{p}\|\widetilde{\Delta}_{j^{\prime}}g\|_{p^{\prime}}$
$\displaystyle\leq C\sum_{j}\sum_{j^{\prime}\geq j-3}\sum_{\ell\geq
j}2^{j-\ell}e_{\ell}(T)2^{j(s_{1}+s_{2}-N(\frac{2}{p}-1))}\|\Delta_{j^{\prime}}f\|_{p}\|\widetilde{\Delta}_{j^{\prime}}g\|_{p}2^{N(\frac{2}{p}-1)j^{\prime}}.$
Then treating it as in the case of $p\geq 2$, we obtain the same inequality
for $s_{1}+s_{2}>N(\frac{2}{p}-1)$. This proves (c). $\blacksquare$
We have a similar result in the weighted Besov spaces with the time.
###### Lemma 3.5
Let $1\leq p,q,q_{1},q_{2}\leq\infty$ with
$\frac{1}{q_{1}}+\frac{1}{q_{2}}=\frac{1}{q}$. Assume that
$f\in\widetilde{L}^{q_{1}}_{T}(\dot{B}^{s_{1}}_{p,1}(\omega)),g\in\widetilde{L}^{q_{2}}_{T}(\dot{B}^{s_{2}}_{p,1})$.
Then there hold
(a) if $s_{2}\leq\frac{N}{p}$, we have
$\displaystyle\|T_{g}f\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s_{1}+s_{2}-\frac{N}{p}}_{p,1}(\omega))}\leq
C\|f\|_{\widetilde{L}^{q_{1}}_{T}(\dot{B}^{s_{1}}_{p,1}(\omega))}\|g\|_{\widetilde{L}^{q_{2}}_{T}(\dot{B}^{s_{2}}_{p,1})};$
(b) if $s_{1}\leq\frac{N}{p}-1$, we have
$\displaystyle\|T_{f}g\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s_{1}+s_{2}-\frac{N}{p}}_{p,1}(\omega))}\leq
C\|f\|_{\widetilde{L}^{q_{1}}_{T}(\dot{B}^{s_{1}}_{p,1}(\omega))}\|g\|_{\widetilde{L}^{q_{2}}_{T}(\dot{B}^{s_{2}}_{p,1})};$
(c) if $s_{1}+s_{2}>N\max(0,\frac{2}{p}-1)$, we have
$\displaystyle\|R(f,g)\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s_{1}+s_{2}-\frac{N}{p}}_{p,1}(\omega))}\leq
C\|f\|_{\widetilde{L}^{q_{1}}_{T}(\dot{B}^{s_{1}}_{p,1}(\omega))}\|g\|_{\widetilde{L}^{q_{2}}_{T}(\dot{B}^{s_{2}}_{p,1})}.$
The following proposition is a direct consequence of Lemma 3.5.
###### Proposition 3.6
Let
$s_{1}\leq\frac{N}{p}-1,s_{2}\leq\frac{N}{p},s_{1}+s_{2}>N\max(0,\frac{2}{p}-1)$,
and $1\leq p,q,q_{1},q_{2}\leq\infty$ with
$\frac{1}{q_{1}}+\frac{1}{q_{2}}=\frac{1}{q}$. Assume that
$f\in\widetilde{L}^{q_{1}}_{T}(\dot{B}^{s_{1}}_{p,1}(\omega))$ and
$g\in\widetilde{L}^{q_{2}}_{T}(\dot{B}^{s_{2}}_{p,1})$. Then there holds
$\displaystyle\|fg\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s_{1}+s_{2}-\frac{N}{p}}_{p,1}(\omega))}\leq
C\|f\|_{\widetilde{L}^{q_{1}}_{T}(\dot{B}^{s_{1}}_{p,1}(\omega))}\|g\|_{\widetilde{L}^{q_{2}}_{T}(\dot{B}^{s_{2}}_{p,1})}.$
From the proof of Lemma 3.4, we can also obtain
###### Proposition 3.7
Let $s_{1}\leq\frac{N}{p}-1,s_{2}<\frac{N}{p},s_{1}+s_{2}\geq
N\max(0,\frac{2}{p}-1)$, and $1\leq p,q,q_{1},q_{2}\leq\infty$ with
$\frac{1}{q_{1}}+\frac{1}{q_{2}}=\frac{1}{q}$. Assume that
$f\in\widetilde{L}^{q_{1}}_{T}(\dot{B}^{s_{1}}_{p,1}(\omega))$ and
$g\in\widetilde{L}^{q_{2}}_{T}(\dot{B}^{s_{2}}_{p,\infty})$. Then there holds
$\displaystyle\|fg\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s_{1}+s_{2}-\frac{N}{p}}_{p,\infty}(\omega))}\leq
C\|f\|_{\widetilde{L}^{q_{1}}_{T}(\dot{B}^{s_{1}}_{p,1}(\omega))}\|g\|_{\widetilde{L}^{q_{2}}_{T}(\dot{B}^{s_{2}}_{p,\infty})}.$
###### Proposition 3.8
Let $s>0$ and $1\leq p,q\leq\infty$. Assume that $F\in
W^{[s]+3,\infty}_{loc}(\mathop{\bf R\kern 0.0pt}\nolimits)$ with $F(0)=0$.
Then for any $f\in
L^{\infty}_{T}(L^{\infty})\cap\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,1}(\omega))$,
we have
$\displaystyle\|F(f)\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,1}(\omega))}\leq
C(1+\|f\|_{L^{\infty}_{T}(L^{\infty})})^{[s]+2}\|f\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,1}(\omega))}.$
Proof. We decompose $F(f)$ as
$\displaystyle
F(f)=\sum_{j^{\prime}\in{\mathbf{Z}}}F(S_{j^{\prime}+1}f)-F(S_{j^{\prime}}f)$
$\displaystyle=\sum_{j^{\prime}\in{\mathbf{Z}}}\Delta_{j^{\prime}}f\int_{0}^{1}F^{\prime}(S_{j^{\prime}}f+\tau\Delta_{j^{\prime}}f)d\tau$
$\displaystyle\triangleq\sum_{j^{\prime}\in{\mathbf{Z}}}\Delta_{j^{\prime}}f\,m_{j^{\prime}}(f),$
where
$m_{j^{\prime}}(f)=\int_{0}^{1}F^{\prime}(S_{j^{\prime}}f+\tau\Delta_{j^{\prime}}f)d\tau$.
Furthermore, we write
$\displaystyle\Delta_{j}F(f)=\sum_{j^{\prime}<j}\Delta_{j}(\Delta_{j^{\prime}}f\,m_{j^{\prime}}(f))+\sum_{j^{\prime}\geq
j}\Delta_{j}(\Delta_{j^{\prime}}f\,m_{j^{\prime}}(f))\triangleq I+II.$
By Lemma 2.3, we have
$\displaystyle\|I\|_{L^{q}_{T}(L^{p})}$
$\displaystyle\leq\sum_{j^{\prime}<j}\|\Delta_{j}(\Delta_{j^{\prime}}f\,m_{j^{\prime}}(f))\|_{L^{q}_{T}(L^{p})}$
$\displaystyle\leq\sum_{j^{\prime}<j}2^{-j|\alpha|}\sup_{|\gamma|=|\alpha|}\|D^{\gamma}\Delta_{j}(\Delta_{j^{\prime}}f\,m_{j^{\prime}}(f))\|_{L^{q}_{T}(L^{p})},$
(3.3)
with $\alpha$ to be determined later. Notice that for $|\gamma|\geq 0$, we
have
$\|D^{\gamma}m_{j^{\prime}}(f)\|_{\infty}\leq
C2^{j^{\prime}|\gamma|}(1+\|f\|_{\infty})^{|\gamma|}\|F^{\prime}\|_{W^{|\gamma|,\infty}},$
from which and (3), it follows that
$\displaystyle 2^{js}\|I\|_{L^{q}_{T}(L^{p})}\leq
C2^{j(s-|\alpha|)}\sum_{j^{\prime}<j}2^{j^{\prime}|\alpha|}\|\Delta_{j^{\prime}}f\|_{L^{q}_{T}(L^{p})}(1+\|f\|_{L^{\infty}_{T}(L^{\infty})})^{|\alpha|}\|F^{\prime}\|_{W^{|\alpha|,\infty}},$
thus, if we take $|\alpha|=[s]+2$, we get by (3.2) that
$\displaystyle\sum_{j}\omega_{j}(T)2^{js}\|I\|_{L^{q}_{T}(L^{p})}$
$\displaystyle\leq
C\sum_{j^{\prime}}2^{j^{\prime}s}\omega_{j^{\prime}}(T)\|\Delta_{j^{\prime}}f\|_{L^{q}_{T}(L^{p})}\sum_{j>j^{\prime}}2^{(j-j^{\prime})(s-|\alpha|+1)}(1+\|f\|_{L^{\infty}_{T}(L^{\infty})})^{|\alpha|}\|F^{\prime}\|_{W^{|\alpha|,\infty}}$
$\displaystyle\leq
C(1+\|f\|_{L^{\infty}_{T}(L^{\infty})})^{[s]+2}\|F^{\prime}\|_{W^{[s]+2,\infty}}\|f\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,1}(\omega))}.$
(3.4)
Now, let us turn to the estimate of $II$. We get by Lemma 2.3 that
$\displaystyle\|II\|_{L^{q}_{T}(L^{p})}$ $\displaystyle\leq
C\sum_{j^{\prime}\geq j}\|\Delta_{j^{\prime}}f\|_{L^{q}_{T}(L^{p})}.$
Then we write
$\displaystyle\sum_{j}\omega_{j}(T)2^{js}\|II\|_{L^{q}_{T}(L^{p})}$
$\displaystyle\leq C\sum_{j}2^{js}\sum_{j^{\prime}\geq
j}\|\Delta_{j^{\prime}}f\|_{L^{q}_{T}(L^{p})}\sum_{j^{\prime}\geq\ell\geq
j}2^{j-\ell}e_{\ell}(T)$
$\displaystyle\quad+C\sum_{j}2^{js}\sum_{j^{\prime}\geq
j}\|\Delta_{j^{\prime}}f\|_{L^{q}_{T}(L^{p})}\sum_{\ell\geq
j,j^{\prime}}2^{j-\ell}e_{\ell}(T),$
from which and a similar argument of (c) in Lemma 3.4, we infer that
$\displaystyle\sum_{j}\omega_{j}(T)2^{js}\|II\|_{L^{q}_{T}(L^{p})}\leq
C\|f\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s}_{p,1}(\omega))},$
from which and (3), we conclude the proof of Proposition 3.8. $\blacksquare$
## 4 Estimates of the linear transport equation
In this section, we study the linear transport equation
$\displaystyle\bigg{\\{}\begin{aligned} &\partial_{t}f+v\cdot\nabla f=g,\\\
&f(0,x)=f_{0}.\end{aligned}\bigg{.}$ (4.1)
###### Proposition 4.1
[14] Let $s\in(-N\min(\frac{1}{p},\frac{1}{p^{\prime}}),1+\frac{N}{p})$,
$1\leq p,r\leq+\infty$, and $s=1+\frac{N}{p}$, if $r=1$. Let $v$ be a vector
field such that $\nabla v\in L^{1}_{T}(\dot{B}^{\frac{N}{p}}_{p,r}\cap
L^{\infty})$. Assume that $f_{0}\in\dot{B}^{s}_{p,r},$ $g\in
L^{1}_{T}(\dot{B}^{s}_{p,r})$ and $f$ is the solution of (4.1). Then there
holds for $t\in[0,T]$,
$\displaystyle\|f\|_{\widetilde{L}^{\infty}_{t}(\dot{B}^{s}_{p,r})}\leq
e^{CV(t)}\bigg{(}\|f_{0}\|_{\dot{B}^{s}_{p,r}}+\int_{0}^{t}e^{-CV(\tau)}\|g(\tau)\|_{\dot{B}^{s}_{p,r}}d\tau\bigg{)},$
where $V(t)\buildrel\hbox{\footnotesize def}\over{=}\int_{0}^{t}\|\nabla
v(\tau)\|_{\dot{B}^{\frac{N}{p}}_{p,r}\cap L^{\infty}}d\tau.$ If $r<+\infty$,
then $f$ belongs to $C([0,T];\dot{B}^{s}_{p,r})$.
###### Proposition 4.2
Let $p\in[1,+\infty]$ and
$s\in(-N\min(\frac{1}{p},\frac{1}{p^{\prime}}),\frac{N}{p}]$. Let $v$ be a
vector field such that $\nabla v\in L^{1}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})$.
Assume that $f_{0}\in\dot{B}^{s}_{p,1},$ $g\in L^{1}_{T}(\dot{B}^{s}_{p,1})$
and $f$ is the solution of (4.1). Then there holds for $t\in[0,T]$,
$\displaystyle\|f\|_{\widetilde{L}^{\infty}_{t}(\dot{B}^{s}_{p,1}(\omega))}\leq
e^{CV(t)}\bigg{(}\|f_{0}\|_{\dot{B}^{s}_{p,1}(\omega)}+\int_{0}^{t}e^{-CV(\tau)}\|g(\tau)\|_{\dot{B}^{s}_{p,1}(\omega)}d\tau\bigg{)},$
where $V(t)\buildrel\hbox{\footnotesize def}\over{=}\int_{0}^{t}\|\nabla
v(\tau)\|_{\dot{B}^{\frac{N}{p}}_{p,1}}d\tau.$
Proof. Applying the operator $\Delta_{j}$ to the transport equation, we obtain
$\displaystyle\partial_{t}\Delta_{j}f+v\cdot\nabla\Delta_{j}f=\Delta_{j}g+[v,\Delta_{j}]\cdot\nabla
f.$ (4.2)
Assume that $p<+\infty$. Multiplying both sides of (4.2) by
$|\Delta_{j}f|^{p-2}\Delta_{j}f$, we get by integrating by parts over
$\mathop{\bf R\kern 0.0pt}\nolimits^{N}$ for the resulting equation that
$\displaystyle\frac{1}{p}\frac{d}{dt}\|\Delta_{j}f\|_{p}^{p}-\frac{1}{p}\int_{\mathop{\bf
R\kern
0.0pt}\nolimits^{N}}|\Delta_{j}f|^{p}\mbox{div}vdx\leq\bigl{(}\|\Delta_{j}g\|_{p}+\|[v,\Delta_{j}]\cdot\nabla
f\|_{p}\bigr{)}\|\Delta_{j}f\|_{p}^{p-1},$
then we have
$\displaystyle\|\Delta_{j}f(t)\|_{p}\leq\|\Delta_{j}f_{0}\|_{p}+\int_{0}^{t}\bigl{(}\|\Delta_{j}g\|_{p}+\|[v,\Delta_{j}]\cdot\nabla
f\|_{p}+\frac{1}{p}\|\mbox{div}v\|_{\infty}\|\Delta_{j}f\|_{p}\bigr{)}d\tau,$
from which, it follows that
$\displaystyle\|f\|_{\widetilde{L}^{\infty}_{t}(\dot{B}^{s}_{p,1}(\omega))}$
$\displaystyle\leq$
$\displaystyle\|f_{0}\|_{\dot{B}^{s}_{p,1}(\omega)}+C\int_{0}^{t}\|\mbox{div}v(\tau)\|_{\dot{B}^{\frac{N}{p}}_{p,1}}\|f(\tau)\|_{\widetilde{L}^{\infty}_{\tau}(\dot{B}^{s}_{p,1}(\omega))}d\tau$
$\displaystyle+\int_{0}^{t}\|g(\tau)\|_{\dot{B}^{s}_{p,1}(\omega)}d\tau+\int_{0}^{t}\sum_{j}\omega_{j}(T)2^{js}\|[v,\Delta_{j}]\cdot\nabla
f(\tau)\|_{p}d\tau,$
from which and Lemma 4.3, we infer that
$\displaystyle\|f\|_{\widetilde{L}^{\infty}_{t}(\dot{B}^{s}_{p,1}(\omega))}\leq\|f_{0}\|_{\dot{B}^{s}_{p,1}(\omega)}+C\int_{0}^{t}\|v(\tau)\|_{\dot{B}^{\frac{N}{p}+1}_{p,1}}\|f(\tau)\|_{\widetilde{L}^{\infty}_{\tau}(\dot{B}^{s}_{p,1}(\omega))}d\tau+\int_{0}^{t}\|g(\tau)\|_{\dot{B}^{s}_{p,1}(\omega)}d\tau.$
Then Gronwall’s lemma applied implies the desired inequality. $\blacksquare$
###### Lemma 4.3
Let
$p\in[1,\infty],s\in(-N\min(\frac{1}{p},\frac{1}{p^{\prime}}),\frac{N}{p}]$.
Assume that $v\in\dot{B}^{\frac{N}{p}+1}_{p,1}$ and
$f\in\dot{B}^{s}_{p,1}(\omega)$. Then there holds
$\displaystyle\sum_{j}\omega_{j}(T)2^{js}\|[v,\Delta_{j}]\cdot\nabla
f\|_{p}\leq
C\|v\|_{\dot{B}^{\frac{N}{p}+1}_{p,1}}\|f\|_{\dot{B}^{s}_{p,1}(\omega)}.$
Proof. Using the Bony’s decomposition, we write
$\displaystyle[v,\Delta_{j}]\cdot\nabla f$ $\displaystyle=$
$\displaystyle[T_{v^{k}},\Delta_{j}]\partial_{k}f+T_{\partial_{k}\Delta_{j}f}v^{k}+R(v^{k},\partial_{k}\Delta_{j}f)$
$\displaystyle-\Delta_{j}(T_{\partial_{k}f}v^{k})-\Delta_{j}R(v^{k},\partial_{k}f).$
Using Lemma 3.4 with $s_{1}=s-1$ and $s_{2}=\frac{N}{p}+1$, we get
$\displaystyle\sum_{j}\omega_{j}(T)2^{js}\|\Delta_{j}(T_{\partial_{k}f}v^{k})\|_{p}\leq
C\|v\|_{\dot{B}^{\frac{N}{p}+1}_{p,1}}\|f\|_{\dot{B}^{s}_{p,1}(\omega)},$
$\displaystyle\sum_{j}\omega_{j}(T)2^{js}\|\Delta_{j}R(v^{k},\partial_{k}f)\|_{p}\leq
C\|v\|_{\dot{B}^{\frac{N}{p}+1}_{p,1}}\|f\|_{\dot{B}^{s}_{p,1}(\omega)}.$
Notice that
$\displaystyle T_{\partial_{k}\Delta_{j}f}^{\prime}v^{k}\triangleq
T_{\partial_{k}\Delta_{j}f}v^{k}+R(v^{k},\partial_{k}\Delta_{j}f)=\sum_{j^{\prime}\geq
j-2}S_{j^{\prime}+2}\Delta_{j}\partial_{k}f\Delta_{j^{\prime}}v^{k},$
then we get by Lemma 2.3 that
$\displaystyle\sum_{j}\omega_{j}(T)2^{js}\|T_{\partial_{k}\Delta_{j}f}^{\prime}v^{k}\|_{p}$
$\displaystyle\leq$ $\displaystyle
C\sum_{j}\omega_{j}(T)2^{js}\|\Delta_{j}\nabla
f\|_{\infty}\sum_{j^{\prime}\geq j-2}\|\Delta_{j^{\prime}}v^{k}\|_{p}$
$\displaystyle\leq$ $\displaystyle
C\sum_{j}\omega_{j}(T)2^{j(s+1+\frac{N}{p})}\|\Delta_{j}f\|_{p}\sum_{j^{\prime}\geq
j}\|\Delta_{j^{\prime}}v^{k}\|_{p}$ $\displaystyle\leq$ $\displaystyle
C\|v\|_{\dot{B}^{\frac{N}{p}+1}_{p,1}}\|f\|_{\dot{B}^{s}_{p,1}(\omega)}.$
Now, we turn to estimate $[T_{v^{k}},\Delta_{j}]\partial_{k}f$. Set
$h(x)=({\cal F}^{-1}\varphi)(x)$, we get by using Taylor’s formula that
$\displaystyle[T_{v^{k}},\Delta_{j}]\partial_{k}f$ $\displaystyle=$
$\displaystyle\sum_{|j^{\prime}-j|\leq
4}[S_{j^{\prime}-1}v^{k},\Delta_{j}]\partial_{k}\Delta_{j^{\prime}}f$
$\displaystyle=$ $\displaystyle\sum_{|j^{\prime}-j|\leq
4}2^{Nj}\int_{\mathop{\bf R\kern
0.0pt}\nolimits^{N}}h(2^{j}(x-y))(S_{j^{\prime}-1}v^{k}(x)-S_{j^{\prime}-1}v^{k}(y))\partial_{k}\Delta_{j^{\prime}}f(y)dy$
$\displaystyle=$ $\displaystyle\sum_{|j^{\prime}-j|\leq
4}2^{(N+1)j}\int_{\mathop{\bf R\kern
0.0pt}\nolimits^{N}}\int_{0}^{1}y\cdot\nabla S_{j^{\prime}-1}v^{k}(x-\tau
y)d\tau\partial_{k}h(2^{j}y)\Delta_{j^{\prime}}f(x-y)dy$
$\displaystyle\qquad\quad+2^{Nj}\int_{\mathop{\bf R\kern
0.0pt}\nolimits^{N}}h(2^{j}(x-y))\partial_{k}S_{j^{\prime}-1}v^{k}(y)\Delta_{j^{\prime}}f(y)dy,$
from which and the Minkowski inequality, we infer that
$\displaystyle\sum_{j}\omega_{j}(T)2^{js}\|[T_{v^{k}},\Delta_{j}]\partial_{k}f\|_{p}$
$\displaystyle\leq$ $\displaystyle
C\sum_{j}\omega_{j}(T)2^{js}\sum_{|j^{\prime}-j|\leq 4}\|\nabla
S_{j^{\prime}-1}v\|_{\infty}\|\Delta_{j^{\prime}}f\|_{p}$ $\displaystyle\leq$
$\displaystyle
C\|v\|_{\dot{B}^{\frac{N}{p}+1}_{p,1}}\|f\|_{\dot{B}^{s}_{p,1}(\omega)}.$
Summing up all the above estimates, we conclude the proof of Lemma 4.3.
$\blacksquare$
## 5 Estimates of the linearized momentum equation
In this section, we study the linearized momentum equation
$\displaystyle\left\\{\begin{array}[]{ll}\partial_{t}u-\mbox{div}(\overline{\mu}\nabla
u)-\nabla((\overline{\lambda}+\overline{\mu})\mbox{div}\,u)=G,\\\
u|_{t=0}=u_{0}.\end{array}\right.$ (5.3)
In what follows, we assume that the viscosity coefficients
$\overline{\lambda}(\rho)$ and $\overline{\mu}(\rho)$ depend smoothly on the
function $\rho$ and there exists a positive constant $c_{1}$ such that
$\displaystyle\overline{\mu}\geq
c_{1},\quad\overline{\lambda}+2\overline{\mu}\geq c_{1}.$
Fix a positive constant $c$ to be chosen later. In this section, the weighted
function $\omega_{k}(t)$ is given by
$\omega_{k}(t)=\sum_{\ell\geq k}2^{k-\ell}e_{\ell}(t),$
with $e_{\ell}(t)=(1-e^{-c2^{2\ell}t})^{\frac{1}{2}}$. It is easy to verify
that the function $e_{\ell}(t)$ satisfies (3.1).
###### Proposition 5.1
Let $q\in[1,\infty]$. Assume that $G\in
L^{1}_{T}(\dot{B}^{s-1}_{p,1}),u_{0}\in\dot{B}^{s-1}_{p,1}$, and
$\rho-\underline{\rho}\in L^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})$. Let
$u$ be a solution of (5.3). Then there hold
(a) If $p\in(1,N],$
$s\in(-N\min(\frac{1}{p},\frac{1}{p^{\prime}})+1,\frac{N}{p}]$, we have
$\displaystyle\|u\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s-1+2/q}_{p,1})}\leq
C\Bigl{(}\|u_{0}\|_{\dot{B}^{s-1}_{p,1}}+\|G(\tau)\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s-1}_{p,1})}+A(T)\|\rho-\underline{\rho}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})}\Bigr{)};$
In addition, if $\rho-\underline{\rho}\in
L^{\infty}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})$, $p\in(1,\infty)$,
$s\in(-N\min(\frac{1}{p},\frac{1}{p^{\prime}})+1,\frac{N}{p}+1]$, then
$\displaystyle\|u\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s-1+2/q}_{p,1})}\leq
C\Bigl{(}\|u_{0}\|_{\dot{B}^{s-1}_{p,1}}+\|G(\tau)\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s-1}_{p,1})}+A(T)\|\rho-\underline{\rho}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s}_{p,1})}\Bigr{)};$
(b) If $p\in(1,N],$
$s\in(-N\min(\frac{1}{p},\frac{1}{p^{\prime}})+1,\frac{N}{p}]$, we have
$\displaystyle\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})}+\|u\|_{\widetilde{L}^{2}_{T}(\dot{B}^{s}_{p,1})}$
$\displaystyle\leq
C\Bigl{(}\|u_{0}\|_{\dot{B}^{s-1}_{p,1}(\omega))}+\|G(\tau)\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s-1}_{p,1}(\omega))}+A(T)\|\rho-\underline{\rho}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})}\Bigr{)}.$
Here $A(T)\buildrel\hbox{\footnotesize
def}\over{=}\bigl{(}1+\|\rho\|_{L^{\infty}_{T}(L^{\infty})}\bigr{)}^{[\frac{N}{p}]+2}$.
Proof. Set $d=\mbox{div}u$ and $w=\textrm{curl}u$. From (5.3), we find that
$(d,w)$ satisfies
$\displaystyle\left\\{\begin{array}[]{ll}\partial_{t}d-\mbox{div}(\overline{\nu}\nabla
d)=\mbox{div}G+F_{1},\\\ \partial_{t}w-\mbox{div}(\overline{\mu}\nabla
w)=\textrm{curl}G+F_{2},\\\ (d,w)|_{t=0}=(\mbox{div}u_{0},\textrm{
curl}u_{0})\triangleq(d_{0},w_{0}),\end{array}\right.$ (5.7)
where $\overline{\nu}=\overline{\lambda}+2\overline{\mu}$ and
$\displaystyle F_{1}=\mbox{div}(\nabla\overline{\mu}\cdot\nabla
u)+\mbox{div}(\nabla(\overline{\lambda}+\overline{\mu})d),$ $\displaystyle
F_{2}^{i,j}=\mbox{div}(\partial_{j}\overline{\mu}\nabla
u^{i}-\partial_{i}\overline{\mu}\nabla u^{j}),\quad i,j=1,\cdots,N.$
Applying the operator $\Delta_{j}$ to (5.7), we obtain
$\displaystyle\left\\{\begin{array}[]{ll}\partial_{t}\Delta_{j}d-\mbox{div}(\overline{\nu}\nabla\Delta_{j}d)=\mbox{div}\Delta_{j}G+\Delta_{j}F_{1}+\mbox{div}[\Delta_{j},\overline{\nu}]\nabla
d,\\\
\partial_{t}\Delta_{j}w-\mbox{div}(\overline{\mu}\nabla\Delta_{j}w)=\textrm{curl}\Delta_{j}G+\Delta_{j}F_{2}+\mbox{div}[\Delta_{j},\overline{\mu}]\nabla
w.\end{array}\right.$
Multiplying the first equation by $|\Delta_{j}d|^{p-2}\Delta_{j}d$, we get by
integrating over $\mathop{\bf R\kern 0.0pt}\nolimits^{N}$ that
$\displaystyle\frac{1}{p}\frac{\operatorname{d}}{\operatorname{d}t}\|\Delta_{j}d\|_{p}^{p}-\int_{\mathop{\bf
R\kern
0.0pt}\nolimits^{N}}\mbox{div}(\overline{\nu}\nabla\Delta_{j}d)|\Delta_{j}d|^{p-2}\Delta_{j}d\operatorname{d}x$
$\displaystyle\quad=\int_{\mathop{\bf R\kern
0.0pt}\nolimits^{N}}\bigl{(}\mbox{div}\Delta_{j}G+\Delta_{j}F_{1}+\mbox{div}[\Delta_{j},\overline{\nu}]\nabla
d\bigr{)}|\Delta_{j}d|^{p-2}\Delta_{j}d\operatorname{d}x$
Lemma 2.4 ensures there exists a positive constant $c_{p}$ depending on
$c_{0},p,N$ such that
$\displaystyle\frac{1}{p}\frac{\operatorname{d}}{\operatorname{d}t}\|\Delta_{j}d\|_{p}^{p}+c_{p}2^{2j}\|\Delta_{j}d\|_{p}^{p}\leq\|\Delta_{j}d\|_{p}^{p-1}\bigl{(}\|\mbox{div}\Delta_{j}G\|_{p}+\|\Delta_{j}F_{1}\|_{p}+\|\mbox{div}[\Delta_{j},\overline{\nu}]\nabla
d\|_{p}\bigr{)}.$
Thus, we have
$\displaystyle\frac{\operatorname{d}}{\operatorname{d}t}\|\Delta_{j}d\|_{p}+c_{p}2^{2j}\|\Delta_{j}d\|_{p}\leq\|\mbox{div}\Delta_{j}G\|_{p}+\|\Delta_{j}F_{1}\|_{p}+\|\mbox{div}[\Delta_{j},\overline{\nu}]\nabla
d\|_{p},$
which implies that
$\displaystyle\|\Delta_{j}d(t)\|_{p}\leq
e^{-c_{p}2^{2j}t}\|\Delta_{j}d_{0}\|_{p}+\int_{0}^{t}e^{-c_{p}2^{2j}(t-\tau)}\bigl{(}\|\mbox{div}\Delta_{j}G\|_{p}+\|\Delta_{j}F_{1}\|_{p}+\|\mbox{div}[\Delta_{j},\overline{\nu}]\nabla
d\|_{p}\bigr{)}\operatorname{d}\tau.$
Similarly, we can obtain
$\displaystyle\|\Delta_{j}w(t)\|_{p}\leq
e^{-c_{p}2^{2j}t}\|\Delta_{j}w_{0}\|_{p}+\int_{0}^{t}e^{-c_{p}2^{2j}(t-\tau)}\bigl{(}\|\textrm{curl}\Delta_{j}G\|_{p}+\|\Delta_{j}F_{2}\|_{p}+\|\mbox{div}[\Delta_{j},\overline{\mu}]\nabla
w\|_{p}\bigr{)}\operatorname{d}\tau.$
From the above two inequalities, we infer that for any $q\in[1,\infty]$ and
$t\in[0,T]$,
$\displaystyle\|\Delta_{j}d(t)\|_{L^{q}_{t}(L^{p})}+\|\Delta_{j}w(t)\|_{L^{q}_{t}(L^{p})}$
$\displaystyle\leq
C2^{-2j/q}c_{j}(T)^{\frac{1}{q}}(\|\Delta_{j}d_{0}\|_{p}+\|\Delta_{j}w_{0}\|_{p})$
$\displaystyle\quad+C2^{-2j/q}c_{j}(T)^{\frac{1}{q}}\bigl{(}\|\mbox{div}\Delta_{j}G\|_{L^{1}_{t}(L^{p})}+\|\Delta_{j}F_{1}\|_{L^{1}_{t}(L^{p})}+\|\mbox{div}[\Delta_{j},\overline{\nu}]\nabla
d\|_{L^{1}_{t}(L^{p})}\bigr{)}$
$\displaystyle\quad+C2^{-2j/q}c_{j}(T)^{\frac{1}{q}}\bigl{(}\|\textrm{curl}\Delta_{j}G\|_{L^{1}_{t}(L^{p})}+\|\Delta_{j}F_{2}\|_{L^{1}_{t}(L^{p})}+\|\mbox{div}[\Delta_{j},\overline{\mu}]\nabla
w\|_{L^{1}_{t}(L^{p})}\bigr{)},$ (5.9)
with $c_{j}(T)=1-e^{c_{p}2^{2j}T}$. Notice that
$\displaystyle
2^{j}\|\Delta_{j}u\|_{p}\sim\|\Delta_{j}d\|_{p}+\|\Delta_{j}w\|_{p},\quad
e_{j}(T)\leq\omega_{j}(T),$
which together with (5) implies that
$\displaystyle\|u\|_{\widetilde{L}^{q}_{T}(\dot{B}^{s-1+2/q}_{p,1})}\leq
C\Bigl{(}\|u_{0}\|_{\dot{B}^{s-1}_{p,1}}+\|G\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s-1}_{p,1})}+\|(F_{1},F_{2})\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s-2}_{p,1})}\Bigr{)}$
$\displaystyle\qquad+C\sum_{j}2^{j(s-2)}\bigl{(}\|\mbox{div}[\Delta_{j},\overline{\nu}]\nabla
d\|_{L^{1}_{T}(L^{p})}+\|\mbox{div}[\Delta_{j},\overline{\mu}]\nabla
w\|_{L^{1}_{T}(L^{p})}\bigr{)},$ (5.10)
and with $c=c_{p}$ in the definition of $e_{k}(t)$,
$\displaystyle\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s-1}_{p,1})}+\|u\|_{\widetilde{L}^{2}_{T}(\dot{B}^{s}_{p,1})}$
$\displaystyle\leq
C\bigl{(}\|u_{0}\|_{\dot{B}^{s-1}_{p,1}(\omega)}+\|G\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s-1}_{p,1}(\omega))}+\|(F_{1},F_{2})\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s-2}_{p,1}(\omega))}\bigr{)}$
$\displaystyle\quad+C\sum_{j}2^{j(s-2)}\omega_{j}(T)\bigl{(}\|\mbox{div}[\Delta_{j},\overline{\nu}]\nabla
d\|_{L^{1}_{T}(L^{p})}+\|\mbox{div}[\Delta_{j},\overline{\mu}]\nabla
w\|_{L^{1}_{T}(L^{p})}\bigr{)}.$ (5.11)
First of all, we deal with the right hand side of (5). From Lemma 2.6 and 2.9,
we infer that
$\displaystyle\|F_{1}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s-2}_{p,1})}$
$\displaystyle\leq$ $\displaystyle C\bigl{(}\|\nabla\overline{\mu}\cdot\nabla
u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s-1}_{p,1})}+\|\nabla(\overline{\lambda}+\overline{\mu})d\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s-1}_{p,1})}\bigr{)}$
(5.12) $\displaystyle\leq$ $\displaystyle
C\bigl{(}\|\overline{\mu}-\overline{\mu}(\underline{\rho})\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}+\|\overline{\lambda}-\overline{\lambda}(\underline{\rho})\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\bigr{)}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})}$
$\displaystyle\leq$ $\displaystyle
CA(T)\|\rho-\underline{\rho}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})}.$
Similarly, we have
$\displaystyle\|F_{2}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s-2}_{p,1})}\leq
CA(T)\|\rho-\underline{\rho}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})}.$
(5.13)
While, we write
$\displaystyle[\Delta_{j},\overline{\nu}]\nabla
d=[\Delta_{j},\overline{\nu}-\overline{\nu}(\underline{\rho})]\nabla
d=\Delta_{j}((\overline{\nu}-\overline{\nu}(\underline{\rho}))\nabla
d)-(\overline{\nu}-\overline{\nu}(\underline{\rho}))\Delta_{j}\nabla d,$
then by Lemma 2.6, Lemma 2.9 we get for $p\in[1,N]$
$\displaystyle\sum_{j}2^{j(s-2)}\|\mbox{div}\Delta_{j}((\overline{\nu}-\overline{\nu}(\underline{\rho}))\nabla
d)\|_{L^{1}_{T}(L^{p})}\leq
CA(T)\|\rho-\underline{\rho}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})},$
$\displaystyle\sum_{j}2^{j(s-2)}\Big{(}\|\overline{\nu}-\overline{\nu}(\underline{\rho})\|_{L^{\infty}_{T}(L^{\infty})}\|\Delta_{j}\mbox{div}\nabla
d\|_{L^{1}_{T}(L^{p})}+\|\mbox{div}(\overline{\nu}-\overline{\nu}(\underline{\rho}))\|_{L^{\infty}_{T}(L^{N})}\|\Delta_{j}\nabla
d\|_{L^{1}_{T}(L^{\frac{pN}{N-p}})}\Big{)}\quad$ $\displaystyle\quad\leq
CA(T)\|\rho-\underline{\rho}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})},$
which imply that
$\displaystyle\sum_{j}2^{j(s-2)}\|\mbox{div}[\Delta_{j},\overline{\nu}]\nabla
d\|_{L^{1}_{T}(L^{p})}\leq
CA(T)\|\rho-\underline{\rho}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})}.$
(5.14)
Similarly, we have
$\displaystyle\sum_{j}2^{j(s-2)}\|\mbox{div}[\Delta_{j},\overline{\mu}]\nabla
w\|_{L^{1}_{T}(L^{p})}\leq
CA(T)\|\rho-\underline{\rho}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})}.$
(5.15)
Then the first inequality of Proposition 5.1 (a) can be deduced from (5) and
(5.12)-(5.15). On the other hand, using Lemma 2.6 and 2.9, we also have
$\displaystyle\|F_{1}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s-2}_{p,1})}+\|F_{2}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s-2}_{p,1})}\leq
CA(T)\|\rho-\underline{\rho}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s}_{p,1})},$
and by Lemma 2.8,
$\displaystyle\sum_{j}2^{j(s-2)}\bigl{(}\|\mbox{div}[\Delta_{j},\overline{\nu}]\nabla
d\|_{L^{1}_{T}(L^{p})}+\|\mbox{div}[\Delta_{j},\overline{\mu}]\nabla
w\|_{L^{1}_{T}(L^{p})}\bigr{)}$ $\displaystyle\quad\leq
CA(T)\|\rho-\underline{\rho}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s}_{p,1})},$
which together with (5) lead to the second inequality of Proposition 5.1 (a).
Next, we deal with the right hand side of (5). From Proposition 3.6 and 3.8,
it follows that
$\displaystyle\|F_{1}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s-2}_{p,1}(\omega))}$
$\displaystyle\leq$ $\displaystyle C\bigl{(}\|\nabla\overline{\mu}\cdot\nabla
u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s-1}_{p,1}(\omega))}+\|\nabla(\overline{\lambda}+\overline{\mu})d\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s-1}_{p,1}(\omega))}\bigr{)}$
(5.16) $\displaystyle\leq$ $\displaystyle
C\bigl{(}\|\overline{\mu}-\overline{\mu}(\underline{\rho})\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}+\|\overline{\lambda}-\overline{\lambda}(\underline{\rho})\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\bigr{)}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})}$
$\displaystyle\leq$ $\displaystyle
CA(T)\|\rho-\underline{\rho}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})}.$
Similarly, we have
$\displaystyle\|F_{2}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s-2}_{p,1}(\omega))}\leq
CA(T)\|\rho-\underline{\rho}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})}.$
(5.17)
Notice that
$\displaystyle[\Delta_{j},\overline{\nu}]\nabla
d=[\Delta_{j},\overline{\nu}-\overline{\nu}(\underline{\rho})]\nabla
d,\quad[\Delta_{j},\overline{\mu}]\nabla
w=[\Delta_{j},\overline{\mu}-\overline{\mu}(\underline{\rho})]\nabla w,$
which together with Lemma 5.2 and Proposition 3.8 ensures that
$\displaystyle\sum_{j}2^{j(s-2)}\omega_{j}(T)\bigl{(}\|\mbox{div}[\Delta_{j},\overline{\nu}]\nabla
d\|_{L^{1}_{T}(L^{p})}+\|\mbox{div}[\Delta_{j},\overline{\mu}]\nabla
w\|_{L^{1}_{T}(L^{p})}\bigr{)}$ $\displaystyle\quad\leq
C\bigl{(}\|\overline{\nu}-\overline{\nu}(\underline{\rho})\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}+\|\overline{\mu}-\overline{\mu}(\underline{\rho})\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\bigr{)}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})}$
$\displaystyle\quad\leq
A(T)\|\rho-\underline{\rho}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})}.$
(5.18)
Summing up (5) and (5.16)-(5), we obtain the inequality of Proposition 5.1
(b). $\blacksquare$
###### Lemma 5.2
Let $p\in[1,N]$ and
$s\in(-N\min(\frac{1}{p},\frac{1}{p^{\prime}}),\frac{N}{p}]$. Assume that
$f\in\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))$ and
$g\in\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})$. Then there holds
$\displaystyle\sum_{j}2^{j(s-1)}\omega_{j}(T)\|{\rm div}[\Delta_{j},f]\nabla
g\|_{L^{1}_{T}(L^{p})}\leq
C\|f\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\|g\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})}.$
Proof. Using the Bony’s decomposition, we write
$\displaystyle[f,\Delta_{j}]\partial_{k}g$ $\displaystyle=$
$\displaystyle[T_{f},\Delta_{j}]\partial_{k}g+T_{\partial_{k}\Delta_{j}g}f+R(f,\partial_{k}\Delta_{j}g)$
$\displaystyle-\Delta_{j}(T_{\partial_{k}g}f)-\Delta_{j}R(f,\partial_{k}g).$
Using Lemma 3.5 (a) and (c) with $s_{1}=\frac{N}{p}$ and $s_{2}=s$, we get
$\displaystyle\sum_{j}\omega_{j}(T)2^{j(s-1)}\|\mbox{div}\Delta_{j}(T_{\partial_{k}g}f)\|_{L^{1}_{T}(L^{p})}\leq
C\|f\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\|g\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})},$
$\displaystyle\sum_{j}\omega_{j}(T)2^{j(s-1)}\|\mbox{div}\Delta_{j}R(f,\partial_{k}g)\|_{L^{1}_{T}(L^{p})}\leq
C\|f\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\|g\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})}.$
Thanks to the proof of Lemma 4.3, we have
$\displaystyle T_{\partial_{k}\Delta_{j}g}^{\prime}f\triangleq
T_{\partial_{k}\Delta_{j}g}f+R(f,\partial_{k}\Delta_{j}g)=\sum_{j^{\prime}\geq
j-2}S_{j^{\prime}+2}\Delta_{j}\partial_{k}g\Delta_{j^{\prime}}f,$
then we get by Lemma 2.3 and (3.2) that
$\displaystyle\sum_{j}\omega_{j}(T)2^{j(s-1)}\|\mbox{div}T_{\partial_{k}\Delta_{j}g}^{\prime}f\|_{L^{1}_{T}(L^{p})}$
$\displaystyle\leq C\sum_{j}\omega_{j}(T)2^{js}\Big{(}\|\Delta_{j}\nabla
g\|_{L^{1}_{T}(L^{\infty})}\sum_{j^{\prime}\geq
j-2}\|\Delta_{j^{\prime}}f\|_{L^{1}_{T}(L^{p})}+\|\Delta_{j}g\|_{L^{1}_{T}(L^{\infty})}\sum_{j^{\prime}\geq
j-2}2^{j^{\prime}}\|\Delta_{j^{\prime}}f\|_{L^{1}_{T}(L^{p})}\Big{)}$
$\displaystyle\leq
C\sum_{j}2^{j(s+\frac{N}{p})}\|\Delta_{j}g\|_{L^{1}_{T}(L^{p})}\sum_{j^{\prime}\geq
j-2}(2^{j}+2^{j^{\prime}})\omega_{j^{\prime}}(T)\|\Delta_{j^{\prime}}f\|_{L^{1}_{T}(L^{p})}$
$\displaystyle\leq
C\|f\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\|g\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})}.$
Set $h(x)=({\cal F}^{-1}\varphi)(x)$. Thanks to the proof of Lemma 4.3, we
have
$\displaystyle[T_{f},\Delta_{j}]\partial_{k}g$ $\displaystyle=$
$\displaystyle\sum_{|j^{\prime}-j|\leq 4}2^{(N+1)j}\int_{\mathop{\bf R\kern
0.0pt}\nolimits^{N}}\int_{0}^{1}y\cdot\nabla S_{j^{\prime}-1}f(x-\tau
y)d\tau\partial_{k}h(2^{j}y)\Delta_{j^{\prime}}g(x-y)dy$
$\displaystyle\qquad\quad+2^{Nj}\int_{\mathop{\bf R\kern
0.0pt}\nolimits^{N}}h(2^{j}(x-y))\partial_{k}S_{j^{\prime}-1}f(y)\Delta_{j^{\prime}}g(y)dy,$
from which and a similar argument of Lemma 3.5 (b), we infer that
$\displaystyle\sum_{j}\omega_{j}(T)2^{j(s-1)}\|\mbox{div}[T_{f},\Delta_{j}]\partial_{k}g\|_{L^{1}_{T}(L^{p})}\leq
C\|f\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\|g\|_{\widetilde{L}^{1}_{T}(\dot{B}^{s+1}_{p,1})}.$
Summing up all the above estimates, we conclude the proof of Lemma 5.2.
$\blacksquare$
To prove the uniqueness of the solution, we also need the following
proposition.
###### Proposition 5.3
Let $p\in[2,N]$. Assume that $G\in
L^{1}_{T}(\dot{B}^{-\frac{N}{p}}_{p,\infty}),u_{0}\in\dot{B}^{-\frac{N}{p}}_{p,\infty}$,
and $\rho-\underline{\rho}\in L^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})$.
Let $u$ be a solution of (5.3). Then there holds
$\displaystyle\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{-\frac{N}{p}+2}_{p,\infty})}+\|u\|_{\widetilde{L}^{2}_{T}(\dot{B}^{-\frac{N}{p}+1}_{p,\infty})}$
$\displaystyle\leq
C\Bigl{(}\|u_{0}\|_{\dot{B}^{-\frac{N}{p}}_{p,\infty}}+\|G(\tau)\|_{\widetilde{L}^{1}_{T}(\dot{B}^{-\frac{N}{p}}_{p,\infty}(\omega))}+A(T)\|\rho-\underline{\rho}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{-\frac{N}{p}+2}_{p,\infty})}\Bigr{)}.$
Proof. We closely follow the proof of Proposition 5.1. From (5), we infer that
$\displaystyle\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{-\frac{N}{p}+2}_{p,\infty})}+\|u\|_{\widetilde{L}^{2}_{T}(\dot{B}^{-\frac{N}{p}+1}_{p,\infty})}$
$\displaystyle\leq
C\Bigl{(}\|u_{0}\|_{\dot{B}^{-\frac{N}{p}}_{p,\infty}(\omega)}+\|G\|_{\widetilde{L}^{1}_{T}(\dot{B}^{-\frac{N}{p}}_{p,\infty}(\omega))}+\|(F_{1},F_{2})\|_{\widetilde{L}^{1}_{T}(\dot{B}^{-\frac{N}{p}-1}_{p,\infty}(\omega))}\Bigr{)}$
$\displaystyle\quad+C\sup_{j\in{\mathbf{Z}}}2^{j(-\frac{N}{p}-1)}\omega_{j}(T)\bigl{(}\|\mbox{div}[\Delta_{j},\overline{\nu}]\nabla
d\|_{L^{1}_{T}(L^{p})}+\|\mbox{div}[\Delta_{j},\overline{\mu}]\nabla
w\|_{L^{1}_{T}(L^{p})}\bigr{)}.$ (5.19)
We use Proposition 3.7 to get
$\displaystyle\|(F_{1},F_{2})\|_{\widetilde{L}^{1}_{T}(\dot{B}^{-\frac{N}{p}-1}_{p,\infty}(\omega))}\leq
CA(T)\|\rho-\underline{\rho}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{-\frac{N}{p}+2}_{p,\infty})}.$
From Lemma 5.4, the second term on the right hand side of (5) is bounded by
$\displaystyle
CA(T)\|\rho-\underline{\rho}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\|u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{-\frac{N}{p}+2}_{p,\infty})}.$
This completes the proof of Proposition 5.3. $\blacksquare$
###### Lemma 5.4
Let $p\in[1,N]$. Assume that
$f\in\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))$ and
$g\in\widetilde{L}^{1}_{T}(\dot{B}^{-\frac{N}{p}+1}_{p,\infty})$. Then there
holds
$\displaystyle\sup_{j\in{\mathbf{Z}}}2^{j(-\frac{N}{p}-1)}\omega_{j}(T)\|{\rm
div}[\Delta_{j},f]\nabla g\|_{L^{1}_{T}(L^{p})}\leq
C\|f\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\|g\|_{\widetilde{L}^{1}_{T}(\dot{B}^{-\frac{N}{p}+1}_{p,\infty})}.$
The proof of Lemma 5.4 is very similar to that of Lemma 5.2. Here we omit its
proof.
## 6 The proof of existence
We set
$\displaystyle
a(t,x)=\frac{\rho(t,x)-\overline{\rho}_{0}}{\overline{\rho}_{0}},\quad\overline{\mu}(\rho)=\frac{\mu(\rho)}{\rho},\quad\overline{\lambda}(\rho)=\frac{\lambda(\rho)}{\rho}.$
Then the system (1.1) reads
$\displaystyle\left\\{\begin{array}[]{ll}\partial_{t}a+u\cdot\nabla a=F,\\\
\partial_{t}u-\mbox{div}(\overline{\mu}\nabla
u)-\nabla((\overline{\lambda}+\overline{\mu})\mbox{div}\,u)=G,\\\
(a,u)|_{t=0}=(a_{0},u_{0}),\end{array}\right.$ (6.4)
with $a_{0}=\frac{\rho_{0}(x)-\overline{\rho}_{0}}{\overline{\rho}_{0}}$ and
$\displaystyle F(a,u)=-(1+a)\mbox{div}\,u,$ $\displaystyle
G(a,u)=-u\cdot\nabla u+\frac{\overline{\rho}_{0}P^{\prime}(\rho)}{\rho}\nabla
a+\frac{\mu(\rho)}{\rho^{2}}\nabla\rho\cdot\nabla
u+\frac{\mu(\rho)+\lambda(\rho)}{\rho^{2}}\nabla\rho\mbox{div}\,u.$
Step 1. The approximate solution sequence
We smooth out the data as follows:
$\displaystyle a^{n}_{0}=S_{n+N}a_{0},\quad u_{0}^{n}=S_{n}u_{0},$
where $N\in{\mathbf{Z}}$ is chosen such that
$\displaystyle\overline{\rho}_{0}(1+a^{n}_{0}(x))\geq\frac{3}{4}c_{0}.$ (6.5)
A standard linearized argument (as in the proof of Theorem 4.2 in [12]) will
ensure that the system (6.4) with the smooth data $(a_{0}^{n},u_{0}^{n})$ has
a solution $(a^{n},u^{n})$ on a time interval $[0,T_{n}]$ for some $T_{n}>0$
such that
$\begin{split}&a^{n}\in
C([0,T_{n}];\dot{B}^{\frac{N}{p}}_{p,1}\cap\dot{B}^{\frac{N}{p}+1}_{p,1})\quad\textrm{and}\\\
&u^{n}\in
C([0,T_{n}];\dot{B}^{\frac{N}{p}-1}_{p,1}\cap\dot{B}^{\frac{N}{p}}_{p,1})\cap
L^{1}([0,T_{n}];\dot{B}^{\frac{N}{p}+1}_{p,1}\cap\dot{B}^{\frac{N}{p}+2}_{p,1}).\end{split}$
(6.6)
In what follows, we also denote by $T_{n}$ the maximal lifespan of the
solution $(a^{n},u^{n})$.
Step 2. Uniform estimates
Let
$E_{0}:=\|a_{0}\|_{\dot{B}^{\frac{N}{p}}_{p,1}}+\|u_{0}\|_{\dot{B}^{\frac{N}{p}-1}_{p,1}}$
and $T\in(0,T_{n})$. We assume that the solutions $(a^{n},u^{n})$ satisfies
the following inequalities for some positive constants $c_{1},C_{0},A_{0}$ and
$\eta$(to be determined later):
(H1) $\overline{\rho}_{0}(1+a_{0}^{n}(t,x))\geq\frac{c_{0}}{2}$ for any
$(t,x)\in[0,T]\times\mathop{\bf R\kern 0.0pt}\nolimits^{N}$;
(H2) $\overline{\mu}^{n}(t,x)\geq
c_{1},\overline{\lambda}^{n}(t,x)+2\overline{\mu}^{n}(t,x)\geq c_{1}$ for any
$(t,x)\in[0,T]\times\mathop{\bf R\kern 0.0pt}\nolimits^{N}$;
(H3)
$\|a^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}+\|u^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}-1}_{p,1})}\leq
C_{0}E_{0};$
(H4)
$\|a^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\leq
A_{0}\eta,\,\|u^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})}+\|u^{n}\|_{\widetilde{L}^{2}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\leq\eta.$
In what follows, we will show that if the conditions (H1) to (H4) are
satisfied for some $T>0$, then they are actually satisfied with strict
inequalities. Since all those conditions depend continuously on the time
variable and are satisfied initially, a standard bootstrap argument will
ensure that (H1) to (H4) are indeed satisfied for $T$.
First of all, we get by Proposition 4.1 that
$\displaystyle\|a^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\leq
e^{CV^{n}(T)}\bigl{(}\|a_{0}\|_{\dot{B}^{\frac{N}{p}}_{p,1}}+\|F^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\bigr{)},$
(6.7)
and by Proposition 5.1, we have
$\displaystyle\|u^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}-1}_{p,1})}$
$\displaystyle\leq$ $\displaystyle
C\|u_{0}\|_{\dot{B}^{\frac{N}{p}-1}_{p,1}}+C\|G^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}-1}_{p,1})}$
(6.8)
$\displaystyle+CA^{n}(T)\|a^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\|u^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})},$
where $V^{n}(t)=\int_{0}^{t}\|\nabla
u^{n}(\tau)\|_{\dot{B}^{\frac{N}{p}}_{p,1}}d\tau$ and
$A^{n}(T)=\bigl{(}1+\|a^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\bigr{)}^{[\frac{N}{p}]+3}$.
For $F^{n}$, we apply Lemma 2.6 to get
$\displaystyle\|F^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\leq\|u^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})}+C\|a^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\|u^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})},$
and for $G^{n}$, we use Lemma 2.6 and 2.9 to get
$\displaystyle\|G^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}-1}_{p,1})}$
$\displaystyle\leq$ $\displaystyle
C\|u^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}-1}_{p,1})}\|u^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})}$
$\displaystyle+CA^{n}(T)\|a^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\bigl{(}T+\|u^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})}\bigr{)}.$
Plugging the above two estimates into (6.7) and (6.8), we obtain
$\displaystyle\|a^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}+\|u^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}-1}_{p,1})}\leq
C_{1}e^{C_{1}V^{n}(T)}(E_{0}+(C_{0}E_{0}+1)\eta)+C_{1}A^{n}(T)C_{0}E_{0}(T+\eta).$
(6.9)
Next, we get by Proposition 4.2 that
$\displaystyle\|a^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\leq
e^{CV^{n}(T)}\bigl{(}\|a_{0}\|_{\dot{B}^{\frac{N}{p}}_{p,1}(\omega)}+\|F^{n}\|_{\widetilde{L}^{1}_{T}\dot{B}^{\frac{N}{p}}_{p,1}(\omega)}\bigr{)},$
(6.10)
and by Proposition 5.1, we have
$\displaystyle\|u^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})}+\|u^{n}\|_{\widetilde{L}^{2}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}$
$\displaystyle\leq
C\Bigl{(}\|u_{0}\|_{\dot{B}^{\frac{N}{p}-1}_{p,1}(\omega))}+\|G^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}-1}_{p,1}(\omega))}+A^{n}(T)\|a^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\|u^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})}\Bigr{)}.$
(6.11)
For $F^{n}$, we have
$\displaystyle\|F^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\leq
2\|F^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\leq
C(1+\|a^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})})\|u^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})},$
and for $G^{n}$, we use Proposition 3.6 and 3.8 to get
$\displaystyle\|G^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}-1}_{p,1}(\omega))}\leq
C\|u^{n}\|_{\widetilde{L}^{2}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}^{2}+CA^{n}(T)\|a^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\bigl{(}T+\|u^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})}\bigr{)}.$
Plugging the above two estimates into (6.10) and (6), we obtain
$\displaystyle\|a^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\leq
e^{C_{1}V^{n}(T)}\bigl{(}\|a_{0}\|_{\dot{B}^{\frac{N}{p}}_{p,1}(\omega)}+C_{2}(1+C_{0}E_{0})\eta\bigr{)},$
(6.12)
$\displaystyle\|u^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})}+\|u^{n}\|_{\widetilde{L}^{2}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\leq
C_{3}\Bigl{(}\|u_{0}\|_{\dot{B}^{\frac{N}{p}-1}_{p,1}(\omega))}+\eta^{2}+A_{0}A^{n}(T)\eta(T+\eta)\Bigr{)}.\quad$
(6.13)
According to the definition of $V^{n}$ and $A^{n}$, we have
$\displaystyle V^{n}(T)\leq\eta,\quad
A^{n}(T)\leq(1+C_{0}E_{0})^{[\frac{N}{p}]+3}.$
Let $C_{0}=4C_{1}$ and $A_{0}=2C_{2}(1+C_{0}E_{0})$. Then we take $\eta$ small
enough such that
$\begin{split}&e^{C_{1}\eta}<\frac{3}{2},\,(C_{0}E_{0}+1)\eta\leq
E_{0},\,C_{1}(C_{0}E_{0}+1)^{[\frac{N}{p}]+3}\eta\leq\frac{1}{16},\\\
&C_{3}\eta\leq\frac{1}{6},\,C_{3}A_{0}(1+C_{0}E_{0})^{[\frac{N}{p}]+3}\eta\leq\frac{1}{6}.\end{split}$
(6.14)
Next, we take $T$ small enough such that
$\displaystyle
C_{1}(C_{0}E_{0}+1)^{[\frac{N}{p}]+3}T\leq\frac{1}{16},\,C_{3}A_{0}(1+C_{0}E_{0})^{[\frac{N}{p}]+3}T\leq\frac{1}{6}.$
(6.15)
and note that $\omega_{k}(0)=0$ and
$(a_{0},u_{0})\in\dot{B}^{\frac{N}{p}}_{p,1}\times\dot{B}^{\frac{N}{p}-1}_{p,1}$,
we can also take $T$ small enough such that
$\displaystyle\|a_{0}\|_{\dot{B}^{\frac{N}{p}}_{p,1}(\omega)}\leq\frac{A_{0}}{12}\eta,\quad\|u_{0}\|_{\dot{B}^{\frac{N}{p}-1}_{p,1}(\omega)}\leq\frac{\eta}{6C_{3}}.$
(6.16)
Then it follows from (6.9) that
$\displaystyle\|a^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}+\|u^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}-1}_{p,1})}\leq\frac{7}{8}C_{0}E_{0},$
and from (6.12) and (6.13), we infer that
$\displaystyle\|a^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1}(\omega))}\leq\frac{7}{8}A_{0}\eta,\,\|u^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})}+\|u^{n}\|_{\widetilde{L}^{2}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}\leq\frac{2}{3}\eta,$
which ensure that (H3) and (H4) are satisfied with strict inequalities for $T$
and $\eta$ satisfying (6.14)-(6.16).
Let $X_{n}(t,x)$ be a solution of
$\displaystyle\frac{d}{dt}X_{n}(t,x)=u^{n}(t,X_{n}(t,x)),\quad X_{n}(0,x)=x,$
and we denote by $X^{-1}_{n}(t,x)$ the inverse of $X_{n}(t,x)$. Then
$a^{n}(t,x)$ can be solved as
$\displaystyle
a^{n}(t,x)=a_{0}^{n}(X^{-1}_{n}(t,x))+\int_{0}^{t}F^{n}(\tau,X_{n}(\tau,X^{-1}_{n}(t,x)))d\tau,$
thus, we have
$\displaystyle\overline{\rho}_{0}(1+a^{n}(t,x))=\rho_{0}^{n}(X^{-1}_{n}(t,x))+\bar{\rho}_{0}\int_{0}^{t}F^{n}(\tau,X_{n}(\tau,X^{-1}_{n}(t,x)))d\tau.$
(6.17)
On the other hand, we have
$\displaystyle\|F^{n}\|_{L^{1}_{T}(L^{\infty})}\leq\|\nabla
u\|_{L^{1}_{T}(L^{\infty})}(1+\|a\|_{L^{\infty}_{T}(L^{\infty})})\leq
C_{4}(1+C_{0}E_{0})\eta.$
We take $\eta$ such that
$\displaystyle C_{4}(1+C_{0}E_{0})\eta<\frac{1}{8}c_{0}.$
Then from (6.17) and (6.5), it follows that
$\displaystyle\overline{\rho}_{0}(1+a^{n}(t,x))\geq\frac{3}{4}c_{0}-\frac{1}{8}c_{0}\geq\frac{5}{8}c_{0},$
that is, (H1) is satisfied with the strict inequality. Finally, take
$\displaystyle
c_{1}=\frac{1}{2}\min(\inf_{|\rho|\leq\overline{\rho}_{0}(1+C_{0}E_{0})}\overline{\mu}(\rho),\inf_{|\rho|\leq\overline{\rho}_{0}(1+C_{0}E_{0})}(\overline{\lambda}(\rho)+2\overline{\mu}(\rho))),$
which ensures that (H2) is satisfied with strict inequality.
Let $T^{*}$ be the supremum of all time $T$ such that (6.15) and (6.16) are
satisfied. We need to prove that $T_{n}\geq T^{*}$. If $T_{n}<T^{*}$, then we
can prove that
$\begin{split}&a^{n}\in\widetilde{L}^{\infty}(0,T_{n};\dot{B}^{\frac{N}{p}}_{p,1}\cap\dot{B}^{\frac{N}{p}+1}_{p,1})\quad\textrm{and}\\\
&u^{n}\in\widetilde{L}^{\infty}(0,T_{n};\dot{B}^{\frac{N}{p}-1}_{p,1}\cap\dot{B}^{\frac{N}{p}}_{p,1})\cap
L^{1}([0,T_{n}];\dot{B}^{\frac{N}{p}+1}_{p,1}\cap\dot{B}^{\frac{N}{p}+2}_{p,1}),\end{split}$
(6.18)
thus, the solution $(a^{n},u^{n})$ can be continued beyond $T^{*}$. Indeed,
from Proposition 4.1, we have
$\displaystyle\|a^{n}\|_{\widetilde{L}^{\infty}_{t}(\dot{B}^{\frac{N}{p}+1}_{p,1})}\leq
e^{CV^{n}(t)}\bigl{(}\|a_{0}^{n}\|_{\dot{B}^{\frac{N}{p}+1}_{p,1}}+\int_{0}^{t}\|F^{n}(\tau)\|_{\dot{B}^{\frac{N}{p}+1}_{p,1}}d\tau\bigr{)},$
(6.19)
and by Proposition 5.1 (a), we have
$\displaystyle\|u^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}+\|u^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}+2}_{p,1})}$
$\displaystyle\leq
C\|u_{0}^{n}\|_{\dot{B}^{\frac{N}{p}}_{p,1}}+C\|G^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}+CA^{n}(T)\|a^{n}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})}\|u^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})},$
(6.20)
On the other hand, we use Lemma 2.5 and the embedding
$\dot{B}^{\frac{N}{p}}_{p,1}\hookrightarrow L^{\infty}$ to get
$\displaystyle\|F^{n}\|_{\dot{B}^{\frac{N}{p}+1}_{p,1}}\leq\|u^{n}\|_{\dot{B}^{\frac{N}{p}+2}_{p,1}}+C\|a^{n}\|_{\dot{B}^{\frac{N}{p}}_{p,1}}\|u^{n}\|_{\dot{B}^{\frac{N}{p}+2}_{p,1}}+C\|a^{n}\|_{\dot{B}^{\frac{N}{p}+1}_{p,1}}\|u^{n}\|_{\dot{B}^{\frac{N}{p}+1}_{p,1}},$
and by Lemma 2.6 and 2.9, we have
$\displaystyle\|G^{n}\|_{\widetilde{L}^{1}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}$
$\displaystyle\leq$ $\displaystyle
C\|u^{n}\|_{\widetilde{L}^{\infty}_{T}\dot{B}^{\frac{N}{p}}_{p,1}}\|u^{n}\|_{\widetilde{L}^{1}_{T}\dot{B}^{\frac{N}{p}+1}_{p,1}}$
$\displaystyle+CA^{n}(T)\|a^{n}\|_{\widetilde{L}^{\infty}_{T}\dot{B}^{\frac{N}{p}+1}_{p,1}}\bigl{(}T+\|u^{n}\|_{\widetilde{L}^{1}_{T}\dot{B}^{\frac{N}{p}+1}_{p,1}}\bigr{)},$
which together with (6.19), (6) and (H3-H4) implies (6.18).
Step 3. Existence of a solution
We will use a compact argument to prove that the approximate sequence
$\\{a^{n},u^{n}\\}_{n\in\mathop{\bf N\kern 0.0pt}\nolimits}$ tends to some
function $(a,u)$ which satisfies the system (6.4) in the sense of
distribution.
Since $\\{u^{n}\\}$ is uniformly bounded in
$L^{1}_{T}(\dot{B}^{\frac{N}{p}+1}_{p,1})\cap
L^{\infty}_{T}(\dot{B}^{\frac{N}{p}-1}_{p,1})$, we get by the interpolation
that $\\{u^{n}\\}_{n\in\mathop{\bf N\kern 0.0pt}\nolimits}$ is also uniformly
bounded in $L^{q}_{T}(\dot{B}^{\frac{N}{p}-1+2/q}_{p,1})$ for any
$q\in[1,\infty]$. By Lemma 2.6, we have
$\displaystyle\|a^{n}\mbox{div}u^{n}\|_{\dot{B}^{\frac{N}{p}-1}_{p,1}}\leq
C\|a^{n}\|_{\dot{B}^{\frac{N}{p}}_{p,1}}\|u^{n}\|_{\dot{B}^{\frac{N}{p}}_{p,1}},$
$\displaystyle\|u^{n}\cdot\nabla a^{n}\|_{\dot{B}^{\frac{N}{p}-1}_{p,1}}\leq
C\|a^{n}\|_{\dot{B}^{\frac{N}{p}}_{p,1}}\|u^{n}\|_{\dot{B}^{\frac{N}{p}}_{p,1}},$
from which and the first equation of the system (6.4), we infer that
$\\{\partial_{t}a^{n}\\}_{n\in\mathop{\bf N\kern 0.0pt}\nolimits}$ is
uniformly bounded in $L^{2}_{T}(\dot{B}^{\frac{N}{p}-1}_{p,1})$. On the other
hand, by Lemma 2.6 and Lemma 2.9, we have
$\displaystyle\|u^{n}\cdot\nabla u^{n}\|_{\dot{B}^{\frac{N}{p}-3/2}_{p,1}}\leq
C\|u^{n}\|_{\dot{B}^{\frac{N}{p}-1}_{p,1}}\|u^{n}\|_{\dot{B}^{\frac{N}{p}+1/2}_{p,1}},$
$\displaystyle\|\frac{\overline{\rho}_{0}P^{\prime}(\rho^{n})}{\rho^{n}}\nabla
a^{n}\|_{\dot{B}^{\frac{N}{p}-1}_{p,1}}\leq
C(\|a^{n}\|_{\infty})\|a^{n}\|_{\dot{B}^{\frac{N}{p}}_{p,1}}(1+\|a^{n}\|_{\dot{B}^{\frac{N}{p}}_{p,1}}),$
$\displaystyle\|\mbox{div}(\overline{\mu}^{n}\nabla
u^{n})\|_{\dot{B}^{\frac{N}{p}-3/2}_{p,1}}+\|\nabla((\overline{\lambda}^{n}+\overline{\mu}^{n})\mbox{div}u^{n})\|_{\dot{B}^{\frac{N}{p}-3/2}_{p,1}}+\|G_{1}^{n}\|_{\dot{B}^{\frac{N}{p}-3/2}_{p,1}}$
$\displaystyle\quad\leq
C(\|a^{n}\|_{\infty})\|a^{n}\|_{\dot{B}^{\frac{N}{p}}_{p,1}}(1+\|a^{n}\|_{\dot{B}^{\frac{N}{p}}_{p,1}})\|u^{n}\|_{\dot{B}^{\frac{N}{p}+1/2}_{p,1}},$
where $G_{1}^{n}\buildrel\hbox{\footnotesize
def}\over{=}\frac{\mu(\rho^{n})}{(\rho^{n})^{2}}\nabla\rho^{n}\cdot\nabla
u^{n}+\frac{\mu(\rho^{n})+\lambda(\rho^{n})}{(\rho^{n})^{2}}\nabla\rho^{n}\mbox{div}\,u^{n}$.
Then, from the second equation of the system (6.4), we infer that
$\\{\partial_{t}u^{n}\\}_{n\in\mathop{\bf N\kern 0.0pt}\nolimits}$ is
uniformly bounded in
$L^{\frac{4}{3}}_{T}(\dot{B}^{\frac{N}{p}-\frac{3}{2}}_{p,1}+\dot{B}^{\frac{N}{p}-1}_{p,1})$.
Let $\\{\chi_{j}\\}_{j\in\mathop{\bf N\kern 0.0pt}\nolimits}$ be a sequence of
smooth functions supported in the ball $B(0,j+1)$ and equal to 1 on $B(0,j)$.
The above proof ensures that for any $j\in\mathop{\bf N\kern 0.0pt}\nolimits$,
$\\{\chi_{j}a^{n}\\}_{n\in\mathop{\bf N\kern 0.0pt}\nolimits}$ is uniformly
bounded in $C^{\frac{1}{2}}([0,T];\dot{B}^{\frac{N}{p}-1}_{p,1})$, and
$\\{\chi_{j}u^{n}\\}_{n\in\mathop{\bf N\kern 0.0pt}\nolimits}$ is uniformly
bounded in
$C^{\frac{1}{4}}([0,T];\dot{B}^{\frac{N}{p}-\frac{3}{2}}_{p,1}+\dot{B}^{\frac{N}{p}-1}_{p,1})$.
Since the embedding
$\dot{B}^{\frac{N}{p}-1}_{p,1}\cap\dot{B}^{\frac{N}{p}}_{p,1}\hookrightarrow\dot{B}^{\frac{N}{p}-1}_{p,1}$
and
$\dot{B}^{\frac{N}{p}-3/2}_{p,1}\cap\dot{B}^{\frac{N}{p}-1}_{p,1}\hookrightarrow\dot{B}^{\frac{N}{p}-3/2}_{p,1}$
are locally compact, by applying Ascoli’s theorem and Cantor’s diagonal
process , there exists some function $(a,u)$ such that for any
$j\in\mathop{\bf N\kern 0.0pt}\nolimits$,
$\displaystyle\begin{split}&\chi_{j}a^{n}\longrightarrow\chi_{j}a\quad\textrm{in}\quad
C([0,T];\dot{B}^{\frac{N}{p}-1}_{p,1}),\\\
&\chi_{j}u^{n}\longrightarrow\chi_{j}u\quad\textrm{in}\quad
C([0,T];\dot{B}^{\frac{N}{p}-\frac{3}{2}}_{p,1}),\end{split}$ (6.21)
as $n$ tends to $\infty$(up to a subsequence). By the interpolation, we also
have
$\displaystyle\begin{split}&\chi_{j}a^{n}\longrightarrow\chi_{j}a\quad\textrm{in}\quad
C([0,T];\dot{B}^{\frac{N}{p}-s}_{p,1}),\quad\forall\,0<s\leq 1,\\\
&\chi_{j}u^{n}\longrightarrow\chi_{j}u\quad\textrm{in}\quad
L^{1}([0,T];\dot{B}^{\frac{N}{p}+s}_{p,1}),\quad\forall\,-\frac{3}{2}\leq
s<1.\end{split}$ (6.22)
Furthermore, we actually have
$\displaystyle(a,u)\in\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})\otimes\Bigl{(}\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}-1}_{p,1})\cap
L^{1}(0,T;\dot{B}^{\frac{N}{p}+1}_{p,1})\Bigr{)},\quad\overline{\rho}_{0}(1+a(t,x))\geq\frac{c_{0}}{2}.$
(6.23)
With (6.21)-(6.23), it is a routine process to verify that $(a,u)$ satisfies
the system (6.4) in the sense of distribution(see also [11]). Finally,
following the argument in [11], we can show that $(a,u)\in
C([0,T];\dot{B}^{\frac{N}{p}}_{p,1})\otimes
C([0,T];\dot{B}^{\frac{N}{p}-1}_{p,1})$.
## 7 The proof of uniqueness
In this section, we prove the uniqueness of the solution. Assume that
$(a^{1},u^{1})\in E^{p}_{T}$ and $(a^{2},u^{2})\in E^{p}_{T}$ are two
solutions of the system (6.4) with the same initial data. Without loss of
generality, we may assume that $a^{1}$ satisfies
$\displaystyle\rho^{1}(t,x)=\overline{\rho}_{0}(1+a^{1}(t,x))\geq\frac{c_{0}}{2}.$
for any $(t,x)\in[0,T]\times\mathop{\bf R\kern 0.0pt}\nolimits^{N}$. Since
$a^{2}\in C([0,T];\dot{B}^{\frac{N}{p}}_{p,1})$ and $\rho^{2}(0,x)\geq c_{0}$,
there exists a positive time $\widetilde{T}\in(0,T]$ such that
$\displaystyle\rho^{2}(t,x)=\overline{\rho}_{0}(1+a^{2}(t,x))\geq\frac{c_{0}}{2}.$
for any $(t,x)\in[0,\widetilde{T}]\times\mathop{\bf R\kern
0.0pt}\nolimits^{N}$. Set $\delta a=a^{1}-a^{2}$ and $\delta u=u^{1}-u^{2}$.
Then $(\delta a,\delta u)$ satisfies
$\displaystyle\left\\{\begin{array}[]{ll}\partial_{t}\delta
a+u^{2}\cdot\nabla\delta a=\delta F-\delta u\cdot\nabla a^{1},\\\
\partial_{t}\delta u-\mbox{div}(\overline{\mu}^{1}\nabla\delta
u)-\nabla((\overline{\lambda}^{1}+\overline{\mu}^{1})\mbox{div}\,\delta
u)=\delta G+\delta H,\\\ (\delta a,\delta u)|_{t=0}=(0,0),\end{array}\right.$
(7.4)
where
$\displaystyle\delta F=F(a^{1},u^{1})-F(a^{2},u^{2}),\quad\delta
G=G(a^{1},u^{1})-G(a^{2},u^{2}),$ $\displaystyle\delta
H=\mbox{div}\bigl{(}(\overline{\mu}^{1}-\overline{\mu}^{2})\nabla
u^{2}\bigr{)}+\nabla\bigl{(}(\overline{\lambda}^{1}-\overline{\lambda}^{2}+\overline{\mu}^{1}-\overline{\mu}^{2})\mbox{div}\,u^{2}\bigr{)},$
with
$\overline{\lambda}^{i}=\overline{\lambda}(a^{i}),\overline{\mu}^{i}=\overline{\mu}(a^{i})$
for $i=1,2$.
In what follows, we set
$U^{i}(t)=\int_{0}^{t}\|u^{i}(\tau)\|_{\dot{B}^{\frac{N}{p}+1}_{p,1}}d\tau$
for $i=1,2$, and denote by $A_{T}$ a constant depending on
$\|a^{1}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}$ and
$\|a^{2}\|_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{p}}_{p,1})}$. Due to
the inclusion relation $E_{T}^{p}\subseteq E^{N}_{T}$, it suffices to prove
the uniqueness of the solution in $E^{N}_{T}$. So, we take $p=N$ in the
sequel.
We apply Proposition 4.1 to get for any $t\in[0,T]$,
$\displaystyle\|\delta a(t)\|_{\dot{B}^{0}_{p,\infty}}\leq
e^{CU^{2}(t)}\int_{0}^{t}\bigl{(}\|\delta
F(\tau)\|_{\dot{B}^{0}_{p,\infty}}+\|\delta u\cdot\nabla
a^{1}(\tau)\|_{\dot{B}^{0}_{p,\infty}}\bigr{)}d\tau.$ (7.5)
By Lemma 2.7, we have
$\displaystyle\|\delta F(\tau)\|_{\dot{B}^{0}_{p,\infty}}+\|\delta
u\cdot\nabla a^{1}(\tau)\|_{\dot{B}^{0}_{p,\infty}}$ $\displaystyle\quad\leq
C\|u^{2}\|_{\dot{B}^{2}_{p,1}}\|\delta
a\|_{\dot{B}^{0}_{p,\infty}}+C(1+\|a^{1}\|_{\dot{B}^{1}_{p,1}})\|\delta
u\|_{\dot{B}^{1}_{p,1}}.$
Plugging it into (7.5), we get by Gronwall’s inequality that
$\displaystyle\|\delta a(t)\|_{\dot{B}^{0}_{p,\infty}}\leq
e^{CU^{2}(t)}\int_{0}^{t}(1+\|a^{1}\|_{\dot{B}^{1}_{p,1}})\|\delta
u\|_{\dot{B}^{1}_{p,1}}d\tau.$ (7.6)
We use Proposition 5.3 to get for any $t\in[0,T]$,
$\displaystyle\|\delta
u(t)\|_{\widetilde{L}^{1}_{t}(\dot{B}^{1}_{p,\infty})}+\|\delta
u(t)\|_{\widetilde{L}^{2}_{t}(\dot{B}^{0}_{p,\infty})}$ $\displaystyle\leq$
$\displaystyle C\int_{0}^{t}\bigl{(}\|\delta
G(\tau)\|_{\dot{B}^{-1}_{p,\infty}(\omega)}+\|\delta
H(\tau)\|_{\dot{B}^{-1}_{p,\infty}}\bigr{)}d\tau$ (7.7)
$\displaystyle+CA_{T}\|a^{1}\|_{\widetilde{L}^{\infty}_{t}(\dot{B}^{1}_{p,1}(\omega))}\|\delta
u\|_{\widetilde{L}^{1}_{t}(\dot{B}^{1}_{p,\infty})}.$
From Lemma 2.7, Proposition 3.7 and 3.8, we infer that for any
$t\in[0,\widetilde{T}]$,
$\displaystyle\|\delta H\|_{\dot{B}^{-1}_{p,\infty}}$ $\displaystyle\leq$
$\displaystyle A_{T}\|u^{2}\|_{\dot{B}^{2}_{p,1}}\|\delta
a\|_{\dot{B}^{0}_{p,\infty}},$ (7.8) $\displaystyle\|\delta
G\|_{\dot{B}^{-1}_{p,\infty}(\omega)}$ $\displaystyle\leq$ $\displaystyle
C\|(u^{1},u^{2})\|_{\dot{B}^{1}_{p,1}}\|\delta
u\|_{\dot{B}^{0}_{p,\infty}}+A_{T}\|a^{1}\|_{\dot{B}^{1}_{p,1}(\omega)}\|\delta
u\|_{\dot{B}^{1}_{p,\infty}}$ (7.9)
$\displaystyle+A_{T}(1+\|u^{2}\|_{\dot{B}^{2}_{p,1}})\|\delta
a\|_{\dot{B}^{0}_{p,\infty}}.$
We take $\widetilde{T}$ small enough such that
$\displaystyle\|(u^{1},u^{2})\|_{\widetilde{L}^{1}_{t}(\dot{B}^{2}_{p,1})\cap\widetilde{L}^{2}_{t}(\dot{B}^{1}_{p,1})}+\|(a^{1},a^{2})\|_{\widetilde{L}^{\infty}_{t}(\dot{B}^{1}_{p,1}(\omega))}\ll
1.$
Thus, plugging (7.8) and (7.9) into (7.7), we infer that for any
$t\in[0,\widetilde{T}]$,
$\displaystyle\|\delta u\|_{\widetilde{L}^{1}_{t}(\dot{B}^{1}_{p,\infty})}\leq
A_{T}\int_{0}^{t}\bigl{(}1+\|(u^{1},u^{2})\|_{\dot{B}^{2}_{p,1}}\bigr{)}\|\delta
a\|_{\dot{B}^{0}_{p,\infty}}d\tau.$ (7.10)
###### Lemma 7.1
[13] Let $s\in\mathop{\bf R\kern 0.0pt}\nolimits$. Then for any $1\leq
p,\rho\leq+\infty$ and $0<\epsilon\leq 1$, we have
$\displaystyle\|f\|_{\widetilde{L}^{\rho}_{T}(\dot{B}^{s}_{p,1})}\leq
C\frac{\|f\|_{\widetilde{L}^{\rho}_{T}(\dot{B}^{s}_{p,\infty})}}{\epsilon}\log\Bigl{(}e+\frac{\|f\|_{\widetilde{L}^{\rho}_{T}(\dot{B}^{s-\epsilon}_{p,\infty})}+\|f\|_{\widetilde{L}^{\rho}_{T}(\dot{B}^{s+\epsilon}_{p,\infty})}}{\|f\|_{\widetilde{L}^{\rho}_{T}(\dot{B}^{s}_{p,\infty})}}\Bigr{)}.$
From Lemma 7.1, it follows that
$\displaystyle\|\delta u\|_{L^{1}_{t}(\dot{B}^{1}_{p,1})}\leq C\|\delta
u\|_{\widetilde{L}^{1}_{t}(\dot{B}^{1}_{p,\infty})}\log\Bigl{(}e+\frac{\|\delta
u\|_{\widetilde{L}^{1}_{t}(\dot{B}^{0}_{p,\infty})}+\|\delta
u\|_{\widetilde{L}^{1}_{t}(\dot{B}^{2}_{p,\infty})}}{\|\delta
u\|_{\widetilde{L}^{1}_{t}(\dot{B}^{1}_{p,\infty})}}\Bigr{)},$
which together with (7.6) and (7.10) yields that for any
$t\in[0,\widetilde{T}]$,
$\displaystyle\|\delta u\|_{\widetilde{L}^{1}_{t}(\dot{B}^{1}_{p,\infty})}\leq
A_{T}\int_{0}^{t}\bigl{(}1+\|(u^{1},u^{2})\|_{\dot{B}^{2}_{p,1}}\bigr{)}\|\delta
u\|_{\widetilde{L}^{1}_{\tau}(\dot{B}^{1}_{p,\infty})}\log\bigl{(}e+C_{T}\|\delta
u\|_{\widetilde{L}^{1}_{\tau}(\dot{B}^{1}_{p,\infty})}^{-1}\bigr{)}d\tau,$
where $C_{T}=\|\delta
u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{0}_{p,\infty})}+\|\delta
u\|_{\widetilde{L}^{1}_{T}(\dot{B}^{2}_{p,\infty})}$. Notice that
$1+\|(u^{1},u^{2})(t)\|_{\dot{B}^{2}_{p,1}}$ is integrable on $[0,T]$, and
$\displaystyle\int_{0}^{1}\frac{dr}{r\log(e+C_{T}r^{-1})}dr=+\infty,$
Osgood lemma applied concludes that $(\delta a,\delta u)=0$ on
$[0,\widetilde{T}]$, and a continuity argument ensures that
$(a^{1},u^{1})=(a^{2},u^{2})$ on $[0,T]$.
## Acknowledgements
Q. Chen and C. Miao were partially supported by the NSF of China under grant
No.10701012, No.10725102. Z. Zhang was partially supported by the NSF of China
under grant No.10601002.
## References
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* [3] D. Bresch, B. Desjardins and Chi-Kun Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems. Comm. Partial Differential Equations, 28(2003), 843–868.
* [4] D. Bresch, B. Desjardins and G. Métivier, Recent mathematical results and open problems about shallow water equations, Analysis and simulation of fluid dynamics, 15–31, Adv. Math. Fluid Mech., Birkhäuser, Basel, 2007.
* [5] M. Cannone, Ondelettes, paraproduits et Navier-Stokes. Nouveaux essais, Diderot éditeurs, Paris, 1995.
* [6] M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations. Handbook of Mathematical fluid Dynamics, Vol. III, North-Holland, Amsterdam, 2004.
* [7] J.-Y. Chemin, Perfect incompressible fluids. Oxford University Press, New York, 1998.
* [8] J.-Y. Chemin, Théorèmes d’unicité pour le système de Navier-Stokes tridimensionnel. J. d’Analyse Math., 77(1999), 27-50.
* [9] J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes. J. Differential Equations, 121(1992), 314-328.
* [10] Q. Chen, C. Miao and Z. Zhang, On the well-posedness for the viscous shallow water equations. SIAM Jour. Math. Anal. (40)2008, 443-474.
* [11] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations. Invent. Math., 141(2000), 579-614.
* [12] R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26(2001), 1183-1233.
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* [15] R. Danchin, Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density . Comm. Partial Differential Equations, 32(2007), 1373-1397.
* [16] H. Fujita and T. Kato, On the Navier-Stokes initial value problem I. Arch. Rational Mech. Anal., 16(1964), 269-315.
* [17] B. Haspot, Cauchy problem for viscous shallow water equations with a term of capillarity, arXiv: 0803.1939v1[math.Ap].
* [18] A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. J. Appl. Math. Mech., 41(1977), 273-282.
* [19] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol.2, Compressible models, Oxford University Press, 1998.
* [20] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Japan Acad. Ser. A Math. Sci. 55(1979), 337-342.
* [21] A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equations. Comm. Partial Differential Equations, 32(2007), 431-452.
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* [23] J. Nash, Le problème de Cauchy pour les équations différentielles d’un fluide général, Bulletin de la Soc. Math. de France, 90(1962),487-497.
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|
arxiv-papers
| 2008-11-26T03:20:23 |
2024-09-04T02:48:58.971240
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Qionglei Chen, Changxing Miao and Zhifei Zhang",
"submitter": "Changxing Miao",
"url": "https://arxiv.org/abs/0811.4215"
}
|
0811.4244
|
# Polymer desorption under pulling: a novel dichotomic phase transition
S. Bhattacharya1, V. G. Rostiashvili1, A. Milchev1,2, and T.A. Vilgis1 1 Max
Planck Institute for Polymer Research, 10 Ackermannweg, 55128 Mainz, Germany
2 Institute for Physical Chemistry, Bulgarian Academy of Sciences, 1113 Sofia,
Bulgaria
###### Abstract
We show that the structural properties and phase behavior of a self-avoiding
polymer chain on adhesive substrate, subject to pulling at the chain end, can
be obtained by means of a Grand Canonical Ensemble (GCE) approach. We derive
analytical expressions for the mean length of the basic structural units of
adsorbed polymer, such as loops and tails, in terms of the adhesive potential
and detachment force, and determine values of the universal exponents which
govern their probability distributions. Most notably, the hitherto
controversial value of the critical adsorption exponent $\phi$ is found to
depend essentially on the interaction between different loops. The chain
detachment transition turns out to be of the first order, albeit dichotomic,
i.e., no coexistence of different phase states exists. These novel theoretical
predictions and the suggested phase diagram of the adsorption-desorption
transformation under external pulling force are verified by means of extensive
Monte Carlo simulations.
###### pacs:
05.50.+q, 68.43.Mn, 64.60.Ak, 82.35.Gh, 62.25.+g
Introduction \- The manipulation of single polymer chains has turned recently
into an important method for understanding their mechanical properties and
characterization of the intermolecular interactions Strick ; Celestini . Such
manipulation is mainly triggered by the progress in atomic force microscopy
(AFM) Rief as well as by the development of optical/magnetic tweezers
technique Bustamante . This rapid development has been followed by theoretical
considerations, based on the mean - field approximation Sevick , which provide
important insight into the mechanism of polymer detachment from adhesive
surfaces under the external force pulling. A comprehensive study by Skvortsov
et al. SKB examines the case of a Gaussian polymer chain. We also note here
the close analogy between the force detachment of adsorbed chain and the
unzipping of a double - stranded DNA. Recently, DNA denaturation and unzipping
have been treated by Kafri et al. Kafri using the Grand Canonical Ensemble
(GCE) approach Poland as well as Duplantier’s analysis of polymer networks of
arbitrary topology Duplantier . An important result for the properties of
adsorbed macromolecule under pulling turns to be the observation Kafri that
the universal exponents (which govern polymer statistics) undergo
renormalization due to excluded volume effects, leading thus to a change of
the order of DNA melting transition from second to first order. In this Letter
we use similar methods to describe the structure and detachment of a single
chain from a sticky substrate when the chain end is pulled by external force.
Single chain adsorption \- Starting with the conventional (i.e., force-free)
adsorption, we recall that an adsorbed chain is build up from loops, trains,
and a free tail. One can treat statistically these basic structural units by
means of the GCE approach where the lengths of the buildings blocks are not
fixed but may rather fluctuate. The GCE-partition function is then given by
$\displaystyle\Xi(z)=\sum_{N=0}^{\infty}\>\Xi_{N}\>z^{N}=\frac{V_{0}(z)\>Q(z)}{1-V(z)U(z)},$
(1)
where $z$ is the fugacity and $U(z)$, $V(z)$, and $Q(z)$ denote the GCE
partition functions of loops, trains and tails, respectively. The building
block adjacent to the tethered chain end is allowed for by $V_{0}(z)=1+V(z)$.
The partition function of the loops is defined as
$U(z)=\sum_{n=1}^{\infty}\>(\mu_{3}z)^{n}/n^{\alpha}$, where $\mu_{3}$ is the
$3d$ connective constant and $\alpha$ is the exponent which governs surface
loops statistics. It is well known that for an isolated loop
$\alpha=1-\gamma_{11}\approx 1.39$ Vanderzande . We will argue below that
$\alpha$ changes value due to the excluded volume interaction between a loop
and the rest of the chain. The train GCE-partition function reads
$V(z)=\sum_{n=1}^{\infty}\>(\mu_{3}wz)^{n}/n^{1-\gamma_{d=2}}$ whereby one
assumes that each adsorbed segment gains an additional statistical weight
$w=\exp(\epsilon)$ with the dimensionless adsorption energy
$\epsilon=\varepsilon/k_{B}T$. Eventually, the GCE partition function for the
chain tail is defined by
$Q(z)=1+\sum_{n=1}^{\infty}\>(\mu_{3}z)^{n}/n^{\beta}$. For an isolated tail
$\beta=1-\gamma_{1}\approx 0.32$ Vanderzande but again the excluded volume
interactions of a tail with the rest of the chain increase the value of
$\beta$. Using the generating function method Rudnick , $\Xi_{N}$ is obtained
as $\Xi_{N}=(z^{*})^{-N}$ where the pole $z^{*}$ is given by the condition
$V(z^{*})U(z^{*})=1$ so that the free energy is $F=k_{B}TN\ln z^{*}$ and the
fraction of adsorbed monomers $n=-\partial\ln z^{*}/\partial\ln w$. In terms
of the so called polylog function, which is defined as
$\Phi(\alpha,z)=\sum_{n=1}^{\infty}\>z^{n}/n^{\alpha}$ Erdelyi , the equation
for $z^{*}$ reads
$\displaystyle\Phi(\alpha,\mu_{3}z^{*})\Phi(\lambda,\mu_{2}wz^{*})=1.$ (2)
A nontrivial solution for $z^{*}$ in terms of $w$ (or the adsorption energy
$\epsilon$) appears at the critical adsorption point (CAP) $w=w_{c}$ where
$w_{c}$ is determined from
$\zeta(\alpha)\Phi(1-\gamma_{d=2},\mu_{2}w_{c}/\mu_{3}))=1$ as
$z^{*}=1/\mu_{3}$ and $\zeta(\alpha)$ is the Riemann function. In the vicinity
of the CAP the solution attains the form
$\displaystyle z^{*}(w)\approx[1-A\>(w-w_{c})^{1/(\alpha-1)}]\mu_{3}^{-1}$ (3)
where $A$ is a constant. Then the average fraction of adsorbed monomers is
$n\propto(\epsilon-\epsilon_{c})^{1/(\alpha-1)-1}$. A comparison with the well
known scaling relationship $n\propto(\epsilon-\epsilon_{c})^{1/\phi-1}$ where
$\phi$ is the so called adsorption (or, crossover) exponent Vanderzande
suggests that
$\displaystyle\phi=\alpha-1$ (4)
This is a result of principal importance. It shows that the crossover exponent
$\phi$, describing polymer adsorption at criticality, is determined by the
exponent $\alpha$ which governs polymer loop statistics! If loops are treated
as isolated objects, then $\alpha=1-\gamma_{11}\approx 1.39$ so that
$\phi=0.39$. In contrast, excluded volume interactions between a loop and the
rest of the chain lead to an increase of $\alpha$ and $\phi$, as we show
below.
Probability distributions of loops and tails \- How does the length
distribution of polymer loops and tails close to the CAP look like? From the
expression for $U(z)$, given above, and eq.(3) we have $P_{\rm
loop}\approx(\mu_{3}z^{*})^{l}/l^{1+\phi}\approx\exp[-c_{1}(\epsilon-\epsilon_{c})^{1/\phi}]/l^{1+\phi}$.
This is valid only for $\epsilon>\epsilon_{c}$ since a solution for eq.(2) for
subcritical values of the adhesive potential $\epsilon$ does not exist.
Nontheless, even in the subcritical region, $\epsilon<\epsilon_{c}$, there are
still monomers which occasionally touch the substrate, creating thus single
loops at the expense of the tail length. The partition function of such a
loop-tail configuration is
$Z_{l-t}=\frac{\mu_{3}^{l}}{l^{1+\phi}}\;\frac{\mu_{3}^{N-l}}{(N-l)^{\beta}}$.
On the other side, the partition function of a tail conformation with no loops
whatsoever (i.e., of a nonadsorbed tethered chain) is
$Z_{t}=\mu_{3}^{N}\;N^{\gamma_{1}-1}$. Thus the probability $P^{<}_{\rm
loop}(l)$ to find a loop of length $l$ next to a tail of length $N-l$ can be
estimated as $P^{<}_{\rm
loop}(l)=\frac{Z_{l-t}}{Z_{t}}\propto\frac{N^{1-\gamma_{1}}}{l^{1+\phi}(N-l)^{\beta}}$,
which is valid at $\epsilon<\epsilon_{c}$. In the vicinity of the CAP,
$\epsilon\approx\epsilon_{c}$, the distribution will be given by an
interpolation between the expressions above. Hence, the overall loop
distribution becomes
$\displaystyle P_{\rm
loop}(l)=\begin{cases}\frac{1}{l^{1+\phi}}\exp\left[-c_{1}(\epsilon-\epsilon_{c})^{1/\phi}\;l\right],\quad&\quad\epsilon>\epsilon_{c}\\\
\frac{A_{1}}{l^{1+\phi}}+\frac{A_{2}N^{1-\gamma_{1}}}{l^{1+\phi}(N-l)^{\beta}},\quad&\quad\epsilon=\epsilon_{c}\\\
\frac{N^{1-\gamma_{1}}}{l^{1+\phi}(N-l)^{\beta}}.\quad&\quad\epsilon<\epsilon_{c}\end{cases}$
(5)
The same reasonings for a tail leads to the distribution
$\displaystyle P_{\rm
tail}(l)=\begin{cases}\frac{1}{l^{\beta}}\exp\left[-c_{1}(\epsilon-\epsilon_{c})^{1/\phi}\;l\right],\quad&\quad\epsilon>\epsilon_{c}\\\
\frac{B_{1}}{l^{\beta}}+\frac{B_{2}N^{1-\gamma_{1}}}{l^{\beta}(N-l)^{1+\phi}},\quad&\quad\epsilon=\epsilon_{c}\\\
\frac{N^{1-\gamma_{1}}}{l^{\beta}(N-l)^{1+\phi}}.\quad&\quad\epsilon<\epsilon_{c}\end{cases}$
(6)
In eqs.(5) - (6) $A_{1},A_{2},B_{1},B_{2}$ are constants. Close to CAP these
distributions are expected to attain a U - shaped form (with two maxima at
$l=1$ and $l\approx N$), as predicted for a Gaussian chain by Gorbunov et al.
Gorbunov . For the average loop length $L$ the GCE-partition function for
loops yields $L=z\partial U(z)/\partial
z|_{z=z^{*}}=\Phi(\alpha-1,\mu_{3}z^{*})/\Phi(\alpha,\mu_{3}z^{*})$. Close to
the CAP, $L$ diverges as $L\propto 1/(\epsilon-\epsilon_{c})^{1/\phi-1}$. The
average tail length $S$ can be obtain as $S=z\partial Q(z)/\partial
z|_{z=z^{*}}=\Phi(\beta-1,\mu_{3}z^{*})/[1+\Phi(\beta,\mu_{3}z^{*})]$. Again,
using the properties of the polylog function, one can show that close to
$\epsilon_{c}$ the average tail length diverges as $S\propto
1/(\epsilon-\epsilon_{c})^{1/\phi}$. Note that this behavior corresponds to a
length of adsorption blob $g\propto 1/(\epsilon-\epsilon_{c})^{1/\phi}$.
Role of interacting loops and tails \- Consider the number of configurations
of a tethered chain in the vicinity of the CAP as an array of loops which end
up with a tail. Using the approach of Kafri et al. Kafri along with
Duplantier’s Duplantier graph theory of polymer networks, one may write the
partition function $Z$ for a chain with ${\cal N}$ building blocks: ${\cal
N}-1$ loops and a tail. Consider a loop of length $M$ while the length of the
rest of the chain is $K$, that is, $M+K=N$. In the limit of $M\gg 1,\;K\gg 1$
(but with $M/K\ll 1$) one can show SBVRAMTV that
$Z\sim\mu_{3}^{M}\>M^{\gamma_{\cal N}^{s}-\gamma_{{\cal
N}-1}^{s}}\>\>\mu_{3}^{K}\>K^{\gamma_{{\cal N}-1}^{s}-1}$ where the surface
exponent $\gamma_{\cal N}^{s}=2-{\cal N}(\nu+1)+\sigma_{1}+\sigma_{1}^{s}$ and
$\sigma_{1},\;\sigma_{1}^{s}$ are critical bulk and surface exponents
Duplantier . The last result indicates that the effective loop exponent
$\alpha$ becomes
$\alpha=\gamma_{{\cal N}-1}^{s}-\gamma_{\cal N}^{s}=\nu+1$ (7)
Thus, $\phi=\alpha-1=\nu=0.588$, in agreement with earlier Monte Carlo
findings Eisenriegler . One should emphasize, however, that the foregoing
derivation is Mean-Field-like ($Z$ appears as a product of loop- and rest-of-
the-chain contributions) which overestimates the interactions and increases
significantly the value of $\alpha$, serving as an upper bound. The value of
$\alpha$, therefore, is found to satisfy the inequality
$1-\gamma_{11}\leq\alpha\leq 1+\nu$, i.e., depending on loop interactions,
$0.39\leq\phi\leq 0.59$.
Adsorption under detaching force \- Using the GCE approach now we treat the
case of self-avoiding polymer chain adsorption in the presence of pulling
force, thus extending the consideration of Gaussian chains by Gorbunov et al.
Skvortsov . Under external detaching force $f$, the tail GCE-partition
function $Q(z)$ in eq. (1) has to be replaced by
$\tilde{Q}(z)=1+\sum_{n=1}^{\infty}[(\mu_{3}z)^{n}/n^{\beta}]\>\int
d^{3}rP_{n}({\bf r})\exp(fr_{\perp}/T)$ where $P_{n}({\bf r})$ is the end-to-
end distance distribution function for a self-avoiding chain DesCloizeaux .
After some straightforward calculations the tail GCE-partition function can be
written as
$\displaystyle\tilde{Q}(z)=1+a_{1}\>{\tilde{f}}^{\theta}\>\Phi(\psi,z\mu_{3}\exp(a_{2}{\tilde{f}}^{1/\nu}))$
(8)
Here the dimensionless force ${\tilde{f}}=fa/k_{B}T$, the exponents
$\psi=1-\nu$, and $\theta=(2+t-3\delta/2)/(\delta-1)$ with
$t=(\beta-3/2+3\nu)/(1-\nu)$ and $\delta=1/(1-\nu)$. The function
$\tilde{Q}(z)$ has a branch point at
$z^{\\#}=\mu_{3}^{-1}\exp(-a_{2}{\tilde{f}}^{1/\nu})$, i.e., $\tilde{Q}(z)\sim
1/(z^{\\#}-z)^{1-\psi}$. One may, therefore, conclude that the total GCE-
partition function $\Xi(z)$ has two singularities on the real axis: the pole
$z^{*}$, and the branch point $z^{\\#}$. It is known (see, e.g., Sec. 2.4.3.
in Rudnick ) that for $N>>1$ the main contributions to $\Xi_{N}$ come from the
pole and the branch singular points, i.e.,
$\displaystyle\Xi_{N}\sim
C_{1}\>(z^{*})^{-N}+\frac{C_{2}}{\Gamma(1-\psi)}\>N^{-\psi}\>(z^{\\#})^{-N}$
(9)
Thus, for large $N$ only the smallest of these points matters. On the other
hand, $z^{*}$ depends on the dimensionless adsorption energy $\epsilon$ only
(or, on $w=\exp(\epsilon)$) whereas $z^{\\#}$ is controlled by the external
force ${\tilde{f}}$. Therefore, in terms of the two control parameters,
$\epsilon$ and ${\tilde{f}}$, the equation
$z^{*}(\epsilon)=z^{\\#}({\tilde{f}})$ determines the critical line of
transition between the adsorbed phase and the force-induced desorbed phase .
In the following this line will be called detachment line. Below it,
$f<f_{D}$, or above, $f>f_{D}$, either $z^{*}$ or $z^{\\#}$, respectively,
contribute to $\Xi_{N}$. The controll parameters, $\epsilon_{D}$ and
${\tilde{f}}_{D}$, which satisfy this equation, denote detachment energy and
detachment force, respectively. On the detachment line the system undergoes a
first-order phase transition. The detachment line itself terminates for
${\tilde{f}}_{D}\rightarrow 0$ in the CAP, $\epsilon_{c}$,
Figure 1: (a) The ’order parameter’, $n$, against the surface potential,
$\epsilon$, for various pulling forces. The chain has length $N$=128. (b)
Variation of $n$ with the pulling force, $f$, for several surface potentials.
where the transition becomes of second order. In the vicinity of the CAP the
detachment force ${\tilde{f}}_{D}$ vanishes as
${\tilde{f}}_{D}\sim(\epsilon-\epsilon_{c})^{\nu/\phi}$. This first order
adsorption-desorption phase transition under pulling force has a clear
dichotomic nature (i.e., it follows an “either - or” scenario): in the
thermodynamic limit $N\rightarrow\infty$ there is no phase coexistence! The
configurations are divided into adsorbed and detached (or stretched)
dichotomic classes. Metastable states are completely absent. Moreover, the
mean loop length $L$ remains finite upon detachment line crossing. The average
tail length $S$, on the contrary, diverges close to the detachment line.
Indeed, at ${\tilde{f}}<{\tilde{f}}_{D}$ the average tail length is given by
$S={\tilde{f}}^{\theta}\Phi(\psi-1,z^{*}(w)/z^{\\#}({\tilde{f}}))/[1+a_{1}\Phi(\psi,z^{*}(w)/z^{\\#}({\tilde{f}}))]$.
At the detachment line, $z^{*}=z^{\\#}$, it diverges as
$S\propto{\tilde{f}}_{D}/({\tilde{f}}_{D}-{\tilde{f}})$.
Figure 2: Plot of the critical detachment force $f_{D}=fa/k_{B}T$ against the
surface potential $\varepsilon/k_{B}T$. Full and empty symbols denote MC and
theoretical results. A double logarithmic plot of $f_{D}$ against
$\epsilon-\epsilon_{c}$ with $\epsilon_{c}=1.67$ is shown in the inset,
yielding a slope of $0.97\pm 0.02$, in agreement with the prediction
$f_{D}\propto(\epsilon-\epsilon_{c})^{\nu/\phi}$. Shaded is shown the same
phase diagram, derived by numeric solution of $z^{*}=z^{\\#}$, which in
dimensional $f$ (right axis) against $T$ (top axis) units appears reentrant.
Reentrant phase behavior \- Recently, it has been realized Mishra that the
detachment line, when represented in terms of dimensional variables, force
$f_{D}$ versus temperature $T$, goes (at a relatively low temperature) through
a maximum, that is, the desorption transition shows reentrant behavior! Below
we demonstrate that this result follows directly from our theory. It can be
seen that the solution of eq.(2) at large values of $\epsilon$ (or, at low
temperature) can be written as $z^{*}\approx{\rm e}^{-\epsilon}/\mu_{3}$ so
that the detachment line, $z^{*}=z^{\\#}$, in terms of dimensionless
parameters is monotonous,
${\tilde{f}}_{D}\propto[\epsilon_{D}-\ln(\mu_{3}/\mu_{2})]^{\nu}$. Note,
however, that the same detachment line, if represented in terms of the
dimensional control parameters, force $f_{D}$ versus temperature $T_{D}$ (with
a fixed dimensional energy $\varepsilon_{0}$), shows a nonmonotonic behavior
$f_{D}=T_{D}[\varepsilon_{0}/T_{D}-\ln(\mu_{3}/\mu_{2})]^{\nu}/a$. This curve
has a maximum at a temperature given by
$T_{D}^{max}=(1-\nu)\varepsilon_{0}/\ln(\mu_{3}/\mu_{2})$.
Monte Carlo Simulation \- We have investigated the force induced desorption of
a polymer by means of extensive Monte Carlo simulations using a coarse grained
off-lattice bead-spring model MC_Milchev of a polymer. Fig. 1a shows the
variation of the order parameter $n$ (average fraction of adsorbed monomers)
with changing adhesive potential $\epsilon$ at fixed pulling force whereas
Fig.1b depicts $n$ vs. force $fa/T$ for various $\epsilon$. The abrupt change
of the order parameter is in close agreement with our theoretical prediction.
Using the threshold values of $f_{D}$ and $\epsilon_{D}$ for critical
adsorption/desorption in the thermodynamic limit $N\rightarrow\infty$, one can
construct the adsorption \- desorption phase diagram for a polymer chain under
pulling shown in Fig.2 which is among the central results of this work. The
detachment lines, obtained from MC data and the numerical solution of
$z^{*}=z^{\\#}$ almost coincide, and the slope of $f_{D}$ vs
$(\epsilon-\epsilon_{c})$ is close to unity, according to the prediction
$f_{D}\propto(\epsilon-\epsilon_{c})^{\nu/\phi}$. Also indicated by the shaded
area in Fig.2 is the reentrant image of the same phase diagram, obtained when
the numerical solution of $z^{*}=z^{\\#}$ is plotted in dimensional units $f$
versus $T$.
Figure 3: (a) Tail length distribution $P(s)$ for different surface potentials
close to $\epsilon_{c}$ in a polymer of length $N=128$ with no pulling force.
In the inset $P(s)$ at $\epsilon=\epsilon_{c}$ (symbols) is compared to the
prediction, Eq. (6) (full line). (b) Average tail length $S$ against
$(\epsilon-\epsilon_{c})/k_{B}T$ plotted for various chain lengths in log-log
coordinates. The slopes of these curves are plotted against $1/N$ in the inset
and extrapolate to $1/\phi$ in the thermodynamic limit $N\rightarrow\infty$.
In Fig. 3a we show the PDF of tail length at different strength of adsorption
in the absence of pulling. This confirms the U - shape of $P(s)$ predicted by
eq.(6). While for $s\rightarrow 1$ the agreement with eq. 6 is perfect, for
$s\rightarrow N$ long tails are slightly overestimated by eq.(6). This small
discrepancy reflects the dominance of our “single loop & tail” approximation -
multiple loops would effectively reduce the tail size. Fig. 3b shows the
divergency of $S$ close to the critical point $\epsilon_{c}$. For chain of
finite length $N$, the tail length divergence at
$\epsilon\rightarrow\epsilon_{c}$ is replaced by a rounding into a plateau
since $S\rightarrow N$ but away from $\epsilon_{c}$ the measured slope
extrapolates to the theoretical prediction $S\propto
1/(\epsilon-\epsilon_{c})^{1/\phi}$. In the presence of pulling force one
observes a remarkable feature of the order parameter probability distribution
Figure 4: Distribution of the order parameter $n$ for a pulling force
$fa/k_{B}T=6.0$ an different strengths of adhesion $\epsilon/k_{B}T$. The
chain length is $N=128$ and the threshold value of the surface potential for
this force is $\epsilon_{D}\approx 6.095\pm 0.03$. The values
$\epsilon/k_{B}T=6.09$ and $\epsilon/k_{B}T=6.10$ are on both sides of the
detachment line, cf. Fig. 2.
\- an absence of two peaks in the vicinity of the critical strength of
adsorption, $\epsilon_{D}\approx 6.095\pm 0.03$, which still keeps the polymer
adsorbed at pulling force $fa/k_{B}T=6.0$ \- Fig. 4. At $\epsilon_{D}$ the
distribution $H(n)$ is flat, indicating huge fluctuations so that any value of
$n$ is equally probable. Close to $\epsilon_{D}$, one observes a clear maximum
in the distribution $H(n)$, indicating a desorbed chain with $n\approx 0.01$
for $\epsilon=6.05$, or a completely adsorbed chain with $n\approx 0.99$ for
$\epsilon=6.15$. This lack of bimodality in the $H(n)$ manifests the
dichotomic nature of the desorption transition which rules out phase
coexistence.
In conclusion, we have demonstrated that a full description of the force
induced polymer chain desorption transition can be derived by means of the GCE
approach, yielding the average size and probability distribution functions of
all basic structural units as well as their variation with changing force or
strength of adhesion. The detachment transition is proved to be of first order
albeit dichotomic in nature thus ruling out phase coexistence. The critical
line of desorption, while monotonous when plotted in dimensionless units of
detachment force against surface potential, becomes non-monotonous in units of
force against temperature, thus outlining a reentrant phase diagram. In
addition, we show that the crossover exponent, $\phi$, governing polymer
behavior at criticality, depends essentially on interactions between different
loops so that $0.39\leq\phi\leq 0.59$. All these predictions appear in very
good agreement with our Monte Carlo computer simulation results.
Acknowledgments We are indebted to A. Skvortsov, L. Klushin, J.-U. Sommer, and
K. Binder for useful discussions during the preparation of this work. A.
Milchev thanks the Max-Planck Institute for Polymer Research in Mainz,
Germany, for hospitality during his visit in the institute. A. Milchev and V.
Rostiashvili acknowledge support from the Deutsche Forschungsgemeinschaft
(DFG), grant No. SFB 625/B4.
## References
* (1) T. Strick, et al. Phys. Today, 54, 46(2001).
* (2) F. Celestini, et al. Phys. Rev. 70, 012801(2008).
* (3) M. Rief, et al., Science, 275, 1295 (1997).
* (4) S. B. Smith,et al., Science, 271, 795 (1996).
* (5) B. J. Haupt,et al. Langmuir, 15, 3868 (1999).
* (6) A. M. Skvortsov, et al., Polymer Sci. A (Moscow) (2009).
* (7) Y. Kafri, et al. Eur. Phys. J. B 27, 135 (2002).
* (8) D. Poland, H.A. Scheraga, J. Chem. Phys. 45, 1456 (1966).
* (9) B. Duplantier, J. Stat. Phys. 54, 581 (1989).
* (10) C. Vanderzande, Lattice Model of Polymers, Cambridge University Press, Cambridge, 1998.
* (11) J.A. Rudnick, G.D. Gaspari, Elements of the random walk, Cambridge University Press, Cambridge, 2004.
* (12) A. Erdélyi, Higher transcendental functions, v.1, N.Y., 1953.
* (13) A. A. Gorbunov, et al. J. Chem. Phys. 114, 5366 (2001).
* (14) S. Bhattacharya et al., submitted for publication.
* (15) E. Eisenriegler, K. Kremer, K. Binder, J. Chem. Phys. 77, 6296 (1982).
* (16) A.A. Gorbunov, A.M. Skvortsov, J. Chem. Phys. 98, 5961 (1993).
* (17) J. des Cloizeaux, G. Jannink, Polymers in Solution, Clarendon Press, Oxford, 1990.
* (18) P.K. Mishra,et al. Europhys. Lett. 69, 102 (2005).
* (19) K. Binder and A. Milchev, J. Computer-Aided Material Design, 9, 33(2002).
|
arxiv-papers
| 2008-11-26T08:28:08 |
2024-09-04T02:48:58.981596
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. Bhattacharya, V. G. Rostiashvili, A. Milchev, T.A. Vilgis",
"submitter": "Vakhtang Rostiashvili",
"url": "https://arxiv.org/abs/0811.4244"
}
|
0811.4262
|
# Restrictions on Transversal Encoded Quantum Gate Sets
Bryan Eastin beastin@nist.gov Emanuel Knill National Institute of Standards
and Technology, Boulder, CO 80305
###### Abstract
Transversal gates play an important role in the theory of fault-tolerant
quantum computation due to their simplicity and robustness to noise. By
definition, transversal operators do not couple physical subsystems within the
same code block. Consequently, such operators do not spread errors within code
blocks and are, therefore, fault tolerant. Nonetheless, other methods of
ensuring fault tolerance are required, as it is invariably the case that some
encoded gates cannot be implemented transversally. This observation has led to
a long-standing conjecture that transversal encoded gate sets cannot be
universal. Here we show that the ability of a quantum code to detect an
arbitrary error on any single physical subsystem is incompatible with the
existence of a universal, transversal encoded gate set for the code.
Quantum computation appears to be intrinsically more powerful than its
classical counterpart. Efficient quantum algorithms have been found for
certain problems that, using the best known classical algorithms, require
resources that scale as a super-polynomial function of the problem size Shor
(1994); Aharonov et al. (2006); Mosca (2008). However, implementing a
computation large enough to take advantage of such scaling properties is a
daunting challenge. Given the difficulty of constructing quantum hardware, it
seems likely that the software for the first quantum computers will need to
incorporate significant amounts of error checking.
As in the classical case, quantum errors are rendered detectable by encoding
the system of interest into a subspace of a larger, typically composite,
system. A quantum code simply specifies which states of a quantum system
correspond to which logical (encoded) information states. Errors that move
states outside of the logical subspace can be detected by measuring the
projector $P$ onto this subspace. Thus, an error $E$ is detectable, in the
sense that it can be discovered or eliminated, if and only if
$\displaystyle PEP\propto P\;.$
Of course, not all errors can be detected; for any nontrivial code there are
operators that act in a nontrivial way within the logical subspace. Most
commonly, quantum codes are designed to permit the detection of independent,
local errors and, as a consequence, are incapable of detecting some errors
that affect many subsystems.
For quantum computation, it is necessary not only to detect errors but also to
apply operators (gates) that transform the logical state of the code. Even
when error processes are local and independent, however, the operations
entailed in computing can generate correlated errors from uncorrelated ones.
Thus, for error detection to be effective, it is important that the logical
operators employed during a quantum computation be designed to limit the
spread of errors. It is particularly important that operators do not spread
errors within code blocks, where a block of a quantum code is defined as a
collection of subsystems for which errors on subsystems in the collection are
detected independently of those on subsystems outside of it. Managing the
spread of errors is the subject of the theory of fault-tolerant quantum
computing Shor (1996); Preskill (1998). One of the primary techniques of this
theory is the use of transversal encoded gates.
We label as “transversal” any partition of the physical subsystems of a code
such that each part contains one subsystem from each code block. Given a
transversal partition of a code, an operator is called transversal if it
exclusively couples subsystems within the same part. Put another way, an
operator is transversal if it couples no subsystem of a code block to any but
the corresponding subsystem in another code block. Transversal operators are
inherently fault tolerant. They can spread errors between code blocks, thereby
increasing the number of locations at which a code block’s error might have
originated, but, since errors on different code blocks are treated
independently, the total number of errors necessary to cause a failure is
unchanged. This is in contrast to non-transversal operators, where, for
example, an encoded gate coupling every subsystem in a code block might
convert an error on a single subsystem into an error on every subsystem of the
code block.
In view of the above, it would be highly desirable to carry out quantum
computations exclusively using transversal encoded gates. To allow for
arbitrary computation, it is necessary that the set of gates employed be
universal, that is, that it be capable of implementing any encoded operator on
the logical state space to arbitrarily high accuracy. However, in spite of
substantial effort, no gate set for a nontrivial quantum code has yet been
found that is both universal and transversal. Consequently, a long-standing
question in quantum information theory is whether there exist nontrivial
quantum codes for which all logical gates can be implemented transversally.
For stabilizer codes, this question has recently been answered in the
negative. Zeng et al. Zeng et al. (2007) showed that transversal unitary
operators are not universal for stabilizer codes on two-level subsystems
(qubits); the companion result for the case of $d$-level subsystems (qudits)
was proven by Chen et al. Chen et al. (2008). In this paper we present a more
general proof based on the structure of the Lie group of transversal unitary
operators. Our result applies to all local-error-detecting quantum codes, that
is, all quantum codes capable of detecting an arbitrary error on any single
subsystem.
An outline of the argument is as follows: The set of logical unitary product
operators, $\mathcal{G}$, is a Lie subgroup of the Lie group of unitary
product operators, $\mathcal{T}$. As a Lie group, $\mathcal{G}$ can be
partitioned into cosets of the connected component of the identity,
$\mathcal{C}$; these cosets form a discrete set, $\mathcal{Q}$. Using the fact
that the Lie algebra of $\mathcal{C}$ is a subalgebra of $\mathcal{T}$, it can
be shown that the connected component of the identity acts trivially for any
local-error-detecting code. This implies that the number of logically distinct
operators implemented by elements of $\mathcal{G}$ is limited to the
cardinality of $\mathcal{Q}$. Due to to the compactness of $\mathcal{T}$, this
number must be finite. A finite number of operators can approximate infinitely
many only up to some fixed accuracy; thus, $\mathcal{G}$, the set of logical
unitary product operators, cannot be universal. Transversal operators may be
viewed as product operators with respect to a transversal partitioning of the
code, so the ability to detect an arbitrary error on a transversal part
implies the non-existence of a universal, transversal encoded gate set.
We begin by exploring the structure of various sets of unitary operators and
subsequently move to our central theorem. The following material relies
heavily on results from topology and the theory of Lie groups. An accessible
introduction to these topics can be found, for example, in Refs. Munkres
(2000); Hall (2004) and on Wikipedia Wik .
Consider a quantum system of finite dimension $d$. The set $\mathcal{U}(d)$ of
unitary operators on a $d$-dimensional quantum system forms a compact,
connected Lie group with a Lie algebra consisting of the Hermitian
operators.111Following the convention in physics, we include a factor of
$\mathrm{i}$ in the mapping between elements of the Lie algebra and Lie group.
Thus, any unitary operator $U\in\mathcal{U}(d)$ satisfies
$\displaystyle U=\mathrm{e}^{\mathrm{i}H}$
for some Hermitian operator $H$.
Now consider a composite quantum system $\mathpzc{Q}$ composed of $n$ physical
subsystems, where the dimension of the $j$th subsystem is $d_{j}$. Let
$\mathcal{T}$ denote the set of all unitary product operators, that is, all
operators of the form
$\displaystyle\bigotimes_{j=1}^{n}U_{j}\;,$
where $U_{j}\in\mathcal{U}(d_{j})$. Being a direct product of a finite number
of compact Lie groups, $\mathcal{T}$ is also a compact Lie group. For the same
reason, $\mathcal{T}$ has a Lie algebra $\mathfrak{t}$ given by the direct sum
of the Lie algebras of the component groups.
Given a quantum code $\mathsf{C}$ on the system $\mathpzc{Q}$, the set of
logical unitary operators on $\mathpzc{Q}$ is defined as the subset of unitary
operators that preserve the code space. In terms of a projector $P$ onto the
code states of $\mathsf{C}$, this is the statement that a unitary operator $U$
is a logical operator if and only if
$\displaystyle(I-P)UP=0\;.$ (1)
Note that $(I-P)UP$ is a continuous function of $U$.
###### Lemma 1.
The set of logical unitary operators forms a group.
###### Proof.
Let $P$ be the projector onto the logical subspace of a quantum code. The set
of logical unitary operators, $\mathcal{L}$, consists of all unitary operators
$U$ satisfying
$\displaystyle PUP=UP\;.$
The set $\mathcal{L}$ fulfills the four requirements of a group: The
multiplication of unitary operators is associative. The identity, $I$, is
contained in $\mathcal{L}$ as
$\displaystyle PIP=P^{2}=P=IP\;.$
The group property of closure is satisfied since
$\displaystyle PUVP=PUPVP=UPVP=UVP$
for any $U,V\in\mathcal{L}$. The inverse $U^{\dagger}$ of any
$U\in\mathcal{L}$ is contained in $\mathcal{L}$ since
$\displaystyle\left(PU^{\dagger}P\right)\left(PUP\right)=\left(PU^{\dagger}\right)\left(UP\right)=P\;,$
which implies that $PU^{\dagger}P$ is the inverse of $PUP$ on the subspace $P$
and therefore that
$\displaystyle
U^{\dagger}(P)=U^{\dagger}(PUPPU^{\dagger}P)=U^{\dagger}UPPU^{\dagger}P=PU^{\dagger}P\;.$
∎
###### Lemma 2.
The logical operators contained in a Lie group of unitary operators form a Lie
subgroup.
###### Proof.
Let $\mathcal{L}$ be the set of logical unitary operators for a given code,
let $\mathcal{A}$ be a Lie group of unitary operators, and let
$\mathcal{B}=\mathcal{A}\cap\mathcal{L}$. Lemma 1 shows that $\mathcal{L}$ is
a group. Because the intersection of two groups is a group, $\mathcal{B}$ is a
subgroup of $\mathcal{A}$. Topologically speaking, $\mathcal{L}$ is a closed
set since, as seen from Eq. 1, it is a preimage of a closed set under a
continuous function. Being a Lie group, $\mathcal{A}$ is also a topologically
closed set, and therefore $B$ is as well. That $\mathcal{B}$ is a Lie subgroup
of $\mathcal{A}$ follows from a theorem by Cartan (See page 3 of Ref. Sepanski
(2007).), which states that a topologically closed subgroup of a Lie group is
a Lie subgroup. ∎
###### Theorem 1.
For any nontrivial local-error-detecting quantum code, the set of logical
unitary product operators is not universal.
###### Proof.
Let $\mathpzc{Q}$, as defined earlier, be a composite quantum system
supporting a local-error-detecting code $\mathsf{C}$. The set of unitary
product operators on $\mathpzc{Q}$ is the compact Lie group that was earlier
denoted by $\mathcal{T}$.
Lemma 2 shows that $\mathcal{G}$, the subset of unitary product operators that
are also logical operators, forms a Lie subgroup of $\mathcal{T}$.
As a Lie group, $\mathcal{G}$ can be partitioned into cosets of the connected
component of the identity, $\mathcal{C}$, where $\mathcal{C}$ is a Lie
subgroup of $\mathcal{G}$. This set of cosets is the quotient group
$\mathcal{Q}=\mathcal{G}/\mathcal{C}$ and constitutes a topologically discrete
group.
Because $\mathcal{C}$ is a connected Lie group, any element $C\in\mathcal{C}$
can be written as
$\displaystyle C=\prod_{k}\mathrm{e}^{\mathrm{i}D_{k}}\;,$
where $D_{k}$ is in $\mathfrak{c}$, the Lie algebra of $\mathcal{C}$. For any
$D\in\mathfrak{c}$ and $\epsilon\in\Re$, the operator
$\mathrm{e}^{\mathrm{i}\epsilon D}$ is also in $\mathcal{C}$ and is,
consequently, a logical gate satisfying
$\displaystyle 0=(I-P)\mathrm{e}^{\mathrm{i}\epsilon D}P\;.$
Since $(I-P)IP=0$, we also have
$\displaystyle 0=\lim_{\epsilon\rightarrow
0}(I-P)\left[\frac{\mathrm{e}^{\mathrm{i}\epsilon
D}-I}{\mathrm{i}\epsilon}\right]P=(I-P)DP$
for all $D\in\mathfrak{c}$.
As $\mathcal{C}$ is a Lie subgroup of the Lie group $\mathcal{T}$, its Lie
algebra $\mathfrak{c}$ must be a subalgebra of $\mathfrak{t}$, the Lie algebra
of $\mathcal{T}$. Consequently, every element $D\in\mathfrak{c}$ can be
written in the form
$\displaystyle D=\sum_{j=1}^{n}\alpha_{j}H_{j}\;,$
where $\alpha_{j}\in\Re$ and $H_{j}$ is a Hermitian operator applied to the
$j$th subsystem. Any local Hermitian operator can be written as a sum over
local error operators, so
$\displaystyle PH_{j}P\propto P\;,$
where $P$ is the projector onto the code space of $\mathsf{C}$.
Combining the preceding three equations yields
$\displaystyle\begin{split}DP=PDP=P\sum_{j=1}^{n}\alpha_{j}H_{j}P=\sum_{j=1}^{n}\alpha_{j}PH_{j}P\propto
P\end{split}$
for all $D\in\mathfrak{c}$, which shows that
$\displaystyle CP=\prod_{k}\mathrm{e}^{\mathrm{i}D_{k}}P\propto P\;.$
Since $C$ is a unitary operator, the constant of proportionality must be one.
Thus, whether it is trivial or not, all operators contained in $\mathcal{C}$
act as the identity on the code space.
Let $\mathcal{F}$ be a set consisting of one representative from each coset of
$\mathcal{C}$ in $\mathcal{G}$. The preceding paragraph shows that every
operator in the group $\mathcal{G}$ acts on the code space as an operator from
$\mathcal{F}$. In other words, for every $G\in\mathcal{G}$,
$\displaystyle GP=FCP=FP$
for some $F\in\mathcal{F}$ and $C\in\mathcal{C}$.
The operators induced by $\mathcal{G}$ on the logical quantum system are
closed under composition and limited in number to the cardinality of
$\mathcal{F}$. The set $\mathcal{F}$ is discrete since its elements are
representatives taken from each of the cosets comprising the discrete group
$\mathcal{Q}=\mathcal{G}/\mathcal{C}$. It follows that $\mathcal{F}$ is also
finite, being a discrete subset of a compact group, namely $\mathcal{T}$.
However, for a nontrivial encoded quantum system, the number of logically
distinct operators is uncountably infinite. As the set of all unitary
operators is a metric space, a finite number of unitary operators cannot
approximate infinitely many to arbitrary precision.222By contrast, the
Solovay-Kitaev theorem Kitaev (1997); Dawson and Nielsen (2006) states that a
universal, and infinite, set of operators can be generated by composition from
certain finite sets of operators. In our case, composition yields nothing new.
Thus, $\mathcal{G}$, the set of logical product operators, is not universal. ∎
Theorem 1 considers only product gates, but the same basic approach can be
applied to the case of transversal gates.
###### Corollary 1.
For any nontrivial local-error-detecting quantum code, the set of transversal,
logical unitary operators is not universal.
###### Proof.
This result follows directly from an application of Theorem 1 in which the
physical subsystems are replaced by transversal parts. Each part contains a
set of physical subsystems that can be coupled by transversal operators.
Transversal operators may therefore be regarded as product operators on the
transversal parts. Theorem 1 thus proves that the set of transversal, logical
unitary operators is not universal for any nontrivial quantum code capable of
detecting an arbitrary error on a single transversal part. For a local-error-
detecting code, the condition that any error on a single transversal part be
detectable is satisfied since this corresponds to a single-subsystem error on
each of the code blocks. ∎
As with any impossibility proof, perhaps the most interesting aspect of
Corollary 1 is how it can be circumvented. The most obvious circumvention, and
an avenue that has been thoroughly explored, is to employ non-unitary
operators Knill (2004); Bravyi and Kitaev (2005); Zhou et al. (2000). The
standard method of achieving universal fault-tolerant quantum computation
takes this approach, making extensive use of measurements and classical feed-
forward during the preparation, testing, and coupling of ancillary states.
Alternatively, one might retain unitarity and instead loosen the requirements
of transversality or universality or even error detection, options that we
discuss in turn.
Among the alternatives listed, non-transversal operators provide the most
promising approach to circumventing Theorem 1. References Zeng et al. (2007);
Chen et al. (2008) discuss the possibility of achieving universality through
the addition of coordinate permutations, which, taken in isolation, are fault
tolerant. Zeng et al. note that the encoded Hadamard gate for the Bacon-Shor
codes Bacon (2006) involves a coordinate permutation and therefore is not
transversal. In fact, for these codes, some sequences of encoded Hadamard and
controlled-NOT gates are not fault tolerant; a single physical gate failure is
capable of producing two errors on a single code block. Strict fault tolerance
is achieved by checking for errors prior to coupling code blocks using a new
transversal partition. Such codes demonstrate that it is sufficient for
individual logical gates to avoid directly coupling subsystems of a code
block. A quantum code for which there existed a universal set of encoded gates
each transversal in isolation would be extremely useful.
Along a different line, we might imagine demanding less than full
universality. Finite groups of operators are already an important component of
schemes for fault-tolerant quantum computing. These schemes typically take
advantage of the existence of codes for which the Clifford gates, a finite
subgroup of all gates, are both sufficient for error detection and
transversally implementable. The Clifford gates are not the only set that can
be implemented transversally, however. It would be interesting to quantify the
maximum size of finite group that is achievable transversally and to
investigate the computational power of the non-Clifford finite gate groups.
Given a local error model, it seems unprofitable to abandon local error
detection entirely. In order to violate the assumptions of our proof, however,
it is sufficient that detection not be deterministic. It might be possible to
find a family of codes satisfying both the universality and transversality
conditions for which the probability of failing to detect an error on a single
subsystem can be made arbitrarily small. The usefulness of such a family of
codes would depend on the scaling of the failure probability with the size of
the code.
In conclusion, we have presented a proof that the ability of a quantum code to
detect arbitrary errors on component subsystems is incompatible with the
existence of a universal, transversal, and unitary encoded gate set. Our proof
makes no assumptions about the dimensions of the quantum subsystems beyond
requiring that they be finite. The quantum system encoded is assumed to be
nontrivial, that is, to have dimension greater than one. The precise structure
of the quantum code and its initialization state are unspecified. Our result
rules out the use of transversal unitary operators with local error detection
as an exclusive means to obtain universality, but it also suggests some
interesting new avenues of investigation.
###### Acknowledgements.
We thank Adam Meier, Scott Glancy, and Yanbao Zhang for their questions and
comments during the development of this proof. Special thanks go to Sergio
Boixo for the discussion that spawned the idea that local error detection and
infinitesimal transversal gates were incompatible. Preliminary investigations
on this topic were funded by National Science Foundation Grant No.
PHY-0653596. This paper is a contribution by the National Institute of
Standards and Technology and, as such, is not subject to US copyright.
## References
* Shor (1994) P. W. Shor, in _Proc. 35th Symp. Found. Comp. Sci._ (1994), p. 124, eprint arXiv:quant-ph/9508027.
* Aharonov et al. (2006) D. Aharonov, V. Jones, and Z. Landau, in _Proc. 38th Symp. Theory Comp._ (2006), p. 427, eprint arXiv:quant-ph/0511096.
* Mosca (2008) M. Mosca, _Quantum algorithms_ , Springer Encyclopedia of Complexity and Systems Science (Springer, 2008), eprint arXiv:0808.0369.
* Shor (1996) P. Shor, in _Proc. 37th Symp. Found. Comp. Sci._ (1996), p. 56, eprint arXiv:quant-ph/9605011.
* Preskill (1998) J. Preskill, in _Introduction to Quantum Computation_ (World Scientific, 1998), eprint arXiv:quant-ph/9712048.
* Zeng et al. (2007) B. Zeng, A. Cross, and I. L. Chuang, _Transversality versus universality for additive quantum codes_ (2007), eprint arXiv:0706.1382.
* Chen et al. (2008) X. Chen, H. Chung, A. W. Cross, B. Zeng, and I. L. Chuang, Phys. Rev. A 78, 012353 (2008), eprint arXiv:0801.2360.
* Munkres (2000) J. R. Munkres, _Topology_ (Prentice Hall, New Jersey, USA, 2000), 2nd ed.
* Hall (2004) B. C. Hall, _Lie Groups, Lie Algebras, and Representations: An Elementary Introduction_ (Springer, New York, USA, 2004).
* (10) URL http://www.wikipedia.org/.
* Sepanski (2007) M. R. Sepanski, _Compact Lie Groups_ (Springer, New York, USA, 2007).
* Kitaev (1997) A. Y. Kitaev, Russ. Math. Surv. 52, 1191 (1997).
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|
arxiv-papers
| 2008-11-26T10:09:31 |
2024-09-04T02:48:58.988088
|
{
"license": "Public Domain",
"authors": "Bryan Eastin and Emanuel Knill",
"submitter": "Bryan Eastin",
"url": "https://arxiv.org/abs/0811.4262"
}
|
0811.4459
|
On leave of absence from the ] Institute of Physics and Electronics, Hanoi,
Vietnam
# Pairing in hot rotating nuclei
N. Quang Hung1 [ nqhung@riken.jp N. Dinh Dang1,2 dang@riken.jp 1) Heavy-Ion
Nuclear Physics Laboratory, RIKEN Nishina Center for Accelerator-Based
Science, 2-1 Hirosawa, Wako City, 351-0198 Saitama, Japan
2) Institute for Nuclear Science and Technique, Hanoi, Vietnam
###### Abstract
Nuclear pairing properties are studied within an approach that includes the
quasiparticle-number fluctuation (QNF) and coupling to the quasiparticle-pair
vibrations at finite temperature and angular momentum. The formalism is
developed to describe non-collective rotations about the symmetry axis. The
numerical calculations are performed within a doubly-folded equidistant
multilevel model as well as several realistic nuclei. The results obtained for
the pairing gap, total energy and heat capacity show that the QNF smoothes out
the sharp SN phase transition and leads to the appearance of a thermally
assisted pairing gap in rotating nuclei at finite temperature. The corrections
due to the dynamic coupling to SCQRPA vibrations and particle-number
projection are analyzed. The effect of backbending of the momentum of inertia
as a function of squared angular velocity is also discussed.
Suggested keywords
###### pacs:
21.60.Jz, 21.60.-n, 24.10.Pa, 24.60.-k
## I INTRODUCTION
Thermal effect on pairing correlations has been extensively studied within the
Bardeen-Cooper-Schrieffer (BCS) theory BCS at finite temperature $T$ (FTBCS
theory). The FTBCS theory predicts a destruction of pairing correlation at a
critical temperature $T_{\rm c}\simeq 0.568\Delta(0)$ [$\Delta(0)$ is the
pairing gap at zero temperature], resulting in a sharp transition from the
superfluid phase to normal one (the SN phase transition) in good agreement
with the experimental findings in macroscopic systems such as metallic
superconductors. However, the BCS theory is valid only when the assumption on
the quasiparticle mean field is good, i.e. when the difference between the
pair correlator $P_{k\sigma}^{\dagger}\equiv
a_{k\sigma}^{\dagger}a_{k-\sigma}^{\dagger}$ and its expectation value
$\langle P_{k\sigma}^{\dagger}\rangle$ is small so that the quadratic term
$(P_{k\sigma}^{\dagger}-\langle P_{k\sigma}^{\dagger}\rangle)^{2}$ is
negligible, where $a_{k\sigma}^{\dagger}$ is the operator that creates a
particle with angular momentum $k$ and spin $\sigma$. For small systems such
as underdoped cuprates, where the coherence lengths (the Cooper-pair sizes)
are very short, the fluctuations $(P_{k\sigma}^{\dagger}-\langle
P_{k\sigma}^{\dagger}\rangle)^{2}$ are no longer small, which invalidate the
quasiparticle mean-field assumption, and break down the BCS theory. As the
result, the gap evolves continuously across $T_{\rm c}$, and persists well
above $T_{\rm c}$ Ding .
Various theoretical studies have been undertaken in the last three decades to
study the effects of thermal fluctuations on pairing in atomic nuclei. Pioneer
papers by Moretto MorettoPLB40 employed the macroscopic Landau theory of
phase transition to treat thermal fluctuations in the pairing field as those
occurring around the most probable value of the pairing gap. The results of
calculations within the uniform model carried out in Ref. MorettoPLB40 show
that the average pairing gap does not collapse as predicted by the FTBCS
theory, but decreases monotonously with increasing $T$, smearing out the sharp
SN phase transition. This approach was later used by Goodman to include the
effects of thermal fluctuations in the Hartree-Fock-Bogoliubov (HFB) theory at
finite temperature GoodmanPRC29 . The static-path approximation (SPA), which
takes into account thermal fluctuations by averaging over all static paths
around the mean field, also shows the non-collapsing pairing gap at finite
temperature DangPLB297 ; DangPRC47 . The recent microscopic approach called
the modified BCS (MBCS) theory DangPRC64 ; DangPRC67 ; DangNPA784 is based on
the secondary Bogoliubov transformation from quasiparticles to the modified
ones to restore the unitary relation for the generalized particle-density
matrix at $T\neq$ 0\. The MBCS theory, for the first time, points out the
quasiparticle-number fluctuation (QNF) as the microscopic origin that causes
the non-collapsing thermal pairing gap in finite small systems. The
predictions of these approaches are in qualitative agreements with the
experimental findings of pairing gaps and heat capacities measured in
underdoped cuprates Ding and extracted from nuclear level densities Schiller
.
While quasiparticles are regarded as independent in all above-mentioned
approaches, the recently proposed FTBCS1 theory with corrections coming from
the QNF and self-consistent quasiparticle random-phase approximation (SCQRPA)
at finite temperature DangHungPRC2008 calculates the quasiparticle occupation
numbers from a set of FTBCS1+SCQRPA equations. Within this approach, which is
called the FTBCS1+SCQRPA and is an extension of the BCS1+SCQRPA developed in
Ref. HungDangPRC76 to finite temperature, the QNF and quantal fluctuations
caused by coupling to SCQRPA vibrations are included into the equations for
the pairing gap and particle number. Under the influence of these SCQRPA
corrections, the temperature functional of the quasiparticle occupation number
deviates from the Fermi-Dirac distribution of independent quasiparticles. The
results obtained within the FTBCS1+SCQRPA for the total energies and heat
capacities agree fairly well with the exact solutions of the Richardson model
Richardson ; Volya at finite temperature, and those obtained within the
finite-temperature quantum Monte Carlo method for the realistic 56Fe nucleus
QMC .
The positive results of the FTBCS1+SCQRPA encourage a further extension of
this approach to include the effect of angular momentum on nuclear pairing so
that it can be applied to study hot rotating nuclei. The rotational phase of
nucleus as a whole, such as that present in spherical nuclei, or that about
the axis of symmetry in deformed nuclei, is known to affect nuclear level
densities. The relationship between this noncollective rotation and pairing
correlations has been the subjects of many theoretical studies. The effect of
thermal pairing on the angular momentum at finite temperature was first
examined by Kammuri in Ref. Kammuri , who included in the FTBCS equations the
effect caused by the projection $M$ of the total angular momentum operator on
the $z$-axis of the laboratory system (or nuclear symmetry axis in the case of
deformed nuclei). It has been pointed out in Ref. Kammuri that, at finite
angular momentum, a system can turn into the superconducting phase at some
intermediate excitation energy (temperature), whereas it remains in the normal
phase at low and high excitation energies. This effect was later confirmed by
Moretto in Refs. MorettoPLB35 ; MorettoNPA185 by applying the FTBCS at finite
angular momentum to the uniform model. It has been shown in these papers that,
apart from the region where the pairing gap decreases with increasing both
temperature $T$ and angular momentum $M$, and vanishes at a given critical
values $T_{\rm c}$ and $M_{\rm c}$, there is a region of $M$, whose values are
slightly higher than $M_{\rm c}$, where the pairing gap reappears at
$T=T_{1}$, increases with $T$ at $T>T_{1}$ to reach a maximum, then decreases
again to vanish at $T\geq T_{2}$. This effect is called anomalous pairing or
thermally assisted pairing correlation. In the recent study of the projected
gaps for even or odd number of particles in ultra-small metallic grains in
Ref. BalianPR317 a similar reappearance of pairing correlation at finite
temperature was also found, which is referred to as the reentrance effect.
Recently, this phenomenon was further studied in Refs. FrauendorfPRB68 ;
SheikhPRC72 by performing the calculations using the exact pairing
eigenvalues embedded in the canonical ensemble at finite temperature and
rotational frequency. The results of Refs. FrauendorfPRB68 ; SheikhPRC72 also
show a manifestation of the reentrance of pairing correlation at finite
temperature. However, different from the results of the FTBCS theory, the
reentrance effect shows up in such a way that the pairing gap reappears at a
given $T=T_{1}$ and remains finite at $T>T_{1}$ due to the strong fluctuations
of the order parameters.
The aim of the present study is to extend the FTBCS1 (FTBCS1+SCQRPA) theory of
Ref. DangHungPRC2008 to finite angular momentum so that both the effects of
angular momentum as well as QNF on nuclear pairing correlation can be studied
simultaneously in a microscopic way. The formalism is applied to a doubly
degenerate equidistant model with a constant pairing interaction $G$ and some
realistic nuclei, namely 20O, 22Ne, and 44Ca.
The paper is organized as follows. The FTBCS1+SCQRPA theory is extended to
include a specified projection $M$ of the total angular momentum on the axis
of quantization in Sec. II. The results of numerical calculations are
discussed in Sec. III. The last section summarizes the paper, where
conclusions are drawn.
## II FORMALISM
### II.1 Model Hamiltonian
We consider a system of $N$ particles interacting via a pairing force with the
parameter $G$, and rotating about the symmetry axis (noncollective rotation)
at an angular velocity (rotational frequency) $\gamma$ with a fixed projection
$M$ (or $K$) of the total angular momentum operator along this axis. For a
spherically symmetric system, it is always possible to make the laboratory-
frame $z$ axis, taken as the axis of quantization, coincide with the body-
fixed one, which is aligned within the quantum mechanical uncertainty with the
direction of the total angular momentum, so that the latter is completely
determined by its $z$-axis projection $M$ alone. As for deformed systems,
where the axis symmetry is the principal (body-fixed) axis, this noncollective
motion is known as the “single-particle” rotation, which takes place when the
angular momenta of individual nucleons are aligned parallel to the symmetry
axis, resulting in an axially symmetric oblate shape rotating about this axis.
Such noncollective motion is also possible in high-$K$ isomers BM , which have
many single-particle orbitals near the Fermi surface with a large and
approximately conserved projection $K$ of individual nucleonic angular momenta
along the symmetry axis. Therefore, without losing generality, further
derivations are carried out below for the pairing Hamiltonian of a spherical
system rotating about the $z$ axis Kammuri ; MorettoPLB35 ; MorettoNPA185 ,
namely
$H=H_{P}-\lambda\hat{N}-\gamma\hat{M}~{},$ (1)
where $H_{P}$ is the well-known pairing Hamiltonian
$H_{P}=\sum_{k}\epsilon_{k}(N_{k}+N_{-k})-G\sum_{k,k^{\prime}}P_{k}^{\dagger}P_{k^{\prime}}~{},\hskip
14.22636ptN_{\pm k}=a_{\pm k}^{\dagger}a_{\pm k}~{},\hskip
14.22636ptP_{k}^{\dagger}=a_{k}^{\dagger}a_{-k}^{\dagger}~{},$ (2)
with $a_{\pm k}^{\dagger}$ ($a_{\pm k}$) denoting the operator that creates
(annihilates) a particle with angular momentum $k$, spin projection $\pm
m_{k}$, and energy $\epsilon_{k}$. For simplicity, the subscripts $k$ are used
to label the single-particle states $|k,m_{k}\rangle$ in the deformed basis
with the positive single-particle spin projections $m_{k}$, whereas the
subscripts $-k$ denote the time-reversal states $|k,-m_{k}\rangle$ ($m_{k}>$
0). The particle number operator $\hat{N}$ and angular momentum $\hat{M}$ can
be expressed in terms of a summation over the single-particle levels:
$\hat{N}=\sum_{k}(N_{k}+N_{-k})~{},\hskip
14.22636pt\hat{M}=\sum_{k}m_{k}(N_{k}-N_{-k})~{},$ (3)
whereas the chemical potential $\lambda$ and angular velocity $\gamma$ are two
Lagrangian multipliers to be determined. For deformed and axially symmetric
systems, the $z$-projection $M$ and spin projections $m_{k}$ should be
identified with the projection $K$ along the body-fixed symmetry axis and spin
projections $\Omega_{k}$, respectively, which are good quantum numbers
MorettoNPA185 .
By using the Bogoliubov transformation from particle operators,
$a_{k}^{\dagger}$ and $a_{k}$, to quasiparticle ones, $\alpha_{k}^{\dagger}$
and $\alpha_{k}$,
$a_{k}^{\dagger}=u_{k}\alpha_{k}^{\dagger}+v_{k}\alpha_{-k}~{},\hskip
14.22636pta_{-k}=u_{k}\alpha_{-k}-v_{k}\alpha_{k}^{\dagger}~{},$ (4)
the Hamiltonian (1) is transformed into the quasiparticle Hamiltonian as
$\mathcal{H}=a+\sum_{k}{b_{k}^{+}\mathcal{N}_{k}^{+}}+\sum_{-k}{b_{k}^{-}\mathcal{N}_{k}^{-}}+\sum_{k}{c_{k}(\mathcal{A}_{k}^{\dagger}+\mathcal{A}_{k}})+\sum_{kk^{\prime}}{d_{kk^{\prime}}\mathcal{A}_{k}^{\dagger}\mathcal{A}_{k^{\prime}}}+\sum_{kk^{\prime}}{g_{k}(k^{\prime})(\mathcal{A}_{k^{\prime}}^{\dagger}\mathcal{N}_{k}+\mathcal{N}_{k}\mathcal{A}_{k^{\prime}})}$
$+\sum_{kk^{\prime}}{h_{kk^{\prime}}(\mathcal{A}_{k}^{\dagger}\mathcal{A}_{k^{\prime}}^{\dagger}+\mathcal{A}_{k^{\prime}}\mathcal{A}_{k})}+\sum_{kk^{\prime}}{q_{kk^{\prime}}\mathcal{N}_{k}\mathcal{N}_{k^{\prime}}}~{},$
(5)
where $\mathcal{N}_{k}^{+}$ and $\mathcal{N}_{k}^{-}$ are the quasiparticle-
number operators, whereas $\mathcal{A}_{k}^{\dagger}$ and $\mathcal{A}_{k}$
are the creation and destruction operators of a pair of time-conjugated
quasiparticles, respectively:
$\displaystyle\mathcal{N}_{k}^{+}=\alpha_{k}^{\dagger}\alpha_{k}~{},\hskip
14.22636pt\mathcal{N}_{k}^{-}=\alpha_{-k}^{\dagger}\alpha_{-k}~{},\hskip
14.22636pt\mathcal{N}_{k}=\mathcal{N}_{k}^{+}+\mathcal{N}_{k}^{-}~{},$ (6)
$\displaystyle\mathcal{A}_{k}^{\dagger}=\alpha_{k}^{\dagger}\alpha_{-k}^{\dagger}~{},\hskip
14.22636pt\mathcal{A}_{k}=(\mathcal{A}_{k}^{\dagger})^{\dagger}~{}.$ (7)
They obey the following commutation relations
$\displaystyle[{\cal A}_{k}~{},~{}{\cal
A}_{k^{\prime}}^{\dagger}]=\delta_{kk^{\prime}}{\cal D}_{k}~{},\hskip
5.69054pt{\rm where}\hskip 5.69054pt{\cal D}_{k}=1-{\cal N}_{k}~{},$ (8)
$\displaystyle[{\cal N}^{\pm}_{k}~{},~{}{\cal
A}_{k^{\prime}}^{\dagger}]=\delta_{kk^{\prime}}{\cal
A}_{k^{\prime}}^{\dagger}~{},\hskip 14.22636pt[{\cal N}^{\pm}_{k}~{},~{}{\cal
A}_{k^{\prime}}]=-\delta_{kk^{\prime}}{\cal A}_{k^{\prime}}~{}.$ (9)
The coefficients $b_{k}^{\pm}$ in Eq. (5) are given as
$b_{k}^{\pm}\equiv b_{k}\mp\gamma
m_{k}=(\epsilon_{k}-\lambda)(u_{k}^{2}-v_{k}^{2})+2Gu_{k}v_{k}\sum_{k^{\prime}}{u_{k^{\prime}}v_{k^{\prime}}}+Gv_{k}^{4}\mp\gamma
m_{k}~{},$ (10)
whereas the expressions for the other coefficients $a$, $b_{k}$, $c_{k}$,
$d_{kk^{\prime}}$, $g_{k}(k^{\prime})$, $h_{kk^{\prime}}$, and
$q_{kk^{\prime}}$ in Eqs. (5) and (10) can be found, e.g., in Refs.
HungDangPRC76 ; HogaasenNP28 ; DangZPA335 .
### II.2 Gap and number equations
We use the exact commutation relations (8) and (9), and follow the same
procedure introduced in Ref. DangHungPRC2008 , which is based on the
variational method
$\frac{\partial\langle\mathcal{H}\rangle}{\partial
u_{k}}+\frac{\partial\langle\mathcal{H}\rangle}{\partial v_{k}}\frac{\partial
v_{k}}{\partial
u_{k}}\equiv\langle[\mathcal{H},\mathcal{A}_{k}^{+}]\rangle=0~{},$ (11)
to minimize the expectation value $\mathcal{H}$ of the pairing Hamiltonian (5)
in the grand canonical ensemble,
$\langle\hat{\cal O}\rangle\equiv\frac{{\rm Tr}[\hat{\cal O}e^{-\beta{\cal
H}}]}{{\rm Tr}e^{-\beta{\cal H}}}~{},$ (12)
with $\langle\hat{\cal O}\rangle$ denoting the ensemble or thermal average of
the operator $\hat{\cal O}$. The following gap equation is obtained, which
formally looks like the one derived in Refs. HungDangPRC76 ; DangHungPRC2008 ,
namely
$\Delta_{k}=\frac{G}{\langle{\cal D}_{k}\rangle}{\sum_{k^{\prime}}\langle{\cal
D}_{k}{\cal D}_{k^{\prime}}\rangle}u_{k^{\prime}}v_{k^{\prime}}~{}.$ (13)
Here
$u_{k}^{2}=\frac{1}{2}\bigg{(}1+\frac{\epsilon^{\prime}_{k}-Gv_{k}^{2}-\lambda}{E_{k}}\bigg{)}~{},\hskip
14.22636ptv_{k}^{2}=\frac{1}{2}\bigg{(}1-\frac{\epsilon^{\prime}_{k}-Gv_{k}^{2}-\lambda}{E_{k}}\bigg{)}~{},$
(14)
with the quasiparticle energies $E_{k}$ defined as
$E_{k}=\sqrt{(\epsilon^{\prime}_{k}-Gv_{k}^{2}-\lambda)^{2}+\Delta_{k}^{2}}~{},$
(15)
where $\epsilon^{\prime}_{k}$ are the renormalized single particle energies:
$\epsilon_{k}^{\prime}=\epsilon_{k}+\frac{G}{\langle{\cal
D}_{k}\rangle}\sum_{k^{\prime}}(u_{k^{\prime}}^{2}-v_{k^{\prime}}^{2})\bigg{(}\langle{\cal
A}_{k}^{\dagger}{\cal A}_{k^{\prime}\neq k}^{\dagger}\rangle+\langle{\cal
A}_{k}^{\dagger}{\cal A}_{k^{\prime}}\rangle\bigg{)}~{}.$ (16)
Notice that the diagonal elements $\langle{\cal A}_{k}^{\dagger}{\cal
A}_{k}^{\dagger}\rangle$ are excluded from all calculations because of the
Pauli’s principle. The Bogoliubov’s coefficients, $u_{k}$ and $v_{k}$, in Eq.
(14) as well as the quasiparticle energy $E_{k}$ in Eq. (15) contain the self-
energy correction $-Gv_{k}^{2}$. It describes the change of the single-
particle energy $\epsilon_{k}$ as a function of the particle number starting
from the constant Hartree-Fock single-particle energy as determined for a
doubly-closed shell nucleus. This self-energy correction is usually discarded
in many nuclear structure calculations, where experimental values or those
obtained within a phenomenological potential such as the Woods-Saxon one are
used for single-particle energies, on the ground that such self-energy
correction is already taken care of in the experimental or phenomenological
single-particle spectra. As all calculations in the present paper use the
constant single-particle levels, determined at $T=$ 0 within the schematic
doubly-folded multilevel equidistant model and within the Woods-Saxon
potentials, we also choose to neglect, for simplicity, the self-energy
correction $-Gv_{k}^{2}$ from the right-hand sides of Eqs. (14) and (15) in
the numerical calculations.
The expectation values $\langle{\cal A}_{k}^{\dagger}{\cal
A}_{k^{\prime}}^{\dagger}\rangle$ and $\langle{\cal A}_{k}^{\dagger}{\cal
A}_{k^{\prime}}\rangle$ in Eq. (16) are called the screening factors. They are
calculated by coupling to the SCQRPA vibrations in the next section. The
quasiparticle-number fluctuation (QNF) is included into the gap equation
following the exact treatment:
$\langle{\cal D}_{k}{\cal D}_{k^{\prime}}\rangle=\langle{\cal
D}_{k}\rangle\langle{\cal D}_{k^{\prime}}\rangle+\delta{\cal
N}_{kk^{\prime}}~{},\hskip 14.22636pt{\rm with}\hskip 14.22636pt\delta{\cal
N}_{kk^{\prime}}=\langle{\cal N}_{k}{\cal N}_{k^{\prime}}\rangle-\langle{\cal
N}_{k}\rangle\langle{\cal N}_{k^{\prime}}\rangle~{}.$ (17)
The term $\delta{\cal N}_{kk^{\prime}}$ can be evaluated by using the mean-
field contraction as
$\delta{\cal N}_{kk^{\prime}}\simeq\delta{\cal
N}_{k}^{2}\delta_{kk^{\prime}}~{},$ (18)
with
$\delta{\cal N}_{k}^{2}=(\delta{\cal N}_{k}^{+})^{2}+(\delta{\cal
N}_{k}^{-})^{2},\hskip 14.22636pt(\delta{\cal N}_{k}^{\pm})^{2}\equiv
n_{k}^{\pm}(1-n_{k}^{\pm})~{},$ (19)
being the QNF for the nonzero angular momentum. The quasiparticle occupation
numbers $n_{k}^{\pm}$ are defined as
$n_{k}^{\pm}=\langle{\cal N}_{k}^{\pm}\rangle~{}.$ (20)
From here, one can rewrite the gap equation (13) as a sum of a level-
independent part, $\Delta$, and a level-dependent part, $\delta\Delta_{k}$,
namely
$\Delta_{k}=\Delta+\delta\Delta_{k}~{},$ (21)
where
$\Delta=G\sum_{k^{\prime}}{u_{k^{\prime}}v_{k^{\prime}}\langle{\cal
D}_{k^{\prime}}\rangle}~{},\hskip
14.22636pt\delta\Delta_{k}=G\frac{\delta\mathcal{N}_{k}^{2}}{\langle{\cal
D}_{k}\rangle}u_{k}v_{k}~{},$ (22)
with
$\langle{\cal D}_{k}\rangle=1-n_{k}^{+}-n_{k}^{-}~{}.$ (23)
Within the quasiparticle mean field, the quasiparticles are independent,
therefore the quasiparticle-occupation numbers (20) can be approximated by the
Fermi-Dirac distribution of non-interacting fermions in the following form
$n_{k}^{\pm}=\frac{1}{{\rm exp}[\beta(E_{k}\mp\gamma m_{k})]+1}~{}.$ (24)
The equations for particle number and total angular momentum are found by
taking the average of the quasiparticle representation of Eq. (3) in the grand
canonical ensemble (12). As the result we obtain
$N\equiv\langle\hat{N}\rangle=2\sum_{k}\bigg{[}v_{k}^{2}\langle{\cal
D}_{k}\rangle+\frac{1}{2}\big{(}1-\langle{\cal
D}_{k}\rangle\big{)}\bigg{]}~{},$ (25)
$M\equiv\langle\hat{M}\rangle=\sum_{k}m_{k}(n_{k}^{+}-n_{k}^{-})~{}.$ (26)
We call the set of equations (21), (25) and (26) as the FTBCS1 equations at
finite angular momentum. By neglecting the QNF (19), as well as the screening
factors $\langle{\cal A}^{\dagger}_{k}{\cal A}_{k^{\prime}}^{\dagger}\rangle$
and $\langle{\cal A}^{\dagger}_{k}{\cal A}_{k^{\prime}}\rangle$, i.e. setting
$\epsilon_{k}^{\prime}=\epsilon_{k}$ in Eq. (16), one recovers from Eqs. (21),
(25) and (26) the well-known FTBCS equations at finite angular momentum
presented in Refs. Kammuri ; MorettoNPA185 .
### II.3 Coupling to the SCQRPA vibrations
#### II.3.1 SCQRPA equations and screening factors
The derivation of the SCQRPA equations at finite temperature and angular
momentum is carried out in the same way as that for $T=0$, and is formally
identical to Eqs. (46), (56) and (57) of Ref. HungDangPRC76 . The only
difference is the expressions for the screening factors
$\langle\mathcal{A}_{k}^{+}\mathcal{A}_{k^{\prime}}^{+}\rangle$ and
$\langle\mathcal{A}_{k}^{+}\mathcal{A}_{k^{\prime}}\rangle$ at the right-hand
side of Eq. (16), which are now the functions of not only the SCQRPA
amplitudes, but also of the expectation values $\langle
Q_{\mu}^{+}Q_{\mu^{\prime}}\rangle$ and $\langle
Q_{\mu}^{+}Q_{\mu^{\prime}}^{+}\rangle$ of the SCQRPA operators. As the
details of the derivation are given in Ref. DangHungPRC2008 , only final
expressions are quoted below. The screening factors are given as
$x_{kk^{\prime}}\equiv\langle\bar{\cal A}_{k}^{\dagger}\bar{\cal
A}_{k^{\prime}}\rangle=\sum_{\mu}{\cal Y}_{k}^{\mu}{\cal
Y}_{k^{\prime}}^{\mu}+\sum_{\mu\mu^{\prime}}\bigg{(}U_{kk^{\prime}}^{\mu\mu^{\prime}}\langle{\cal
Q}^{\dagger}_{\mu}{\cal
Q}_{\mu^{\prime}}\rangle+Z_{kk^{\prime}}^{\mu\mu^{\prime}}\langle{\cal
Q}^{\dagger}_{\mu}{\cal Q}_{\mu^{\prime}}^{\dagger}\rangle\bigg{)}~{},$ (27)
$y_{kk^{\prime}}\equiv\langle\bar{\cal A}_{k}^{\dagger}\bar{\cal
A}_{k^{\prime}}^{\dagger}\rangle=\sum_{\mu}{\cal Y}_{k}^{\mu}{\cal
X}_{k^{\prime}}^{\mu}+\sum_{\mu\mu^{\prime}}\bigg{(}U_{kk^{\prime}}^{\mu\mu^{\prime}}\langle{\cal
Q}^{\dagger}_{\mu}{\cal
Q}_{\mu^{\prime}}^{\dagger}\rangle+Z_{kk^{\prime}}^{\mu\mu^{\prime}}\langle{\cal
Q}^{\dagger}_{\mu}{\cal Q}_{\mu^{\prime}}\rangle\bigg{)}~{},$ (28)
where
$\bar{\cal A}_{k}^{\dagger}=\frac{{\cal A}_{k}^{\dagger}}{\sqrt{\langle{\cal
D}_{k}\rangle}}~{},\hskip 14.22636pt\bar{\cal A}_{k}=[\bar{\cal
A}_{k}^{\dagger}]^{\dagger}~{},\hskip
14.22636pt{U}_{kk^{\prime}}^{\mu\mu^{\prime}}={\cal X}_{k}^{\mu}{\cal
X}_{k^{\prime}}^{\mu^{\prime}}+{\cal Y}_{k^{\prime}}^{\mu}{\cal
Y}_{k}^{\mu^{\prime}}~{},\hskip
14.22636pt{Z}_{kk^{\prime}}^{\mu\mu^{\prime}}={\cal X}_{k}^{\mu}{\cal
Y}_{k^{\prime}}^{\mu^{\prime}}+{\cal Y}_{k}^{\mu^{\prime}}{\cal
X}_{k^{\prime}}^{\mu}~{},$ (29)
with ${\cal X}_{k}^{\mu}$ and ${\cal Y}_{k}^{\mu}$ being the amplitudes of the
SCQRPA operators 111Although at finite angular momentum, the expectation value
$\langle[{\cal B}_{k},{\cal
B}_{k^{\prime}}^{\dagger}]\rangle=\delta_{kk^{\prime}}(n_{k}^{-}-n_{k}^{+})$,
where ${\cal B}^{\dagger}_{k}\equiv\alpha_{k}^{\dagger}\alpha_{-k}$, becomes
finite as $n_{k}^{-}\neq n_{k}^{+}$, the scattering operators ${\cal B}_{k}$
and ${\cal B}_{k}^{\dagger}$ do not contribute to the QRPA because they
commute with operators ${\cal A}_{k}^{\dagger}$, ${\cal A}_{k}$, and ${\cal
N}_{k}$.
${\cal Q}_{\mu}^{\dagger}=\sum_{k}({\cal X}_{k}^{\mu}\bar{\cal
A}_{k}^{\dagger}-{\cal Y}_{k}^{\mu}\bar{\cal A}_{k})~{},\hskip 14.22636pt{\cal
Q}_{\mu}=[{\cal Q}_{\mu}^{\dagger}]^{\dagger}~{}.$ (30)
The expectation values of $\langle{\cal Q}_{\mu}^{\dagger}{\cal
Q}_{\mu^{\prime}}\rangle$ and $\langle{\cal Q}_{\mu}^{\dagger}{\cal
Q}_{\mu^{\prime}}^{\dagger}\rangle$ are found as
$\langle{\cal Q}_{\mu}^{\dagger}{\cal Q}_{\mu^{\prime}}\rangle=\sum_{k}{\cal
Y}_{k}^{\mu}{\cal
Y}_{k}^{\mu^{\prime}}+\sum_{kk^{\prime}}(U_{kk^{\prime}}^{\mu\mu^{\prime}}x_{kk^{\prime}}-W_{kk^{\prime}}^{\mu\mu^{\prime}}y_{kk^{\prime}})~{},$
(31) $\langle{\cal Q}_{\mu}^{\dagger}{\cal
Q}_{\mu^{\prime}}^{\dagger}\rangle=-\sum_{k}{\cal Y}_{k}^{\mu}{\cal
X}_{k}^{\mu^{\prime}}+\sum_{kk^{\prime}}(U_{kk^{\prime}}^{\mu\mu^{\prime}}y_{kk^{\prime}}-W_{kk^{\prime}}^{\mu\mu^{\prime}}x_{kk^{\prime}})~{},$
(32)
where
$W^{\mu\mu^{\prime}}_{kk^{\prime}}={\cal X}_{k}^{\mu}{\cal
Y}_{k^{\prime}}^{\mu^{\prime}}+{\cal Y}_{k^{\prime}}^{\mu}{\cal
X}_{k}^{\mu^{\prime}}~{}.$ (33)
From Eqs. (27), (28), (31) and (32), the set of exact equations for the
screening factors is obtained in the form
$\sum_{k_{1}k^{\prime}_{1}}\bigg{[}\delta_{kk_{1}}\delta_{k^{\prime}k_{1}^{\prime}}-\sum_{\mu\mu^{\prime}}\big{(}U_{kk^{\prime}}^{\mu\mu^{\prime}}U_{k_{1}k_{1}^{\prime}}^{\mu\mu^{\prime}}-Z_{kk^{\prime}}^{\mu\mu^{\prime}}W_{k_{1}k_{1}^{\prime}}^{\mu\mu^{\prime}}\big{)}\bigg{]}x_{k_{1}k_{1}^{\prime}}+\sum_{k_{1}k_{1}^{\prime}\mu\mu^{\prime}}\big{(}U_{kk^{\prime}}^{\mu\mu^{\prime}}W_{k_{1}k_{1}^{\prime}}^{\mu\mu^{\prime}}-Z_{kk^{\prime}}^{\mu\mu^{\prime}}U_{k_{1}k_{1}^{\prime}}^{\mu\mu^{\prime}}\big{)}y_{k_{1}k_{1}^{\prime}}$
$=\sum_{\mu}{\cal Y}_{k}^{\mu}{\cal
Y}_{k^{\prime}}^{\mu}+\sum_{k^{\prime\prime}\mu\mu^{\prime}}{\cal
Y}_{k^{\prime\prime}}^{\mu}\big{(}U_{kk^{\prime}}^{\mu\mu^{\prime}}{\cal
Y}_{k^{\prime\prime}}^{\mu^{\prime}}-Z_{kk^{\prime}}^{\mu\mu^{\prime}}{\cal
X}_{k^{\prime\prime}}^{\mu^{\prime}}\big{)}~{},$ (34)
$\sum_{k_{1}k_{1}^{\prime}\mu\mu^{\prime}}\big{(}U_{kk^{\prime}}^{\mu\mu^{\prime}}W_{k_{1}k_{1}^{\prime}}^{\mu\mu^{\prime}}-Z_{kk^{\prime}}^{\mu\mu^{\prime}}U_{k_{1}k_{1}^{\prime}}^{\mu\mu^{\prime}}\big{)}x_{k_{1}k_{1}^{\prime}}+\sum_{k_{1}k^{\prime}_{1}}\bigg{[}\delta_{kk_{1}}\delta_{k^{\prime}k_{1}^{\prime}}-\sum_{\mu\mu^{\prime}}\big{(}U_{kk^{\prime}}^{\mu\mu^{\prime}}U_{k_{1}k_{1}^{\prime}}^{\mu\mu^{\prime}}-Z_{kk^{\prime}}^{\mu\mu^{\prime}}W_{k_{1}k_{1}^{\prime}}^{\mu\mu^{\prime}}\big{)}\bigg{]}y_{k_{1}k_{1}^{\prime}}$
$=\sum_{\mu}{\cal Y}_{k}^{\mu}{\cal
X}_{k^{\prime}}^{\mu}+\sum_{k^{\prime\prime}\mu\mu^{\prime}}{\cal
Y}_{k^{\prime\prime}}^{\mu}\big{(}Z_{kk^{\prime}}^{\mu\mu^{\prime}}{\cal
Y}_{k^{\prime\prime}}^{\mu^{\prime}}-U_{kk^{\prime}}^{\mu\mu^{\prime}}{\cal
X}_{k^{\prime\prime}}^{\mu^{\prime}}\big{)}~{}.$ (35)
#### II.3.2 Quasiparticle occupation numbers
The quasiparticle occupation numbers (20) are calculated by coupling to the
SCQRPA phonons making use of the method of double-time Green’s functions
Bogoliubov ; Zubarev . By representing the Hamiltonian (5) in the effective
form as
$H_{eff}=\sum_{k}b_{k}^{+}{\cal N}_{k}^{+}+\sum_{-k}b_{k}^{-}{\cal
N}_{k}^{-}+\sum_{k^{\prime}}q_{kk^{\prime}}{\cal N}_{k}{\cal
N}_{k^{\prime}}+\sum_{\mu}\omega_{\mu}{\cal Q}_{\mu}^{\dagger}{\cal
Q}_{\mu}+\sum_{k\mu}V_{k}^{\mu}{\cal N}_{k}({\cal Q}_{\mu}^{\dagger}+{\cal
Q}_{\mu})~{}.$ (36)
with $\omega_{\mu}$ denoting the phonon energies (eigenvalues of the SCQRPA
equations) and the vertex $V_{k}^{\mu}$ given as
$V_{k}^{\mu}=\sum_{k^{\prime}}g_{k}(k^{\prime})\sqrt{\langle{\cal
D}_{k^{\prime}}\rangle}({\cal X}_{k^{\prime}}^{\mu}+{\cal
Y}_{k^{\prime}}^{\mu})~{},$ (37)
we introduce the following double-time Green’s functions for the quasiparticle
propagations
$G_{\pm k}(t-t^{\prime})=\langle\langle\alpha_{\pm k}(t);\alpha^{\dagger}_{\pm
k}(t^{\prime})\rangle\rangle~{},$ (38)
as well as those corresponding to quasiparticle$\otimes$phonon couplings
${\Gamma}_{{\pm k}\mu}^{--}(t-t^{\prime})=\langle\langle\alpha_{\pm k}(t){\cal
Q}_{\mu}(t);\alpha^{\dagger}_{\pm k}(t^{\prime})\rangle\rangle~{},\hskip
14.22636pt{\Gamma}_{{\pm k}\mu}^{-+}(t-t^{\prime})=\langle\langle\alpha_{\pm
k}(t){\cal Q}_{\mu}^{\dagger}(t);\alpha^{\dagger}_{\pm
k}(t^{\prime})\rangle\rangle~{}.$ (39)
Following the same procedure in Ref. DangHungPRC2008 , we obtain the final
equations for the quasiparticle Green’s functions $G_{\pm k}(E)$ in the
following form
$G_{\pm k}(E)=\frac{1}{2\pi}\frac{1}{E-E_{k}^{\pm}-M_{k}^{\pm}(E)}~{},$ (40)
where
$E_{k}^{\pm}=b_{k}^{\pm}+q_{kk}~{},$ (41) $M_{k}^{\pm}(E=\omega\pm
i\varepsilon)=M_{k}^{\pm}(\omega)\mp i\gamma_{k}^{\pm}(\omega)~{},$ (42)
$M_{k}^{\pm}(\omega)=\sum_{\mu}(V_{k}^{\mu})^{2}\bigg{[}\frac{(1-n_{k}^{\pm}+\nu_{\mu})(\omega-
E_{k}^{\pm}-\omega_{\mu})}{(\omega-
E_{k}^{\pm}-\omega_{\mu})^{2}+\varepsilon^{2}}+\frac{(n_{k}^{\pm}+\nu_{\mu})(\omega-
E_{k}^{\pm}+\omega_{\mu})}{(\omega-
E_{k}^{\pm}+\omega_{\mu})^{2}+\varepsilon^{2}}\bigg{]}~{},$ (43)
$\gamma_{k}^{\pm}(\omega)=\varepsilon\sum_{\mu}(V_{k}^{\mu})^{2}\bigg{[}\frac{1-n_{k}^{\pm}+\nu_{\mu}}{(\omega-
E_{k}^{\pm}-\omega_{\mu})^{2}+\varepsilon^{2}}+\frac{n_{k}^{\pm}+\nu_{\mu}}{(\omega-
E_{k}^{\pm}+\omega_{\mu})^{2}+\varepsilon^{2}}\bigg{]}~{}.$ (44)
In Eqs. (42) – (44), the imaginary parts $\gamma^{\pm}_{k}(\omega)$ ($\omega$
real) of the analytic continuation of $M_{k}^{\pm}(E)$ into the complex energy
describe the damping of quasiparticle excitations due to coupling to SCQRPA
vibrations, $\nu_{\mu}=\langle{\cal Q}_{\mu}^{+}{\cal Q}_{\mu}\rangle$ is the
phonon occupation number, and $\varepsilon$ is a sufficient small parameter.
These results allow to find the spectral intensities $J_{k}^{\pm}(\omega)$
from the relations $J_{k}^{\pm}(\omega)=i[G_{\pm
k}(\omega+i\varepsilon)-G_{\pm k}(\omega-i\varepsilon)]/[\exp(\beta\omega)+1]$
in the form
$J_{k}^{\pm}(\omega)=\frac{1}{\pi}\frac{\gamma_{k}^{\pm}(\omega)(e^{\beta\omega}+1)^{-1}}{[\omega-
E_{k}^{\pm}-M_{k}^{\pm}(\omega)]^{2}+[\gamma_{k}^{\pm}(\omega)]^{2}}~{},$ (45)
and, finally, the quasiparticle occupation numbers (20) as
$n_{k}^{\pm}=\int_{-\infty}^{\infty}J_{k}^{\pm}(\omega)d\omega~{}.$ (46)
In the limit of quasiparticle damping $\gamma_{k}^{\pm}(\omega)\rightarrow 0$,
$n_{k}^{\pm}$ can be approximated with the Fermi-Dirac distribution
$n_{k}^{\pm}\simeq\frac{1}{{\rm exp}(\beta\widetilde{E}_{k}^{\pm})+1}~{},$
(47)
where $\widetilde{E}_{k}^{\pm}$ are the solutions of the equations for the
poles of the quasiparticle Green’s functions $G_{\pm k}(\omega)$ (40), namely
$\widetilde{E}_{k}^{\pm}-E_{k}^{\pm}-M_{k}^{\pm}(\widetilde{E}_{k}^{\pm})=0~{}.$
(48)
The particle-number violation inherent in the BCS-based theories still causes
some quantal fluctuation of particle number starting from $T=$ 0\. This defect
can be removed by carrying out a proper particle-number projection (PNP).
Among different methods of PNP, the Lipkin-Nogami (LN) prescription (LN-PNP)
LN is widely used because of its simplicity. This method has been implemented
into the FTBCS1 and FTBCS1+SCQRPA in Ref. DangHungPRC2008 , and the ensuing
approaches are called the FTLN1 and FTLN1+SCQRPA, respectively. Their
extension to $M\neq$ 0 is straightforward. It is easy to see that, in the
nonrotating limit ($\gamma=$ 0), one has $b_{k}^{+}=b_{k}^{-}=b_{k}$ from Eq.
(10), $n_{k}^{+}=n_{k}^{-}$ from Eqs. (46), and all above-derived formalism
reduces to that presented in Ref. DangHungPRC2008 .
## III NUMERICAL RESULTS
### III.1 Ingredients of numerical calculations
The numerical calculations are carried out within a schematic model as well as
several nuclei with realistic single-particle spectra. For the schematic
model, we use the one with $N$ particles distributed over $\Omega=N$ doubly-
folded equidistant levels. These levels interact via an attractive pairing
force with the constant parameter $G$. When the interaction is switched off,
all the lowest $\Omega/2$ levels are filled up with $N$ particles so that each
of them is occupied by two particles with the spin projections equal to $\pm
m_{k}$ ($k$=1,…, $\Omega$, and $m_{k}=$ 1/2, 3/2, … , $\Omega-1/2$). The
single-particle energies $\epsilon_{k}$ are measured from the middle of the
spectrum as $\epsilon_{k}=\epsilon[k-(\Omega+1)/2]$ so that the energies of
the lower 5 levels are negative, whereas those of the upper 5 levels are
positive. The results obtained for $N=$ 10, $\epsilon=$ 1 MeV, and $G=$ 0.5
MeV are analyzed in the present paper.
As for the realistic nuclei, we carry out the calculations for neutrons in 20O
and 44Ca, whereas the contribution of proton and neutron components to nuclear
pairing is studied for the well-deformed 22Ne nucleus, where a backbending of
moment of inertia as a function of the square of angular velocity was detected
SzantoPRL42 . The calculations use the single-particle energies generated at
$T=$ 0 within deformed Woods-Saxon potentials. For the slightly axially
deformed 20O, the multipole deformation parameters $\beta_{2}$, $\beta_{3}$,
$\beta_{4}$, $\beta_{5}$, and $\beta_{6}$ are chosen to be equal to 0.03, 0.0,
-0.108, 0.0, and -0.003, respectively. For 22Ne, the axial deformation is
rather strong with these parameters taking the values equal to 0.326, 0.0,
0.225, 0.0, and 0.011, respectively. For the spherical 44Ca, all the
deformation parameters $\beta_{i}$ are set to be equal to zero. Other
parameters of Woods-Saxon potentials are taken from Table 1 of Ref. CwiokCPC46
. The neutron single-particle spectrum for 20O includes all levels up to the
shell closure with $N=$ 20 (between around -25.84 MeV and 0.49 MeV), from
which two orbitals, $1d_{3/2}$ and $1d_{1/2}$, are unbound. These unbound
states have been shown to have a large contribution to pairing correlations in
20-22O isotopes BetanNPA771 . The neutron single-particle spectrum for 44Ca
include all bound states between around -35.6 MeV and -1.05 MeV, up to the
$2p_{1/2}$ orbital of the closed shell with $N=$ 50\. The single-particle
spectra for 22Ne consist of all 11 proton bound states between -30.23
$\leq\epsilon_{p}\leq$ -0.156 MeV, and 12 neutron ones between -29.834
$\leq\epsilon_{n}\leq$ -0.742 MeV. The values of pairing interaction parameter
$G$ are chosen so that the pairing gaps $\Delta(T=0,M=0)$ obtained at zero
temperature and zero angular momentum match the experimental values extracted
from the odd-even mass differences for these nuclei, namely,
$\Delta(0,0)\simeq$ 4 MeV for protons in 22Ne, and 3, 2, and 3 MeV for
neutrons in 20O, 44Ca, 22Ne, respectively.
The numerical calculations are carried out within the FTBCS and FTBCS1 for the
level-weighted pairing gap $\bar{\Delta}=\sum_{k}{\Delta_{k}}/\Omega$ as
functions of temperature $T$, angular momentum $M$, and angular velocity
$\gamma$. The effect caused by coupling to SCQRPA vibrations is analyzed by
studying the total energy ${\cal E}=\langle H\rangle$ and heat capacity
$C=\partial{\cal E}/\partial T$, whereas the backbending is studied by
considering the momentum of inertia as a function of $\gamma^{2}$ as $T$
varies.
### III.2 Results within the doubly-folded multilevel equidistant model
Figure 1: (Color online) Level-weighted pairing gaps $\bar{\Delta}$ as
functions of $T$ at various $M$ [(a), (d)], and as functions of $M$ [(b), (e)]
and $\gamma$ [(c), (f)] at several $T$ for $N$ = 10, $G$ = 0.5 MeV obtained
within the FTBCS (left) and FTBCS1 (right).
Shown in Figs. 1 (a) and 1 (d) are the level-weighted pairing gaps
$\bar{\Delta}$ as functions of $T$ at various $M$, whereas the dependence of
$\bar{\Delta}$ on $M$ at several $T$ is displayed in Figs. 1 (b) and 1(e).
Finally, Figs. 1 (c) and 1 (f) show the gaps $\bar{\Delta}$ as functions of
the angular velocity $\gamma$ at various $T$. All the results are obtained for
the system with $N$ = 10 and $G$ = 0.5 MeV, from which the left panels are the
predictions by the FTBCS theory, whereas the right panels are those by the
FTBCS1 one. It is clearly seen from Figs. 1 (a) and 1 (b) that the FTBCS gap
decreases with increasing $T$ ($M$) at $M=$ 0 ($T=$ 0) up to a certain
critical value $T_{\rm c}=$ 0.77 MeV ($M_{\rm c}=$ 8$\hbar$), where the FTBCS
gap collapses. The collapse of the pairing gap at $M=M_{\rm c}$ (at $T=$ 0)
was proposed by Mottelson and Valatin as being caused by the Coriolis force,
which breaks the Cooper pairs MV . This feature remains with the FTBCS gap as
a function of $M$, when $T\neq$ 0, but with decreasing $M_{\rm c}(T)<M_{\rm
c}$ as $T$ increases beyond 0.6$T_{\rm c}$. As for the behavior of the FTBCS
gap as a function of $T$, one notices that, at $M$ slightly larger than
$M_{\rm c}$, the so-called thermally assisted pairing correlation takes place,
in which the pairing gap is zero at $T\leq T_{1}$, increases at $T>T_{1}$ to
reach a maximum, then decreases again to vanish at $T\geq T_{2}$ [See. Fig. 1
(a) for $M/M_{\rm c}\geq$ 1]. This interesting phenomenon was predicted and
explained, for the first time, by Moretto in Refs. MorettoPLB35 ;
MorettoNPA185 by applying the FTBCS to the uniform model. At $M/M_{\rm c}>$
1.1, no FTBCS pairing gap remains.
Different from the FTBCS predictions, the results obtained within the FTBCS1
include the effect caused by the QNF. As one can see in Fig. 1 (d), while, in
the region of low temperature $T<T_{\rm c}$, the FTBCS1 and FTBCS gaps for
different $M$ are rather similar, they are qualitatively different at
$T>T_{\rm c}$. Here, the QNF, which is rather strong at high $T$, causes a
monotonous decrease of the FTBCS1 gap $\bar{\Delta}$ as $T$ increases. This
FTBCS1 gap never collapses even at very high $T$. Instead all the values of
the FTBCS1 gap obtained at various $M$ seem to saturate at a value of around
2.25 MeV at $T>$ 5 MeV. This feature shows that, the effect of angular
momentum on reducing the pairing correlation is significant only at low $T$.
In the high temperature region, the QNF leads to a persistence of the pairing
correlation in a rotating system. Compared to the FTBCS theory, when the QNF
is neglected, the effect of thermally assisted pairing correlation also takes
place at $M/M_{\rm c}>1.1$. However, the FTBCS1 gap is now zero at $T\leq
T_{1}$, reappears at $T=T_{1}$, and remains finite at $T>T_{1}$. This result
is found in qualitative agreement with those obtained in the calculations of
the canonical gap of in Ref. FrauendorfPRB68 , where the reappearance of the
pairing gap at finite $T$ and $\gamma$ is related to the strong fluctuations
of order parameter in the canonical ensemble of small systems such as metal
clusters and nuclei. In the present paper, we point out the QNF as the
microscopic origin of this effect. Comparisons between the FTBCS1 gaps and the
canonical ones obtained at ($T\neq$ 0, $M=$ 0) and ($T=$ 0, $M$ and/or
$\gamma\neq$ 0) are discussed later, in Sec. III.5.
The QNF has a similar effect on the behavior of the pairing gap $\bar{\Delta}$
as a function of angular momentum. As low $T$, when the QNF is still
negligible, the FTBCS and FTBCS1 gaps as functions of $M$ are similar. They
both decreases as $M$ increases, and collapse at $M=M_{\rm c}$ and at $M$
slightly higher than $M_{\rm c}$ for 0 $<T/T_{\rm c}\leq$ 0.2, contrary to the
trend within the FTBCS theory, where $M_{\rm c}(T)$ decreases as $T/T_{\rm c}$
increases above 0.6 discussed above [Compare Figs. 1 (b) and 1 (e)]. At
$T/T_{\rm c}=$ 0.8, e.g., the collapsing points of the FTBCS and FTBCS1 gaps
are $M/M_{\rm c}\simeq$ 0.85, and 2.9, respectively.
The FTBCS and FTBCS1 pairing gaps are displayed in Figs. 1 (c) and 1 (f) as
functions of angular velocity $\gamma$ at various $T$. For $T/T_{\rm c}\leq$
0.2, the pairing gap undergoes a backbending, which will be discussed in the
Sec. III.4. At $T/T_{\rm c}>$ 0.2 no backbending is seen for the pairing gaps.
This result agrees with those obtained in calculations within the finite-
temperature Hartree-Fock-Bogoliubov cranking (FTHFBC) theory, which is applied
to the two-level model in Ref. GoodmanNPA352 . Within the FTBCS1, the pairing
gaps at large $M$ become enhanced with $T$, in agreement with the results
obtained within an exactly solvable model for a single $f_{7/2}$ shell in Ref.
SheikhPRC72 .
### III.3 Results for realistic nuclei
Figure 2: (Color online) Same as Fig. 1 but for neutrons in 20O using $G=$
1.04 MeV.
Shown in Fig. 2 are the level-weighted pairing gaps as functions of $T,M$ and
$\gamma$ obtained within the FTBCS and FTBCS1 theories for neutrons in 20O.
The values of $T_{\rm c}$ (at $M=$ 0) and $M_{\rm c}$ (at $T=$ 0) are found
equal to 1.57 MeV and 4$\hbar$, respectively. Compared to the case with
schematic model discussed in the previous section, the difference is that no
thermally assisted pairing correlation appears within the FTBCS for 20O. All
the FTBCS gaps behave similarly as functions of $T$ with increasing $M$. At a
given value of $M$, they decrease with increasing both $T$, and collapse at
some values $T_{\rm c}(M)<T_{\rm c}$. A similar behavior is seen for the gaps
as functions of $M$ at a given value of $T$. Here the critical value $M_{\rm
c}(T)$ for the angular momentum, at which the gap collapses is found
decreasing with increasing $T$ so that $M_{\rm c}(T)<M_{\rm c}$ [See Figs. 2
(a) and 2 (b)]. Meanwhile, the temperature dependence of the FTBCS1 gap in
Fig. 2 (d) shows a clear manifestation of the thermally assisted pairing gap.
As $M$ increases up to $M/M_{\rm c}\simeq$ 0.8, the gap decreases monotonously
with increasing $T$ up to $T\simeq$ 1.5$T_{\rm c}$, higher than which the gap
seems to be rather stable against the variation of $T$. At $M/M_{\rm c}\geq$
0.9, the reentrance of thermal pairing starts to show up as the enhancement of
the tail at $T>T_{\rm c}$. When $M/M_{\rm c}$ becomes equal to or larger than
1, the gap completely vanishes at low $T$, but reappears starting from a
certain value of $T$, above which the gap increases with $T$ and reach a
saturation at high $T$. At $T\geq$ 3 MeV, all the gaps obtained at different
values of $M$ seem to coalesce to limiting value around 0.7 – 0.8 MeV. At a
given value of $T$ in the region $T/T_{\rm c}\leq$ 0.7, as shown in Fig. 2
(e), the pairing gaps decrease steeply with increasing $T$ and all collapse at
the same value $M_{\rm c}$. This difference compared to the FTBCS theory comes
from the presence of the QNF. At $T/T_{\rm c}\geq$ 0.8, the QNF becomes
stronger, which pushes up the collapsing point to $M_{\rm c}(T)=2M_{\rm c}$.
One can also sees some oscillation occurring in the region between 0.8 $\leq
M/M_{\rm c}\leq$ 1.4 because of the shell structure. The collapsing point
might be shifted even further to higher $M$ with increasing $T$, but at too
high $T$ the temperature dependence of single-particle energies becomes
significant so that the use of the spectrum obtained at $T=$ 0 is no longer
valid BrackBonche .
The pairing gaps as functions of angular velocity $\gamma$ obtained at various
$T$ within the FTBCS and FTBCS1 theories are plotted in Figs. 2 (c) and 2 (f),
respectively. As $E_{k}$, $\gamma$ and $m_{k}$ are positive, at $T=0$, the
quasiparticle occupation number $n_{k}^{-}$ is always zero, whereas
$n_{k}^{+}$ is a step function of $E_{k}-\gamma m_{k}$, which is zero if
$E_{k}>\gamma m_{k}$ and 1 if $E_{k}\leq\gamma m_{k}$. As the result, the
FTBCS and FTBCS1 gaps decrease with increasing $\gamma$ in a stepwise manner
up to a critical value $\gamma_{\rm c}$, where they vanish. At $T\neq$ 0, the
Fermi-Dirac distribution replaces the step function, which washes out the
stepwise manner in the behavior of the gaps as functions of the $\gamma$. Here
again, once can see that, at $T/T_{\rm c}>$ 0.8, the QNF is so strong that the
collapse of the FTBCS1 gap is completely smoothed out [Fig. 2 (f)].
Figure 3: (Color online) Same as Fig. 1 but for neutrons in 44Ca using $G=$
0.48 MeV.
The level-weighted pairing gaps $\bar{\Delta}$ for neutrons in 44Ca shown in
Fig. 3 have a similar behavior as as functions of $T$, $M$ and $\gamma$ with
the values of $T_{\rm c}$ and $M_{\rm c}$ are found to be 1.07 MeV and 8
$\hbar$, respectively. The thermally assisted pairing gap appears at $M/M_{\rm
c}>$ 1.0 but the high-$T$ tail is much depleted due to a weaker QNF in a
heavier system compared to that in 20O.
Figure 4: (Color online) Level-weighted pairing gaps as functions of $T$ at
various $M$ obtained within the FTBCS (left) and FTBCS1 (right) for neutrons
[(a), (c)], and protons [(b), (d)] in 22Ne using $G_{n}=$ 1.0 MeV and $G_{p}=$
1.32 MeV.
The well deformed nucleus 22Ne has both neutron and proton open shells,
therefore the gap and two number equations for protons ($p$) and neutrons
($n$) are simultaneously solved together with one equation for the total
angular momentum $M=M_{p}+M_{n}$ to obtain the pairing gaps $\Delta_{p}$ and
chemical potential $\lambda_{p}$ for protons, the corresponding quantities,
$\Delta_{n}$ and $\lambda_{n}$, for neutrons, as well as the angular velocity
$\gamma$ of the entire nucleus MorettoNPA216 . The level-weighted pairing gaps
as functions of $T$ at several $M$ obtained for neutrons and protons in 22Ne
are shown in Fig. 4. The FTBCS neutron gaps become depleted with increasing
$M$, and completely disappears at $M>3\hbar$. As a function of $T$, the FTBCS
neutron gaps decrease as $T$ increases and collapse at $T_{\rm c}(M)$, which
decreases from $T_{\rm c}(M=0)\simeq$ 1.7 MeV to $T_{\rm c}(M=3\hbar)\simeq$
1.1 MeV. The FTBCS1 gaps obtained at $M<$ 3 $\hbar$ never collapse, but
gradually decrease with increasing $T>T_{\rm c}(M=0)$, and remains a finite
value of around 0.4 MeV at $T$ as high as 4 MeV. At $M=$ 4 $\hbar$, whereas
there is no FTBCS gap, the thermally assisted pairing gap appears within the
FTBCS1 theory at $T>$ 0, increases with $T$ to reach a maximum at $T\sim$ 1.5
MeV, then decreases slowly to reach the same high-$T$ limit of around 0.4 MeV
at $T\simeq$ 4 MeV. The situation is the similar for the proton pairing gaps,
where the effect of thermally assisted pairing correlation takes place at $M>$
8 $\hbar$ with the rather stable values of the gap against $T>$ 3 MeV.
### III.4 Backbending
Figure 5: (Color online) Moment of inertia as a function of the square
$\gamma^{2}$ of angular velocity $\gamma$ obtained within the FTBCS (left) and
FTBCS1 (right) at various $T$ for $N=10$ [(a), e)], neutrons in 20O [(b), (f)]
and 44Ca [(c), (g)], and the whole 22Ne nucleus (including both proton and
neutron gaps) [(d), (h)].
For an object that rotates about a fixed symmetry axis, its moment of inertia
${\cal J}$ is found as the total angular momentum $M$ divided by the angular
velocity $\gamma$, i.e. ${\cal J}=M/\gamma$. The backbending phenomenon is
most easily demonstrated by the behavior of ${\cal J}$ as a function of the
square $\gamma^{2}$. This curve first increases with $\gamma^{2}$ up to a
certain region of $\gamma^{2}$, where the increase suddenly becomes very
steep, and the curve even bends backward. This phenomenon is understood as the
consequence of the no-crossing rule in the region of band crossing LL . The SN
phase transition has been suggested as one of microscopic interpretations of
backbending MV .
The values of the moment of inertia ${\cal J}$, obtained at various $T$ within
the schematic model as well as realistic nuclei, is plotted in Fig. 5. In the
schematic model, one can see in Figs. 5 (a) and 5 (e) a sharp backbending,
which takes place at very low temperatures, $T/T_{\rm c}\leq 0.2$. As the QNF
is negligible in this temperature region, the predictions by the FTBCS and
FTBCS1 theories are almost identical. As $T$ increases, the moment of inertia
obtained within the FTBCS changes abruptly to reach the rigid-body value,
generating a cusp, whereas, under the effect of QNF, the value obtained within
the FTBCS1 theory gradually approaches the rigid-body value in such a way that
the cusp is smoothed out. While no signature of backbending is seen in the
results obtained in 20O [Figs. 5 (c) and 5 (f)] and 44Ca [Figs. 5 (d) and
5(g)], backbending can be seen in 22Ne [Figs. 5 (d) and 5 (h)] at $T\leq$ 0.4
MeV in agreement with the experimental data reported in Ref. SzantoPRL42 .
### III.5 Corrections due to SCQRPA and particle-number projection
Figure 6: (Color online) Level-weighted pairing gap $\bar{\Delta}$ and moment
of inertia ${\cal J}$ for $N=$ 10 with $G=$ 0.5 MeV [$\varepsilon=$ 0.1 MeV in
Eqs. (43) and (44)]. (a1) – (c1): $\bar{\Delta}$ vs temperature $T$ at
different angular momenta $M$. (a2) – (c4): Results obtained at different
values of $T$, namely, (a2) – (c2): $\bar{\Delta}$ vs $M$; (a3) – (c3):
$\bar{\Delta}$ vs angular velocity $\gamma$; (a4) – c4): ${\cal J}$ vs
$\gamma^{2}$. The dotted, thin solid, thick solid, thin dash-dotted, thick
dash-dotted lines are results obtained within the FTBCS, FTBCS1,
FTBCS1+SCQRPA, FTLN1, FTLN1+SCQRPA, respectively. The solid lines with circles
and boxes in (a1) and (a3) correspond to two definitions $\Delta_{\rm
c}^{(1)}$ and $\Delta_{\rm c}^{(2)}$ of the canonical gaps at $T=0$,
respectively (See Appendix A). In (a2) the dashed lines connecting the
discrete values of the corresponding canonical gaps at $T=0$ are drawn to
guide the eye.
Shown in Fig. 6 are the level-weighted pairing gaps and moment of inertia,
obtained within the schematic model with $N=$ 10, where predictions offered by
several approaches, namely the FTBCS, FTBCS1, FTBCS1 + SCQRPA, FTLN1, and
FTLN1 + SCQRPA, are collected. In the same figure, the canonical gaps
$\Delta_{\rm C}^{(1)}$ and $\Delta_{\rm C}^{(2)}$ obtained at ($T\neq$ 0, $M=$
0) [Fig. 6 (a1)], ($T=0$, $M\neq$ 0) [Fig. 6 (a2)], and ($T=0$, $\gamma\neq$
0) [Fig. 6 (a3)], are also shown (See Appendix A for the detailed discussion
of the canonical results).
As seen from Figs. 6, the effect due to the SCQRPA corrections on the pairing
gap increases with $M$. At $M/M_{\rm c}\leq$ 0.8 it is rather weak, causing
only a slight enhancement of the gap at 1.2 $<T\leq$ 2 MeV as compared with
the FTBCS1 results [Figs. 6 (a1) and 6 (b1)]. However, it becomes important at
$M>1.2M_{\rm c}$ [Figs. 6 (c1), 6 (a2) – 6 (c2)], or $\gamma>$ 0.2 MeV$/\hbar$
(at $T\geq 0.8T_{\rm c}$) [Figs. 6 (b3) and 6 (c3)]. In particular, the
reappearance of the thermal gap at $M\geq 1.1M_{\rm c}$ is significantly
enhanced by the SCQRPA corrections [Figs. 6 (c1) and 6 (c2)]. For the moment
of inertia [Figs. 6 (a4) – 6 (c4)], the SCQRPA corrections are important only
at low $T$ and $\gamma<$ 0.25 MeV$/\hbar$. At $T>$ 1 MeV, the predictions by
all the approximations for ${\cal J}$ saturate to the rigid-body value.
As compared to the predictions by the FTBCS1 and FTBCS1+SCQRPA, the
corrections due to LN-PNP are important only at low $T$ and $M$. As the
result, the gap is pushed up to be closer to the canonical results at $T\leq
T_{\rm c}$ and $M=0$ [Fig. 6 (a1)]. This feature is well-known and has been
discussed within the present approach at $M=$ 0 in Ref. DangHungPRC2008 . At
$M\neq$ 0 ($\gamma\neq$ 0), the effect due to LN-PNP is noticeable in the gaps
as functions of $M$ (or $\gamma$) only at [$M\leq$ 1.2$M_{\rm c}$ ($\gamma<$
0.2 MeV$/\hbar$), $T<$ 1 MeV], otherwise the FTLN1 (FTLN1+SCQRPA) results are
hardly distinguishable from the FTBCS1 (FTBCS1+SCQRPA) ones [Figs. 6 (b2), 6
(c2), 6 (b3), and 6 (c3)]. Consenquently, for the moment of inertia, the LN-
PNP corrections to the FTBCS1 (FTBCS1+SCQRPA) results are important only at
($T<T_{\rm c}$, $\gamma<\gamma_{\rm c}$) [Figs. 6 (a4) – 6 (c4)]. In
particular, the results at $T=$ 0 [Fig. 6 (a4)], where the BCS1 coincides with
the BCS, show that, backbending becomes less pronounced within the SCQRPA and
LN-SCQRPA. For this reason, the corrections due to LN-PNP are omitted in the
results obtained for realistic nuclei below.
Shown in Figs. 7 and 8 are the pairing gaps, total energies and heat
capacities as functions of $T$ obtained at $M/M_{\rm c}=$ 0, 0.4, and 0.8
within the FTBCS, FTBCS1 and FTBCS1 + SCQRPA for realistic nuclei, 20O and
44Ca. The SCQRPA corrections are significant for the total energy in the light
nucleus, 20O, due to the important contributions of the screening factors (27)
and (28) HungDangPRC76 ; DangHungPRC2008 . In medium 44Ca nucleus, the effect
of SCQRPA corrections on the total energy is weaker. The corrections due to LN
particle-number projection have a similar effect as that discussed above for
the schematic model, but with much reduced magnitudes, so they are not shown
in these figures. With increasing $M$ the pairing gap decreases. As the
result, the total energy becomes larger but the relative effect of the SCQRPA
correction does not change. For the heat capacity, as has been reported in
Ref. DangHungPRC2008 , the spike at $T_{\rm c}$ obtained within the FTBCS
theory, which serves as the signature of the sharp SN phase transition, is
smeared out within the FTBCS1 theory into a bump in the temperature region
around $T_{\rm c}$. With increasing $M$, this bump becomes depleted further.
Finally, the SQRPA corrections erase all the traces of the sharp SN phase
transition in the model case as well as realistic nuclei.
Figure 7: (Color online) Level-weighted pairing gaps $\bar{\Delta}$, total
energies ${\cal E}$, and heat capacities $C$ as functions of temperature $T$
for three values of angular momentum $M$ obtained within the FTBCS (dotted
lines), FTBCS1 (thin solid lines) and FTBCS1 + SCQRPA (thick solid lines) for
neutrons in 20O with $G=$ 1.04 MeV ($\varepsilon=$ 0.1 MeV). Figure 8: (Color
online) Same as in Fig. 7 but for neutrons in 44Ca with $G=$ 0.48 MeV
($\varepsilon=$ 0.1 MeV).
## IV CONCLUSIONS
The present work extends the FTBCS1 (FTBCS1 + SCQRPA) theory to finite angular
momentum to study the pairing properties of hot nuclei, which rotate
noncollectively about the symmetry axis. The FTBCS1 theory includes the
quasiparticle number fluctuation whereas the FTBCS1 + SCQRPA also takes into
account the correction due to dynamic coupling to SCQRPA vibrations. The
proposed extension is tested within the doubly degenerate equidistant model
with $N=10$ particles as well as some realistic (spherical and deformed)
nuclei, 20O, 22Ne, and 44Ca. The numerical calculations were carried out
within the FTBCS, FTBCS1, and FTBCS1 + SCQRPA for the pairing gap, total
energy, and heat capacity as functions of temperature $T$, total angular
momentum $M$, and angular velocity $\gamma$. The corrections due to the
Lipkin-Nogami particle-number projection are also discussed. The analysis of
the numerical results- allows us to draw the following conclusions:
1\. The proposed approach is able to reproduce the effect of thermally
assisted pairing correlation that takes place in the schematic model within
the FTBCS theory, according to which a finite pairing gap can reappear within
a given temperature interval, $T_{1}<T<T_{2}$ ($T_{1}>0$), while it is zero
beyond this interval. However, this phenomenon does not show up in realistic
nuclei under consideration.
2\. Under the effect of QNF, the paring gaps obtained within the FTBCS1 at
different values $M$ of angular momentum decrease monotonously as $T$
increases, and do not collapse even at hight $T$ in the schematic model as
well as realistic nuclei. The effect of thermally assisted pairing correlation
is seen in all the cases, but in such a way that the pairing gap reappears at
a given $T_{1}>$ 0 and remains finite at $T>T_{1}$, in qualitative agreement
with the canonical results of Ref. FrauendorfPRB68 .
3\. The backbending of the moment of inertia is found in the schematic model
and in 22Ne in the low temperature region, whereas it is washed out with
increasing temperature. This effect does not occur in 20O and 44Ca, in
consistent with existing experimental data and results of other theoretical
approaches.
4\. The effect caused by the corrections due to the dynamic coupling to SCQRPA
vibrations on the pairing gaps, total energies, and heat capacities is found
to be significant in the region around the critical temperature $T_{\rm c}$ of
the SN phase transition and/or at large angular momentum $M$ (or angular
velocity $\gamma$). It is larger in lighter systems. As the result, all the
signatures of the sharp SN phase transition are smoothed out in both schematic
model and realistic nuclei. The SCQRPA corrections also significantly enhance
the reappearance of the thermal gap at finite angular momentum. On the other
hand, the effect caused by the corrections due to PNP is important only at
temperatures below $T_{\rm c}$, and at quite low angular momentum. In
particular, it makes backbending less pronoucned at $T=$ 0.
Still the fluctuations due to violation of angular-momentum conservation are
not implemented in the present extension. We hope that further studies in this
direction will shed light on this issue.
###### Acknowledgements.
The authors thank L.G. Moretto (Berkeley) for suggestions, which led to the
present study. Fruitful discussions with S. Frauendorf (Notre Dame), and P.
Schuck (Orsay) are acknowledged. NQH is a RIKEN Asian Program Associate. The
numerical calculations were carried out using the FORTRAN IMSL Library by
Visual Numerics on the RIKEN Super Combined Cluster (RSCC) system.
## Appendix A On the comparison with canonical results
Figure 9: (Color online) (a) Canonical moment of inertia vs $\gamma^{2}$; (b):
Absolute values $|\langle{\cal E}\rangle_{\rm C}-{\cal E}_{\rm m.f.}|$ (solid
line) and $|{\cal E}_{\rm unc.}|$ (dotted line) vs $\gamma$; (c):
$[\Delta_{\rm C}^{(1)}]^{2}$ vs $\gamma$ for the schematic model with $N=$ 10
and $G=$ 0.5 MeV at $T=$ 0\.
It has been shown in Sec. III.2 that the FTBCS1 (FTBCS1+SCQRPA) produces
results in qualitative agreement with the canonical ones of Ref.
FrauendorfPRB68 , in particular, the reappearance of the thermal gap at
$M\neq$ 0\. However, it is important to make clear the difference between the
predictions by BCS-based approaches and the canonical results. As a matter of
fact, the $z$-projection $M$ of the total angular momentum within the FTBCS
(FTBCS1) approach is temperature-independent. At $T$ varies, by solving the
FTBCS (FTBCS1) equations, the angular velocity $\gamma$ is defined as a
Lagrangian multiplier so that $M$, being the thermal average of the total
angular momentum within the grand canonical ensemble (12), remains unchanged.
In this way, within the FTBCS (FTBCS1), the angular velocity $\gamma$ varies
with $T$, whereas $M$ does not. Similar to that for choosing the chemical
potential $\lambda$ to preserve the (grand-canonical ensemble) average
particle-number $N$, this contraint is physically reasonable when the total
angular momentum is conserved as in the noncollective rotation of spherical
systems or rotation of axially symmetric systems about the symmetry axis, as
has been discussed in Sec. II.1.
On the contrary, the canonical results in Ref. FrauendorfPRB68 are obtained
by embedding the eigenvalues $E_{\nu,i}(\gamma)=E_{\nu}-\gamma M_{\nu,i}$ in
the canonical ensemble with the partition function
$Z(\beta,\gamma)=\sum_{\nu,i}e^{-\beta E_{\nu,i}(\gamma)}~{}.$ (49)
Here $E_{\nu}$ denote the eigenvalues of the $\nu$th state with seniority
$\nu$ at $\gamma=$ 0, whereas $M_{\nu,i}$ are the z-projections of angular
momenta of $\nu$ nucleons. While the eigenvalues $E_{\nu}$ are obtained by
separately diagonalizing the pairing Hamiltonian $H_{P}$ in Eq. (2), the
rotational part $\Phi_{\nu}=\sum_{i}\exp(\beta\gamma M_{\nu,i})$ of the
partition function $Z(\beta,\gamma)$ is calculated following Ref. Kuzmenko .
The resulting canonical average value $\langle M(\beta,\gamma)\rangle_{\rm
C}=\beta Z(\beta,\gamma)^{-1}\partial{Z(\beta,\gamma)}/\partial{\gamma}$ of
angular momentum, therefore, varies with $T$. On the other hand, the angular
velocity $\gamma$ just plays the role of an independent parameter, therefore,
does not depend on $T$. By the same reason, each canonical average value
$\langle M(\beta,\gamma)\rangle_{\rm C}$ corresponds to a single value of
$\gamma$, i.e. the canonical moment of inertia ${\cal J}_{\rm C}$ undergoes no
backbending, as shown in Fig. 9 (a).
Because of this principal difference, a quantitative comparison between the
FTBCS (FTBCS1) results, and the canonical ones as functions of $M$ (or
$\gamma$) at $T\neq$ 0 unfortunately turns out to be impossible. To establish
a meaningful correspondence, one needs to know the exact eigenvalues of the
ground state as well as all excited states of the pairing problem described by
Hamiltonian (1) so that, by embedding the eigenvalues in the grand canonical
ensemble, $\gamma$ becomes a function of $T$ in such a way to keep $\langle
M(\beta,\gamma)\rangle_{\rm C}$ always equal to $M$. To our knowledge, this
problem still remains unsolved. One may also try to estimate the results
within the microcanonical ensemble. However, here one faces a problem of
extracting the nuclear temperature, which is rather ambiguous at low level
density (small $N$) within the schematic model under consideration Sumaryada ;
DangHung , whereas the extension of exact solution of the pairing problem to
$T\neq$ 0 is unpracticable at $N\geq$ 16.
Therefore, in the present paper, we can only compare the predictions of our
approach with the canonical results as functions of temperature $T$ at $M=$ 0,
or as functions of $M$ (or angular velocity $\gamma$) at $T=$ 0\. For this
purpose, and given several definitions of the “effective” gaps existing in
literature, we choose to employ in the present paper two definitions of the
canonical gaps, $\Delta_{\rm C}^{(1)}$ and $\Delta_{\rm C}^{(2)}$. They should
be understood as effective ones since a gap per se, which is a mean-field
concept, does not exist in the exact solutions of the pairing problem.
The canonical gap $\Delta_{\rm C}^{(1)}$ is defined from the pairing energy
${\cal E}_{\rm pair}$ of the system as
$[\Delta_{C}^{(1)}]^{2}={-G{\cal E}_{\rm pair}}~{},\hskip 8.53581pt{\cal
E}_{\rm pair}\equiv\langle{\cal E}\rangle_{\rm C}-{\cal E}_{\rm m.f.}-{\cal
E}_{\rm unc.}~{},\hskip 8.53581pt{\cal E}_{\rm m.f.}\equiv
2\sum_{k}\epsilon_{k}f_{k}~{},\hskip 8.53581pt{\cal E}_{\rm
unc.}\equiv-{G}\sum_{k}f_{k}^{2}~{}.$ (50)
Here $\langle{\cal E}\rangle_{\rm C}$ is the total energy within the canonical
ensemble with the partition function $Z(\beta,\gamma)$ given by Eq. (49) of a
system rotating at angular velocity $\gamma$, or with the partition function
$Z(\beta,0)$ at $M=$ 0\. The term ${\cal E}_{\rm m.f.}$ denotes the energy of
the single-particle motion described by the first term at the right-hand side
of the pairing Hamiltonian $H_{P}$ in Eq. (2). Functions $f_{k}$ are
occupation numbers of $k$th orbitals within the canonical ensemble. The energy
${\cal E}_{\rm m.f.}$ becomes that of the mean-field once the single-particle
occupation numbers $f_{k}$ are replaced with those describing the Fermi-Dirac
distributions of independent particles. The energy ${\cal E}_{\rm unc.}$ comes
from the uncorrelated single-particle configurations caused by the pairing
interaction in Hamiltonian (2). Therefore, by subtracting the term ${\cal
E}_{\rm m.f.}+{\cal E}_{\rm unc.}$ from the total energy $\langle{\cal
E}\rangle_{\rm C}$, one obtains the result that corresponds to the energy due
to pure pairing correlations. The definition (50) is very similar to that
given in Ref. Delft . It is, however, different from the canonical gap
$\Delta_{\rm C}^{(2)}$, which is used in Refs. FrauendorfPRB68 . The latter is
defined as
$[\Delta_{\rm C}^{(2)}]^{2}={-G\big{[}\langle{\cal E}\rangle_{\rm
C}-\langle{\cal E}(G=0)\rangle_{\rm C}\big{]}}~{},$ (51)
where $\langle{\cal E}(G=0)\rangle_{C}$ is the total canonical energy
$\langle{\cal E}\rangle_{\rm C}$ at $G=$ 0.
The canonical gaps $\Delta_{\rm C}^{(1)}$ and $\Delta_{\rm C}^{(2)}$ are shown
in Figs. 6 (a1), 6 (a2), and 6 (a3) as functions of temperature $T$ (at $M=$
0), angular momentum $M$ (at $T=$ 0), and angular velocity $\gamma$ (at $T=$
0), respectively. It is seen from these figures that the difference between
the two canonical gaps $\Delta_{\rm C}^{(1)}$ and $\Delta_{\rm C}^{(2)}$ is
rather significant at large $T$ for $M=$ 0, and at large $M$ (or $\gamma$) for
$T=$ 0\. The reason is rather simple since the definition (50) of $\Delta_{\rm
C}^{(1)}$ is rather similar to that for the BCS gap. As a matter of fact, by
replacing the canonical single-particle occupation numbers $f_{k}$ with the
Bogoliubov’s coefficients $v_{k}^{2}$, and the total energy $\langle{\cal
E}\rangle_{\rm C}$ with that obtained within the BCS theory, the gap
$\Delta_{\rm C}^{(1)}$ reduces to the usual BCS gap. Meanwhile, by doing so
with $\Delta_{\rm C}^{(2)}$, the energy $\langle{\cal E}(G=0)\rangle_{\rm C}$
just reduces to the Hartree-Fock energy, leaving the uncorrelated energy
${\cal E}_{\rm unc.}$ out of the definition. Consequently, as functions of
$T$, the gaps predicted by the BCS-based approaches under consideration agree
better with the canonical gap $\Delta_{\rm C}^{(1)}$ than with $\Delta_{\rm
C}^{(2)}$ [Fig. 6 (a1)].
As functions of angular velocity $\gamma$, both the squared values (50) and
(51) of the canonical gaps undergo a stepwise decrease with increasing
$\gamma$. The step occurs whenever the state of the lowest energy changes from
$\nu-2$ to $\nu$, causing a stepwise increase of $\langle
M(\beta,\gamma)\rangle_{\rm C}$ FrauendorfPRB68 . Therefore, for $N=$ 10, the
pairs are gradually broken in 5 steps with a corresponding stepwise increase
of seniority $\nu$ from 0 to 10 by two units in each step. However, Fig. 9 (b)
shows that the absolute value of the uncorrelated energy ${\cal E}_{\rm
unc.}$, which enters in the definition (50) of the gap $\Delta_{\rm C}^{(1)}$,
becomes larger than that of the difference ${\cal E}_{\rm C}-{\cal E}_{\rm
m.f.}$ already at the second step, leading to $[\Delta_{\rm C}^{(1)}]^{2}<$ 0
[Fig. 9 (c)], i.e. an imaginary value for $\Delta_{\rm C}^{(1)}$. As the
result, instead of collapsing as $\Delta_{\rm C}^{(2)}$ in 5 steps at a rather
large value of $M$ (or $\gamma$), the canonical gap $\Delta_{\rm C}^{(1)}$
collapses in two steps at a value of $M$ (or $\gamma$) much closer to $M_{\rm
c}$ (or $\gamma_{\rm c}$) for the BCS gap [Figs. 6 (a2) and 6 (a3)]. Once
again, this makes the gaps predicted by the BCS-based approaches as functions
of $M$ (or $\gamma$) agree better with the canonical gap $\Delta_{\rm
C}^{(1)}$, rather than with $\Delta_{\rm C}^{(2)}$ [Figs. 6 (a2) and 6 (a3)].
## References
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|
arxiv-papers
| 2008-11-27T03:18:18 |
2024-09-04T02:48:58.999588
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "N. Quang Hung and N. Dinh Dang",
"submitter": "Nguyen Quang Hung",
"url": "https://arxiv.org/abs/0811.4459"
}
|
0811.4460
|
# Transgression and twisted anomaly cancellation formulas on odd dimensional
manifolds
Yong Wang
###### Abstract
We compute the transgressed forms of some modularly invariant characteristic
forms, which are related to the twisted elliptic genera. We study the
modularity properties of these secondary characteristic forms and relations
among them. We also get some twisted anomaly cancellation formulas on some odd
dimensional manifolds.
Subj. Class.: Differential geometry; Algebraic topology
MSC: 58C20; 57R20; 53C80
Keywords: Transgression; elliptic genera; cancellation formulas
## 1 Introduction
In 1983, the physicists Alvarez-Gaumé and Witten [AW] discovered the
”miraculous cancellation” formula for gravitational anomaly which reveals a
beautiful relation between the top components of the Hirzebruch
$\widehat{L}$-form and $\widehat{A}$-form of a $12$-dimensional smooth
Riemannian manifold. Kefeng Liu [Li] established higher dimensional
”miraculous cancellation” formulas for $(8k+4)$-dimensional Riemannian
manifolds by developing modular invariance properties of characteristic forms.
These formulas could be used to deduce some divisibility results. In [HZ1],
[HZ2], for each $(8k+4)$-dimensional smooth Riemannian manifold, a more
general cancellation formula that involves a complex line bundle was
established. This formula was applied to ${\rm spin}^{c}$ manifolds, then an
analytic Ochanine congruence formula was derived. For $(8k+2)$ and
$(8k+6)$-dimensional smooth Riemannian manifolds, F. Han and X. Huang [HH]
obtained some cancellation formulas.
On the other hand, motivated by the Chern-Simons theory, in [CH], Qingtao Chen
and Fei Han computed the transgressed forms of some modularly invariant
characteristic forms, which are related to the elliptic genera. They studied
the modularity properties of these secondary characteristic forms and
relations among them. They also got an anomaly cancellation formula for
$11$-dimensional manifold. Thus a nature question is to get some twisted
modular forms by transgression and some twisted anomaly cancellation formulas
for odd dimensional manifolds. In this paper, we compute the transgressed
forms of some modularly invariant characteristic forms, which are related to
the ”twisted” elliptic genera. We study the modularity properties of these
secondary characteristic forms and relations among them. We also get some
twisted anomaly cancellation formulas on some odd dimensional manifolds. We
hope that these new geometric invariants of connections with modularity
properties obtained here could be applied somewhere.
This paper is organized as follows: In Section 2, we review some knowledge on
characteristic forms and modular forms that we are going to use. In Section 3,
for $(4k-1)$ dimensional manifolds, we apply the Chern-Simons transgression to
characteristic forms with modularity properties which are related to the
”twisted” elliptic genera and obtain some interesting secondary characteristic
forms with modularity properties. We also get two twisted cancellation
formulas for $11$-dimensional manifolds. In Section 4, for $(4k+1)$
dimensional manifolds, by transgression, we again obtain some interesting
secondary characteristic forms with modularity properties. As a corollary, we
get a twisted cancellation formula for $9$-dimensional manifolds.
## 2 characteristic forms and modular forms
The purpose of this section is to review the necessary knowledge on
characteristic forms and modular forms that we are going to use.
2.1 characteristic forms. Let $M$ be a Riemannian manifold. Let $\nabla^{TM}$
be the associated Levi-Civita connection on $TM$ and
$R^{TM}=(\nabla^{TM})^{2}$ be the curvature of $\nabla^{TM}$. Let
$\widehat{A}(TM,\nabla^{TM})$ and $\widehat{L}(TM,\nabla^{TM})$ be the
Hirzebruch characteristic forms defined respectively by (cf. [Z])
$\widehat{A}(TM,\nabla^{TM})={\rm
det}^{\frac{1}{2}}\left(\frac{\frac{\sqrt{-1}}{4\pi}R^{TM}}{{\rm
sinh}(\frac{\sqrt{-1}}{4\pi}R^{TM})}\right),$
$\widehat{L}(TM,\nabla^{TM})={\rm
det}^{\frac{1}{2}}\left(\frac{\frac{\sqrt{-1}}{2\pi}R^{TM}}{{\rm
tanh}(\frac{\sqrt{-1}}{4\pi}R^{TM})}\right).$ $None$
Let $E$, $F$ be two Hermitian vector bundles over $M$ carrying Hermitian
connection $\nabla^{E},\nabla^{F}$ respectively. Let $R^{E}=(\nabla^{E})^{2}$
(resp. $R^{F}=(\nabla^{F})^{2}$) be the curvature of $\nabla^{E}$ (resp.
$\nabla^{F}$). If we set the formal difference $G=E-F$, then $G$ carries an
induced Hermitian connection $\nabla^{G}$ in an obvious sense. We define the
associated Chern character form as
${\rm ch}(G,\nabla^{G})={\rm tr}\left[{\rm
exp}(\frac{\sqrt{-1}}{2\pi}R^{E})\right]-{\rm tr}\left[{\rm
exp}(\frac{\sqrt{-1}}{2\pi}R^{F})\right].$ $None$
For any complex number $t$, let
$\wedge_{t}(E)={\bf C}|_{M}+tE+t^{2}\wedge^{2}(E)+\cdots,~{}S_{t}(E)={\bf
C}|_{M}+tE+t^{2}S^{2}(E)+\cdots$
denote respectively the total exterior and symmetric powers of $E$, which live
in $K(M)[[t]].$ The following relations between these operations hold,
$S_{t}(E)=\frac{1}{\wedge_{-t}(E)},~{}\wedge_{t}(E-F)=\frac{\wedge_{t}(E)}{\wedge_{t}(F)}.$
$None$
Moreover, if $\\{\omega_{i}\\},\\{\omega_{j}^{\prime}\\}$ are formal Chern
roots for Hermitian vector bundles $E,F$ respectively, then
${\rm ch}(\wedge_{t}(E))=\prod_{i}(1+e^{\omega_{i}}t).$ $None$
Then we have the following formulas for Chern character forms,
${\rm ch}(S_{t}(E))=\frac{1}{\prod_{i}(1-e^{\omega_{i}}t)},~{}{\rm
ch}(\wedge_{t}(E-F))=\frac{\prod_{i}(1+e^{\omega_{i}}t)}{\prod_{j}(1+e^{\omega^{\prime}_{j}}t)}.$
$None$
If $W$ is a real Euclidean vector bundle over $M$ carrying a Euclidean
connection $\nabla^{W}$, then its complexification $W_{\bf C}=W\otimes{\bf C}$
is a complex vector bundle over $M$ carrying a canonical induced Hermitian
metric from that of $W$, as well as a Hermitian connection $\nabla^{W_{\bf
C}}$ induced from $\nabla^{W}$. If $E$ is a vector bundle (complex or real)
over $M$, set $\widetilde{E}=E-{\rm dim}E$ in $K(M)$ or $KO(M)$.
2.2 Some properties about the Jacobi theta functions and modular forms
We first recall the four Jacobi theta functions are defined as follows( cf.
[Ch]):
$\theta(v,\tau)=2q^{\frac{1}{8}}{\rm sin}(\pi
v)\prod_{j=1}^{\infty}[(1-q^{j})(1-e^{2\pi\sqrt{-1}v}q^{j})(1-e^{-2\pi\sqrt{-1}v}q^{j})],$
$None$ $\theta_{1}(v,\tau)=2q^{\frac{1}{8}}{\rm cos}(\pi
v)\prod_{j=1}^{\infty}[(1-q^{j})(1+e^{2\pi\sqrt{-1}v}q^{j})(1+e^{-2\pi\sqrt{-1}v}q^{j})],$
$None$
$\theta_{2}(v,\tau)=\prod_{j=1}^{\infty}[(1-q^{j})(1-e^{2\pi\sqrt{-1}v}q^{j-\frac{1}{2}})(1-e^{-2\pi\sqrt{-1}v}q^{j-\frac{1}{2}})],$
$None$
$\theta_{3}(v,\tau)=\prod_{j=1}^{\infty}[(1-q^{j})(1+e^{2\pi\sqrt{-1}v}q^{j-\frac{1}{2}})(1+e^{-2\pi\sqrt{-1}v}q^{j-\frac{1}{2}})],$
$None$
where $q=e^{2\pi\sqrt{-1}\tau}$ with $\tau\in\textbf{H}$, the upper half
complex plane. Let
$\theta^{\prime}(0,\tau)=\frac{\partial\theta(v,\tau)}{\partial v}|_{v=0}.$
$None$
Then the following Jacobi identity (cf. [Ch]) holds,
$\theta^{\prime}(0,\tau)=\pi\theta_{1}(0,\tau)\theta_{2}(0,\tau)\theta_{3}(0,\tau).$
$None$
Denote $SL_{2}({\bf Z})=\left\\{\left(\begin{array}[]{cc}\ a&b\\\
c&d\end{array}\right)\mid a,b,c,d\in{\bf Z},~{}ad-bc=1\right\\}$ the modular
group. Let $S=\left(\begin{array}[]{cc}\ 0&-1\\\
1&0\end{array}\right),~{}T=\left(\begin{array}[]{cc}\ 1&1\\\
0&1\end{array}\right)$ be the two generators of $SL_{2}(\bf{Z})$. They act on
H by $S\tau=-\frac{1}{\tau},~{}T\tau=\tau+1$. One has the following
transformation laws of theta functions under the actions of $S$ and $T$ (cf.
[Ch]):
$\theta(v,\tau+1)=e^{\frac{\pi\sqrt{-1}}{4}}\theta(v,\tau),~{}~{}\theta(v,-\frac{1}{\tau})=\frac{1}{\sqrt{-1}}\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}e^{\pi\sqrt{-1}\tau
v^{2}}\theta(\tau v,\tau);$ $None$
$\theta_{1}(v,\tau+1)=e^{\frac{\pi\sqrt{-1}}{4}}\theta_{1}(v,\tau),~{}~{}\theta_{1}(v,-\frac{1}{\tau})=\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}e^{\pi\sqrt{-1}\tau
v^{2}}\theta_{2}(\tau v,\tau);$ $None$
$\theta_{2}(v,\tau+1)=\theta_{3}(v,\tau),~{}~{}\theta_{2}(v,-\frac{1}{\tau})=\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}e^{\pi\sqrt{-1}\tau
v^{2}}\theta_{1}(\tau v,\tau);$ $None$
$\theta_{3}(v,\tau+1)=\theta_{2}(v,\tau),~{}~{}\theta_{3}(v,-\frac{1}{\tau})=\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}e^{\pi\sqrt{-1}\tau
v^{2}}\theta_{3}(\tau v,\tau).$ $None$
Differentiating the above transformation formulas, we get that
$\theta^{\prime}(v,\tau+1)=e^{\frac{\pi\sqrt{-1}}{4}}\theta^{\prime}(v,\tau),$
$\theta^{\prime}(v,-\frac{1}{\tau})=\frac{1}{\sqrt{-1}}\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}e^{\pi\sqrt{-1}\tau
v^{2}}(2\pi\sqrt{-1}\tau v\theta(\tau v,\tau)+\tau\theta^{\prime}(\tau
v,\tau));$
$\theta^{\prime}_{1}(v,\tau+1)=e^{\frac{\pi\sqrt{-1}}{4}}\theta_{1}^{\prime}(v,\tau),$
$\theta^{\prime}_{1}(v,-\frac{1}{\tau})=\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}e^{\pi\sqrt{-1}\tau
v^{2}}(2\pi\sqrt{-1}\tau v\theta_{2}(\tau v,\tau)+\tau\theta^{\prime}_{2}(\tau
v,\tau));$ $\theta^{\prime}_{2}(v,\tau+1)=\theta_{3}^{\prime}(v,\tau),$
$\theta^{\prime}_{2}(v,-\frac{1}{\tau})=\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}e^{\pi\sqrt{-1}\tau
v^{2}}(2\pi\sqrt{-1}\tau v\theta_{1}(\tau v,\tau)+\tau\theta^{\prime}_{1}(\tau
v,\tau));$ $\theta^{\prime}_{3}(v,\tau+1)=\theta_{2}^{\prime}(v,\tau),$
$\theta^{\prime}_{3}(v,-\frac{1}{\tau})=\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}e^{\pi\sqrt{-1}\tau
v^{2}}(2\pi\sqrt{-1}\tau v\theta_{3}(\tau v,\tau)+\tau\theta^{\prime}_{3}(\tau
v,\tau))$ $None$
Therefore
$\theta^{\prime}(0,-\frac{1}{\tau})=\frac{1}{\sqrt{-1}}\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}\tau\theta^{\prime}(0,\tau).$
$None$
Definition 2.1 A modular form over $\Gamma$, a subgroup of $SL_{2}({\bf Z})$,
is a holomorphic function $f(\tau)$ on H such that
$f(g\tau):=f\left(\frac{a\tau+b}{c\tau+d}\right)=\chi(g)(c\tau+d)^{k}f(\tau),~{}~{}\forall
g=\left(\begin{array}[]{cc}\ a&b\\\ c&d\end{array}\right)\in\Gamma,$ $None$
where $\chi:\Gamma\rightarrow{\bf C}^{\star}$ is a character of $\Gamma$. $k$
is called the weight of $f$.
Let
$\Gamma_{0}(2)=\left\\{\left(\begin{array}[]{cc}\ a&b\\\
c&d\end{array}\right)\in SL_{2}({\bf Z})\mid c\equiv 0~{}({\rm
mod}~{}2)\right\\},$ $\Gamma^{0}(2)=\left\\{\left(\begin{array}[]{cc}\ a&b\\\
c&d\end{array}\right)\in SL_{2}({\bf Z})\mid b\equiv 0~{}({\rm
mod}~{}2)\right\\},$ $\Gamma_{\theta}=\left\\{\left(\begin{array}[]{cc}\
a&b\\\ c&d\end{array}\right)\in SL_{2}({\bf Z})\mid\left(\begin{array}[]{cc}\
a&b\\\ c&d\end{array}\right)\equiv\left(\begin{array}[]{cc}\ 1&0\\\
0&1\end{array}\right){\rm or}\left(\begin{array}[]{cc}\ 0&1\\\
1&0\end{array}\right)~{}({\rm mod}~{}2)\right\\}$
be the three modular subgroups of $SL_{2}({\bf Z})$. It is known that the
generators of $\Gamma_{0}(2)$ are $T,~{}ST^{2}ST$, the generators of
$\Gamma^{0}(2)$ are $STS,~{}T^{2}STS$ and the generators of $\Gamma_{\theta}$
are $S,~{}T^{2}$ (cf.[Ch]).
If $\Gamma$ is a modular subgroup, let ${\mathcal{M}}_{{\bf R}}(\Gamma)$
denote the ring of modular forms over $\Gamma$ with real Fourier coefficients.
Writing $\theta_{j}=\theta_{j}(0,\tau),~{}1\leq j\leq 3,$ we introduce six
explicit modular forms (cf. [Li]),
$\delta_{1}(\tau)=\frac{1}{8}(\theta_{2}^{4}+\theta_{3}^{4}),~{}~{}\varepsilon_{1}(\tau)=\frac{1}{16}\theta_{2}^{4}\theta_{3}^{4},$
$\delta_{2}(\tau)=-\frac{1}{8}(\theta_{1}^{4}+\theta_{3}^{4}),~{}~{}\varepsilon_{2}(\tau)=\frac{1}{16}\theta_{1}^{4}\theta_{3}^{4},$
$\delta_{3}(\tau)=\frac{1}{8}(\theta_{1}^{4}-\theta_{2}^{4}),~{}~{}\varepsilon_{3}(\tau)=-\frac{1}{16}\theta_{1}^{4}\theta_{2}^{4}.$
They have the following Fourier expansions in $q^{\frac{1}{2}}$:
$\delta_{1}(\tau)=\frac{1}{4}+6q+\cdots,~{}~{}\varepsilon_{1}(\tau)=\frac{1}{16}-q+\cdots,$
$\delta_{2}(\tau)=-\frac{1}{8}-3q^{\frac{1}{2}}+\cdots,~{}~{}\varepsilon_{2}(\tau)=q^{\frac{1}{2}}+\cdots,$
$\delta_{3}(\tau)=-\frac{1}{8}+3q^{\frac{1}{2}}+\cdots,~{}~{}\varepsilon_{3}(\tau)=-q^{\frac{1}{2}}+\cdots,$
where the $"\cdots"$ terms are the higher degree terms, all of which have
integral coefficients. They also satisfy the transformation laws,
$\delta_{2}(-\frac{1}{\tau})=\tau^{2}\delta_{1}(\tau),~{}~{}~{}~{}~{}~{}\varepsilon_{2}(-\frac{1}{\tau})=\tau^{4}\varepsilon_{1}(\tau),$
$None$
$\delta_{2}(\tau+1)=\delta_{3}(\tau),~{}~{}~{}~{}~{}~{}\varepsilon_{2}(\tau+1)=\varepsilon_{3}(\tau).$
$None$
Lemma 2.2 ([Li]) $\delta_{1}(\tau)$ (resp. $\varepsilon_{1}(\tau)$) is a
modular form of weight $2$ (resp. $4$) over $\Gamma_{0}(2)$,
$\delta_{2}(\tau)$ (resp. $\varepsilon_{2}(\tau)$) is a modular form of weight
$2$ (resp. $4$) over $\Gamma^{0}(2)$, while $\delta_{3}(\tau)$ (resp.
$\varepsilon_{3}(\tau)$) is a modular form of weight $2$ (resp. $4$) over
$\Gamma_{\theta}(2)$ and moreover ${\mathcal{M}}_{{\bf R}}(\Gamma^{0}(2))={\bf
R}[\delta_{2}(\tau),\varepsilon_{2}(\tau)]$.
## 3 Transgressed forms and modularities on $4k-1$ dimensional manifolds
Let $M$ be a $4k-1$ dimensional Riemannian manifold and $\xi$ be a rank two
real oriented Euclidean vector bundle over $M$ carrying with a Euclidean
connection $\nabla^{\xi}$. Set
$\Theta_{1}(T_{C}M,\xi_{C})=\bigotimes_{n=1}^{\infty}S_{q^{n}}(\widetilde{T_{C}M})\otimes\bigotimes_{m=1}^{\infty}\wedge_{q^{m}}(\widetilde{T_{C}M}-2\widetilde{\xi_{C}})\otimes\bigotimes_{r=1}^{\infty}\wedge_{q^{r-\frac{1}{2}}}(\widetilde{\xi_{C}})\otimes\bigotimes_{s=1}^{\infty}\wedge_{-q^{s-\frac{1}{2}}}(\widetilde{\xi_{C}}),$
$\Theta_{2}(T_{C}M,\xi_{C})=\bigotimes_{n=1}^{\infty}S_{q^{n}}(\widetilde{T_{C}M})\otimes\bigotimes_{m=1}^{\infty}\wedge_{-q^{m-\frac{1}{2}}}(\widetilde{T_{C}M}-2\widetilde{\xi_{C}})\otimes\bigotimes_{r=1}^{\infty}\wedge_{q^{r-\frac{1}{2}}}(\widetilde{\xi_{C}})\otimes\bigotimes_{s=1}^{\infty}\wedge_{q^{s}}(\widetilde{\xi_{C}}),$
$\Theta_{3}(T_{C}M,\xi_{C})=\bigotimes_{n=1}^{\infty}S_{q^{n}}(\widetilde{T_{C}M})\otimes\bigotimes_{m=1}^{\infty}\wedge_{q^{m-\frac{1}{2}}}(\widetilde{T_{C}M}-2\widetilde{\xi_{C}})\otimes\bigotimes_{r=1}^{\infty}\wedge_{q^{r}}(\widetilde{\xi_{C}})\otimes\bigotimes_{s=1}^{\infty}\wedge_{-q^{s-\frac{1}{2}}}(\widetilde{\xi_{C}}).$
$None$
Let $c=e(\xi,\nabla^{\xi})$ be the Euler form of $\xi$ canonically associated
to $\nabla^{\xi}$. Set
$\Phi_{L}(\nabla^{TM},\nabla^{\xi},\tau)=\frac{\widehat{L}(TM,\nabla^{TM})}{{\rm
cosh}^{2}(\frac{c}{2})}{\rm
ch}(\Theta_{1}(T_{C}M,\xi_{C}),\nabla^{\Theta_{1}(T_{C}M,\xi_{C})}),$
$\Phi_{W}(\nabla^{TM},\nabla^{\xi},\tau)={\widehat{A}(TM,\nabla^{TM})}{\rm
cosh}(\frac{c}{2}){\rm
ch}(\Theta_{2}(T_{C}M,\xi_{C}),\nabla^{\Theta_{2}(T_{C}M,\xi_{C})}),$
$\Phi_{W}^{\prime}(\nabla^{TM},\nabla^{\xi},\tau)={\widehat{A}(TM,\nabla^{TM})}{\rm
cosh}(\frac{c}{2}){\rm
ch}(\Theta_{3}(T_{C}M,\xi_{C}),\nabla^{\Theta_{3}(T_{C}M,\xi_{C})}).$ $None$
Let $\\{\pm 2\pi\sqrt{-1}x_{j}|~{}1\leq j\leq 2k-1\\}$ and $\\{\pm
2\pi\sqrt{-1}u\\}$ be the Chern roots of $T_{C}M$ and $\xi_{C}$ respectively
and $c=2\pi\sqrt{-1}u.$ Through direct computations, we get (cf. [HZ2])
$\Phi_{L}(\nabla^{TM},\nabla^{\xi},\tau)=\sqrt{2}^{4k-1}\left\\{\left(\prod_{j=1}^{2k-1}x_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(x_{j},\tau)}\frac{\theta_{1}(x_{j},\tau)}{\theta_{1}(0,\tau)}\right)\frac{\theta_{1}^{2}(0,\tau)}{\theta_{1}^{2}(u,\tau)}\frac{\theta_{3}(u,\tau)}{\theta_{3}(0,\tau)}\frac{\theta_{2}(u,\tau)}{\theta_{2}(0,\tau)}\right\\};$
$None$
$\Phi_{W}(\nabla^{TM},\nabla^{\xi},\tau)=\left(\prod_{j=1}^{2k-1}x_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(x_{j},\tau)}\frac{\theta_{2}(x_{j},\tau)}{\theta_{2}(0,\tau)}\right)\frac{\theta_{2}^{2}(0,\tau)}{\theta_{2}^{2}(u,\tau)}\frac{\theta_{3}(u,\tau)}{\theta_{3}(0,\tau)}\frac{\theta_{1}(u,\tau)}{\theta_{1}(0,\tau)};$
$None$
$\Phi_{W}^{\prime}(\nabla^{TM},\nabla^{\xi},\tau)=\left(\prod_{j=1}^{2k-1}x_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(x_{j},\tau)}\frac{\theta_{3}(x_{j},\tau)}{\theta_{3}(0,\tau)}\right)\frac{\theta_{3}^{2}(0,\tau)}{\theta_{3}^{2}(u,\tau)}\frac{\theta_{1}(u,\tau)}{\theta_{1}(0,\tau)}\frac{\theta_{2}(u,\tau)}{\theta_{2}(0,\tau)}.$
$None$
Consider the following function defined on ${\bf C}\times{\bf H}$,
$f_{\Phi_{L}}(z,\tau)=z\frac{\theta^{\prime}(0,\tau)}{\theta(z,\tau)}\frac{\theta_{1}(z,\tau)}{\theta_{1}(0,\tau)},$
$f_{\Phi_{W}}(z,\tau)=z\frac{\theta^{\prime}(0,\tau)}{\theta(z,\tau)}\frac{\theta_{2}(z,\tau)}{\theta_{2}(0,\tau)},$
$f_{\Phi_{W}^{\prime}}(z,\tau)=z\frac{\theta^{\prime}(0,\tau)}{\theta(z,\tau)}\frac{\theta_{3}(z,\tau)}{\theta_{3}(0,\tau)}.$
Applying the Chern-Weil theory, we can express
$\Phi_{L},~{}\Phi_{W},\Phi_{W}^{\prime}$ as follows:
$\Phi_{L}(\nabla^{TM},\nabla^{\xi},\tau)=\sqrt{2}^{4k-1}{\rm
det}^{\frac{1}{2}}\left(f_{\Phi_{L}}(\frac{R^{TM}}{4\pi^{2}},\tau)\right){\rm
det}^{\frac{1}{2}}\left(\frac{\theta_{1}^{2}(0,\tau)}{\theta_{1}^{2}(\frac{R^{\xi}}{4\pi^{2}},\tau)}\frac{\theta_{3}(\frac{R^{\xi}}{4\pi^{2}},\tau)}{\theta_{3}(0,\tau)}\frac{\theta_{2}(\frac{R^{\xi}}{4\pi^{2}},\tau)}{\theta_{2}(0,\tau)}\right);$
$None$ $\Phi_{W}(\nabla^{TM},\nabla^{\xi},\tau)={\rm
det}^{\frac{1}{2}}\left(f_{\Phi_{W}}(\frac{R^{TM}}{4\pi^{2}},\tau)\right){\rm
det}^{\frac{1}{2}}\left(\frac{\theta_{2}^{2}(0,\tau)}{\theta_{2}^{2}(\frac{R^{\xi}}{4\pi^{2}},\tau)}\frac{\theta_{3}(\frac{R^{\xi}}{4\pi^{2}},\tau)}{\theta_{3}(0,\tau)}\frac{\theta_{1}(\frac{R^{\xi}}{4\pi^{2}},\tau)}{\theta_{1}(0,\tau)}\right);$
$None$ $\Phi_{W}^{\prime}(\nabla^{TM},\nabla^{\xi},\tau)={\rm
det}^{\frac{1}{2}}\left(f_{\Phi_{W}^{\prime}}(\frac{R^{TM}}{4\pi^{2}},\tau)\right){\rm
det}^{\frac{1}{2}}\left(\frac{\theta_{3}^{2}(0,\tau)}{\theta_{3}^{2}(\frac{R^{\xi}}{4\pi^{2}},\tau)}\frac{\theta_{1}(\frac{R^{\xi}}{4\pi^{2}},\tau)}{\theta_{1}(0,\tau)}\frac{\theta_{2}(\frac{R^{\xi}}{4\pi^{2}},\tau)}{\theta_{2}(0,\tau)}\right).$
$None$
Let $E$ be a vector bundle and $f$ be a power series with constant term $1$.
Let $\nabla_{t}^{E}$ be deformed connection given by
$\nabla_{t}^{E}=(1-t)\nabla_{0}^{E}+t\nabla_{1}^{E}$ and
$R^{E}_{t},~{}t\in[0,1],$ denote the curvature of $\nabla_{t}^{E}$.
$f^{\prime}(t)$ is the power series obtained from the derivative of $f(x)$
with respect to $x$. $\omega$ is a closed form. Recall the trivial
modification of Theorem 2.2 in [CH],
Lemma 3.1( [CH])
${\rm det}^{\frac{1}{2}}(f(R^{E}_{1}))\omega-{\rm
det}^{\frac{1}{2}}(f(R^{E}_{0}))\omega=d\int_{0}^{1}\frac{1}{2}{\rm
det}^{\frac{1}{2}}(f(R^{E}_{t}))\omega{\rm
tr}\left[\frac{d\nabla_{t}^{E}}{dt}\frac{f^{\prime}(R^{E}_{t})}{f(R^{E}_{t})}\right]dt.$
$None$
Now we let $E=TM$ and $A=\nabla_{1}^{TM}-\nabla_{0}^{TM}$, then by Lemma 3.1,
we have
$\Phi_{L}(\nabla_{1}^{TM},\nabla^{\xi},\tau)-\Phi_{L}(\nabla_{0}^{TM},\nabla^{\xi},\tau)$
$=\frac{1}{8\pi^{2}}d\int_{0}^{1}\Phi_{L}(\nabla_{t}^{TM},\nabla^{\xi},\tau){\rm
tr}\left[A\left(\frac{1}{\frac{R_{t}^{TM}}{4\pi^{2}}}-\frac{\theta^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}+\frac{\theta_{1}^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta_{1}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}\right)\right]dt.$
$None$
We define
$CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)$
$:=\frac{\sqrt{2}}{8\pi^{2}}\int_{0}^{1}\Phi_{L}(\nabla_{t}^{TM},\nabla^{\xi},\tau){\rm
tr}\left[A\left(\frac{1}{\frac{R_{t}^{TM}}{4\pi^{2}}}-\frac{\theta^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}+\frac{\theta_{1}^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta_{1}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}\right)\right]dt.$
$None$
which is in $\Omega^{\rm odd}(M,{\bf C})[[q^{\frac{1}{2}}]].$ Since $M$ is
$4k-1$ dimensional,
$\\{CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)}$
represents an element in $H^{4k-1}(M,{\bf C})[[q^{\frac{1}{2}}]]$. Similarly,
we can compute the transgressed forms for $\Phi_{W},~{}\Phi_{W}^{\prime}$
respectively and define
$CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)$
$:=\frac{1}{8\pi^{2}}\int_{0}^{1}\Phi_{W}(\nabla_{t}^{TM},\nabla^{\xi},\tau){\rm
tr}\left[A\left(\frac{1}{\frac{R_{t}^{TM}}{4\pi^{2}}}-\frac{\theta^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}+\frac{\theta_{2}^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta_{2}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}\right)\right]dt;$
$None$
$CS\Phi_{W}^{\prime}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)$
$:=\frac{1}{8\pi^{2}}\int_{0}^{1}\Phi_{W}^{\prime}(\nabla_{t}^{TM},\nabla^{\xi},\tau){\rm
tr}\left[A\left(\frac{1}{\frac{R_{t}^{TM}}{4\pi^{2}}}-\frac{\theta^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}+\frac{\theta_{3}^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta_{3}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}\right)\right]dt,$
$None$
which also lie in $\Omega^{\rm odd}(M,{\bf C})[[q^{\frac{1}{2}}]]$ and their
top components represent elements in $H^{4k-1}(M,{\bf C})[[q^{\frac{1}{2}}]]$.
As pointed in [CH], the equality (3.10) and the modular invariance properties
of $\Phi_{L}(\nabla_{0}^{TM},\nabla^{\xi},\tau)$ and
$\Phi_{L}(\nabla_{1}^{TM},\nabla^{\xi},\tau)$ are not enough to guarantee that
$CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)$ is a modular
form. However we have the following results.
Theorem 3.2 Let $M$ be a $4k-1$ dimensional manifold and
$\nabla_{0}^{TM},~{}\nabla_{1}^{TM}$ be two connections on $TM$ and $\xi$ be a
two dimensional oriented Euclidean real vector bundle with a Euclidean
connection $\nabla^{\xi}$, then we have
1)
$\\{CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma_{0}(2)$;
$\\{CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma^{0}(2);$
$\\{CS\Phi_{W}^{\prime}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma_{\theta}(2).$
2) The following equalities hold,
$\\{CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},-\frac{1}{\tau})\\}^{(4k-1)}=(2\tau)^{2k}\\{CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)},$
$CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau+1)=CS\Phi_{W}^{\prime}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau).$
Proof. By (2.12)-(2.17), we have
$z\frac{\theta^{\prime}(0,-\frac{1}{\tau})}{\theta(z,-\frac{1}{\tau})}\frac{\theta_{1}(z,-\frac{1}{\tau})}{\theta_{1}(0,-\frac{1}{\tau})}=(\tau
z)\frac{\theta^{\prime}(0,{\tau})}{\theta(\tau z,{\tau})}\frac{\theta_{2}(\tau
z,{\tau})}{\theta_{2}(0,{\tau})};$ $None$
$\frac{1}{z}-\frac{\theta^{\prime}(z,-\frac{1}{\tau})}{\theta(z,-\frac{1}{\tau})}+\frac{\theta_{1}^{\prime}(z,-\frac{1}{\tau})}{\theta_{1}(z,-\frac{1}{\tau})}=\tau\left(\frac{1}{\tau
z}-\frac{\theta^{\prime}(\tau z,{\tau})}{\theta(\tau
z,{\tau})}+\frac{\theta_{2}^{\prime}(\tau z,{\tau})}{\theta_{2}(\tau
z,{\tau})}\right);$ $None$
$\frac{\theta_{1}^{2}(0,-\frac{1}{\tau})}{\theta_{1}^{2}(u,-\frac{1}{\tau})}\frac{\theta_{3}(u,-\frac{1}{\tau})}{\theta_{3}(0,-\frac{1}{\tau})}\frac{\theta_{2}(u,-\frac{1}{\tau})}{\theta_{2}(0,-\frac{1}{\tau})}=\frac{\theta_{2}^{2}(0,{\tau})}{\theta_{2}^{2}(\tau
u,{\tau})}\frac{\theta_{3}(\tau
u,{\tau})}{\theta_{3}(0,{\tau})}\frac{\theta_{1}(u\tau,{\tau})}{\theta_{1}(0,{\tau})}.$
$None$
Note that we only take $(4k-1)$-component, so by (3.6)-(3.8),(3.11), (3.12),
(3.14)-(3.16), we can get
$\\{CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},-\frac{1}{\tau})\\}^{(4k-1)}=(2\tau)^{2k}\\{CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)},$
$None$
Similarly we can show that
$CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},{\tau}+1)=CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau),$
$\\{CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},-\frac{1}{\tau})\\}^{(4k-1)}=(\frac{\tau}{2})^{2k}\\{CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)},$
$CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},{\tau}+1)=CS\Phi_{W}^{\prime}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau),$
$\\{CS\Phi_{W}^{\prime}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},-\frac{1}{\tau})\\}^{(4k-1)}=(\tau)^{2k}\\{CS\Phi_{W}^{\prime}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)},$
$CS\Phi_{W}^{\prime}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},{\tau}+1)=CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau).$
$None$
From (3.17) and (3.18), we can get
$\\{CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},{\tau})\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma_{0}(2).$ Similarly we can prove
that
$\\{CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma^{0}(2)$ and
$\\{CS\Phi_{W}^{\prime}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma_{\theta}(2).$ $\Box$
Let $M$ be a compact oriented smooth $3$-dimensional manifold, then our
transgressed forms are same as transgressed forms in the untwisted case which
have been computed in [CH]. From Theorem 3.2, we can imply some twisted
cancellation formulas for odd dimensional manifolds. For example, let $M$ be
$11$ dimensional and $k=3$. We have that
$\\{CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(11)}$
is a modular form of weight $6$ over $\Gamma_{0}(2),$
$\\{CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(11)}$
is a modular form of weight $6$ over $\Gamma^{0}(2)$ and
$\\{CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},-\frac{1}{\tau})\\}^{(11)}=(2\tau)^{6}\\{CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(11)}.$
By Lemma 2.2, we have
$\\{CS\Phi_{W}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(11)}=z_{0}(8\delta_{2})^{3}+z_{1}(8\delta_{2})\varepsilon_{2},$
$None$
and by (2.19) and Theorem 3.2,
$\\{CS\Phi_{L}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(11)}=2^{6}[z_{0}(8\delta_{1})^{3}+z_{1}(8\delta_{1})\varepsilon_{1}].$
$None$
By comparing the $q^{\frac{1}{2}}$-expansion coefficients in (3.19), we get
$z_{0}=-\left\\{\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}_{t}){\rm
cosh}(\frac{c}{2}){\rm tr}\left[A\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{8\pi{\rm
tan}{\frac{R^{TM}_{t}}{4\pi}}}\right)\right]dt\right\\}^{(11)},$ $None$
$z_{1}=\left\\{\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}_{t}){\rm
cosh}(\frac{c}{2})\left({\rm
ch}(T_{C}M,\nabla^{T_{C}M}_{t})-3(e^{c}+e^{-c}-2)\right)\right.$
$\left.\times{\rm tr}\left[A\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{8\pi{\rm
tan}{\frac{R^{TM}_{t}}{4\pi}}}\right)\right]dt\right.+$
$\left.\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}_{t}){\rm cosh}(\frac{c}{2}){\rm
tr}\left[A\left(-\frac{1}{2\pi}{\rm
sin}\frac{R^{TM}_{t}}{4\pi}+61\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{8\pi{\rm
tan}{\frac{R^{TM}_{t}}{4\pi}}}\right)\right)\right]dt\right\\}^{(11)}.$ $None$
Plugging (3.21) and (3.22) into (3.20) and comparing the constant terms of
both sides, we obtain that
$\left\\{\int_{0}^{1}\frac{\sqrt{2}\widehat{L}(TM,\nabla^{TM}_{t})}{{\rm
cosh}^{2}{\frac{c}{2}}}{\rm
tr}\left[A\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{4\pi{\rm
sin}{\frac{R^{TM}_{t}}{2\pi}}}\right)\right]\right\\}^{(11)}=2^{3}(2^{6}z_{0}+z_{1}),$
so we have the following $11$-dimensional analogue of the twisted miraculous
cancellation formula.
Corollary 3.3 The following equality holds
$\left\\{\int_{0}^{1}\frac{\sqrt{2}\widehat{L}(TM,\nabla^{TM}_{t})}{{\rm
cosh}^{2}{\frac{c}{2}}}{\rm
tr}\left[A\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{4\pi{\rm
sin}{\frac{R^{TM}_{t}}{2\pi}}}\right)\right]\right\\}^{(11)}$
$=8\left\\{\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}_{t}){\rm
cosh}(\frac{c}{2})\left({\rm
ch}(T_{C}M,\nabla^{T_{C}M}_{t})-3(e^{c}+e^{-c}-2)\right)\right.$
$\left.\times{\rm tr}\left[A\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{8\pi{\rm
tan}{\frac{R^{TM}_{t}}{4\pi}}}\right)\right]dt\right.+$
$\left.\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}_{t}){\rm cosh}(\frac{c}{2}){\rm
tr}\left[A\left(-\frac{1}{2\pi}{\rm
sin}\frac{R^{TM}_{t}}{2\pi}-3\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{8\pi{\rm
tan}{\frac{R^{TM}_{t}}{4\pi}}}\right)\right)\right]dt\right\\}^{(11)}.$ $None$
Next we consider the transgression of
$\Phi_{L}(\nabla^{TM},\nabla^{\xi},\tau),~{}\Phi_{W}(\nabla^{TM},\nabla^{\xi},\tau)$,
$\Phi_{W}^{\prime}(\nabla^{TM},\nabla^{\xi},\tau)$ about $\nabla^{\xi}$. Let
$\nabla_{1}^{\xi},~{}\nabla_{0}^{\xi}$ be two Euclidean connections on $\xi$
and $B=\nabla_{1}^{\xi}-\nabla_{0}^{\xi}$. By (3.6)-(3.9), we have
$\Phi_{L}(\nabla^{TM},\nabla_{1}^{\xi},\tau)-\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\tau)$
$=\frac{1}{8\pi^{2}}d\int_{0}^{1}\Phi_{L}(\nabla^{TM},\nabla_{t}^{\xi},\tau){\rm
tr}\left[B\left(\frac{\theta^{\prime}_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}+\frac{\theta_{3}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{3}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}-2\frac{\theta_{1}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{1}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}\right)\right]dt.$
$None$
We define
$CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)$
$:=\frac{\sqrt{2}}{8\pi^{2}}\int_{0}^{1}\Phi_{L}(\nabla^{TM},\nabla_{t}^{\xi},\tau){\rm
tr}\left[B\left(\frac{\theta^{\prime}_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}+\frac{\theta_{3}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{3}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}-2\frac{\theta_{1}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{1}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}\right)\right]dt.$
$None$
which is in $\Omega^{\rm odd}(M,{\bf C})[[q^{\frac{1}{2}}]].$ Since $M$ is
$4k-1$ dimensional,
$\\{CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)}$
represents an element in $H^{4k-1}(M,{\bf C})[[q^{\frac{1}{2}}]]$. Similarly,
we can compute the transgressed forms for $\Phi_{W},~{}\Phi_{W}^{\prime}$
respectively and define
$CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)$
$:=\frac{1}{8\pi^{2}}\int_{0}^{1}\Phi_{W}(\nabla^{TM},\nabla_{t}^{\xi},\tau){\rm
tr}\left[B\left(\frac{\theta^{\prime}_{3}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{3}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}+\frac{\theta_{1}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{1}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}-2\frac{\theta_{2}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}\right)\right]dt,$
$None$
$CS\Phi_{W}^{\prime}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)$
$:=\frac{1}{8\pi^{2}}\int_{0}^{1}\Phi_{W}^{\prime}(\nabla^{TM},\nabla_{t}^{\xi},\tau){\rm
tr}\left[B\left(\frac{\theta^{\prime}_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}+\frac{\theta_{1}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{1}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}-2\frac{\theta_{3}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{3}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}\right)\right]dt,$
$None$
which also lie in $\Omega^{\rm odd}(M,{\bf C})[[q^{\frac{1}{2}}]]$ and their
top components represent elements in $H^{4k-1}(M,{\bf C})[[q^{\frac{1}{2}}]]$.
Similarly we have
Theorem 3.4 Let $M$ be a $4k-1$ dimensional manifold and $\nabla^{TM}$ be a
connection on $TM$ and $\xi$ be a two dimensional oriented Euclidean real
vector bundle with two Euclidean connections $\nabla_{1}^{\xi}$,
$\nabla_{0}^{\xi}$, then we have
1)
$\\{CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma_{0}(2)$;
$\\{CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma^{0}(2);$
$\\{CS\Phi_{W}^{\prime}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma_{\theta}(2).$
2) The following equalities hold,
$\\{CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)}=(2\tau)^{2k}\\{CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)},$
$CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)=CS\Phi_{W}^{\prime}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau).$
Proof. By (3.14),(3.16) and
$\frac{\theta_{2}^{\prime}(z,-\frac{1}{\tau})}{\theta_{2}(z,-\frac{1}{\tau})}+\frac{\theta_{3}^{\prime}(z,-\frac{1}{\tau})}{\theta_{3}(z,-\frac{1}{\tau})}-2\frac{\theta_{1}^{\prime}(z,-\frac{1}{\tau})}{\theta_{1}(z,-\frac{1}{\tau})}=\tau\left(\frac{\theta_{1}^{\prime}(\tau
z,{\tau})}{\theta_{1}(\tau z,{\tau})}+\frac{\theta_{3}^{\prime}(\tau
z,{\tau})}{\theta_{3}(\tau z,{\tau})}-2\frac{\theta_{2}^{\prime}(\tau
z,{\tau})}{\theta_{2}(\tau z,{\tau})}\right),$ $None$
we can get
$\\{CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},-\frac{1}{\tau})\\}^{(4k-1)}=(2\tau)^{2k}\\{CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{TM},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)},$
$None$
Similarly we can show that
$CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},{\tau}+1)=CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau),$
$\\{CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},-\frac{1}{\tau})\\}^{(4k-1)}=(\frac{\tau}{2})^{2k}\\{CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)},$
$CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},{\tau}+1)=CS\Phi_{W}^{\prime}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau),$
$\\{CS\Phi_{W}^{\prime}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},-\frac{1}{\tau})\\}^{(4k-1)}=(\tau)^{2k}\\{CS\Phi_{W}^{\prime}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)},$
$CS\Phi_{W}^{\prime}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},{\tau}+1)=CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau).$
$None$
From (3.28) and (3.29), we can get
$\\{CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},{\tau})\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma_{0}(2).$ Similarly we can prove
that
$\\{CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma^{0}(2)$ and
$\\{CS\Phi_{W}^{\prime}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(4k-1)}$
is a modular form of weight $2k$ over $\Gamma_{\theta}(2).$ $\Box$
Let $M$ be a compact oriented smooth $3$-dimensional manifold, we have
$\displaystyle CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)$
$\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{8\pi^{2}}\int_{0}^{1}\Phi_{L}(\nabla^{TM},\nabla_{t}^{\xi},\tau){\rm
tr}\left[B\left(\frac{\theta^{\prime}_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}+\frac{\theta_{3}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{3}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}-2\frac{\theta_{1}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{1}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}\right)\right]dt$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi^{2}}\int_{0}^{1}{\rm
tr}\left[B\left(\frac{\theta^{\prime}_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{2}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}+\frac{\theta_{3}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{3}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}-2\frac{\theta_{1}^{\prime}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}{\theta_{1}(\frac{R_{t}^{\xi}}{4\pi^{2}},\tau)}\right)\right]dt$
$\displaystyle=$ $\displaystyle\frac{1}{8\pi^{4}}\frac{\partial}{\partial
z}\left(\frac{\theta^{\prime}_{2}(z,\tau)}{\theta_{2}(z,\tau)}+\frac{\theta_{3}^{\prime}(z,\tau)}{\theta_{3}(z,\tau)}-2\frac{\theta_{1}^{\prime}(z,\tau)}{\theta_{1}(z,\tau)}\right)|_{z=0}\int_{0}^{1}{\rm
tr}[BR_{t}^{\xi}]dt.$
Since $\frac{\partial}{\partial
z}\left(\frac{\theta^{\prime}_{2}(z,\tau)}{\theta_{2}(z,\tau)}+\frac{\theta_{3}^{\prime}(z,\tau)}{\theta_{3}(z,\tau)}-2\frac{\theta_{1}^{\prime}(z,\tau)}{\theta_{1}(z,\tau)}\right)|_{z=0}$
is a modular form of weight $2$ over $\Gamma_{0}(2),$ then it is a scalar
multiple of $\delta_{1}(\tau)$. Direct computations show
$\frac{\partial}{\partial
z}\left(\frac{\theta^{\prime}_{2}(z,\tau)}{\theta_{2}(z,\tau)}+\frac{\theta_{3}^{\prime}(z,\tau)}{\theta_{3}(z,\tau)}-2\frac{\theta_{1}^{\prime}(z,\tau)}{\theta_{1}(z,\tau)}\right)|_{z=0}=2\pi^{2}+O(q^{\frac{1}{2}}),$
so
$\frac{\partial}{\partial
z}\left(\frac{\theta^{\prime}_{2}(z,\tau)}{\theta_{2}(z,\tau)}+\frac{\theta_{3}^{\prime}(z,\tau)}{\theta_{3}(z,\tau)}-2\frac{\theta_{1}^{\prime}(z,\tau)}{\theta_{1}(z,\tau)}\right)|_{z=0}=8\pi^{2}\delta_{1}(\tau).$
By (4.15) in [CH], we have
$CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)=\frac{1}{2\pi^{2}}\delta_{1}(\tau){\rm
tr}\left[B[\nabla_{0}^{\xi},\nabla_{1}^{\xi}]+\frac{2}{3}B\wedge B\wedge
B\right].$ $None$
Similarly, we obtain that
$CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)=\frac{1}{8\pi^{2}}\delta_{2}(\tau){\rm
tr}\left[B[\nabla_{0}^{\xi},\nabla_{1}^{\xi}]+\frac{2}{3}B\wedge B\wedge
B\right],$ $None$
$CS\Phi^{\prime}_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)=\frac{1}{8\pi^{2}}\delta_{3}(\tau){\rm
tr}\left[B[\nabla_{0}^{\xi},\nabla_{1}^{\xi}]+\frac{2}{3}B\wedge B\wedge
B\right].$ $None$
Let $M$ be $11$ dimensional and $k=3$. We have that
$\\{CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(11)}$
is a modular form of weight $6$ over $\Gamma_{0}(2),$
$\\{CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(11)}$
is a modular form of weight $6$ over $\Gamma^{0}(2)$ and
$\\{CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},-\frac{1}{\tau})\\}^{(11)}=(2\tau)^{6}\\{CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(11)}.$
By Lemma 2.2, we have
$\\{CS\Phi_{W}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(11)}=z_{0}(8\delta_{2})^{3}+z_{1}(8\delta_{2})\varepsilon_{2},$
$None$
and by (2.19) and Theorem 3.4,
$\\{CS\Phi_{L}(\nabla^{TM},\nabla_{0}^{\xi},\nabla_{1}^{\xi},\tau)\\}^{(11)}=2^{6}[z_{0}(8\delta_{1})^{3}+z_{1}(8\delta_{1})\varepsilon_{1}].$
$None$
By comparing the $q^{\frac{1}{2}}$-expansion coefficients in (3.33), we get
$z_{0}=\left\\{\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}){\rm
cos}(\frac{R_{t}^{\xi}}{4\pi}){\rm tr}\left[\frac{B}{8\pi}{\rm
tan}\frac{R_{t}^{\xi}}{4\pi}\right]dt\right\\}^{(11)},$ $None$
$z_{1}=\left\\{\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}){\rm
cos}(\frac{R_{t}^{\xi}}{4\pi})\left(3{\rm
ch}(\xi_{C},\nabla^{\xi_{C}}_{t})-{\rm
ch}(T_{C}M,\nabla^{T_{C}M})+77\right)\right.$ $\left.\times{\rm
tr}\left[\frac{B}{8\pi}{\rm
tan}\frac{R_{t}^{\xi}}{4\pi}\right]dt+\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}){\rm
cos}(\frac{R_{t}^{\xi}}{4\pi}){\rm tr}\left[\frac{3B}{2\pi}{\rm
sin}\frac{R_{t}^{\xi}}{2\pi}\right]dt\right\\}^{(11)}.$ $None$
Plugging (3.35) and (3.36) into (3.34) and comparing the constant terms of
both sides, we obtain that
Corollary 3.5 The following equality holds
$\left\\{\int_{0}^{1}\frac{\widehat{L}(TM,\nabla^{TM})}{{\rm
cos}^{2}(\frac{R_{t}^{\xi}}{4\pi})}{\rm tr}\left[B{\rm
tan}\frac{R_{t}^{\xi}}{4\pi}\right]dt\right\\}^{(11)}$
$=16\sqrt{2}\pi\left\\{\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}){\rm
cos}(\frac{R_{t}^{\xi}}{4\pi})\left(3{\rm
ch}(\xi_{C},\nabla^{\xi_{C}}_{t})-{\rm
ch}(T_{C}M,\nabla^{T_{C}M})+13\right)\right.$ $\left.\times{\rm
tr}\left[\frac{B}{8\pi}{\rm
tan}\frac{R_{t}^{\xi}}{4\pi}\right]dt+\int_{0}^{1}\widehat{A}(TM,\nabla^{TM}){\rm
cos}(\frac{R_{t}^{\xi}}{4\pi}){\rm tr}\left[\frac{3B}{2\pi}{\rm
sin}\frac{R_{t}^{\xi}}{2\pi}\right]dt\right\\}^{(11)}.$ $None$
## 4 Transgressed forms and modularities on $4k+1$ dimensional manifolds
Let $M$ be a $4k+1$ dimensional Riemannian manifold. Set
$\Theta_{1}(T_{C}M+\xi_{C},\xi_{C})=\bigotimes_{n=1}^{\infty}S_{q^{n}}(\widetilde{T_{C}M+\xi_{C}})\otimes\bigotimes_{m=1}^{\infty}\wedge_{q^{m}}(\widetilde{T_{C}M+\xi_{C}}-2\widetilde{\xi_{C}})$
$~{}~{}~{}~{}~{}~{}~{}\otimes\bigotimes_{r=1}^{\infty}\wedge_{q^{r-\frac{1}{2}}}(\widetilde{\xi_{C}})\otimes\bigotimes_{s=1}^{\infty}\wedge_{-q^{s-\frac{1}{2}}}(\widetilde{\xi_{C}}),$
$\Theta_{2}(T_{C}M+\xi_{C},\xi_{C})=\bigotimes_{n=1}^{\infty}S_{q^{n}}(\widetilde{T_{C}M+\xi_{C}})\otimes\bigotimes_{m=1}^{\infty}\wedge_{-q^{m-\frac{1}{2}}}(\widetilde{T_{C}M+\xi_{C}}-2\widetilde{\xi_{C}})$
$~{}~{}~{}~{}~{}~{}\otimes\bigotimes_{r=1}^{\infty}\wedge_{q^{r-\frac{1}{2}}}(\widetilde{\xi_{C}})\otimes\bigotimes_{s=1}^{\infty}\wedge_{q^{s}}(\widetilde{\xi_{C}}),$
$\Theta_{3}(T_{C}M+\xi_{C},\xi_{C})=\bigotimes_{n=1}^{\infty}S_{q^{n}}(\widetilde{T_{C}M+\xi_{C}})\otimes\bigotimes_{m=1}^{\infty}\wedge_{q^{m-\frac{1}{2}}}(\widetilde{T_{C}M+\xi_{C}}-2\widetilde{\xi_{C}})$
$~{}~{}~{}~{}~{}~{}\otimes\bigotimes_{r=1}^{\infty}\wedge_{q^{r}}(\widetilde{\xi_{C}})\otimes\bigotimes_{s=1}^{\infty}\wedge_{-q^{s-\frac{1}{2}}}(\widetilde{\xi_{C}}).$
$None$
Define
$\displaystyle\widetilde{\Phi_{L}}(\nabla^{TM},\nabla^{\xi},\tau)$
$\displaystyle=$ $\displaystyle\widehat{L}(TM,\nabla^{TM})\frac{{\rm
cosh}(\frac{c}{2})}{{\rm sinh}(\frac{c}{2})}$ $\displaystyle\cdot\left({\rm
ch}(\Theta_{1}(T_{C}M+\xi_{C},C^{2}))-\frac{{\rm
ch}(\Theta_{1}(T_{C}M+\xi_{C},\xi_{C}))}{{\rm cosh}^{2}(\frac{c}{2})}\right),$
$\displaystyle\widetilde{\Phi_{W}}(\nabla^{TM},\nabla^{\xi},\tau)$
$\displaystyle=$ $\displaystyle{\widehat{A}(TM,\nabla^{TM})}\frac{1}{2{\rm
sinh}(\frac{c}{2})}\left({\rm ch}(\Theta_{2}(T_{C}M+\xi_{C},C^{2}))\right.$
$\displaystyle\left.-{\rm cosh}(\frac{c}{2}){\rm
ch}(\Theta_{2}(T_{C}M+\xi_{C},\xi_{C}))\right)$
$\displaystyle\widetilde{\Phi^{\prime}_{W}}(\nabla^{TM},\nabla^{\xi},\tau)$
$\displaystyle=$ $\displaystyle{\widehat{A}(TM,\nabla^{TM})}\frac{1}{2{\rm
sinh}(\frac{c}{2})}\left({\rm ch}(\Theta_{3}(T_{C}M+\xi_{C},C^{2}))\right.$
$\displaystyle\left.-{\rm cosh}(\frac{c}{2}){\rm
ch}(\Theta_{3}(T_{C}M+\xi_{C},\xi_{C}))\right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(4.2)$
Through direct computations, we get (cf. [HH])
$\displaystyle\widetilde{\Phi_{L}}(\nabla^{TM},\nabla^{\xi},\tau)$
$\displaystyle=$
$\displaystyle\frac{\sqrt{2}^{4k+1}}{\pi\sqrt{-1}}\left(\prod_{j=1}^{2k}x_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(x_{j},\tau)}\frac{\theta_{1}(x_{j},\tau)}{\theta_{1}(0,\tau)}\right)\frac{\theta^{\prime}(0,\tau)}{\theta(u,\tau)}$
$\displaystyle\cdot\left(\frac{\theta_{1}(u,\tau)}{\theta_{1}(0,\tau)}-\frac{\theta_{1}(0,\tau)}{\theta_{1}(u,\tau)}\frac{\theta_{3}(u,\tau)}{\theta_{3}(0,\tau)}\frac{\theta_{2}(u,\tau)}{\theta_{2}(0,\tau)}\right)$
$\displaystyle\widetilde{\Phi_{W}}(\nabla^{TM},\nabla^{\xi},\tau)$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi\sqrt{-1}}\left(\prod_{j=1}^{2k}x_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(x_{j},\tau)}\frac{\theta_{2}(x_{j},\tau)}{\theta_{2}(0,\tau)}\right)\frac{\theta^{\prime}(0,\tau)}{\theta(u,\tau)}$
$\displaystyle\cdot\left(\frac{\theta_{2}(u,\tau)}{\theta_{2}(0,\tau)}-\frac{\theta_{2}(0,\tau)}{\theta_{2}(u,\tau)}\frac{\theta_{3}(u,\tau)}{\theta_{3}(0,\tau)}\frac{\theta_{1}(u,\tau)}{\theta_{1}(0,\tau)}\right)$
$\displaystyle\widetilde{\Phi_{W}}(\nabla^{TM},\nabla^{\xi},\tau)$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi\sqrt{-1}}\left(\prod_{j=1}^{2k}x_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(x_{j},\tau)}\frac{\theta_{3}(x_{j},\tau)}{\theta_{3}(0,\tau)}\right)\frac{\theta^{\prime}(0,\tau)}{\theta(u,\tau)}$
$\displaystyle\cdot\left(\frac{\theta_{3}(u,\tau)}{\theta_{3}(0,\tau)}-\frac{\theta_{3}(0,\tau)}{\theta_{3}(u,\tau)}\frac{\theta_{2}(u,\tau)}{\theta_{2}(0,\tau)}\frac{\theta_{1}(u,\tau)}{\theta_{1}(0,\tau)}\right)~{}~{}~{}~{}~{}~{}(4.3)$
Applying the Chern-Weil theory and Lemma 3.1 again, we can transgress
$\widetilde{\Phi_{L}},~{}\widetilde{\Phi_{W}},\widetilde{\Phi_{W}^{\prime}}$
about $\nabla_{TM}$ and define transgressed forms as follows:
$CS\widetilde{\Phi_{L}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)$
$:=\frac{\sqrt{2}}{8\pi^{2}}\int_{0}^{1}\widetilde{\Phi_{L}}(\nabla_{t}^{TM},\nabla^{\xi},\tau){\rm
tr}\left[A\left(\frac{1}{\frac{R_{t}^{TM}}{4\pi^{2}}}-\frac{\theta^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}+\frac{\theta_{1}^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta_{1}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}\right)\right]dt.$
$None$
which is in $\Omega^{\rm odd}(M,{\bf C})[[q^{\frac{1}{2}}]].$ Since $M$ is
$4k+1$ dimensional,
$\\{CS\widetilde{\Phi_{L}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k+1)}$
represents an element in $H^{4k+1}(M,{\bf C})[[q^{\frac{1}{2}}]]$. Similarly,
we can define
$CS\widetilde{\Phi_{W}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)$
$:=\frac{1}{8\pi^{2}}\int_{0}^{1}\widetilde{\Phi_{W}}(\nabla_{t}^{TM},\nabla^{\xi},\tau){\rm
tr}\left[A\left(\frac{1}{\frac{R_{t}^{TM}}{4\pi^{2}}}-\frac{\theta^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}+\frac{\theta_{2}^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta_{2}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}\right)\right]dt;$
$None$
$CS\widetilde{\Phi_{W}^{\prime}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)$
$:=\frac{1}{8\pi^{2}}\int_{0}^{1}\widetilde{\Phi_{W}^{\prime}}(\nabla_{t}^{TM},\nabla^{\xi},\tau){\rm
tr}\left[A\left(\frac{1}{\frac{R_{t}^{TM}}{4\pi^{2}}}-\frac{\theta^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}+\frac{\theta_{3}^{\prime}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}{\theta_{3}(\frac{R_{t}^{TM}}{4\pi^{2}},\tau)}\right)\right]dt.$
$None$
Using the same discussions as Theorem 3.2, we obtain
Theorem 4.1 Let $M$ be a $4k+1$ dimensional manifold and
$\nabla_{0}^{TM},~{}\nabla_{1}^{TM}$ be two connections on $TM$ and $\xi$ be a
two dimensional oriented Euclidean real vector bundle with a Euclidean
connection $\nabla^{\xi}$, then we have
1)
$\\{CS\widetilde{\Phi_{L}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k+1)}$
is a modular form of weight $2k+2$ over $\Gamma_{0}(2)$;
$\\{CS\widetilde{\Phi_{W}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k+1)}$
is a modular form of weight $2k+2$ over $\Gamma^{0}(2);$
$\\{CS\widetilde{\Phi_{W}^{\prime}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k+1)}$
is a modular form of weight $2k+2$ over $\Gamma_{\theta}(2).$
2) The following equalities hold,
$\\{CS\widetilde{\Phi_{L}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},-\frac{1}{\tau})\\}^{(4k+1)}=(2\tau)^{2k+2}\\{CS\widetilde{\Phi_{W}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(4k+1)},$
$CS\widetilde{\Phi_{W}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau+1)=CS\widetilde{\Phi_{W}^{\prime}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau).$
Let $M$ be $9$ dimensional and $k=2$. We have that
$\\{CS\widetilde{\Phi_{L}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(9)}$
is a modular form of weight $6$ over $\Gamma_{0}(2),$
$\\{CS\widetilde{\Phi_{W}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(9)}$
is a modular form of weight $6$ over $\Gamma^{0}(2)$ and
$\\{CS\widetilde{\Phi_{L}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},-\frac{1}{\tau})\\}^{(9)}=(2\tau)^{6}\\{CS\widetilde{\Phi_{W}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(9)}.$
By Lemma 2.2, we have
$\\{CS\widetilde{\Phi_{W}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(9)}=z_{0}(8\delta_{2})^{3}+z_{1}(8\delta_{2})\varepsilon_{2},$
$None$
and by (2.19) and Theorem 4.1,
$\\{CS\widetilde{\Phi_{L}}(\nabla_{0}^{TM},\nabla_{1}^{TM},\nabla^{\xi},\tau)\\}^{(9)}=2^{6}[z_{0}(8\delta_{1})^{3}+z_{1}(8\delta_{1})\varepsilon_{1}].$
$None$
By comparing the $q^{\frac{1}{2}}$-expansion coefficients in (4.7), we get
$z_{0}=-\left\\{\int_{0}^{1}\frac{\widehat{A}(TM,\nabla^{TM}_{t})}{2{\rm
sinh}(\frac{c}{2})}(1-{\rm cosh}\frac{c}{2}){\rm
tr}\left[A\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{8\pi{\rm
tan}{\frac{R^{TM}_{t}}{4\pi}}}\right)\right]dt\right\\}^{(9)},$ $None$
$z_{1}=\left\\{-\int_{0}^{1}\frac{\widehat{A}(TM,\nabla^{TM}_{t})}{2{\rm
sinh}(\frac{c}{2})}(1-{\rm cosh}\frac{c}{2}){\rm tr}\left[\frac{A}{2\pi}{\rm
sin}\frac{R^{TM}_{t}}{2\pi}\right]dt\right.$
$+\int_{0}^{1}\frac{\widehat{A}(TM,\nabla^{TM}_{t})}{2{\rm
sinh}(\frac{c}{2})}{\rm
tr}\left[A\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{8\pi{\rm
tan}{\frac{R^{TM}_{t}}{4\pi}}}\right)\right]$ $\left.\cdot\left((1-{\rm
cosh}\frac{c}{2})({\rm ch}(T_{C}M,\nabla_{t}^{T_{C}M})+61)+(1+2{\rm
cosh}\frac{c}{2})(e^{c}+e^{-c}-2)\right)dt\right\\}^{(9)}.$ $None$
Plugging (4.9) and (4.10) into (4.8) and comparing the constant terms of both
sides, we obtain that
Corollary 4.2 The following equality holds
$\left\\{\int_{0}^{1}\sqrt{2}\widehat{L}(TM,\nabla^{TM}_{t})\frac{{\rm
sinh}\frac{c}{2}}{{\rm cosh}{\frac{c}{2}}}{\rm
tr}\left[A\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{4\pi{\rm
sin}{\frac{R^{TM}_{t}}{2\pi}}}\right)\right]\right\\}^{(9)}$
$=8\left\\{-\int_{0}^{1}\frac{\widehat{A}(TM,\nabla^{TM}_{t})}{2{\rm
sinh}(\frac{c}{2})}(1-{\rm cosh}\frac{c}{2}){\rm tr}\left[\frac{A}{2\pi}{\rm
sin}\frac{R^{TM}_{t}}{2\pi}\right]dt\right.$
$+\int_{0}^{1}\frac{\widehat{A}(TM,\nabla^{TM}_{t})}{2{\rm
sinh}(\frac{c}{2})}{\rm
tr}\left[A\left(\frac{1}{2R_{t}^{TM}}-\frac{1}{8\pi{\rm
tan}{\frac{R^{TM}_{t}}{4\pi}}}\right)\right]$ $\left.\cdot\left((1-{\rm
cosh}\frac{c}{2})({\rm ch}(T_{C}M,\nabla_{t}^{T_{C}M})-3)+(1+2{\rm
cosh}\frac{c}{2})(e^{c}+e^{-c}-2)\right)dt\right\\}^{(9)}.$ $None$
Acknowledgement This work was supported by NSFC No. 10801027\.
References
[AW] L. Alvarez-Gaumé, E. Witten, Graviational anomalies, Nucl. Phys. B234
(1983), 269-330.
[Ch] K. Chandrasekharan, Elliptic Functions, Spinger-Verlag, 1985.
[CH] Q. Chen, F. Han, Elliptic genera, transgression and loop space Chern-
Simons form, arXiv:0605366.
[HZ1]F. Han, W. Zhang, ${\rm Spin}^{c}$-manifold and elliptic genera, C. R.
Acad. Sci. Paris Serie I., 336 (2003), 1011-1014.
[HZ2] F. Han, W. Zhang, Modular invariance, characteristic numbers and eta
Invariants, J. Diff. Geom. 67 (2004), 257-288.
[HH] F. Han, X. Huang, Even dimensional manifolds and generalized anomaly
cancellation formulas , Trans. AMS 359 (2007), No. 11, 5365-5382.
[Li] K. Liu, Modular invariance and characteristic numbers. Commu.Math. Phys.
174 (1995), 29-42.
[Z] W. Zhang, Lectures on Chern-weil Theory and Witten Deformations. Nankai
Tracks in Mathematics Vol. 4, World Scientific, Singapore, 2001.
School of Mathematics and Statistics , Northeast Normal University, Changchun,
Jilin 130024, China ;
E-mail: wangy581@nenu.edu.cn
|
arxiv-papers
| 2008-11-27T01:26:46 |
2024-09-04T02:48:59.010908
|
{
"license": "Public Domain",
"authors": "Yong Wang",
"submitter": "Wang Yong",
"url": "https://arxiv.org/abs/0811.4460"
}
|
0811.4476
|
# Conservation laws of the Haldane-Shastry type spin chains
Junpeng Cao Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, People's
Republic of China Peng He Beijing National Laboratory for Condensed Matter
Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190,
People's Republic of China Yuzhu Jiang Beijing National Laboratory for
Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences,
Beijing 100190, People's Republic of China Yupeng Wang∗ Beijing National
Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy
of Sciences, Beijing 100190, People's Republic of China
###### Abstract
A systematic method to construct the complete set of conserved quantities of
the Haldane-Shastry type spin chains is proposed. The hidden relationship
between the Yang-Baxter relation and the conservation laws of the long-range
interacting integrable models is exposed explicitly. An integrable anisotropic
Haldane-Shastry model is also constructed.
###### pacs:
02.30.Ik, 75.10.Jm
A typical characteristic of the integrable models is that each of them
possesses a complete set of conservation laws. For an integrable system with a
finite number of degrees of freedom, the number of linearly independent
conserved quantities is exactly the same as that of the degrees of freedom,
while for a continuous integrable model, there is an infinite number of
conserved quantities. It is well known that the conservation laws of the
integrable models with short-range interactions are tightly related to the
Yang-Baxter relation. The conserved quantities can be obtained from the
derivatives of the transfer matrix of the system tak ; bax . On the other
hand, there is another class of integrable models with $r^{-2}$ type
interaction potentials which are called as the Calogero-Sutherland (CS) model
cs in the continuous case and the Haldane-Shastry (HS) model hs-1 ; hs-2 in
the lattice case. These models belong to the long-range interacting ones and
have many applications in the fields of two-dimensional fractional quantum
Hall effect and fractional statistics st . Because of the importance, these
models have been studied extensively hs2 ; h21 ; h12 ; h2 ; s2 ; hs3 ; hs4 ;
hs5 ; hs6 ; hs7 ; hs8 . Several people have tried to construct the
conservation laws of these models. For example, by using the Dunkl operators
dun , Polychronakos pol1 constructed the invariants of motion of the CS
continuous model. Nevertheless, the demonstration of the integrability of the
HS Lattice model is still a challenge problem. Borrowed Polychronakos's idea,
Fowler and Minahan fow proposed a set of conserved quantities for the HS
model. However, there are still some debates cyb ; pol2 on this construction
because the HS Hamiltonian appears in the third level of invariants, and
should act on some magnon states to erase the unwanted terms.
The main difficulty of the demonstration of the complete integrability of HS
type models is lack of a systematic method to construct the complete set of
conservation laws. Recalling the case of nearest neighbor interacting
integrable models, an interesting question may arise: Is there any intrinsic
relationship between the HS model and the Yang-Baxter relation cyb ? In this
Letter, we expose this intrinsic relationship exactly and develop a systematic
method to construct the complete set of conservation laws of HS model. This
provides us not only a deep understanding to this important model but also a
general method to construct other integrable models with long-range couplings.
Our starting point is the following Lax operator
$\displaystyle L_{0j}(u)=1+\frac{\eta}{u}{\bf\sigma}_{0}\cdot{\bf S}_{j},$ (1)
where $u$ is the spectral parameter; $\eta$ is the crossing parameter;
${\bf\sigma}_{0}$ is the auxiliary Pauli matrix and ${\bf S}_{j}$ is the
spin-$1/2$ operator on site $j$. It is well known that the integrability of
the Heisenberg spin chain model is related to the monodromy matrix
$T_{0}(u)=L_{01}(u)\cdots L_{0N}(u)$ which satisfies the Yang-Baxter relation
$L_{12}(u_{1}-u_{2})T_{1}(u_{1})T_{2}(u_{2})=T_{2}(u_{2})T_{1}(u_{1})L_{12}(u_{1}-u_{2}).$
(2)
Define the transfer matrix $t(u)=tr_{0}T_{0}(u)$. From Eq.(2), one can prove
that the transfer matrices with different spectral parameters are mutually
commutative, i.e., $[t(u),t(v)]=0$. Therefore, $t(u)$ serves as the generating
function of the conserved quantities of the corresponding system. The first
order derivative of logarithm of the transfer matrix gives the Hamiltonian of
the Heisenberg spin chain
$\displaystyle H_{H}=\frac{1}{2\eta}\left.\frac{\partial}{\partial u}\ln
t(u)\right|_{u=\frac{\eta}{2}}+(\frac{1}{\eta^{2}}-\frac{1}{4})N=\sum_{j=1}^{N}{\bf
S}_{j}\cdot{\bf S}_{j+1}.$ (3)
In fact, the inhomogeneous Lax operator $L_{0j}(u-\delta_{j})$ with a site-
dependent shift $\delta_{j}$ to the spectral parameter $u$ and the
inhomogeneous transfer matrix $\tilde{T}_{0}(u)\equiv
L_{01}(u-\delta_{1})\cdots L_{0N}(u-\delta_{N})$ also satisfy the Yang-Baxter
relation Eq.(2). Consider the classical expansion of the Lax operator
$L_{0j}(u-\delta_{j})=1+\eta{\cal L}_{0j}(u-\delta_{j})$ and define the
classical monodromy matrix ${\cal T}_{0}(u)=\sum_{j=1}^{N}{\cal
L}_{0j}(u-\delta_{j})$. We find that they satisfy the following classical
Yang-Baxter relation skly
$[{\cal T}_{1}(u_{1}),{\cal T}_{2}(u_{2})]=[{\cal T}_{1}(u_{1})+{\cal
T}_{2}(u_{2}),{\cal L}_{12}(u_{1}-u_{2})].$ (4)
The Eq.(4) ensures that the functional $\tau(u)\equiv tr_{0}{{\cal
T}_{0}}^{2}(u)/4$ with different spectral parameters are mutually commutative,
i.e., $[{\tau}(u),{\tau}(v)]=0$. Therefore, ${\tau}(u)$ can be treated as the
generating functional of a series of conserved quantities which can be written
out explicitly as
${\tau}(u)=\sum_{j=1}^{N}\frac{3}{8(u-\delta_{j})^{2}}+\sum_{j=1}^{N}\frac{h_{j}}{u-\delta_{j}},$
(5)
where
$\displaystyle h_{j}=\sum_{l=1,\neq j}^{N}\frac{1}{\delta_{j}-\delta_{l}}{\bf
S}_{j}\cdot{\bf S}_{l}.$ (6)
The operators $h_{j}$ are nothing but the Gaudin operators gau associated
with the Heisenberg spin chain. It is easy to prove that the Gaudin operators
commute with each other, i.e., $[h_{j},h_{k}]=0$. This allows us to construct
the mutually commutative operators $I_{n}=\sum_{j=1}^{N}{h_{j}}^{n}$ for
arbitrary $n$. If we choose one of $I_{n}$ as the Hamiltonian, $\\{I_{n}\\}$
form a set of conserved quantities and the Hamiltonian describes an integrable
system. For a translational invariant lattice $\delta_{j}=j$ and $N\to\infty$,
we find
$\displaystyle H_{ISE}$ $\displaystyle=$
$\displaystyle\lim_{N\to\infty}\left[-\sum_{j=1}^{N}h_{j}^{2}+\frac{\pi^{2}}{16}N\right]$
(7) $\displaystyle=$
$\displaystyle{\sum_{l<j}}^{\prime}\frac{1}{(j-l)^{2}}{\bf S}_{j}\cdot{\bf
S}_{l}.$
$H_{ISE}$ is just the Hamiltonian of inverse square exchanging (ISE) model.
The prime in the summation means that $j$ and $l$ in the summation take values
from negative to positive infinity. Obviously, operators $h_{j}=\sum_{l\neq
j}^{\prime}(j-l)^{-1}{\bf S}_{j}\cdot{\bf S}_{l}$ and their arbitrary
combinations commute with the Hamiltonian (7) and form a set of the conserved
quantities. In addition, the model (7) has another set of conserved quantities
$h_{j}^{\prime}=\sum_{l\neq j}^{\prime}{\bf S}_{j}\cdot{\bf S}_{l}$ by simply
checking $[H_{S},h_{j}^{\prime}]=0$. We note that (7) with arbitrary
$\delta_{j}$ gives a disordered integrable system. Now it is clear that there
is an intrinsic relationship between the short-range coupling Heisenberg model
and the long-range coupling ISE model, i.e., they share the common Yang-Baxter
equation. This provides us a powerful method to construct new integrable
models with long-range interactions from the known solutions of the Yang-
Baxter equation or to obtain the conservation laws of the predicted integrable
models with $r^{-2}$ type potentials. For example, from the Lax operator of
the anisotropic XXZ Heisenberg spin chain tak we have the following mutually
commutative Gaudin operators
$\displaystyle h_{j}=\sum_{l=1,\neq
j}^{N}\left[\frac{S_{j}^{x}S_{l}^{x}+S_{j}^{y}S_{l}^{y}}{\sin(\delta_{j}-\delta_{l})}+\cot(\delta_{j}-\delta_{l})S_{j}^{z}S_{l}^{z}\right].$
(8)
For the equally spaced $\delta_{j}=\pi j/N$, we obtain
$\displaystyle H_{AHS}$ $\displaystyle=$
$\displaystyle-\sum_{j=1}^{N}h_{j}^{2}-\frac{1}{4}\left(\sum_{j=1}^{N}S_{j}^{z}\right)^{2}+\frac{1}{16}N(N^{2}-N+1)$
(9) $\displaystyle=$
$\displaystyle\sum_{l<j}^{N}\frac{\cos\frac{\pi}{N}(j-l)(S_{j}^{x}S_{l}^{x}+S_{j}^{y}S_{l}^{y})+S_{j}^{z}S_{l}^{z}}{\sin^{2}\frac{\pi}{N}(j-l)}.$
Notice that in the anisotropic XXZ spin chain, the total spin is no longer a
good quantum number but $\sum_{l=1}^{N}S_{l}^{z}$ is indeed a conserved
quantity which commutes with the $h_{j}$ in Eq.(8) and the $H_{AHS}$ in
Eq.(9). Therefore, $H_{AHS}$ can be treated as the Hamiltonian of an
anisotropic HS model. In fact, the ISE Hamiltonian (7) is the limiting case of
the anisotropic HS model (9), i.e.,
$H_{ISE}=\lim_{N\to\infty}{\pi^{2}}/{N^{2}}H_{AHS}$. By putting $N\to\infty$
and $\delta_{m}=im$, where $i$ is the imaginary unit, we readily obtain the
hyperbolic version of this integrable Hamiltonian
$\displaystyle
H_{HAHS}={\sum_{l<j}}^{\prime}\frac{\cosh(j-l)(S_{j}^{x}S_{l}^{x}+S_{j}^{y}S_{l}^{y})+S_{j}^{z}S_{l}^{z}}{\sinh^{2}(j-l)}.$
(10)
Motivated by these findings, we introduce the following local operators
$\displaystyle h_{j}=\sum_{l=1,\neq j}^{N}f(\delta_{j}-\delta_{l}){\bf
S}_{j}\cdot{\bf S}_{l},$ (11)
and look for the solutions of $[h_{j},h_{k}]=0$, where
$f(\delta_{j}-\delta_{l})$ is a function to be determined. For simplicity, we
denote $f(\delta_{j}-\delta_{l})\equiv f_{jl}$. With the relation $[{\bf
S}_{j}\cdot{\bf S}_{l},{\bf S}_{j}\cdot{\bf S}_{k}]=i{\bf S}_{j}\cdot({\bf
S}_{l}\times{\bf S}_{k})$ for $l\neq j\neq k$, we obtain
$\displaystyle[h_{j},h_{k}]=i\sum_{l=1,\neq
j,k}^{N}(f_{jl}f_{kj}-f_{jl}f_{kl}+f_{jk}f_{kl}){\bf S}_{j}\cdot({\bf
S}_{l}\times{\bf S}_{k}).$ (12)
Therefore, the constraint for $[h_{j},h_{k}]=0$ is the solution of
$f_{jl}f_{kj}-f_{jl}f_{kl}+f_{jk}f_{kl}=0$. After some simple algebra, we find
three sets of solutions:
(i) The first solution is $f(x)=x^{-1}$. This solution just gives Eq.(6) and
thus the ISE Hamiltonian (7).
(ii) The second solution is $f(x)=\cot(x)\pm i$. The operators $h_{j}$ and
$h_{j}^{\dagger}$ take the forms
$\displaystyle h_{j}=\sum_{l=1,\neq
j}^{N}\left[\cot(\delta_{j}-\delta_{l})+i\right]{\bf S}_{j}\cdot{\bf S}_{l},$
$\displaystyle h_{j}^{\dagger}=\sum_{l=1,\neq
j}^{N}\left[\cot(\delta_{j}-\delta_{l})-i\right]{\bf S}_{j}\cdot{\bf S}_{l}.$
(13)
Both $\\{h_{j}\\}$ and $\\{h_{j}^{\dagger}\\}$ are not hermitian for real
$\delta_{j}$ but each of them form a set of mutually commutative operators,
though $h_{j}$ and $h_{k}^{\dagger}$ do not commute with each other,
$[h_{j},h_{k}^{{\dagger}}]=2\sum_{l=1,\neq
j,k}^{N}\left[\cot(\delta_{j}-\delta_{k})+i\right]{\bf S}_{j}\cdot({\bf
S}_{l}\times{\bf S}_{k})$. A key problem is to construct a hermitian
Hamiltonian from those non-hermitian operators. Fortunately, we find that
$I_{2}\equiv\sum_{j=1}^{N}h_{j}^{2}=\sum_{j=1}^{N}{h_{j}^{\dagger}}^{2}$ is a
hermitian operator. For $\delta_{j}=\pi j/N$, the HS Hamiltonianhs-1 ; hs-2
can be derived as
$\displaystyle H_{HS}$ $\displaystyle=$
$\displaystyle-\sum_{j=1}^{N}h_{j}^{2}+\frac{i}{2}(N-4)\sum_{j=1}^{N}h_{j}+\frac{1}{16}N(3N^{2}-6N-5)$
(14) $\displaystyle=$
$\displaystyle\sum_{j<l}^{N}\frac{1}{\sin^{2}\frac{\pi}{N}(j-l)}{\bf
S}_{j}\cdot{\bf S}_{l}.$
An obvious fact is that $[H_{HS},h_{j}]=[H_{HS},h_{j}^{\dagger}]=0$. We
readily have two independent sets of conserved hermitian quantities for
$H_{HS}$:
$\displaystyle{I_{j}}^{+}=\frac{1}{2}(h_{j}+h_{j}^{\dagger})=\sum_{l=1,\neq
j}^{N}\cot\frac{\pi}{N}(j-l){\bf S}_{j}\cdot{\bf S}_{l},$
$\displaystyle{I_{j}}^{-}=\frac{1}{2i}(h_{j}-h_{j}^{\dagger})=\sum_{l=1,\neq
j}^{N}{\bf S}_{j}\cdot{\bf S}_{l}.$ (15)
For the spin half system, we know that the number of degrees of freedom of
each site is two. The two linearly independent conserved quantities
$I_{j}^{\pm}$ clearly show that the HS Hamiltonian is completely integrable.
Actually, we can also define the classical operators ${\cal
L}_{0j}(u-\delta_{j})=[\cot(u-\delta_{j})+i]{\bf S}_{0}\cdot{\bf S}_{j}$ and
${\cal T}_{0}(u)=\sum_{j=1}^{N}{\cal L}_{0j}(u-\delta_{j})$ for this solution.
They satisfy the following deformed classical Yang-Baxter relation
$\displaystyle[{\cal T}_{1}(u),{\cal T}_{2}(v)]=[{\cal L}_{21}(v-u),{\cal
T}_{1}(u)]+[{\cal T}_{2}(v),{\cal L}_{12}(u-v)].$ (16)
Define a functional $\tau(u)=tr_{0}{\cal T}_{0}^{2}(u)$, which can be written
out explicitly as
$\displaystyle{\tau}(u)=\sum_{j=1}^{N}\left[\frac{3\cos
2(u-\delta_{j})}{8\sin^{2}(u-\delta_{j})}+\frac{3}{4}i\cot(u-\delta_{j})\right]$
$\displaystyle\qquad\quad+\sum_{j=1}^{N}\left[\cot(u-\delta_{j})+i\right]h_{j}.$
(17)
From the deformed Yang-Baxter relation (16), we can prove that the functional
$\tau(u)$ with different spectral parameters are mutually commutative and thus
can be treated as the generation functional of a integrable system. Obviously,
for $\delta_{j}=j\pi/N$, $[H_{HS},\tau(u)]=0$, implying that the conserved
quantities of the HS model can also be generated by $\tau(u)$.
(iii) The third solution is $f(x)=\coth(x)\pm 1$ and
$h_{j}^{\pm}=\sum_{l=1,\neq j}^{N}\left[\coth(\delta_{j}-\delta_{l})\pm
1\right]{\bf S}_{j}\cdot{\bf S}_{l}$. With the same procedure, we find that
the Inozemtsev Hamiltonian hs2
$\displaystyle H_{I}={\sum_{l<j}}^{\prime}\frac{1}{\sinh^{2}(j-l)}{\bf
S}_{j}\cdot{\bf S}_{l}$ (18)
can be easily derived from $\sum_{j}^{\prime}{h_{j}^{+}}^{2}$ or
$\sum_{j}^{\prime}{h_{j}^{-}}^{2}$ by taking $\delta_{j}=j$ and $N\to\infty$.
The operators $h_{j}^{\pm}=\sum_{l\neq j}^{\prime}\left[\coth(j-l)\pm
1\right]{\bf S}_{j}\cdot{\bf S}_{l}$ span the complete space of the conserved
hermitian quantities and the Inozemtsev model (18) is also completely
integrable.
Similarly, we can construct the conservation laws of the $SU(M)$-invariant HS
and Inozemtsev models. In these cases, the generating operators $h_{j}$ take
the form:
$\displaystyle h_{j}=\sum_{l=1,\neq j}^{N}f(\delta_{j}-\delta_{l})(P_{jl}-c),$
(19)
where $P_{ij}$ is the $SU(M)$ spin permutation operator and $c$ is a constant.
One can easily check that
$\displaystyle[h_{j},h_{k}]=\sum_{l=1,\neq
j,k}^{N}(f_{jl}f_{kj}-f_{jl}f_{kl}+f_{jk}f_{kl})P_{jl}(P_{kj}-P_{kl}).$ (20)
The constraint $[h_{j},h_{k}]=0$ gives the same solutions of $f(x)$ as those
in the (i) - (iii). The $SU(M)$ HS Hamiltonian can be constructed from the
linear combination of $\sum_{j=1}^{N}{I_{j}^{+}}^{2}$ and
$\sum_{j=1}^{N}I_{j}^{+}$, whose explicit form reads
$\displaystyle
H_{SU(M)}=\sum_{l<j}^{N}\frac{P_{jl}}{\sin^{2}\frac{\pi}{N}(j-l)}.$ (21)
In conclusion, we establish the intrinsic relationship between the inverse
square potential spin chain models and the Yang-Baxter relation. This provides
us a powerful method to construct new integrable models with long-range
couplings from the solutions of the Yang-Baxter relation obtained from the
systems with short-range interactions. As an example, an integrable
anisotropic HS spin chain model is derived. The complete sets of conservation
laws of the ISE, HS and Inozemtsev models are constructed in quite simple
forms.
This work was supported by the National Natural Science Foundation of China,
the Knowledge Innovation Project of Chinese Academy of Sciences, and the
National Program for Basic Research of MOST.
* Email: yupeng@aphy.iphy.ac.cn
## References
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* (2) R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982).
* (3) F. Calogero, J. Math. Phys. 10, 2191 (1969); 10, 2197 (1969); B. Sutherland, J. Math. Phys. 12, 246 (1971); 12, 250 (1971); Phys. Rev. A 4, 2019 (1971); 5, 1372 (1972).
* (4) F. D. M. Haldane, Phys. Rev. Lett. 60, 635 (1988).
* (5) B. S. Shastry, Phys. Rev. Lett. 60 639 (1988).
* (6) A. P. Polychronakos, Nucl. Phys. B 324, 597 (1989); F.D.M. Haldane, Phys. Rev. Lett. 67 937 (1991); Z.N.C. Ha, Nucl. Phys. B 435, 604 (1995).
* (7) V. I. Inozemtsev, J. Stat. Phys. 59, 1143 (1989).
* (8) B. S. Shastry and B. Sutherland, Phys. Rev. Lett. 70, 4029 (1993).
* (9) B. Sutherland and B. S. Shastry, Phys. Rev. Lett. 71, 5 (1993).
* (10) F. D. M. Haldane, Phys. Rev. Lett. 66, 1529 (1991).
* (11) B. S. Shastry, Phys. Rev. Lett. 69, 164 (1992).
* (12) H. Kiwata and Y. Akutsu, J. Phys. Soc. Jpn. 61, 1441 (1992).
* (13) N. Kawakami, Phys. Rev. B 46, 1005 (1992); 46, 3191 (1992).
* (14) Z. N. C. Ha and F. D. M. Haldane, Phys. Rev. B 46, 9359 (1992).
* (15) V. I. Inozemtsev, Commun. Math. Phys. 148, 359 (1992).
* (16) B. D. Simons and B. L. Altschuler, Phys. Rev. B 50, 1102 (1994).
* (17) D. Bernard, V. Pasquier and D. Serban, Europhys. Lett. 30, 301 (1995).
* (18) C. F. Dunkl, Trans. Amer. Math. Soc. 311, 167 (1989).
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* (21) J. C. Talstra, F. D. M. Haldane, J. Phys. A: Math. Gen. 28, 2369 (1995).
* (22) A. P. Polychronakos, Phys. Rev. Lett. 70, 2329 (1993).
* (23) E. K. Sklyanin, J. Sov. Math. 47, 2473 (1989).
* (24) M. Gaudin, J. Phys. (Paris) 37, 1087 (1976).
|
arxiv-papers
| 2008-11-27T08:02:24 |
2024-09-04T02:48:59.018140
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Junpeng Cao, Peng He, Yuzhu Jiang, Yupeng Wang",
"submitter": "Junpeng Cao",
"url": "https://arxiv.org/abs/0811.4476"
}
|
0811.4477
|
# Generalized second law of thermodynamics in warped DGP braneworld
Ahmad Sheykhi 1,2111sheykhi@mail.uk.ac.ir and Bin Wang 3222wangb@fudan.edu.cn
1Department of Physics, Shahid Bahonar University, P.O. Box 76175, Kerman,
Iran
2Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),
Maragha, Iran
3 Department of Physics, Fudan University, Shanghai 200433, China
###### Abstract
We investigate the validity of the generalized second law of thermodynamics on
the $(n-1)$-dimensional brane embedded in the $(n+1)$-dimensional bulk. We
examine the evolution of the apparent horizon entropy extracted through
relation between gravitational equation and the first law of thermodynamics
together with the matter field entropy inside the apparent horizon. We find
that the apparent horizon entropy extracted through connection between gravity
and the first law of thermodynamics satisfies the generalized second law of
thermodynamics. This result holds regardless of whether there is the intrinsic
curvature term on the brane or a cosmological constant in the bulk. The
observed satisfaction of the generalized second law provides further support
on the thermodynamical interpretation of gravity based on the profound
connection between gravity and thermodynamics.
## I Introduction
Inspired by the profound connection between the black hole physics and
thermodynamics, there has been some deep thinking on the relation between
gravity and thermodynamics in general for a long time. The pioneer work was
done by Jacobson who showed that the gravitational Einstein equation can be
derived from the relation between the horizon area and entropy, together with
the Clausius relation $\delta Q=T\delta S$ Jac . Further studies on the
connection between gravity and thermodynamics has been investigated in various
gravity theories Elin ; Cai1 ; Cai11 ; Pad1 ; Pad2 ; Pad3 ; Pad4 ; Pad5 ; Pad6
. In the cosmological context, attempts to disclose the connection between
Einstein gravity and thermodynamics were carried out in Cai2 ; Cai3 ; CaiKim ;
Fro1 ; Fro2 ; Fro3 ; Fro4 ; Fro5 ; verlinde1 ; verlinde2 ; verlinde3 ;
verlinde4 . It was shown that the differential form of the Friedmann equation
in the FRW universe can be written in the form of the first law of
thermodynamics on the apparent horizon. The profound connection provides a
thermodynamical interpretation of gravity which makes it interesting to
explore the cosmological properties through thermodynamics.
Investigations on the deep connection between gravity and thermodynamics has
recently been extended to braneworld scenarios Cai4 ; Shey1 ; Shey2 . The
motivating idea for disclosing the connection between the thermodynamics and
gravity in braneworld is to get deeper understanding on the entropy of the
black hole in braneworld. In the braneworld scenarios, gravity on the brane
does not obey Einstein theory, thus the usual area formula for the black hole
entropy does not hold on the brane. The exact analytic black hole solutions on
the brane have not been found so far, so that the relation between the
braneworld black hole horizon entropy and its geometry is not known. We expect
that the connection between gravity and thermodynamics in the braneworld can
shed some lights on understanding these problems. There are two main pictures
in the braneworld scenario. In the first picture which we refer as the
Randall-Sundrum II model (RS II), a positive tension 3-brane embedded in an
5-dimensional AdS bulk and the cross over between 4D and 5D gravity is set by
the AdS radius RS1 ; RS2 ; Bin . In this case, the extra dimension has a
finite size. In another picture which is based on the work of Dvali,
Gabadadze, Porrati (DGP model)DGP ; DG , a 3-brane is embedded in a spacetime
with an infinite-size extra dimension, with the hope that this picture could
shed new light on the standing problem of the cosmological constant as well as
on supersymmetry breaking DGP ; Wit . The recovery of the usual gravitational
laws in this picture is obtained by adding to the action of the brane an
Einstein-Hilbert term computed with the brane intrinsic curvature. The
obtained connection between gravity and thermodynamics in the braneworld shows
that the connection is general and not just an accident in Einstein gravity.
The correspondence of the gravitational field equation describing the gravity
in the bulk to the first law of thermodynamics on the boundary, the apparent
horizon also sheds the light on holography, since the Friedmann equation
persists the information in the bulk and the first law of thermodynamics on
the apparent horizon contains the information on the boundary. The holographic
description of braneworld scenarios and the enropy function on the brane have
also been explored in Padi1 ; Padi2 ; Sav ; James ; wang0 .
Besides showing the universality of the connection between gravity and
thermodynamics by expressing the gravitational field equation into the first
law of thermodynamics on the apparent horizon in different spacetimes, it is
of great interest to examine other thermodynamical principles if the
thermodynamical interpretation of gravity from this correspondence is a
generic feature. This is especially interesting in the braneworld. In the
braneworld, the entropy was extracted through writing the gravitational
equation into the first law of thermodynamics on the apparent horizonShey1 ;
Shey2 . Whether this derived entropy satisfies general thermodynamical
principles is another interesting way to examine the correctness of the
thermodynamical interpretation of the gravity and the validity of the
connection between gravity and thermodynamics. In this paper we are going to
study the generalized second law of thermodynamics by investigating the
evolution of the apparent horizon entropy deduced through the connection
between gravity and the first law of thermodynamics together with the matter
fields entropy inside the apparent horizon. The generalized second law of
thermodynamics is a universal principle governing the universe. Recently the
generalized second law of thermodynamics in the accelerating universe
enveloped by the apparent horizon has been studied extensively in wang1 ;
wang2 ; Shey3 . For other gravity theories, the generalized second law has
also been studied in akbar1 ; akbar2 . In this work we will explore the
generalized second law of thermodynamics in braneworld scenarios, regardless
of whether there is the intrinsic curvature term on the brane or a
cosmological constant in the bulk. If the thermodynamical interpretation of
gravity is correct, the deduced apparent horizon entropy from the connection
between gravitational equation and the first law of thermodynamics should
satisfy the generalized second law.
Our starting point is the $n$-dimensional homogenous and isotropic FRW
universe on the brane with the metric
$ds^{2}={h}_{\mu\nu}dx^{\mu}dx^{\nu}+\tilde{r}^{2}d\Omega_{n-2}^{2},$ (1)
where $\tilde{r}=a(t)r$, $x^{0}=t,x^{1}=r$, the two dimensional metric
$h_{\mu\nu}$=diag $(-1,a^{2}/(1-kr^{2}))$ and $d\Omega_{n-2}$ is the metric of
$(n-2)$-dimensional unit sphere. The dynamical apparent horizon which is the
marginally trapped surface with vanishing expansion, is determined by the
relation $h^{\mu\nu}\partial_{\mu}\tilde{r}\partial_{\nu}\tilde{r}=0$, which
implies that the vector $\nabla\tilde{r}$ is null on the apparent horizon
surface. The apparent horizon was argued as a causal horizon for a dynamical
spacetime and is associated with gravitational entropy and surface gravity
Hay2 ; Hay3 ; Bak . For the FRW universe the apparent horizon radius reads
$\tilde{r}_{A}=\frac{1}{\sqrt{H^{2}+k/a^{2}}}.$ (2)
The associated surface gravity on the apparent horizon can be defined as
$\kappa=\frac{1}{\sqrt{-h}}\partial_{a}\left(\sqrt{-h}h^{ab}\partial_{ab}\tilde{r}\right),$
(3)
thus one can easily express the surface gravity on the apparent horizon
$\kappa=-\frac{1}{\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(4)
The associated temperature on the apparent horizon can be expressed in the
form
$T_{h}=\frac{|\kappa|}{2\pi}=\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(5)
where $\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}<1$ ensures that the
temperature is positive. Recently the Hawking radiation on the apparent
horizon has been observed in cao which gives more solid physical implication
of the temperature associated with the apparent horizon.
## II GSL of Thermodynamics in RS II braneworld
Let us start with the Randall-Sundrum (RS II) model in which no intrinsic
curvature term on the brane is included in the action. The Friedmann equation
for $(n-1)$-dimensional brane embedded in $(n+1)$-dimensional bulk in the RS
II model can be written Shey1
$H^{2}+\frac{k}{a^{2}}-\frac{2\kappa_{n+1}^{2}\Lambda_{n+1}}{n(n-1)}-\frac{\mathcal{C}}{a^{n}}=\frac{\kappa_{n+1}^{4}}{4(n-1)^{2}}\rho^{2}.$
(6)
where
$\kappa_{n+1}^{2}=8\pi
G_{n+1}\,,\quad\Lambda_{n+1}=-\frac{n(n-1)}{2\kappa_{n+1}^{2}\ell^{2}},$ (7)
$\Lambda_{n+1}$ is the $(n+1)$-dimensional bulk cosmological constant,
$H=\dot{a}/a$ is the Hubble parameter on the brane, and we assume the matter
content on the brane is in the form of the perfect fluid in homogenous and
isotropic universe,
$T_{\mu\nu}=(\rho+P)u_{\mu}u_{\nu}+Pg_{\mu\nu},$ (8)
where $u^{\mu}$, $\rho$ and $P$ are the perfect fluid velocity
($u^{\mu}u_{\nu}=-1$), energy density and pressure, respectively. Hereafter we
assume that the brane cosmological constant is zero (if it does not vanish,
one can absorb it in the stress-energy tensor of perfect fluid on the brane).
The constant $\mathcal{C}$ comes from the $(n+1)$-dimensional bulk Weyl
tensor. Here we are interested in the flat (Minkowskian) and conformally flat
(AdS) bulk spacetimes, so that the bulk Weyl tensor vanishes and thus we set
$\mathcal{C}=0$ in the following discussions.
### II.1 Brane embedded in Minkowski bulk
We begin with the simplest case, namely the Minkowski bulk, in which
$\Lambda_{n+1}=0$. We can rewrite the Friedmann equation (6) in the simple
form
$H^{2}+\frac{k}{a^{2}}=\frac{\kappa_{n+1}^{4}}{4(n-1)^{2}}\rho^{2}.$ (9)
In terms of the apparent horizon radius, we can rewrite the Friedmann equation
(9) on the brane as
$\frac{1}{\tilde{r}_{A}}=\frac{4\pi G_{n+1}}{n-1}\rho,$ (10)
where we have used Eq. (7). Now, differentiating equation (10) with respect to
the cosmic time and using the continuity equation
$\dot{\rho}+(n-1)H(\rho+P)=0,$ (11)
we get
$\dot{\tilde{r}}_{A}=4\pi G_{n+1}H(\rho+P){\tilde{r}_{A}^{2}}.$ (12)
One can see from the above equation that $\dot{\tilde{r}}_{A}>0$ provided that
the dominant energy condition, $\rho+P>0$, holds. In our previous work Shey1 ,
we showed that the Friedmann equation can be written in the form of the first
law of thermodynamics on the apparent horizon of the brane
$dE=T_{h}dS_{h}+WdV,$ (13)
where $W=(\rho-P)/2$ is the matter work density Hay2 ; Hay3 , $E=\rho V$ is
the total energy of the matter inside the $(n-1)$-sphere of radius
$\tilde{r}_{A}$ on the brane, where $V=\Omega_{n-1}\tilde{r}_{A}^{n-1}$ is the
volume enveloped by $(n-1)$\- dimensional sphere with the area of apparent
horizon $A=(n-1)\Omega_{n-1}\tilde{r}_{A}^{n-2}$ and
$\Omega_{n-1}=\frac{\pi^{(n-1)/2}}{\Gamma((n+1)/2)}$. Using this procedure we
extracted an expression for the entropy at the apparent horizon on the brane
Shey1
$\displaystyle S_{h}=\frac{2\Omega_{n-1}\tilde{r}_{A}^{n-1}}{4G_{n+1}}.$ (14)
It is worth noting that the entropy obeys the area formula of horizon in the
bulk (the factor $2$ comes from the $\mathbb{Z}_{2}$ symmetry in the bulk).
This is due to the fact that because of the absence of the negative
cosmological constant in the bulk, no localization of gravity happens on the
brane. As a result, the gravity on the brane is still $(n+1)$-dimensional. Let
us now turn to find out $T_{h}\dot{S_{h}}$:
$T_{h}\dot{S_{h}}=\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)\frac{d}{dt}\left(\frac{2\Omega_{n-1}\tilde{r}_{A}^{n-1}}{4G_{n+1}}\right).$
(15)
After some simplification and using Eq. (12) we get
$T_{h}\dot{S_{h}}=(n-1)\Omega_{n-1}H(\rho+P){\tilde{r}_{A}}^{n-1}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(16)
As we argued above the term
$\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)$ is positive to
ensure $T_{h}>0$, however, in the accelerating universe the dominant energy
condition may violate, $\rho+P<0$. This indicates that the second law of
thermodynamics ,$\dot{S_{h}}\geq 0$, does not hold. Then the question arises,
“will the generalized second law of thermodynamics,
$\dot{S_{h}}+\dot{S_{m}}\geq 0$, can be satisfied on the brane?” The entropy
of matter fields inside the apparent horizon, $S_{m}$, can be related to its
energy $E=\rho V$ and pressure $P$ in the horizon by the Gibbs equation Pavon2
$T_{m}dS_{m}=d(\rho V)+PdV=Vd\rho+(\rho+P)dV,$ (17)
where $T_{m}$ is the temperature of the energy inside the horizon. We limit
ourselves to the assumption that the thermal system bounded by the apparent
horizon remains in equilibrium so that the temperature of the system must be
uniform and the same as the temperature of its boundary. This requires that
the temperature $T_{m}$ of the energy inside the apparent horizon should be in
equilibrium with the temperature $T_{h}$ associated with the apparent horizon,
so we have $T_{m}=T_{h}$Pavon2 . This expression holds in the local
equilibrium hypothesis. If the temperature of the fluid differs much from that
of the horizon, there will be spontaneous heat flow between the horizon and
the fluid and the local equilibrium hypothesis will no longer hold. Therefore
from the Gibbs equation (17) we can obtain
$T_{h}\dot{S_{m}}=(n-1)\Omega_{n-1}{\tilde{r}_{A}}^{n-2}\dot{\tilde{r}}_{A}(\rho+P)-(n-1)\Omega_{n-1}{\tilde{r}_{A}}^{n-1}H(\rho+P).$
(18)
To check the generalized second law of thermodynamics, we have to examine the
evolution of the total entropy $S_{h}+S_{m}$. Adding equations (16) and (18),
we get
$T_{h}(\dot{S_{h}}+\dot{S_{m}})=\frac{1}{2}(n-1)\Omega_{n-1}{\tilde{r}_{A}}^{n-2}\dot{\tilde{r}}_{A}(\rho+P)=\frac{A}{2}(\rho+P)\dot{\tilde{r}}_{A}.$
(19)
where $A>0$ is the area of apparent horizon. Substituting
$\dot{\tilde{r}}_{A}$ from Eq. (12) into (19) we get
$T_{h}(\dot{S_{h}}+\dot{S_{m}})=2\pi
G_{n+1}A{\tilde{r}_{A}}^{2}H(\rho+P)^{2}.$ (20)
The right hand side of the above equation cannot be negative throughout the
history of the universe. Hence we have $\dot{S_{h}}+\dot{S_{m}}\geq 0$ which
guarantees that the generalized second law of thermodynamics is fulfilled in a
region enclosed by the apparent horizon on the brane embedded in the Minkowski
bulk.
### II.2 Brane embedded in AdS bulk
In the previous subsection we assumed that the bulk cosmological constant is
absent and hence we saw that no localization of gravity happens on the brane.
Let us now leave that assumption by taking $\Lambda_{n+1}<0$, which is the
case of the real RS II braneworld scenario. Using Eq. (7) the Friedmann
equation (6) can be written as
$\sqrt{H^{2}+\frac{k}{a^{2}}+\frac{1}{\ell^{2}}}=\frac{4\pi
G_{n+1}}{n-1}\rho.$ (21)
In terms of the apparent horizon radius we have
$\rho=\frac{n-1}{4\pi
G_{n+1}}\sqrt{\frac{1}{{\tilde{r}_{A}}^{2}}+\frac{1}{\ell^{2}}}.$ (22)
Taking the derivative of the above equation with respect to the cosmic time
and using the continuity equation (11), one gets
$\dot{\tilde{r}}_{A}=\frac{4\pi}{\ell}G_{n+1}H(\rho+P){\tilde{r}_{A}}^{2}\sqrt{{\tilde{r}_{A}}^{2}+\ell^{2}}.$
(23)
The entropy expression associated with the apparent horizon in the RS II
braneworld with negative bulk cosmological constant can be obtained as Cai4 ;
Shey1
$S_{h}=\frac{(n-1)\ell\Omega_{n-1}}{2G_{n+1}}{\displaystyle\int^{\tilde{r}_{A}}_{0}\frac{\tilde{r}_{A}^{n-2}}{\sqrt{\tilde{r}_{A}^{2}+\ell^{2}}}d\tilde{r}_{A}}.$
(24)
After the integration we have
$S_{h}=\frac{2\Omega_{n-1}{\tilde{r}_{A}}^{n-1}}{4G_{n+1}}\times{}_{2}F_{1}\left(\frac{n-1}{2},\frac{1}{2},\frac{n+1}{2},-\frac{{\tilde{r}_{A}}^{2}}{\ell^{2}}\right),$
(25)
where ${}_{2}F_{1}(a,b,c,z)$ is the hypergeometric function. It is worth
noticing when $\tilde{r}_{A}\ll\ell$, which physically means that the size of
the extra dimension is very large if compared with the apparent horizon
radius, one recovers the area formula for the entropy on the brane given in
Eq. (14). This is an expected result since in this regime we have a quasi-
Minkowski bulk and we have shown in the previous subsection that for a RS II
brane embedded in the Minkowski bulk, the entropy on the brane follows the
$(n+1)$-dimensional area formula in the bulk. Next we turn to calculate
$T_{h}\dot{S_{h}}$:
$\displaystyle T_{h}\dot{S_{h}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)\frac{d}{dt}\left[\frac{\Omega_{n-1}{\tilde{r}_{A}}^{n-1}}{2G_{n+1}}\times{}_{2}F_{1}\left(\frac{n-1}{2},\frac{1}{2},\frac{n+1}{2},-\frac{{\tilde{r}_{A}}^{2}}{\ell^{2}}\right)\right]$
(26) $\displaystyle=$
$\displaystyle\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)\frac{(n-1)\ell\Omega_{n-1}}{2G_{n+1}}\frac{{\tilde{r}_{A}}^{n-2}\dot{\tilde{r}}_{A}}{\sqrt{\tilde{r}_{A}^{2}+\ell^{2}}}.$
Using Eq. (23), after some simplification we obtain again
$T_{h}\dot{S_{h}}=(n-1)\Omega_{n-1}H(\rho+P){\tilde{r}_{A}}^{n-1}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(27)
Adding equation (27) with Gibbs equation (18), we reach
$T_{h}(\dot{S_{h}}+\dot{S_{m}})=\frac{1}{2}(n-1)\Omega_{n-1}{\tilde{r}_{A}}^{n-2}\dot{\tilde{r}}_{A}(\rho+P)=\frac{A}{2}(\rho+P)\dot{\tilde{r}}_{A}.$
(28)
Substituting $\dot{\tilde{r}}_{A}$ from Eq. (23) into (28) we get
$T_{h}(\dot{S_{h}}+\dot{S_{m}})=\frac{2\pi}{\ell}G_{n+1}A{\tilde{r}_{A}}^{2}\sqrt{\tilde{r}_{A}^{2}+\ell^{2}}\
H(\rho+P)^{2}.$ (29)
The right hand side of the above equation cannot be negative throughout the
history of the universe, which means that $\dot{S_{h}}+\dot{S_{m}}\geq 0$
always holds. This indicates that the generalized second law of thermodynamics
is fulfilled in the RS II braneworld embedded in the AdS bulk.
## III GSL of Thermodynamics in DGP braneworld
In the previous section, we have studied the validity of the generalized
second law of thermodynamics in RS II braneworlds embedded in
$(n+1)$-dimensional Minkowski and AdS bulks. In this section, we would like to
extend the discussion to the DGP braneworld in which the intrinsic curvature
term on the brane is included in the action. The generalized Friedmann
equation for the DGP model is given by Shey1
$\epsilon\sqrt{H^{2}+\frac{k}{a^{2}}-\frac{2\kappa_{n+1}^{2}\Lambda_{n+1}}{n(n-1)}-\frac{\mathcal{C}}{a^{n}}}=-\frac{\kappa_{n+1}^{2}}{4\kappa_{n}^{2}}(n-2)(H^{2}+\frac{k}{a^{2}})+\frac{\kappa_{n+1}^{2}}{2(n-1)}\rho,$
(30)
where $\kappa_{n}^{2}=8\pi G_{n}$ and $\epsilon=\pm 1$. For later convenience
we choose $\epsilon=1$. Taking the limit $\kappa_{n}\to\infty$, while keeping
$\kappa_{n+1}$ finite, the equation (30) reduces to the Friedmann equation (6)
in the RS II braneworld. Again, we are interested in studying DGP braneworlds
embedded in the Minkowski and AdS bulks, and we set $\mathcal{C}=0$.
### III.1 Brane embedded in Minkowski bulk
In the Minkowski bulk, $\Lambda_{n+1}=0$, and the Friedmann equation (30)
reduces to the form
$\sqrt{H^{2}+\frac{k}{a^{2}}}=-\frac{\kappa_{n+1}^{2}}{4\kappa_{n}^{2}}(n-2)(H^{2}+\frac{k}{a^{2}})+\frac{\kappa_{n+1}^{2}}{2(n-1)}\rho.$
(31)
In terms of the apparent horizon radius, we can rewrite this equation in the
form
$\rho=\frac{(n-1)(n-2)}{2\kappa_{n}^{2}}\frac{1}{{\tilde{r}_{A}}^{2}}+\frac{2(n-1)}{\kappa_{n+1}^{2}}\frac{1}{{\tilde{r}_{A}}}.$
(32)
Now, differentiating equation (32) with respect to the cosmic time and using
the continuity equation we get
$\dot{\tilde{r}}_{A}=4\pi{\tilde{r}_{A}^{2}}H(\rho+P)\left(\frac{n-2}{2G_{n}\tilde{r}_{A}}+\frac{1}{G_{n+1}}\right)^{-1}.$
(33)
One can see from the above equation that $\dot{\tilde{r}}_{A}>0$ provided that
the dominant energy condition, $\rho+P>0$, holds. However this is not always
the case in an accelerating universe. The entropy expression associated with
the apparent horizon in the pure DGP braneworld can be obtained from the
connection between Friedmann equation and the first law of thermodynamics on
the apparent horizon Shey1
$\displaystyle S_{h}$ $\displaystyle=$
$\displaystyle\frac{(n-1)\Omega_{n-1}{\tilde{r}_{A}}^{n-2}}{4G_{n}}+\frac{2\Omega_{n-1}{\tilde{r}_{A}}^{n-1}}{4G_{n+1}}.$
(34)
It is interesting to note that in this case the entropy can be regarded as a
sum of two area formulas; one (the first term) corresponds to the gravity on
the brane and the other (the second term) corresponds to the gravity in the
bulk. This indeed reflects the fact that there are two gravity terms in the
action of DGP model. Next we turn to calculate $T_{h}\dot{S_{h}}$:
$\displaystyle T_{h}\dot{S_{h}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)\frac{d}{dt}\left(\frac{(n-1)\Omega_{n-1}{\tilde{r}_{A}}^{n-2}}{4G_{n}}+\frac{2\Omega_{n-1}{\tilde{r}_{A}}^{n-1}}{4G_{n+1}}\right)$
(35) $\displaystyle=$
$\displaystyle\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)(n-1)\Omega_{n-1}\left(\frac{(n-2){\tilde{r}_{A}}^{n-3}}{4G_{n}}+\frac{{\tilde{r}_{A}}^{n-2}}{2G_{n+1}}\right)\dot{\tilde{r}}_{A}.$
Using Eq. (33), we obtain
$T_{h}\dot{S_{h}}=(n-1)\Omega_{n-1}H(\rho+P){\tilde{r}_{A}}^{n-1}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(36)
To check the generalized second law of thermodynamics, we have to examine the
evolution of the total entropy $S_{h}+S_{m}$. Combining equations (36) with
Gibbs equation (18), again we get
$T_{h}(\dot{S_{h}}+\dot{S_{m}})=\frac{A}{2}(\rho+P)\dot{\tilde{r}}_{A}.$ (37)
Substituting $\dot{\tilde{r}}_{A}$ from Eq. (33) into (37) we get
$T_{h}(\dot{S_{h}}+\dot{S_{m}})=2\pi
A{\tilde{r}_{A}}^{2}H(\rho+P)^{2}\left(\frac{n-2}{2G_{n}\tilde{r}_{A}}+\frac{1}{G_{n+1}}\right)^{-1}.$
(38)
The right hand side of the above equation is always positive throughout the
history of the universe. Therefore the generalized second law of
thermodynamics $\dot{S_{h}}+\dot{S_{m}}\geq 0$ is fulfilled in the DGP
braneworld embedded in the Minkowski bulk.
### III.2 Brane embedded in AdS bulk
For the AdS bulk with $\Lambda_{n+1}<0$, we can write the Friedmann equation
(30) in the form
$\sqrt{H^{2}+\frac{k}{a^{2}}+\frac{1}{\ell^{2}}}=-\frac{\kappa_{n+1}^{2}}{4\kappa_{n}^{2}}(n-2)(H^{2}+\frac{k}{a^{2}})+\frac{\kappa_{n+1}^{2}}{2(n-1)}\rho,$
(39)
where we have used Eq. (7). In terms of the apparent horizon radius, this
equation can be rewritten into
$\rho=\frac{(n-1)(n-2)}{2\kappa_{n}^{2}}\frac{1}{{\tilde{r}_{A}}^{2}}+\frac{2(n-1)}{\kappa_{n+1}^{2}}\sqrt{\frac{1}{{\tilde{r}_{A}}^{2}}+\frac{1}{\ell^{2}}}.$
(40)
If one takes the derivative of the equation (40) with respect to the cosmic
time, after using Eqs. (7) and (11), one gets
$\dot{\tilde{r}}_{A}=4\pi{\tilde{r}_{A}^{2}}H(\rho+P)\left(\frac{n-2}{2G_{n}\tilde{r}_{A}}+\frac{\ell}{G_{n+1}\sqrt{{\tilde{r}_{A}}^{2}+\ell^{2}}}\right)^{-1}.$
(41)
When the dominant energy condition holds, $\dot{\tilde{r}}_{A}>0$. The entropy
expression associated with the apparent horizon of the DGP brane embedded in
the AdS bulk can be extracted through relating the Friedmann equation to the
first law of thermodynamics on the apparent horizon Shey1
$\displaystyle S_{h}$ $\displaystyle=$
$\displaystyle(n-1)\Omega_{n-1}{\displaystyle\int^{\tilde{r}_{A}}_{0}\left(\frac{(n-2){\tilde{r}_{A}}^{n-3}}{4G_{n}}+\frac{\ell}{2G_{n+1}}\frac{{\tilde{r}_{A}}^{n-2}}{\sqrt{{\tilde{r}_{A}}^{2}+\ell^{2}}}\right)d\tilde{r}_{A}}.$
(42)
After integration it reads
$S_{h}=\frac{(n-1)\Omega_{n-1}{\tilde{r}_{A}}^{n-2}}{4G_{n}}+\frac{2\Omega_{n-1}{\tilde{r}_{A}}^{n-1}}{4G_{n+1}}\times{}_{2}F_{1}\left(\frac{n-1}{2},\frac{1}{2},\frac{n+1}{2},-\frac{{\tilde{r}_{A}}^{2}}{\ell^{2}}\right).$
(43)
Again it is interesting to see that in the warped DGP brane model embedded in
the AdS bulk, the entropy associated with the apparent horizon on the brane
has two parts. The first part follows the $n$-dimensional area law on the
brane and the second part is the same as the entropy expression obtained in RS
II model. The calculation of $T_{h}\dot{S_{h}}$ yields
$\displaystyle T_{h}\dot{S_{h}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)(n-1)\Omega_{n-1}\left[\frac{(n-2){\tilde{r}_{A}}^{n-3}}{4G_{n}}+\frac{\ell}{2G_{n+1}}\frac{{\tilde{r}_{A}}^{n-2}}{\sqrt{{\tilde{r}_{A}}^{2}+\ell^{2}}}\right]\dot{\tilde{r}}_{A}.$
(44)
Substituting $\dot{\tilde{r}}_{A}$ from Eq. (41), we obtain
$T_{h}\dot{S_{h}}=(n-1)\Omega_{n-1}H(\rho+P){\tilde{r}_{A}}^{n-1}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(45)
Adding equation (45) to (18), one gets
$T_{h}(\dot{S_{h}}+\dot{S_{m}})=\frac{A}{2}(\rho+P)\dot{\tilde{r}}_{A}.$ (46)
Inserting $\dot{\tilde{r}}_{A}$ from Eq. (41) into (46) we reach
$T_{h}(\dot{S_{h}}+\dot{S_{m}})=2\pi
A{\tilde{r}_{A}}^{2}H(\rho+P)^{2}\left[\frac{n-2}{2G_{n}\tilde{r}_{A}}+\frac{\ell}{G_{n+1}\sqrt{{\tilde{r}_{A}}^{2}+\ell^{2}}}\right]^{-1},$
(47)
which cannot be negative throughout the history of the universe and hence the
general second law of thermodynamics, $\dot{S_{h}}+\dot{S_{m}}\geq 0$, is
always protected on the DGP brane embedded in the AdS bulk.
## IV Summary and discussions
To conclude, we have investigated the validity of the generalized second law
of thermodynamics for the $(n-1)$-dimensional brane embedded in the
$(n+1)$-dimensional bulk. In the braneworld, the apparent horizon entropy was
extracted through the relation between the Friedmann equation to the first law
of thermodynamics Shey1 . We have examined the time evolution of the derived
apparent horizon entropy together with the entropy of matter fields enclosed
inside the apparent horizon on the brane. We have shown that the extracted
apparent horizon entropy through the connection between Friedmann equation and
the first law of thermodynamics satisfies the generalized second law of
thermodynamics, regardless of whether there is the intrinsic curvature term on
the brane or a cosmological constant in the bulk. The validity of the
generalized second law of thermodynamics on the brane further supports the
thermodynamical interpretation of gravity and provides more confidence on the
profound connection between gravity and thermodynamics.
Finally, we must mention that as one can see from Eqs. (16), (27), (36) and
(45), the variation of the horizon entropy takes the same form in different
braneworlds. This fact sheds the light on holography. The details of the shape
of the Hubble parameter (or the Friedmann equation) differ in different
braneworld models; this is the bulk effect. However, when they project on the
boundary, on the horizon entropy, these differences are simplified, which just
hid in the $H$, $\dot{\tilde{r}}_{A}$ etc, while the information on the
boundary (the horizon entropy) evolves in the same form. This is similar to
the topological black holes, different topology will not change the entropy
form, $S=A/4$, on the black hole horizon.
###### Acknowledgements.
We thank the anonymous referee for constructive comments. This work has been
supported financially by Research Institute for Astronomy and Astrophysics of
Maragha, Iran. The work of B. W. was partially supported by NNSF of China,
Shanghai Science and Technology Commission and Shanghai Education Commission.
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|
arxiv-papers
| 2008-11-27T08:10:51 |
2024-09-04T02:48:59.023266
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Sheykhi and Bin Wang",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/0811.4477"
}
|
0811.4478
|
# Generalized second law of thermodynamics in Gauss-Bonnet braneworld
Ahmad Sheykhi 1,2111sheykhi@mail.uk.ac.ir and Bin Wang 3222wangb@fudan.edu.cn
1Department of Physics, Shahid Bahonar University, P.O. Box 76175, kerman,
Iran
2Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),
Maragha, Iran
3 Department of Physics, Fudan University, Shanghai 200433, China
###### Abstract
We investigate the validity the generalized second law of thermodynamics in a
general braneworld model with curvature correction terms on the brane and in
the bulk, respectively. Employing the derived entropy expression associated
with the apparent horizon, we examine the time evolution of the total entropy,
including the derived entropy of the apparent horizon and the entropy of the
matter fields inside the apparent horizon. We show that the generalized second
law of thermodynamics is fulfilled on the $3$-brane embedded in the $5$D
spacetime with curvature corrections.
It was first pointed out in Jac that the hyperbolic second order partial
differential Einstein equation has a predisposition to the first law of
thermodynamics. This profound connection between the first law of
thermodynamics and the gravitational field equations has been extensively
observed in various gravity theories Elin ; Cai1 ; Pad . Recently the study on
the connection between thermodynamics and gravity has been generalized to the
cosmological situations Cai2 ; Cai3 ; CaiKim ; Fro ; Wang , where it was shown
that the differential form of the Friedmann equation on the apparent horizon
in the FRW universe can be rewritten in the form of the first law of
thermodynamics. The extension of this connection has also been carried out in
the braneworld cosmology Cai4 ; Shey1 ; Shey2 . The deep connection between
the gravitational equation describing the gravity in the bulk and the first
law of thermodynamics on the apparent horizon reflects some deep ideas of
holography.
Besides examining the validity of the thermodynamical interpretation of
gravity by expressing the gravitational field equations into the first law of
thermodynamics on the apparent horizon in different spacetimes, it is of great
interest to examine other thermodynamical principles if the thermodynamical
interpretation of gravity really holds and is a generic feature. This is
especially interesting in the braneworld. In the braneworld, the gravity is no
longer Einstein gravity so that the horizon entropy does not satisfy the usual
area law. The entropy on the apparent horizon in the braneworld was extracted
through connecting the gravitational equation to the first law of
thermodynamics on the apparent horizon Shey1 ; Shey2 . Whether this obtained
entropy satisfies general thermodynamical principles is a question to be
asked. In this work we are going to investigate the generalized second law of
thermodynamics by examining the evolution of the apparent horizon entropy
deduced through the connection between gravity and the first law of
thermodynamics together with the matter fields’ entropy inside the apparent
horizon. If the thermodynamical interpretation of gravity is correct, the
extracted apparent horizon entropy from the profound connection between
gravitational equations and the first law of thermodynamics should satisfy the
generalized second law of thermodynamics.
The generalized second law of thermodynamics is an important principle in
governing the nature. Recently the generalized second law in the accelerating
universe enveloped by the apparent horizon has been investigated in wang1 ;
wang2 . Using the general expression of temperature at apparent horizon of FRW
universe, it has been shown that the generalized second law holds in Einstein,
Gauss-Bonnet and more general lovelock gravity akbar . Other studies on the
generalized second law of thermodynamics have been done in Pavon2 ; Other ;
Setare . Here we will consider a general braneworld models with correction
terms, such as a 4D scalar curvature from induced gravity on the brane, and a
5D Gauss-Bonnet curvature term in the bulk. With these correction terms,
especially including a Gauss-Bonnet correction to the 5D action, we have the
most general action with second-order field equations in 5D lovelock , which
provides the most general model for the braneworld scenarios RS ; DGP . In an
effective action approach to the string theory, the Gauss-Bonnet term
corresponds to the leading order quantum corrections to gravity, its presence
guarantees a ghost-free actionzwiebach . We will first adopt the method
developed in Shey2 to extract the entropy expression associated with the
apparent horizon from the first law of thermodynamics in Gauss-Bonnet
braneworld. We will check the time evolution of the total entropy, including
the entropy of the apparent horizon together with the entropy of the matter
fields inside the apparent horizon.
We start with the following action
$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{1}{2{\kappa_{5}}^{2}}\int{d^{5}x\sqrt{-{g}}\left({R}-2\Lambda+\alpha\mathcal{L}_{GB}\right)}$
$\displaystyle+\frac{1}{2{\kappa_{4}}^{2}}\int{d^{4}x\sqrt{-\widetilde{g}}\widetilde{R}}+\int{d^{4}x\sqrt{-\widetilde{g}}(\mathcal{L}_{m}-2\lambda)},$
where $\Lambda<0$ is the bulk cosmological constant and
$\displaystyle{\mathcal{L}}_{GB}=R^{2}-4R^{AB}R_{AB}+R^{ABCD}R_{ABCD},$ (1)
is the Gauss-Bonnet correction term. Quantities on the brane are expressed
with tilde to be distinguished from those in the bulk. $\kappa_{4}$ and
$\kappa_{5}$ are the gravitational constants on the brane and in the bulk and
can be written respectively as
$\kappa_{4}^{2}=8\pi G_{4},\ \ \kappa_{5}^{2}=8\pi G_{5}.$ (2)
The last term in the action corresponds to the matter content.
$\mathcal{L}_{m}$ is the Lagrangian density of the brane matter fields, and
$\lambda$ is the brane tension. Hereafter we assume that the brane
cosmological constant is zero (if it does not vanish, one can absorb it to the
stress-energy tensor of matter fields on the brane). We assume that there are
no sources in the bulk other than $\Lambda$.
The field equations can be obtained by varying the action with respect to the
bulk metric $g_{AB}$. The result is
$G_{AB}+\Lambda g_{AB}+2\alpha H_{AB}={\kappa_{5}}^{2}T_{AB}\delta(y),$
where we have assumed that the brane is located at $y=0$ with $y$ being the
coordinate in the extra dimension, and
$H_{AB}=RR_{AB}-2R_{A}{}^{C}R_{BC}-2R^{CD}R_{ACBD}+R_{A}{}^{CDE}R_{BCDE}-\textstyle{1\over
4}g_{AB}{\cal L}_{GB},$ (3)
is the second-order Lovelock tensor. The energy-momentum tensor can be
decomposed into two parts
${T}_{AB}=\widetilde{T}_{AB}-\frac{1}{{\kappa_{4}}^{2}}\widetilde{G}_{AB}$,
where $\widetilde{T}_{AB}$ is the energy-momentum tensor describing the matter
confined to the 3-brane which we assume in the form of a perfect fluid,
$\widetilde{T}_{AB}=(\rho+P)u_{A}u_{B}+P\widetilde{g}_{AB}$, in the homogenous
and isotropic universe on the brane, where $u^{A}$, $\rho$, and $P$, are the
fluid velocity ($u^{A}u_{A}=-1$), energy density and pressure respectively.
The possible contribution of nonzero brane tension $\lambda$ will be assumed
to be included in $\rho$ and $P$. A homogeneous and isotropic brane at fixed
coordinate position $y=0$ in the bulk is described by the line element
$ds^{2}=-N^{2}(t,y)dt^{2}+A^{2}(t,y)\gamma_{ij}dx^{i}dx^{j}+B^{2}(t,y)dy^{2},$
where $\gamma_{ij}$ is a maximally symmetric $3$-dimensional metric for the
surface ($t$=const., $y$=const.), whose spatial curvature is parameterized by
$k=-1,0,1$. On every hypersurface ($y$=const), we have the metric of a FRW
cosmological model. The metric coefficient $N$ is chosen so that, $N(t,0)=1$
and $t$ is the cosmic time on the brane. The presence of the four-dimensional
curvature scalar in the gravitational action does not affect the bulk
equations. If we define
$\Phi=\frac{1}{N^{2}}\frac{\dot{A}^{2}}{A^{2}}-\frac{1}{B^{2}}\frac{A^{\prime\,2}}{A^{2}}+\frac{k}{A^{2}}$
(4)
then the field equation reduces to germani
$\displaystyle\frac{\dot{A}}{A}\frac{N^{\prime}}{N}+\frac{A^{\prime}}{A}\frac{\dot{B}}{B}-\frac{\dot{A}^{\prime}}{A}=0,$
$\displaystyle\Phi+2\alpha\Phi^{2}+\frac{1}{\ell^{2}}-\frac{\mathcal{C}}{A^{4}}=0,$
(5)
where $\ell^{2}\equiv-6/\Lambda$ and $\mathcal{C}$ is an integration constant
which is related to the mass of the bulk black hole. In the above equations
dot and prime denote derivatives with respect to $t$ and $y$, respectively.
Integrating the $(00)$ component of the field equation across the brane and
imposing $\mathbb{Z}_{2}$ symmetry, we have the jump across the brane kofin
$\displaystyle\frac{2{\kappa_{4}}^{2}}{{\kappa_{5}}^{2}}\left[1+4\alpha\left(H^{2}+\frac{k}{a^{2}}-\frac{A^{\prime\,2}_{+}}{3a^{2}b^{2}}\right)\right]\frac{A^{\prime}_{+}}{ab}=-\frac{\kappa^{2}_{4}}{3}\rho+H^{2}+\frac{k}{a^{2}},$
(6)
where $a=A(t,0)$, $b=B(t,0)$ and $2A^{\prime}_{+}=-2A^{\prime}_{-}$ is the
discontinuity of the first derivative. $H=\dot{a}/a$ is the Hubble parameter
on the brane. Inserting $\Phi$ into Eq. (6) we can obtain the generalized
Friedmann equation on the brane kofin
$\displaystyle\epsilon\frac{2{\kappa_{4}}^{2}}{{\kappa_{5}}^{2}}\left[1+\frac{8}{3}\alpha\left(H^{2}+{k\over
a^{2}}+{\Phi_{0}\over 2}\right)\right]\left(H^{2}+{k\over
a^{2}}-\Phi_{0}\right)^{1/2}=-\frac{\kappa^{2}_{4}}{3}\rho+H^{2}+{k\over
a^{2}},$ (7)
where $\Phi_{0}=\Phi(t,0)$ and $\epsilon=\pm 1$. For later convenience we
choose $\epsilon=-1$. This Friedmann equation contains various special cases
discussed extensively in the literature. The DGP braneworld is the limiting
case when $\alpha=0$, while the RS II braneworld can be reproduced in the
limit $\kappa_{4}\to\infty$ and $\alpha=0$. The pure Gauss-Bonnet braneworld
is the case with $\kappa_{4}\to\infty$. To have further understanding about
the nature of the apparent horizon we rewrite more explicitly, the metric of
homogenous and isotropic FRW universe on the brane in the form
$ds^{2}={h}_{\mu\nu}dx^{\mu}dx^{\nu}+\tilde{r}^{2}d{\Omega_{2}}^{2},$ (8)
where $\tilde{r}=a(t)r$, $x^{0}=t,x^{1}=r$, the two-dimensional metric
$h_{\mu\nu}$=diag $(-1,a^{2}/(1-kr^{2}))$, and $d\Omega_{2}$ is the metric of
two-dimensional unit sphere. Then, the dynamical apparent horizon, a
marginally trapped surface with vanishing expansion, is determined by the
relation $h^{\mu\nu}\partial_{\mu}\tilde{r}\partial_{\nu}\tilde{r}=0$, which
implies that the vector $\nabla\tilde{r}$ is null on the apparent horizon
surface. The apparent horizon has been argued to be a causal horizon for a
dynamical spacetime and is associated with gravitational entropy and surface
gravity Hay2 ; Bak . The explicit evaluation of the apparent horizon for the
FRW universe gives the apparent horizon radius
$\tilde{r}_{A}=\frac{1}{\sqrt{H^{2}+k/a^{2}}}.$ (9)
The associated surface gravity on the apparent horizon can be defined as
$\kappa=\frac{1}{\sqrt{-h}}\partial_{a}\left(\sqrt{-h}h^{ab}\partial_{ab}\tilde{r}\right).$
(10)
Then one can easily show that the surface gravity at the apparent horizon of
FRW universe can be written as
$\kappa=-\frac{1}{\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(11)
The associated temperature on the apparent horizon can then be defined as
$T_{h}=\frac{|\kappa|}{2\pi}=\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(12)
where $\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}<1$ ensures that the
temperature is positive. Recently the connection between temperature on the
apparent horizon and the Hawking radiation has been observed in cao . Hawking
radiation is an important quantum phenomenon of black hole, which is closely
related to the existence of event horizon of black hole. The cosmological
event horizon of de Sitter space has the Hawking radiation with thermal
spectrum as well. Using the tunneling approach proposed by Parikh and Wilczek,
the authors of cao showed that there is indeed a Hawking radiation with a
finite temperature, for locally defined apparent horizon of Friedmann-
Robertson-Walker universe with any spatial curvature. This gives more solid
physical implication of the temperature associated with the apparent horizon.
Now we turn to apply these results to investigate the validity of the
generalized second law of thermodynamics in the Gauss-Bonnet braneworld
scenario. We apply the approach we developed in Shey1 to find the expression
of entropy associated with the horizon geometry in Gauss-Bonnet braneworld.
First of all, we assume that there is no black hole in the bulk and thus
$\mathcal{C}=0$. Inserting this condition into Eq. (5), we get
$\Phi=\Phi_{0}=\frac{1}{4\alpha}\left(-1+\sqrt{1-\frac{8\alpha}{\ell^{2}}}\right)=\rm{const}.$
This condition also implies $\alpha<\ell^{2}/8$. In terms of the apparent
horizon radius, we can rewrite the Friedmann equation (7) into
$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\frac{3}{8\pi
G_{4}}\frac{1}{{\tilde{r}_{A}}^{2}}+\frac{3}{4\pi
G_{5}}\left(\frac{1}{{\tilde{r}_{A}}^{2}}-\Phi_{0}\right)^{1/2}\times\left[1+\frac{8}{3}\alpha\left(\frac{1}{{\tilde{r}_{A}}^{2}}+{\Phi_{0}\over
2}\right)\right].$ (13)
where we have used Eq. (2). Now, differentiating Eq. (13) with respect to the
cosmic time and using the continuity equation
$\dot{\rho}+3H(\rho+P)=0,$ (14)
we get
$\dot{\tilde{r}}_{A}=4\pi
H(\rho+P){\tilde{r}_{A}^{2}}\left[\frac{1}{G_{4}\tilde{r}_{A}}+\frac{1}{G_{5}\sqrt{1-\Phi_{0}\tilde{r}_{A}^{2}}}+\frac{4\alpha}{G_{5}\tilde{r}_{A}^{2}}\left(\frac{2-\Phi_{0}\tilde{r}_{A}^{2}}{\sqrt{1-\Phi_{0}\tilde{r}_{A}^{2}}}\right)\right]^{-1}.$
(15)
One can see from the above equation that $\dot{\tilde{r}}_{A}>0$ provided that
the dominant energy condition, $\rho+P>0$, holds. In our previous work Shey2 ,
we showed that the Friedmann equation in a general braneworld model with
curvature correction terms can be written in the form of the first law of
thermodynamics on the apparent horizon of the brane,
$dE=T_{h}dS_{h}+WdV,$ (16)
where $W=(\rho-P)/2$ is the matter work density Hay2 which is regarded as the
work done by the change of the apparent horizon, $E=\rho V$ is the total
energy of the matter fields on the brane inside a 3-ball with the volume
$V=\Omega_{3}\tilde{r}_{A}^{3}$. The area of the apparent horizon is the area
of the ball on a 2-sphere of radius $\tilde{r}_{A}$, which is expressed as
$A=3\Omega_{3}\tilde{r}_{A}^{2}$ TT . Using the first law we can extract the
expression for the entropy on the apparent horizon in the general Gauss-Bonnet
braneworld Shey2
$\displaystyle S_{h}$ $\displaystyle=$
$\displaystyle\frac{3\Omega_{3}}{2G_{4}}{\displaystyle\int^{\tilde{r}_{A}}_{0}\tilde{r}_{A}d\tilde{r}_{A}}+\frac{3\Omega_{3}}{2G_{5}}{\displaystyle\int^{\tilde{r}_{A}}_{0}\frac{\tilde{r}_{A}^{2}d\tilde{r}_{A}}{\sqrt{1-\Phi_{0}\tilde{r}_{A}^{2}}}}$
(17) $\displaystyle+$
$\displaystyle\frac{6\alpha\Omega_{3}}{G_{5}}{\displaystyle\int^{\tilde{r}_{A}}_{0}\frac{2-\Phi_{0}\tilde{r}_{A}^{2}}{\sqrt{1-\Phi_{0}\tilde{r}_{A}^{2}}}d\tilde{r}_{A}}.$
The explicit form of the entropy can be obtained by integrating (17), which
reads
$\displaystyle S_{h}$ $\displaystyle=$
$\displaystyle\frac{3\Omega_{3}{\tilde{r}_{A}}^{2}}{4G_{4}}+\frac{2\Omega_{3}{\tilde{r}_{A}}^{3}}{4G_{5}}\times{}_{2}F_{1}\left(\frac{3}{2},\frac{1}{2},\frac{5}{2},\Phi_{0}{\tilde{r}_{A}}^{2}\right)$
(18)
$\displaystyle+\frac{6\alpha\Omega_{3}{\tilde{r}_{A}}^{3}}{G_{5}}\left[\Phi_{0}\times{}_{2}F_{1}\left(\frac{3}{2},\frac{1}{2},\frac{5}{2},\Phi_{0}{\tilde{r}_{A}}^{2}\right)+\frac{\sqrt{1-\Phi_{0}\tilde{r}_{A}^{2}}}{{\tilde{r}_{A}}^{2}}\right],$
where ${}_{2}F_{1}(a,b,c,z)$ is a hypergeometric function. The expression
looks complicated. But its physical meaning is clear and includes various
special cases Shey2 . Some remarks are as follows. The first term in (18) has
the form of Bekenstein-Hawking entropy of the 4D Einstein gravity. In fact,
the Einstein-Hilbert term ($\tilde{R}$) on the brane contributes to this area
term. The second term is due to the Einstein-Hilbert term ($R$) in the bulk.
The Gauss-Bonnet correction term in the bulk contributes to the last term. In
the limit $\alpha\rightarrow 0$, this expression reduces to the entropy of
apparent horizon in warped DGP braneworld embedded in a $AdS_{5}$ bulk, while
in the limit $\alpha\rightarrow 0$ and $\Phi_{0}\rightarrow 0$ (or equality
$\ell\rightarrow\infty$), it reduces to the entropy of apparent horizon in
pure DGP braneworld with a Minkowskian bulk Shey1 . Furthermore, taking the
limit $\alpha\rightarrow 0$ and $G_{4}\rightarrow\infty$, while keeping
$G_{5}$ finite, the first and the last terms in (18) vanish and we obtain the
entropy associated with the apparent horizon in RSII braneworld Shey1 .
Finally, keeping $\alpha$ finite, and taking the limit
$G_{4}\rightarrow\infty$ and $\Phi_{0}\rightarrow 0$, one can extract from Eq.
(18) the entropy associated with the apparent horizon on the brane in the
Gauss-Bonnet braneworld with a Minkowskian bulk
$\displaystyle
S_{h}=\frac{2\Omega_{3}{\tilde{r}_{A}}^{3}}{4G_{5}}\left(1+\frac{12\alpha}{{\tilde{r}_{A}}^{2}}\right).$
(19)
This gives an expression of horizon entropy in the Gauss-Bonnet gravity GB ;
Cai2 ; Cai3 ; CaiKim ; Cvetic . This is an expected result. Indeed, because of
the absence of scalar curvature term on the brane and the negative
cosmological constant in the bulk, no localization of gravity happens on the
brane. As a result, the gravity on the brane is still $5$D Gauss-Bonnet
gravity and the brane looks like a domain wall moving in a Minkowski
spacetime. Therefore, the entropy of apparent horizon on the brane still obeys
the entropy formula in the bulk. Note that the factor $2$ in the entropy comes
from the $\mathbb{Z}_{2}$ symmetry of the bulk. These results give a self-
consistency check for the entropy expression (18).
Next we turn to find out $T_{h}\dot{S_{h}}$. It is a matter of calculation to
show that
$T_{h}\dot{S_{h}}=\frac{3\Omega_{3}}{4\pi}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)\left[\frac{1}{G_{4}}+\frac{\tilde{r}_{A}}{G_{5}\sqrt{1-\Phi_{0}\tilde{r}_{A}^{2}}}+\frac{4\alpha}{G_{5}\tilde{r}_{A}}\left(\frac{2-\Phi_{0}\tilde{r}_{A}^{2}}{\sqrt{1-\Phi_{0}\tilde{r}_{A}^{2}}}\right)\right]\dot{\tilde{r}_{A}}.$
(20)
Substituting Eq. (15) and doing some simplifications, we obtain
$T_{h}\dot{S_{h}}=3\Omega_{3}H(\rho+P){\tilde{r}_{A}}^{3}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(21)
As we argued above that the physical positive temperature $T_{h}$ requires the
term $\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)$ to be
positive. However, in the accelerating universe the dominant energy condition
may be violated, $\rho+P<0$, which can make the second law of thermodynamics
,$\dot{S_{h}}\geq 0$, do not hold in general. Then the question arises, “will
the generalized second law of thermodynamics, $\dot{S_{h}}+\dot{S_{m}}\geq 0$,
still holds on the brane?” According to the generalized second law the entropy
of matter fields inside the horizon together with the entropy on the apparent
horizon cannot decrease with time. The entropy of the matter fields inside the
apparent horizon, $S_{m}$, can be related to its energy $E=\rho V$ and
pressure $P$ in the horizon by the Gibbs equation Pavon2
$T_{m}dS_{m}=d(\rho V)+PdV=Vd\rho+(\rho+P)dV,$ (22)
where $T_{m}$ is the temperature of the energy inside the horizon. We limit
ourselves to the assumption that the thermal system enveloped by the apparent
horizon remains in equilibrium so that the temperature of the system must be
uniform and the same as the temperature of its boundary. This requires that
the temperature $T_{m}$ of the energy inside the apparent horizon should be in
equilibrium with the temperature $T_{h}$ associated with the apparent horizon,
so we have $T_{m}=T_{h}$. This expression holds in the local equilibrium
hypothesis. If the temperature of the fluid differs much from that of the
horizon, there will be spontaneous heat flow between the horizon and the fluid
inside and the local equilibrium hypothesis will no longer hold. Therefore
from the Gibbs equation (22) we can obtain
$T_{h}\dot{S_{m}}=(n-1)\Omega_{n-1}{\tilde{r}_{A}}^{n-2}\dot{\tilde{r}}_{A}(\rho+P)-(n-1)\Omega_{n-1}{\tilde{r}_{A}}^{n-1}H(\rho+P).$
(23)
To check the generalized second law of thermodynamics, we have to examine the
evolution of the total entropy $S_{h}+S_{m}$. Adding equations (21) and (23),
we get
$T_{h}(\dot{S_{h}}+\dot{S_{m}})=\frac{3\Omega_{3}}{2}{\tilde{r}_{A}}^{2}\dot{\tilde{r}}_{A}(\rho+P)=\frac{A}{2}(\rho+P)\dot{\tilde{r}}_{A}.$
(24)
Substituting $\dot{\tilde{r}}_{A}$ from Eq. (15) into (24) we get
$T_{h}(\dot{S_{h}}+\dot{S_{m}})=2\pi
A{\tilde{r}_{A}}^{2}H(\rho+P)^{2}\left[\frac{1}{G_{4}\tilde{r}_{A}}+\frac{1}{G_{5}\sqrt{1-\Phi_{0}\tilde{r}_{A}^{2}}}+\frac{4\alpha}{G_{5}\tilde{r}_{A}^{2}}\left(\frac{2-\Phi_{0}\tilde{r}_{A}^{2}}{\sqrt{1-\Phi_{0}\tilde{r}_{A}^{2}}}\right)\right]^{-1}.$
(25)
It is worth noticing that $1-\Phi_{0}\tilde{r}_{A}^{2}>0$, therefore the right
hand side of the above equation cannot be negative throughout the history of
the universe. Hence we have $\dot{S_{h}}+\dot{S_{m}}\geq 0$ indicating that
the generalized second law of thermodynamics is fulfilled in a general
braneworld model with curvature correction terms in a region enclosed by the
apparent horizon.
In summary, we have investigated the validity of the generalized second law of
thermodynamics in a general braneworld model with curvature correction terms,
such as a 4D scalar curvature from induced gravity on the brane, and a 5D
Gauss-Bonnet curvature term in the bulk. Following the method developed in
Shey2 we have extracted the entropy associated with the apparent horizon in a
general Gauss-Bonnet braneworld by relating the gravitational equation to the
first law of thermodynamics on the apparent horizon. We have examined the
total entropy evolution with time, including the derived apparent horizon
entropy and the entropy of matter fields inside the apparent horizon on the
brane. We have shown that the generalized second law of thermodynamics on the
$3$-brane embedded in the $5$D spacetime with curvature correction terms is
fulfilled throughout the history of the universe. The satisfaction of the
generalized second law of thermodynamics provides further confidence on the
thermodynamical interpretation of gravity.
###### Acknowledgements.
This work has been supported financially by Research Institute for Astronomy
and Astrophysics of Maragha, Iran. The work of B. W. was partially supported
by NNSF of China and Shanghai Science and Technology Commission and Shanghai
Education Commission.
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|
arxiv-papers
| 2008-11-27T08:14:03 |
2024-09-04T02:48:59.029132
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Sheykhi and Bin Wang",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/0811.4478"
}
|
0811.4600
|
# Climate Sensitivity and the Response of Temperature to CO2
Arthur P. Smith apsmith@altenergyaction.org
###### Abstract
A review of some of the evidence for the IPCC’s conclusion that doubling CO2
levels will warm Earth significantly, in contrast to the claims of a recent
articlemonckton . Simply looking at raw temperature and CO2 data over the past
150 years gives a transient response of roughly 2 K per doubling, in good
agreement with IPCC conclusions based on far more extensive analysis. The 0.58
K of Ref. monckton, is very unlikely.
###### pacs:
92.60.Ry,92.70.Gt,92.70.Np
## I Introduction
As Spencer Weart has recently notedweart , many scientifically trained people
seek a “straightforward calculation” of anthropogenic global warming, but “the
nature of the climate system inevitably betrays a lover of simple answers.”
The consensus estimates of the climate science community, expressed by the
Intergovernmental Panel on Climate Change (IPCC) after years of struggling
with these complexities, is that doubling atmospheric CO2will result in a
steady-state temperature increase (“equilibrium climate sensitivity”) of 2 to
4.5 K, with a best estimate of 3 K, and very unlikely to be less than 1.5
KIPCCAR4WG1 .
In a recent article in Physics and Societymonckton , Christopher Monckton
presents a number of arguments that lead him to conclude that the IPCC
estimate is wrong, and the correct value for sensitivity should be close to
0.58 K, well within the “very unlikely” range stated by the IPCC. I have
listed elsewhereSmithMoncktonErrors over 100 errors of fact or logic,
misinterpretations, invalid reasoning, or misleading statements in Monckton’s
article. In many cases these invalid claims are not original with Monckton,
but have been addressed repeatedly in the past. For example he uses at least
ten of the “hottest skeptic arguments” found at skepticalscience.com: “It’s
the sun”, “It’s cooling”, surface temperatures or models are unreliable, and
so forth. Original with Monckton are a handful of numerical errors or
misinterpretations of others’ work; I would refer those interested in such
details to the list I have collectedSmithMoncktonErrors ; those who spot other
errors are invited to contact me with details to be added to the list.
The central originality of Monckton’s article, which I will address in the
following, is his breakdown of climate sensitivity into three components, and
his further reasoning about those components. Monckton’s breakdown is implicit
in much of the discussion about forcings and feedbacks in the IPCC
reportIPCCAR4WG1 and review papers such as the one by Bony et alBonyFeedback
. However, it must be emphasized that this breakdown into forcing, base
response, and feedback factors is artificial and strictly valid only in a
perturbative sense. It is useful for understanding what is going on, but it is
not how sensitivity is actually determined. The IPCC consensus on sensitivity
comes from model calculations and attribution studies that look at the
response of the full climate system to changes in greenhouse gases, without
breaking that response into a linearized “base” response and separate
feedbacks. That is, the numerical values for base response and feedbacks, and
to a lesser degree for forcings, the central concerns of Monckton’s article,
are an output from the numerical models, not an input. They are a guide to
understanding what is going on, and little more. So while Monckton’s breakdown
is useful in the sense of being more explicit than is typically done, it is
not in any sense a replication of the manner in which sensitivity is
determined by the IPCC, since he does not derive his numbers from climate
models. In fact in his treatment of the 2007 IPCC numbers he manages to get
both forcing and feedbacks off by 10 to 20%, while coming to roughly the right
final valueSmithMoncktonErrors .
## II Sensitivity Reconsidered, Reconsidered
But it is the “reconsidered” sections of Monckton’s article which merit the
most attention. The entire “radiative forcing reconsidered” section argues not
about forcing at all, but about the temperature changes expected from
forcings. Monckton ends by claiming he can divide the 3.7 W/m2 forcing from
doubling CO2 by a factor of 3 because a certain temperature response is low –
but this has nothing to do with the forcing at all, which is completely
determined by the underlying physics. If anything, this section is an argument
about feedbacks – but it is not phrased in that way, and so at the least is
highly confusing.
Monckton culls a figure from the latest IPCC report (Monckton’s figure 4, IPCC
AR4 WG1 figure 9.1IPCCAR4WG1 ) which is discussed at length in section
9.2.2.1, “Spatial and Temporal Patterns of Response”, but then misinterprets
it. The image is based on estimates of forcing changes from 1890 to 1999. The
strongest pattern is in the greenhouse gas image, because that is where the
largest forcing change has occured. But the surface and low-altitude warming
(or cooling) patterns are essentially the same across all the forcings – as
the IPCC discussion puts it: “Solar forcing results in a general warming of
the atmosphere with a pattern of surface warming that is similar to that
expected from greenhouse gas warming, but in contrast to the response to
greenhouse warming, the simulated solar-forced warming extends throughout the
atmosphere.” The spatial pattern of response between different forcings
differs only in the contrast between lower atmosphere and upper atmosphere:
for solar forcing the warming happens everywhere, while for greenhouse forcing
the upper atmosphere (stratosphere) cools while the surface and lower
atmosphere warm. This differential in temperature change is readily observed,
as Monckton’s figure 6 shows: warming at the surface, and cooling in the
stratosphere. This is observational proof that the sun cannot be behind recent
warming.
The tropical mid-troposphere “hot spot” that Monckton highlights is not a
“fingerprint” of greenhouse gases: it is well known to be a consequence of
higher water vapor levels in a warmer world, whatever the cause of the
warming. As warm air rises, it cools almost adiabatically – this is known as
the “lapse rate”, and stability of the atmosphere ensures that temperatures
fall no faster than this rate with altitude. When air holding water vapor
rises and cools, some of the water condenses and releases heat, resulting in
warmer air at a given altitude, and a lower lapse rate. The strongest effect
should show up in the tropical mid-troposphereSanterTropics , hence a “hot
spot”. The observations are still being disputed, as even Monckton admits by
refering to the wind-based measurements of Allen et al. Whatever the
measurements and theory sort themselves out to on this, note again that
tropical mid-troposphere temperature trends are not a signature of greenhouse
gases, and this whole argument has no bearing on CO2 forcing. Monckton has not
made any case for arbitrarily dividing the forcing by 3 as in his equation 17.
That would require drastically changing the spectroscopic properties of
atmospheric constituents, for which there is certainly no justification in the
arguments presented.
Figure 1: The gray curve is annual average temperature anomaly from the Hadley
centerHadCrut3 , and in red is the corresponding 21-point smoothed curve. The
other three lines plot the logarithm of CO2 concentrations measured at Mauna
LoaMaunaLoa and the 20-year smoothed Law Dome measurementsLawDome multiplied
by appropriate sensitivity-to-doubling numbers and adjusted to a 0 average for
the HadCRUT3 baseline period 1961 to 1990. Other than the sensitivity number
there is no free parameter here; this is entirely derived from observations.
In his second “reconsidered” section, Monckton’s errors of logic and
interpretation are spread thickest, on a matter for which there is again no
real dispute. Similar to (in fact much more so than for) the forcing, the
“base” response of the climate system is tightly constrained by the
spectroscopic properties and temperature profile of the atmosphere, and is
easily calculated in any model. The Soden and Held (2006) review paper refered
to by Bony et alBonyFeedback provides two tables which show this base or
“Planck” response as calculated from a variety of models. The range of what
amounts to the inverse of Monckton’s $\kappa$ parameter is from 3.13 to 3.28
W/K m^2, with a mean of 3.22 and standard deviation of 0.04, or just over 1%.
There is almost no uncertainty over the value of this number, despite the
confusion Monckton fosters.
Finally, on the feedback factor f, and in particular the sum b of the feedback
parameters, Monckton argues that the individual feedbacks must be too high
because adding them plus their standard deviations leads to instability. He
also quotes two papers that suggest that water vapor and cloud feedbacks are
overstated by the climate models. The argument of the first “reconsidered”
section on forcings, while completely irrelevant to forcings, if it had any
validity would reinforce the suggestion of a reduced value for feedbacks.
Nevertheless, Monckton here does the most mystifying thing in his entire
article – he is “prudent and conservative” and retains the same (excessively
high) value for the feedback sum b he has been using throughout the article.
But it is the feedbacks that are the most scientifically uncertain issues, by
far the hardest things for modeling to get right, and the reason you cannot
determine climate sensitivity from a one-page simple straightforward
calculation. Getting the complex water vapor, cloud, lapse rate (convection
and latent heat) and other responses to temperature changes sorted out is what
makes climate modeling so tough. The feedbacks are the source of almost the
entire uncertainty range in the IPCC’s estimate of climate sensitivity (which
also relies on measurements of recent and ancient climate). Somehow Monckton
treats the least certain quantity of the three in his breakdown as the most
certain, while wildly reducing the values of the other two. His final estimate
(equation 30) is not believable on these and other grounds, including his many
other errorsSmithMoncktonErrors .
## III Simpler Evidence
The simplest evidence I have seen that feedbacks are likely to be positive
comes not from calculations but from measurements, dependent on the IPCC’s
radiative forcing calculations (table 2.12IPCCAR4WG1 ) where the forcings from
1750 to 2005 due to all sources except CO2 almost cancel out (the net total is
about 10% of the CO2 effect, with considerable uncertainty). That means an
estimate of transient climate response to increased CO2 can be found by
plotting the historically measured atmospheric CO2 values on the same chart as
the historical temperature anomalies, over the past 150 years as is done in
Figure 1.
The best fit to observed temperature is for a transient response of about 2K
per doubling of CO2. This compares well with the IPCC range of transient
climate response of between 1 K and 3.5 K (see section 9.6.2.3 of IPCC AR4 WG1
IPCCAR4WG1 . The fact that the 20th century rise in temperatures was of almost
exactly the expected size is pretty strong evidence that the IPCC’s transient
and equilibrium climate sensitivity numbers match reality. In particular, the
equilibrium sensitivity is unlikely to be as low as the 1.2 K found with no
feedbacks, and nowhere near the 0.58 K that Monckton claims.
## IV Acknowledgments
I am grateful to several friends and colleagues for comments; this work has
received no funding or support from any source.
## References
* (1) Christopher Monckton: “Climate Sensitivity Reconsidered”, Physics and Society, 37, issue 3, p. 6 (2008)
* (2) Spencer Weart: “Simple Question, Simple Answer … Not”, RealClimate, 8 September 2008, http://www.realclimate.org/index.php/archives/2008/09/simple-question-simple-answer-no/ \- see also Weart’s history website: “The Discovery of Global Warming”, http://www.aip.org/history/climate/
* (3) IPCC, 2007: Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change [Solomon, S., D. Qin, M. Manning, Z. Chen, M. Marquis, K.B. Averyt, M.Tignor and H.L. Miller (eds.)]. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA.
* (4) Arthur Smith, “A detailed list of the errors in Monckton’s July 2008 Physics and Society article”, http://altenergyaction.org/Monckton.html
* (5) S. R. Bony et al.: “How well do we understand and evaluate climate change feedback processes?”, Journal of Climate 19, p. 3445 (2006).
* (6) B. D. Santer et al., “Amplification of Surface Temperature Trends and Variability in the Tropical Atmosphere” Science 309, 1551 (2005).
* (7) Hadley Center temperature series: http://hadobs.metoffice.com/hadcrut3/diagnostics/global/nh+sh/annual
* (8) Mauna Loa CO2 series: ftp://ftp.cmdl.noaa.gov/ccg/co2/trends/co2_annmean_mlo.txt
* (9) Law Dome CO2 series: http://cdiac.ornl.gov/ftp/trends/co2/lawdome.combined.dat
|
arxiv-papers
| 2008-11-27T17:53:34 |
2024-09-04T02:48:59.039294
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Arthur P. Smith",
"submitter": "Arthur Smith",
"url": "https://arxiv.org/abs/0811.4600"
}
|
0811.4665
|
# Rapid Formation of Icy Super-Earths and the Cores of Gas Giant Planets
Scott J. Kenyon Smithsonian Astrophysical Observatory, 60 Garden Street,
Cambridge, MA 02138 e-mail: skenyon@cfa.harvard.edu Benjamin C. Bromley
Department of Physics, University of Utah, 201 JFB, Salt Lake City, UT 84112
e-mail: bromley@physics.utah.edu
###### Abstract
We describe a coagulation model that leads to the rapid formation of super-
Earths and the cores of gas giant planets. Interaction of collision fragments
with the gaseous disk is the crucial element of this model. The gas entrains
small collision fragments, which rapidly settle to the disk midplane.
Protoplanets accrete the fragments and grow to masses $\gtrsim$ 1 M⊕ in $\sim$
1 Myr. Our model explains the mass distribution of planets in the Solar System
and predicts that super-Earths form more frequently than gas giants in low
mass disks.
planetary systems – solar system: formation – planets and satellites:
formation
## 1 INTRODUCTION
Collisional cascades play a central role in planet formation. In current
theory, planets grow from collisions and mergers of km-sized planetesimals in
a gaseous disk. As planets grow, they stir leftover planetesimals along their
orbits to high velocities. Eventually, collisions among planetesimals produce
smaller fragments instead of larger, merged objects. Continued stirring leads
to a cascade of destructive collisions which grinds the leftovers to dust.
This process (i) explains the masses of terrestrial planets (Kenyon & Bromley,
2006) and Kuiper belt objects (Kenyon et al., 2008) and (ii) produces debris
disks similar to those observed around nearby main sequence stars (Wyatt,
2008).
Numerical simulations of icy planet formation suggest the cascade limits the
masses of growing protoplanets to $\sim$ 0.01 M⊕ (Kenyon & Bromley, 2008,
hereafter KB08). This mass is much smaller than the core mass, $\gtrsim$ 0.1–1
M⊕, required for a protoplanet to accrete gas and become a gas giant planet
(Pollack et al., 1996; Inaba et al., 2003; Alibert et al., 2005). Unless icy
protoplanets can accrete collision fragments before the fragments are ground
to dust, these protoplanets cannot grow into gas giant planet cores. Thus,
finding a mechanism to halt the cascade is essential to form gas giant
planets.
Here, we describe how interactions between the fragments and the gaseous disk
can halt the cascade. In our picture, the gas traps small fragments with sizes
of 0.1 mm to 1 m and prevents them from colliding at large velocities. These
fragments then settle rapidly to the disk midplane, where protoplanets can
accrete them. For a broad range of initial conditions, analytic results and
detailed numerical simulations demonstrate that this process yields 1–10 M⊕
cores in 1–2 Myr.
We develop the analytic theory in §2 and derive the conditions needed for
protoplanets to accrete collision fragments and grow to masses of $\sim$ 1 M⊕
in 1–2 Myr. We confirm these estimates in §3 with detailed numerical
calculations. We conclude with a brief discussion in §4.
## 2 PHYSICAL MODEL
The crucial element of our model is the interaction of collision fragments
with the gaseous disk. Fragments larger than the ‘stopping radius’
$r_{s}\approx$ 0.5–2 m at 5–10 AU (Weidenschilling, 1977; Rafikov, 2004),
orbit with the growing protoplanets, independently of the gas. Destructive
collisions among these fragments fuel the collisional cascade. However, the
gas entrains particles with radii $r\lesssim r_{s}$. These fragments orbit
with the gas; thus, their velocity dispersions are small and independent of
massive protoplanets. By trapping small collision fragments, the gas halts the
collisional cascade.
The gas also allows protoplanets to accrete the debris. When the collisional
cascade begins, the mass in leftover planetesimals is $\sim$ 1–10 M⊕. The
cascade grinds all of this mass into small fragments which are trapped by the
gas. Most of the trapped fragments fall through the gas into the midplane of
the disk, where growing protoplanets accrete them. Protoplanets that accrete
$\sim$ 0.1–1 M⊕ before the gas dissipates ($\sim$ 3–10 Myr; Hartmann et al.,
1998; Haisch, Lada, & Lada, 2001; Kennedy & Kenyon, 2009) become gas giants.
Thus, our model yields gas giant cores if (i) the collisional cascade produces
fragments fast enough, (ii) the fragments quickly settle to the midplane, and
(iii) the largest protoplanets rapidly accrete the fragments.
To examine whether this physical model leads to cores with masses of $\sim$ 1
M⊕, we consider the growth of planets in a disk of gas and icy objects around
a star with mass M⋆. Material at a distance $a$ from the central star orbits
with angular frequency $\Omega$ and has surface densities $\Sigma_{s}$
(solids) and $\Sigma_{g}$ (gas). We adopt a solid-to-gas ratio of 1:100 and
$\Sigma_{s}=\Sigma_{s,0}~{}x_{m}~{}a^{-3/2}$, where $\Sigma_{s,0}$ = 2.5 g
cm-2 at 5 AU and $x_{m}$ is a scale factor.
Forming icy protoplanets is the first step in our model. In an ensemble of 1
km planetesimals, collisional growth yields a few 1000 km objects –
‘oligarchs’ – that contain an ever-increasing fraction of the mass in solids
(Ida & Makino, 1993; Wetherill & Stewart, 1993; Rafikov, 2003). From numerical
simulations of planet growth at 30–150 AU, the timescale to produce an
oligarch around a solar-type star is (KB08)
$t_{1000}\sim 10^{5}~{}x_{m}^{-1.15}~{}\left(\frac{a}{\rm
5~{}AU}\right)^{3}~{}{\rm yr}~{}.$ (1)
Thus, oligarchs form at 5 AU before the gas dissipates.
Once oligarchs form, collisions among leftover planetesimals produce copious
amounts of fragments. In the high velocity limit, the collision time for a
planetesimal of mass $M$ in a swarm of icy planetesimals with mass $M$, radius
$r$, and surface density $\Sigma$ is $t_{c}$ $\approx
M/(\Sigma~{}\pi~{}r^{2}~{}\Omega)$ (Goldreich, Lithwick, & Sari, 2004). Thus,
$t_{c}\approx 10^{5}~{}x_{m}^{-1}\left(\frac{r}{\rm
1~{}km}\right)\left(\frac{a}{\rm 5~{}AU}\right)^{3/2}~{}{\rm yr}~{}.$ (2)
Collisions among planetesimals produce debris at a rate $\dot{M}\approx
N~{}\delta M~{}t_{c}^{-1}$, where $N$ is the number of planetesimals of mass
$M$ and $\delta M$ is the mass in fragments produced in a single collision. In
an annulus of width $\delta a$ at distance $a$ from the central star,
$N\approx 2\pi~{}\Sigma~{}a~{}\delta a/M$. If $\sim$ 10% of the mass in each
collision is converted into fragments
$\dot{M_{f}}\approx 4\times 10^{-7}~{}x_{m}^{2}~{}\left(\frac{\rm
5~{}AU}{a}\right)^{7/2}\left(\frac{\delta a}{\rm
0.2~{}AU}\right)\left(\frac{\rm 1~{}km}{r}\right)~{}M_{\oplus}~{}{\rm
yr^{-1}}~{},$ (3)
where we have set the width of the annulus equal to the width of the ‘feeding
zone’ for a 0.1 M⊕ protoplanet (Lissauer, 1987). Thus, disks with
$x_{m}\gtrsim$ 1–2 produce fragments at a rate sufficient to form $\gtrsim$ 1
M⊕ cores in 1–2 Myr.
Most of the mass in fragments settles quickly to the disk midplane. For a
settling time $t_{s}\approx 4x_{m}({\rm 1~{}m}/r)$ yr (Chiang & Goldreich,
1997), fragments with $r\gtrsim$ 0.1 mm reach the midplane on the collision
timescale of $\sim 10^{5}$ yr (Eq. 2). For a size distribution $n(m)$ $\propto
m^{q}$ with $q$ = $-1$ to $-0.8$ (Dohnanyi, 1969; Holsapple & Housen, 2007),
66% to 99% of the total mass in fragments with $r\lesssim$ 1 m settles to the
midplane in $\lesssim 10^{5}$ yr at 5–10 AU.
Oligarchs rapidly accrete fragments in the midplane. The maximum accretion
rate for an oligarch with $M_{o}\sim$ 0.01 M⊕ at 5 AU is $\sim 5\times
10^{-6}$ M⊕ yr-1 (Rafikov, 2004). This maximum rate yields 5 M⊕ cores in 1
Myr. At the onset of the cascade, our simulations suggests oligarchs at 5 AU
accrete at a rate
$\dot{M_{o}}\approx
10^{-6}\left(\frac{M_{o}}{0.01~{}M_{\oplus}}\right)^{2/3}~{}\left(\frac{M_{f}}{1~{}M_{\oplus}}\right)~{}M_{\oplus}~{}{\rm
yr^{-1}}~{},$ (4)
where $M_{f}$ is the total mass in fragments in a feeding zone with width
$\delta a\approx$ 0.2 AU at 5 AU. Thus, protoplanets likely reach masses
$\sim$ 1 M⊕ in 1–2 Myr.
These analytic estimates confirm the basic aspects of our model. In a gaseous
disk with $\Sigma_{g}\approx$ 250 g cm-2, the gas halts the collisional
cascade. Collision fragments entrained by the gas rapidly settle to the
midplane. Protoplanets with masses $\sim$ 0.01 M⊕ can accrete collision
fragments rapidly and grow to masses $\sim$ 1 M⊕ before the gas dissipates in
3–10 Myr.
## 3 NUMERICAL MODEL
To explore this picture in more detail, we calculate the formation of cores
with our hybrid multiannulus coagulation–$n$-body code (Bromley & Kenyon,
2006). In previous calculations, we followed the evolution of objects with
$r\gtrsim r_{s}$; collision fragments with $r\lesssim r_{s}$ were removed by
the collisional cascade (KB08). Here, we include the evolution of small
fragments entrained by the gas. We follow Brauer et al. (2008a) and calculate
the scale height of small particles with $r<r_{s}$ as $H=\alpha H_{g}/[{\rm
min}(St,0.5)(1+St)]$, where $H_{g}$ is the scale height of the gas (Kenyon &
Hartmann, 1987, KB08), $\alpha$ is the turbulent viscosity of the gas, and
$St=r\rho_{s}\Omega/c_{s}\rho_{g}$. In this expression for the Stokes number
($St$), $c_{s}$ is the sound speed of the gas, $\rho_{g}$ is the gas density,
and $\rho_{s}$ is the mass density of a fragment. We assume small particles
have vertical velocity $v=H\Omega$ and horizontal velocity $h=1.6v$.
Protoplanets accrete fragments at a rate $n\sigma v_{rel}$, where $n$ is the
number density of fragments, $\sigma$ is the cross-section (including
gravitational focusing), and $v_{rel}$ is the relative velocity (e.g., Kenyon
& Luu, 1998, Appendix A.2). Although this approximation neglects many details
of the motion of particles in the gas (Brauer et al., 2008a), it approximates
the dynamics and structure of the fragments reasonably well and allows us to
calculate accretion of fragments by much larger oligarchs.
Using the statistical approach of Safronov (1969), we evolve the masses and
orbits of planetesimals in a set of concentric annuli with widths $\delta
a_{i}$ at distances $a_{i}$ from the central star (KB08). The calculations use
realistic cross-sections (including gravitational focusing) to derive
collision rates (Spaute et al., 1991) and a Fokker-Planck algorithm to derive
gravitational stirring rates (Ohtsuki, Stewart, & Ida, 2002). When large
objects reach a mass $M_{pro}$, we ‘promote’ them into an $n$-body code
(Bromley & Kenyon, 2006). This code follows the trajectories of individual
objects and includes algorithms to allow interactions between the massive
$n$-bodies and less massive objects in the coagulation code.
To assign collision outcomes, we use the ratio of the center of mass collision
energy $Q_{c}$ and the energy needed to eject half the mass of a pair of
colliding planetesimals to infinity $Q_{d}^{*}$. We adopt
$Q_{d}^{*}=Q_{b}r^{\beta_{b}}+Q_{g}\rho_{g}r^{\beta_{g}}$ (Benz & Asphaug,
1999), where $Q_{b}r^{\beta_{b}}$ is the bulk component of the binding energy,
$Q_{g}\rho_{g}r^{\beta_{g}}$ is the gravity component of the binding energy,
and $\rho_{g}$ is the mass density of a planetesimal. The mass of a merged
pair is $M_{1}+M_{2}-M_{ej}$, where the mass ejected in the collision is
$M_{ej}=0.5(M_{1}+M_{2})(Q_{c}/Q_{d}^{*})^{9/8}$ (Kenyon & Luu, 1999).
Consistent with recent $n$-body simulations, we consider two sets of
fragmentation parameters $f_{i}$. Strong planetesimals have $f_{S}$ = ($Q_{b}$
= 1, $10^{3}$, or $10^{5}$ erg g-1, $\beta_{b}\approx$ 0, $Q_{g}$ = 1.5 erg
g-1 cm-1.25, $\beta_{g}$ = 1.25; KB08, Benz & Asphaug, 1999). Weaker
planetesimals have $f_{W}$ = ($Q_{b}$ = $2\times 10^{5}$ erg g-1 cm0.4,
$\beta_{b}\approx-0.4$, $Q_{g}$ = 0.22 erg g-1 cm-1.3, $\beta_{g}$ = 1.3;
Leinhardt & Stewart, 2008).
Our initial conditions are appropriate for a disk around a young star (e.g.
Dullemond & Dominik, 2005; Ciesla, 2007a; Garaud, 2007; Brauer et al., 2008b).
We consider systems of 32 annuli with $a_{i}$ = 5–10 AU and $\delta
a_{i}/a_{i}$ = 0.025. The disk is composed of small planetesimals with radii
ranging from $r_{min}=r_{s}\approx$ 0.5–2 m (Rafikov, 2004) to $r_{0}$ = 1 km,
10 km, or 100 km and an initial mass distribution $n_{i}(M_{ik})\propto
M_{ik}^{-0.17}$. The mass ratio between adjacent bins is
$\delta=M_{ik+1}/M_{ik}$ = 1.4–2 (e.g., Kenyon & Luu, 1998, KB08). Each bin
has the same initial eccentricity $e_{0}=10^{-4}$ and inclination
$i_{0}=e_{0}/2$.
For each combination of $r_{0}$, $f_{i}$, and $x_{m}$ = 1–5, we calculate the
growth of oligarchs with two different approaches to grain accretion. In
models without grain accretion, fragments with $r\lesssim r_{min}$ are ‘lost’
to the grid. Oligarchs cannot accrete these fragments; their masses stall at
$M\lesssim$ 0.1 M⊕. In models with grain accretion, we track the abundances of
fragments with 0.1 mm $\lesssim r\lesssim r_{min}$ which settle to the disk
midplane on short timescales. Oligarchs can accrete these fragments; they grow
rapidly at rates set by the production of collision fragments.
For the gaseous disk, we adopt $\alpha=10^{-4}$, an initial surface density,
$\Sigma_{g,0}$ = 100 $\Sigma_{s,0}~{}x_{m}~{}a^{-3/2}$, and a depletion time
$t_{g}$ = 3 Myr. The surface density at later times is
$\Sigma_{g,t}=\Sigma_{g,0}~{}e^{-t/t_{g}}$. We ignore the migration of
protoplanets from torques between the gas and the planet (Lin & Papaloizou,
1986; Ward, 1997). Alibert et al. (2005) show that migration enhances growth
of protoplanets; thus our approach underestimates the growth time. We also
ignore the radial drift of fragments coupled to the gas. Depending on the
internal structure of the disk, fragments can drift inward, drift outward, or
become concentrated within local pressure maxima or turbulent eddies
(Weidenschilling, 1977; Haghighipour & Boss, 2003; Inaba & Barge, 2006; Masset
et al., 2006; Ciesla, 2007b; Kretke & Lin, 2007; Kato et al., 2008). Here, our
goal is to provide a reasonable first estimate for the growth rates of
protoplanets. We plan to explore the consequences of radial drift in
subsequent papers.
## 4 RESULTS
Fig. 1 shows mass histograms at 1–10 Myr for coagulation calculations without
grain accretion using $r_{0}$ = 1 km and the strong fragmentation parameters
($f_{S}$). After the first oligarchs with $M\sim$ 0.01 M⊕ form at $\sim$ 0.1
Myr, the collisional cascade starts to remove leftover planetesimals from the
grid. Independent of $Q_{b}$, the cascade removes $\sim$ 50% of the initial
mass of the grid in $\sim$ 4 $x_{m}^{-1.25}$ Myr. As the cascade proceeds,
growth of the largest oligarchs stalls at a maximum mass $M_{o,max}\approx$
0.1 $x_{m}$ M⊕.
These results depend weakly on $r_{0}$. The time to produce the first oligarch
with $r\sim$ 1000 km increases with $r_{0}$, $t_{1000}\sim
0.1~{}x_{m}^{-1.25}~{}(r_{0}/{\rm 10~{}km})^{1/2}$ Myr. Calculations with
larger $r_{0}$ tend to produce larger oligarchs at 10 Myr: $M_{o,max}\approx$
1 M⊕ (2 M⊕) for $r_{0}$ = 10 km (100 km). In $\sim$ 50 calculations, none
produce cores with $M_{o,max}\gtrsim$ 1 M⊕ on timescales of $\lesssim$ 10 Myr.
For $r_{0}\lesssim$ 100 km, our results depend on $f_{i}$. In models with
$r_{0}$ = 1 km and 10 km, the $f_{W}$ fragmentation parameters yield oligarchs
with smaller maximum masses, $M_{o,max}\approx$ 0.3–0.6 M⊕. Because leftover
planetesimals with $r\sim$ 1–10 km fragment more easily, the cascade begins
(and growth stalls) at smaller collision velocities when oligarchs are less
massive (Kenyon et al., 2008).
Calculations with grain accretion produce cores rapidly. Fig. 2 shows results
at 1–10 Myr for calculations with $r_{0}$ = 1 km and the $f_{S}$ fragmentation
parameters. As the first oligarchs reach masses of $\sim$ 0.01 M⊕ at 0.1 Myr,
the cascade generates many small collision fragments with $r\sim$ 1 mm to 1 m.
These fragments rapidly settle to the disk midplane and grow to sizes of 0.1–1
m. When the cascade has shattered $\sim$ 25% of the leftover planetesimals,
oligarchs begin a second phase of runaway growth by rapidly accreting small
particles in the midplane. For calculations with $x_{m}$ = 1–5, it takes
$\sim$ 1–2 $x_{m}^{-1.25}$ Myr to produce at least one core with
$M_{o}\gtrsim$ 1–5 M⊕. Thus, cores form before the gas dissipates.
These results depend on $r_{0}$. For $r_{0}$ = 10 km, fragmentation produces
small grains 2–3 times more slowly than calculations with $r_{0}$ = 1 km.
These models form cores more slowly, in 5–10 $x_{m}^{-1.25}$ Myr instead of
1–2 $x_{m}^{-1.25}$ Myr. For models with $r_{0}$ = 100 km, fragmentation
yields a negligible mass in small grains. Thus, cores never form in $\lesssim$
10–20 Myr.
The timescales to form cores also depend on $f_{i}$. Calculations with the
$f_{W}$ parameters form cores 10% to 20% faster than models with the $f_{S}$
parameters.
## 5 CONCLUSIONS
Gaseous disks are a crucial element in the formation of the cores of gas giant
planets. The gas traps small collision fragments and halts the collisional
cascade. Once fragments settle to the disk midplane, oligarchs accrete the
fragments and grow to masses $\gtrsim$ 1 M⊕ in 1–3 Myr.
Our model predicts two outcomes for icy planet formation. Oligarchs that form
before (after) the gas disk dissipates reach maximum masses $\gtrsim$ 1 M⊕
($\lesssim$ 0.01–0.1 M⊕). Setting the timescale to form a 1000 km oligarch
(Eq. 1) equal to the gas dissipation timescale $t_{g}$ yields a boundary
between these two types of icy protoplanets at $a_{g}\approx$ 15
$x_{m}^{0.4}(t_{g}/{\rm 3~{}Myr})^{1/3}$ AU. We expect massive cores at
$a\lesssim a_{g}$ and low mass icy protoplanets at $a\gtrsim a_{g}$.
This prediction has a clear application to the Solar System. Recent dynamical
calculations suggest that the Solar System formed with four gas giants at 5–15
AU and an ensemble of Pluto-mass and smaller objects beyond 20 AU (Morbidelli
et al., 2008). For a protosolar disk with $x_{m}^{1.2}~{}(t_{g}/{\rm
3~{}Myr})$ $\approx$ 1, our model explains this configuration. Disks with
these parameters are also common in nearby star-forming regions (Andrews &
Williams, 2005). Thus, our results imply planetary systems like our own are
common.
Our model yields a large range of final masses for massive icy cores.
Protoplanets that grow to a few M⊕ well before the gas dissipates can accrete
large amounts of gas from the disk and become gas giants (Pollack et al.,
1996; Alibert et al., 2005). Protoplanets that grow more slowly cannot accrete
much gas and become icy ‘super-Earths’ with much lower masses (Kennedy et al.,
2006; Kennedy & Kenyon, 2008). For solar-type stars with $t_{g}\approx$ 3 Myr,
our results suggest that gas giants (super-Earths) are more likely in disks
with $x_{m}\gtrsim$ 1.5 ($x_{m}\lesssim$ 1.5) at 5–10 AU.
Testing this prediction requires (i) extending our theory to a range of
stellar masses and (ii) more detections of massive planets around lower mass
stars. We plan to explore the consequences of our model for other stellar
masses in future papers. Larger samples of planetary systems will test the
apparent trend that gas giants (super-Earths) are much more common around
solar-type (lower mass) stars (e.g., Cumming et al., 2008; Forveille et al.,
2008). Comparing the results of our planned numerical calculations with these
additional observations will yield a clear test of our model.
We acknowledge a generous allotment, $\sim$ 25 cpu years, of computer time on
the 1024 cpu Dell Xeon cluster ‘cosmos’ at the Jet Propulsion Laboratory
through funding from the NASA Offices of Mission to Planet Earth, Aeronautics,
and Space Science. We thank M. Werner for his strong support of this project.
We also acknowledge use of $\sim$ 10 cpu years on the CfA cluster ‘hydra.’
Advice and comments from T. Currie, M. Geller, G. Kennedy, and an anonymous
referee greatly improved our presentation. Portions of this project were
supported by the NASA TPF Foundation Science Program, through grant
NNG06GH25G.
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Figure 1: Mass histograms at 1 Myr (left panel), 3 Myr (center panel), and 10
Myr (right panel) for coagulation calculations without grain accretion using
the $f_{S}$ fragmentation parameters at 5 AU. Magenta histograms plot median
results for 25 calculations with $x_{m}$ = 1; blue histograms show median
results for 25 calculations with $x_{m}$ = 5. Independent of disk mass,
calculations without grain accretion yield planets with maximum masses
$\lesssim$ 1 M⊕ in 10 Myr. Figure 2: As in Fig. 1 for calculations with
grain accretion. When large oligarchs can accrete fragments trapped by the
gas, disks with $x_{m}\gtrsim$ 1 produce gas giant cores in 3–10 Myr.
|
arxiv-papers
| 2008-11-28T19:50:03 |
2024-09-04T02:48:59.045257
|
{
"license": "Public Domain",
"authors": "Scott J. Kenyon and Benjamin C. Bromley",
"submitter": "Scott J. Kenyon",
"url": "https://arxiv.org/abs/0811.4665"
}
|
0811.4719
|
# Work distributions in the $T=0$ Random Field Ising Model
Xavier Illa Department of Applied Physics, Helsinki University of Technology,
P.O.Box 1100, FIN-02015 HUT, Finland Josep Maria Huguet Departament de
Física Fonamental, Universitat de Barcelona
Martí i Franquès 1, Facultat de Física, 08028 Barcelona, Catalonia Eduard
Vives Departament d’Estructura i Constituents de la Matèria, Universitat de
Barcelona
Martí i Franquès 1 , Facultat de Física, 08028 Barcelona, Catalonia
###### Abstract
We perform a numerical study of the three-dimensional Random Field Ising Model
at $T=0$. We compare work distributions along metastable trajectories obtained
with the single-spin flip dynamics with the distribution of the internal
energy change along equilibrium trajectories. The goal is to investigate the
possibility of extending the Crooks fluctuation theorem G.E.Crooks (1999) to
zero temperature when, instead of the standard ensemble statistics, one
considers the ensemble generated by the quenched disorder. We show that a
simple extension of Crooks fails close to the disordered induced equilibrium
phase transition due to the fact that work and internal energy distributions
are very asymmetric.
###### pacs:
75.60.Ej, 75.50.Lk, 81.30.Kf, 81.40.Jj
††preprint: EV/25-07-2008
## I Introduction
The $T=0$ Random Field Ising Model is a prototype model for the study of
collective phenomena in disordered systems. Although it neglects thermal
fluctuations, it contains essential competitions between the quenched
disorder, the ferromagnetic interaction and the external applied field. The
model can be numerically studied from two different points of view: on the one
hand, the exact ground state calculation Hartmann and Young (2001); Middleton
and D.S.Fisher (2002); Dukovski and Machta (2003); Wu and Machta (2005)
provides an approach to the equilibrium phase diagram. On the other the use of
a local relaxation dynamics based on single spin-flips provides a good
framework for the understanding of avalanche dynamics and hysteresis Sethna et
al. (1993); J.P.Sethna et al. (2005), which is closer to experimental
observations. In this sense, the model is a good workbench for the comparison
of equilibrium and out-of-equilibrium trajectories.
A number of non-equilibrium work theorems Sevick et al. (2008) have received a
lot of attention in the last 10 years, particularly after the work of
Jarzynski in 1997 C.Jarzynski (1997); Jarzynski (2006). These theorems relate
in different ways the distribution of work performed on a system which is
driven out of equilibrium to some equilibrium thermodynamic properties. One
example is the original Jarzinsky’s equality: $\langle e^{-\beta
W}\rangle=e^{-\beta\Delta F}$, where $W$ is the work performed on the system
that has been driven (out-of-equilibrium) by varying an external control
parameter $H$ changing from $H(0)$ to $H(1)$, and $\Delta F$ is the free
energy difference between two states $0$ and $1$ that correspond to the
equilibrium states at $H(0)$ and $H(1)$. The average $\langle\cdot\rangle$
should be understood as obtained after many repetitions of the driving
process. The system is assumed to be in contact with a heat bath at
temperature $\beta^{-1}$ which generates an statistical ensemble of copies of
the system which is the source of work fluctuations. Most of these theorems
have been experimentally verified. Collin et al. (2005); Trepagnier et al.
(2004)
The goal of this paper is to investigate how such kind of theorems can be
extended to systems at $T=0$. In such a case, although equilibrium states and
out-of-equilibrium trajectories are well defined, there are no thermal
fluctuations. A priori it seems that there is no statistical ensemble over
which one can define probability distributions or averages. The idea we want
to test is whether or not, for systems with quenched disorder, the thermal
ensemble can be substituted by the ensemble of different realizations of
disorder and still some work theorems can be applied.
In order to perform the $T\rightarrow 0$ limit we choose a different but
related work theorem which is Crooks fluctuations theorem G.E.Crooks (1999);
Crooks (2000). The advantage will be that it allows to derive an equality that
can be extrapolated to the $T\rightarrow 0$ limit. This theorem can be written
as:
$\frac{P_{F}(W)}{P_{R}(-W)}=e^{\beta(W-\Delta F)}$ (1)
In this case $P_{F}(W)$ and $P_{R}(-W)$ are the probability densities
corresponding to the out-of-equilibrium work performed on a system driven
forward from $H(0)$ to $H(1)$ and reversely from $H(1)$ to $H(0)$. The Crooks
fluctuation theorem has been formulated under several assumptions: the driven
systems must be finite, classical and coupled to a thermal bath. Moreover, the
dynamics should be stochastic, Markovian and reversible and the entropy
production should be odd under time reversal. Some of these assumptions are
clearly not accomplished at the work at hand. For instance the thermal bath is
at $T=0$ and the dynamics is deterministic. Nevertheless we have been a little
speculative and investigated the possibility of extending the Crooks
fluctuation theorem to $T=0$ for systems with quenched disorder.
From Eq. (1) one can derive that the value $W^{*}$ for which the two
probability densities are equal, i.e. $P_{F}(W^{*})=P_{R}(-W^{*})$ satisfies
$W^{*}=\Delta F$. This result is particularly suitable to be investigated at
$T\rightarrow 0$. Note that in the low temperature limit, the only possibility
to have a crossing point of the probability densities at $W^{*}$ is that
$W^{*}=\Delta F(T\rightarrow 0)=\Delta U$, so that the diverging behaviour of
$\beta$ might be cancelled to get a finite limit
$P_{F}(W^{*})/P_{R}(-W^{*})\rightarrow 1$.
## II Model
For our investigation we consider the Random Field Ising Model. A set of $N$
spin variables $S_{i}=\pm 1$ with $i=1,\cdots N$ is defined on a regular 3D
cubic lattice with linear size $L$ ($N=L\times L\times L$). We are interested
in following the response of the order parameter (magnetization)
$M=\sum_{i=1}^{N}S_{i}$ as a function of the external driving field $H$. The
Hamiltonian (magnetic enthalpy) of the model is defined as:
${\cal H}=U-HM$ (2) $U=-\sum_{ij}S_{i}S_{j}-\sum_{i=1}^{N}h_{i}S_{i}$ (3)
where the first sum in Eq. 3, extending over nearest neigbours pairs, accounts
for a ferromagnetic exchange interaction, the second term in Eq. 3 includes
the interaction with the quenched disorder and the second term in Eq. 2
accounts for the interaction with the external field $H$ that will be used as
the driving parameter. The local fields $h_{i}$ are independent random
variables, quenched, Gaussian distributed with zero mean and standard
deviation $\sigma$. All along the paper we will use small letters
($u=\frac{U}{N}$,$m=\frac{M}{N}\ldots$) to refer to intensive magnitudes.
The equilibrium properties of this model can be studied at $T=0$. For a given
realization of the random fields $\\{h_{i}\\}$ and a given value of $H$, the
exact ground state can be found in a polynomial time Middleton (2001).
Moreover optimized algorithms can also be used to obtain the full sequence of
ground-sates as a function of $H$ between the two saturated states
$\\{S_{i}=-1\\}$ at $H=-\infty$ and $\\{S_{i}=1\\}$ at $H=\infty$ C.Frontera
et al. (2000). In the thermodynamic limit the 3D model exhibits a first order
phase transition at $H=0$ and for $\sigma<\sigma_{c}^{eq}=2.27$ Hartmann and
Young (2001); Middleton and D.S.Fisher (2002). Although finite systems do not
exhibit a true phase transition, there is a region of $\sigma$ and $H$ where
the correlation length is of the same order as the system size. This leads to
collective effects and correlations involving all the spins of the system.
Concerning the non-equilibrium trajectories, the model has been intensively
studied by using the so called single-spin flip metastable dynamics at $T=0$
which can be understood as a $T\rightarrow 0$ Glauber dynamics. It consists in
adiabatically sweeping the external field and relaxing individual spins
according to their local energy, i.e. spins should align with the local field
$F_{i}=H+h_{i}+\sum_{k}S_{k}$ (4)
where the sum extends over the neigbours of spin $i$. Note that the reversal
of a spin at a given field $H$ may induce the reversal of some of its
neigbours at the same field $H$. Such collective events are the so called
avalanches that end when all the spins are stable. Only after the avalanches
have finished, the external field is varied again. It has been shown that,
during a monotonous driving of the external field, no reverse spin-flips may
occur and, moreover, the model exhibits the so called abelian property: i.e.
the final states do not depend on the order in which the unstable spins are
flipped. In the thermodynamic limit, the system with this metastable dynamics
also shows a critical point at $\sigma_{c}^{met}=2.21$ Sethna et al. (1993);
F.J.Pérez-Reche and E.Vives (2003), below which the metastable trajectory
exhibits a discontinuity.
Note that avalanches are the source of energy dissipation Ortín and Goicoechea
(1998); X.Illa et al. (2005). If we think about an increasing field
trajectory, the triggering spin of each avalanche flips at the field $H$ that
corresponds to $F_{i}=0$. This spin, therefore, flips without energy loss
$\Delta{\cal H}=\Delta U-H\Delta M=0$. But the subsequent unstable spins flip
(at a fixed external field $H$) with an external force $F_{i}>0$ giving raise
to an energy loss (associated to each individual spin flip) $Q=-2F_{i}<0$ with
$Q=\Delta U-W$. For the decreasing field branches, $F_{i}<0$ but the energy
loss is then $Q=2F_{i}<0$ since the spins flip from $1$ to $-1$.
Note that in this discussion we are using the standard definition of work that
is not a state function. It is computed as a sum over the ($k=1,2\ldots$)
spins that flip along the trajectory
$W=\sum_{k}H_{k}\Delta M_{k},$ (5)
where $\Delta M_{k}=2$. Recently there has been a discussion Vilar and Rubi
(2008); A.Imparato and L.Peliti (2007); J.Horowitz and C.Jarzynski (2008);
Peliti (2008) on which is the most suitable definition of work to be used in
such non-equilibrium work theorems. Already from the initial Jarzynski works
C.Jarzynski (1997), it was proposed that if the system is driven by
controlling the external field, the convenient definition of work is the
integral over the trajectory $V=\int MdH$. Note that along a metastable
trajectory , the two definitions of work are related by $W=\Delta(HM)-V$.
Without going into the discussion, we will numerically test here the two
possibilities. In Sec. III we will test whether the crossing point $W^{*}$ of
the histograms $P_{F}(W)$ and $P_{R}(-W)$ satisfies
$W^{*}=\langle\Delta U\rangle$ (6)
and, in Sec. IV we will test whether the crossing point $V^{*}$ of the
histograms $P_{F}(V)$ and $P_{R}(-V)$ satisfies the corresponding equation:
$V^{*}=\langle\Delta U-\Delta(HM)\rangle$ (7)
The test of the two hypothesis will require slightly different strategies (as
indicated in Fig. 1) since $V$ can not be computed for trajectories starting
at saturation ($H\rightarrow\pm\infty$).
Figure 1: Schematic representation of the two different strategies that have
been used. (a) corresponds to the first strategy in which the starting point
is the saturated state $0$ ($m=-1$,$H=-\infty$), and (b) corresponds to the
second strategy, starting from the equilibrium state $0$ at a finite field
$H_{0}$. Dashed lines correspond to metastable trajectories and the continuum
line indicates the equilibrium states.
In the first strategy (Sec. III), we will compute forward trajectories
starting from the state $0$, saturated with $m=-1$ at $H=-\infty$ and sweep
the field adiabatically from $-\infty$ to $H_{1}$ until the metastable state
$A$ is reached. Then we will compute the equilibrium state $1$ corresponding
to the field $H_{1}$ and perform the reverse trajectory from $1$ to $0$ with
the metastable dynamics. We will compute $\Delta u=(U_{1}-U_{0})/N$ and the
following works (using the ’standard’ definition):
$\displaystyle w_{0\rightarrow 1}$ $\displaystyle=$
$\displaystyle\frac{W_{0\rightarrow
1}}{N}=\frac{1}{N}\int_{0}^{A}HdM+\frac{1}{N}\int_{A}^{1}HdM=\frac{1}{N}\int_{0}^{A}HdM+\frac{1}{N}H(M_{1}-M_{A})$
(8) $\displaystyle w_{1\rightarrow 0}$ $\displaystyle=$
$\displaystyle\frac{W_{1\rightarrow 0}}{N}=\frac{1}{N}\int_{1}^{0}HdM$ (9)
where $M_{A}$ and $M_{1}$ are the magnetizations of states $A$ and $1$. The
integrals are computed along the trajectories schematically represented in
Fig. 1(a) using Eq. 5. Note also that the second integral in Eq. 8 can be
computed since the process $A\rightarrow 1$ takes place at constant field
$H_{1}$.
In the second strategy (Sec. IV), we will start from a computed equilibrium
state at $H_{0}$, perform a metastable trajectory until the state $A$ is
reached at $H_{1}$. Then we will compute the equilibrium state $1$ at $H_{1}$
and perform a metastable trajectory driving back until reaching the state $B$
at $H_{0}$. In this case we will use the alternative definition of work:
$\displaystyle v_{0\rightarrow 1}=v_{0\rightarrow A}=\frac{V_{0\rightarrow
1}}{N}$ $\displaystyle=$ $\displaystyle\frac{V_{0\rightarrow
A}}{N}=\frac{1}{N}\int_{0}^{A}MdH,$ (10) $\displaystyle v_{1\rightarrow
0}=v_{1\rightarrow B}=\frac{V_{1\rightarrow 0}}{N}$ $\displaystyle=$
$\displaystyle\frac{V_{1\rightarrow B}}{N}=\frac{1}{N}\int_{1}^{B}MdH.$ (11)
The integrals are computed along the trajectories schematically represented in
Fig. 1(b) using $V=\sum_{k}M_{k}\Delta H_{k}$ where $k$ accounts for the
sequence of interavalanche field increments $\Delta H_{k}$ occuring at
constant $M_{k}$ along the trajectory.
.
Figure 2: (Color on line) Example of histograms corresponding to
$P_{F}(w_{0\rightarrow 1})$, $P_{R}(-w_{1\rightarrow 0})$ and $P(\Delta u)$.
The arrow pointing downwards indicates the position of the maximum of
$P(\Delta u)$ and the arrow pointing upwards the average value $\langle\Delta
u\rangle$. The example corresponds to a system with $\sigma=4$, $H_{1}=-0.35$
and $L=12$. Histograms have been obtained by cumulating data corresponding to
$4\times 10^{5}$ realizations of disorder.
## III First strategy
Fig. 2 shows histograms corresponding to the distributions of $w_{0\rightarrow
1}$, $\Delta u$ and $-w_{1\rightarrow 0}$ for $\sigma=4$, $H_{1}=-0.35$ and a
system size $L=12$. The computed histograms give an estimation of the
corresponding probability densities $P_{F}(w_{0\rightarrow 1})$,
$P_{R}(-w_{1\rightarrow 0})$ and $P(\Delta u)$. Four important points should
be realized from this example: (i) For this value of $\sigma$ and field
$H_{1}$, the densities look symmetric and Gaussian. This will not be the case
when $H_{1}$ and $\sigma$ are close to the disorder induced phase transition
at $\sigma_{c}^{eq}$ and $H_{1}=0$. (ii) Second, the distribution of $\Delta
u$ has a width similar to those of the out-of-equilibrium works. This is
different from what happens in the analysis of Crooks fluctuation theorem at
finite $T$. Typically, equilibrium thermal fluctuations are much smaller than
work fluctuations. (iii) For increasing systems size the histograms become
narrower and then the crossing points are harder to locate (iv) Finally, note
that for this case the hypothesis that we are testing is fulfilled: the peak
in $P(\Delta u)$ as well as the average $\langle\Delta u\rangle$ coincide with
the crossing point $w^{*}$ of $P_{F}(w_{0\rightarrow 1})$ and
$P_{R}(-w_{1\rightarrow 0})$.
Figure 3: (Color on line) Example of histograms corresponding to
$P_{F}(w_{0\rightarrow 1})$, $P_{R}(-w_{1\rightarrow 0})$ and $P(\Delta u)$.
The arrow pointing downwards indicates the position of the maximum of
$P(\Delta u)$ and the arrow pointing upwards the average value $\langle\Delta
u\rangle$. The example corresponds to a system with $\sigma=3$, $H_{1}=0.1$
and $L=12$. The histograms have been computed by cumulating data corresponding
to $1.6\times 10^{6}$ realizations of disorder. The inset shows a detailed
view of the crossing point of the $P_{F}(w_{0\rightarrow 1})$ and
$P_{R}(-w_{1\rightarrow 0})$ histograms. Figure 4: Schematic representation
of the case $\sigma\lesssim\sigma_{c}^{eq}$ and $H_{1}>0$. Dashed lines
correspond to metastable trajectories, continuous lines to the equilibrium
trajectory and the dashed-dotted line to the non typical equilibrium states
that are responsible for the high asymmetry of the distribution
$P_{R}(-w_{1\rightarrow 0})$.
Fig. 3 shows histograms corresponding to $P(w_{0\rightarrow 1})$, $P(\Delta
u)$ and $P(-w_{1\rightarrow 0})$ for $\sigma=3$, $H_{1}=0.1$ and a system size
$L=12$. Note than in this case, the work distributions as well as the energy
distribution are very asymmetric. This is because although in this case we
expect $M_{1}>0$ since $H_{1}>0$, a certain non vanishing fraction of the
equilibrium states still has $M_{1}<0$, as schematically indicated in Fig.4.
In other words, for certain realizations of disorder, the state $1$ turns out
to be non-typical and previous to the equilibrium transition from $M<0$ to
$M>0$. Consequently, the distribution of reverse works $w_{1\rightarrow 0}$
widens enormously. As can be seen the proposed equality $w^{*}=\Delta u$
clearly fails in this case. The relative error is $|w^{*}-\langle\Delta
u\rangle|/|\langle\Delta u\rangle|\sim 0.25$.
Figure 5: (Color on line) Comparison of the crossing point of the histograms
$w^{*}$ ($\Box$) with $\langle\Delta u\rangle$ ($\Circle$) as a function of
$H_{1}$, corresponding to (a) $\sigma=4$ and (b) $\sigma=3$ for different
values of $L$ as indicated by the legend. For the case $L=12$ and $\sigma=3$
the dashed line indicated the position of the maximum of $P(\Delta u)$. Data
for $L=18$ ($L=24$) has been shifted 0.3 (0.6) units along the vertical axis
in order to clarify the picture.
We have performed a detailed study of the deviation of $w^{*}$ from
$\langle\Delta u\rangle$ for different values of $\sigma$, $H_{1}$ and $L$.
Examples of the results are presented in Fig. 5. Note that the data
corresponding to the crossing point $w^{*}$ exhibits increasing error bars
with increasing $H_{1}$. This is because the histograms of
$P_{F}(w_{0\rightarrow 1})$, $P_{R}(-w_{1\rightarrow 0})$ become more and more
separated and the finding of the intersection requires more and more
statistics. (This problem is accentuated for larger values of $H_{1}$ and for
larger system sizes). One can conclude therefore than the proposed
extrapolation of Crooks theorem, given by the equality $w^{*}=\langle\Delta
u\rangle$ fails in the region of $\sigma$ and $H_{1}$ where the system
exhibits a collective behaviour (i.e. the correlation length is similar to the
system size) because of the proximity of the equilibrium phase transition.
Consistently, we observe in Fig. 5(b) (comparing the data corresponding to
$L=12$ and $L=18$ at $H_{1}\sim 0$) that the region of breakdown becomes
smaller when the system size $L$ is larger. As expected, increasing the system
size with fixed correlation length decreases the collective effects.
## IV Second strategy
Fig. 6 shows histograms corresponding to the distributions of $v_{0\rightarrow
1}$ and $-v_{1\rightarrow 0}$ for the case $L=12$, $\sigma=3$, $H_{1}=-10$ and
$H_{2}=-0.4$. As can be seen the intersection point $v^{*}$ agrees very well
with the average value of $\Delta u-\Delta(Hm)$. In this case we have also
studied if this agreement is equally good in other regions of the phase
diagram.
Figure 6: (Color on line) Example of histograms corresponding to
$P_{F}(v_{0\rightarrow 1})$, $P_{R}(-v_{1\rightarrow 0})$ and $P\left(\Delta
u-\Delta(Hm)\right)$. Data corresponds to $H_{0}=-10.0$,$H_{1}=-0.4$,
$\sigma=3$ and $L=12$. Histograms are computed by cumulating $10^{5}$
realizations of disorder. The arrows pointing down (up) indicate the peak
(average value) of the distribution $P(\Delta u-\Delta(HM))$. Figure 7:
(Color on line) Comparison of the crossing point $v^{*}$ ($\Box$) of the
histograms $P_{F}(v_{0\rightarrow 1})$ and $P_{R}(-v_{1\rightarrow 0})$ with
$\langle\Delta u-\Delta(Hm)\rangle$ ($\Circle$) as a function of $H_{2}$,
corresponding to $\sigma=3$ and $L=12$. Data corresponds to averages over
$10^{5}$ realizations of disorder
Fig 7 shows a comparison of $\langle\Delta u\rangle$ and
$v^{*}+\langle\Delta(Hm)\rangle$. On the right hand side, the value of $v^{*}$
has not been obtained by a direct location of the intersection of the
histograms but locating the mid point between the positions of the maxima of
the histograms corresponding to $P_{F}(v_{0\rightarrow 1})$ and
$P_{R}(-v_{1\rightarrow 0})$ which are far separated but very symmetric. As
can be seen, as occurs with strategy 1, the agreement
$v^{*}\simeq\langle\Delta u-\Delta(Hm)\rangle$ fails when $H_{2}$ approaches
the phase transition region.
## V Summary and conclusion
We have investigated the possibility of extending some work fluctuation
theorems to $T=0$ for systems with quenched disorder. In this case, thermal
averages should be substituted by disorder averages. We have proposed an
hypothesis based on the extrapolation of Crooks theorem to the $T\rightarrow
0$ limit and we have numerically tested its validity. The hypothesis is that
if one considers the distributions of non-equilibrium work corresponding to a
forward and backwards trajectory, the crossing point $w^{*}$ of
$P_{F}(w_{0\rightarrow 1})$ and $P_{R}(-w_{1\rightarrow 0})$, is equal to the
average of the internal energy difference $\Delta u=u_{1}-u_{2}$. The
investigation has been done by considering two strategies, based on the two
possible definitions of work that have been discussed in the literature:
$w=\int Hdm$ and $v=\int mdH$.
The reported numerical investigation indicates that, for both strategies, the
formulated hypothesis is valid when the system does not behave collectively.
Thus, far from the equilibrium critical point, where correlations are small
compared to system size, the distribution $p(\Delta u)$ is very symmetric and
the average (equilibrium) value $\langle\Delta u\rangle$ can be obtained from
the crossing point $w^{*}$ of the distributions of the non equilibrium works.
When the system is close to the critical point a definitive conclusion can not
be stated. Data for small $L$ suggest that when the correlation length
approaches the system size $L$, the distribution $p(\Delta u)$ is very
assymetric and there is no connection between the equilibrium value
$\langle\Delta u\rangle$ and the crossing point $w^{*}$ of the non-equilibrium
work distributions. A bigger computational effort (not affordable at present)
would be needed in order to compute the accurate histograms for $L>24$. The
presented results may provide some clues for future investigations in order to
extend work fluctuation theorems.
###### Acknowledgements.
This work has received financial support from CICyT (Spain), project
MAT2007-61200, CIRIT (Catalonia), project 2005SGR00969, EU Marie Curie RTN
MULTIMAT, Contract No. MRTN-CT-2004-5052226 and the Academy of Finland. X.I.
acknowledges the hospitality of ECM Department (Universitat de Barcelona)
where the work at hand was to a large degree completed. We also acknowledge
fruitful discussions with F. Ritort, A. Planes, and M.J. Alava.
## References
* G.E.Crooks (1999) G.E.Crooks, Phys. Rev. E 60, 2721 (1999).
* Hartmann and Young (2001) A. K. Hartmann and A. P. Young, Phys. Rev. B 64, 214419 (2001).
* Middleton and D.S.Fisher (2002) A. A. Middleton and D.S.Fisher, Phys. Rev. B 65, 134411 (2002).
* Dukovski and Machta (2003) I. Dukovski and J. Machta, Phys. Rev. B 67, 014413 (2003).
* Wu and Machta (2005) Y. Wu and J. Machta, Physical Review Letters 95, 137208 (2005).
* Sethna et al. (1993) J. P. Sethna, K. Dahmen, S. Kartha, J. A. Krumhansl, B. W. Roberts, and J. D. Shore, Phys. Rev. Lett. 70, 3347 (1993).
* J.P.Sethna et al. (2005) J.P.Sethna, K.Dahmen, and O.Perković, _The Science of Hysteresis_ (Elsevier Inc., 2005), vol. 2, chap. 2, pp. 107–179.
* Sevick et al. (2008) E. Sevick, R. Prabhakar, Stephen, and R. Williams, Annual Review of Physical Chemistry 59, 603 (2008).
* C.Jarzynski (1997) C.Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).
* Jarzynski (2006) C. Jarzynski, Phys. Rev. E 73, 046105 (2006).
* Collin et al. (2005) D. Collin, F. Ritort, C. Jarzynski, S. Smith, I. T. Jr, and C. Bustamante, Nature 437, 231 (2005).
* Trepagnier et al. (2004) E. Trepagnier, C. Jarzynski, F. Ritort, G. Crooks, C. Bustamante, and J. Liphardt, PNAS 101, 15038 (2004).
* Crooks (2000) G. E. Crooks, Phys. Rev. E 61, 2361 (2000).
* Middleton (2001) A. A. Middleton, Phys. Rev. Lett. 88, 017202 (2001).
* C.Frontera et al. (2000) C.Frontera, J.Goicoechea, J.Ortín, and E.Vives, J. Comp. Phys. 160, 117 (2000).
* F.J.Pérez-Reche and E.Vives (2003) F.J.Pérez-Reche and E.Vives, Phys. Rev. B 67, 134421 (2003).
* Ortín and Goicoechea (1998) J. Ortín and J. Goicoechea, Phys. Rev. B 58, 5628 (1998).
* X.Illa et al. (2005) X.Illa, J.Ortín, and E.Vives, Phys. Rev. B 71, 184435 (2005).
* Vilar and Rubi (2008) J. M. G. Vilar and J. M. Rubi, Phys. Rev. Lett. 100, 020601 (2008).
* A.Imparato and L.Peliti (2007) A.Imparato and L.Peliti, Comptes Rendus Physique 8, 556 (2007).
* J.Horowitz and C.Jarzynski (2008) J.Horowitz and C.Jarzynski, Phys. Rev. Lett. 101, 098901 (2008).
* Peliti (2008) L. Peliti, Phys. Rev. Lett. 101, 098903 (2008).
|
arxiv-papers
| 2008-11-28T13:53:16 |
2024-09-04T02:48:59.052396
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xavier Illa, Josep Maria Huguet, Eduard Vives",
"submitter": "Xavier Illa",
"url": "https://arxiv.org/abs/0811.4719"
}
|
0812.0202
|
# Intermittent Peel Front Dynamics and the Crackling Noise in an Adhesive Tape
Jagadish Kumar1 Rumi De2 G. Ananthakrishna1 1Materials Research Centre, Indian
Institute of Science, Bangalore 560012, India
2 Department of Materials and Interfaces, Weizmann Institute of Science,
Rehovot 76100, Israel
###### Abstract
We report a comprehensive investigation of a model for peeling of an adhesive
tape along with a nonlinear time series analysis of experimental acoustic
emission signals in an effort to understand the origin of intermittent peeling
of an adhesive tape and its connection to acoustic emission. The model
represents the acoustic energy dissipated in terms of Rayleigh dissipation
functional that depends on the local strain rate. We show that the nature of
the peel front exhibits rich spatiotemporal patterns ranging from smooth,
rugged and stuck-peeled configurations that depend on three parameters,
namely, the ratio of inertial time scale of the tape mass to that of the
roller, the dissipation coefficient and the pull velocity. The stuck-peeled
configurations are reminiscent of fibrillar peel front patterns observed in
experiments. We show that while the intermittent peeling is controlled by the
peel force function, the model acoustic energy dissipated depends on the
nature of the peel front and its dynamical evolution. Even though the acoustic
energy is a fully dynamical quantity, it can be quite noisy for a certain set
of parameter values suggesting the deterministic origin of acoustic emission
in experiments. To verify this suggestion, we have carried out a dynamical
analysis of experimental acoustic emission time series for a wide range of
traction velocities. Our analysis shows an unambiguous presence of chaotic
dynamics within a subinterval of pull speeds within the intermittent regime.
Time series analysis of the model acoustic energy signals is also found to be
chaotic within a subinterval of pull speeds. Further, the model provides
insight into several statistical and dynamical features of the experimental AE
signals including the transition from burst type acoustic emission to
continuous type with increasing pull velocity and the connection between
acoustic emission and stick-slip dynamics. Finally, the model also offers an
explanation for the recently observed feature that the duration of the slip
phase can be less than that of the stick phase.
###### pacs:
83.60.Df, 05.45.-a, 05.45.Tp, 62.20.Mk
## I Introduction
Adhesive tapes are routinely used in a variety of situations including daily
usage as stickers, in packing and sealing. Yet, day-to-day experiences like
intermittent peeling of an adhesive tape and the origin of the accompanying
audible noise have remained ill understood. This may be partly attributed to
the fact that adhesion is a highly interdisciplinary subject involving diverse
but interrelated physical phenomena such as intermolecular forces of
attraction at the interface, mechanics of contact, debonding and rupture,
visco-plastic deformation and fracture Kendall00 , and frictional dissipation
which operates during peeling Kendall00 ; Urbakh04 ; Persson . Yet another
reason is that most information on adhesion is obtained from quasistatic or
near stationary conditions. Apart from scientific interest, understanding the
intermittent peel or the stick-slip process has relevance to industrial
applications as well. For example, optimizing production schedules that
involve pasting or peeling of an adhesive tapes at a rapid pace in an assembly
line requires a good understanding of stick-slip dynamics. Moreover, insight
into the time dependent and dynamical aspects of adhesion is expected to be
important in design of adhesives with versatile properties required in variety
of applications, in understanding the mechanisms leading to the failure of
adhesive joints as also in understanding biologically relevant systems such as
the geckoJagota07 or reorientation dynamics of cells Rumi07 .
Adhesion tests are essentially fracture tests designed to study adherence of
solids and generally involve normal pulling off and peeling. Such experiments
can be performed under quasistatic or near-stationary and nonequilibrium
conditions as well. The latter kind of experiments demonstrate the rate
dependence of adhesive properties. It is this rate dependence and the inherent
nonlinearity that leads to a variety of instabilities. These kinds of peeling
experiments are comparatively easy to setup in a laboratory. Moreover, the set
up also allows one to record unusually long force waveforms and AE signals
that should be helpful in extracting useful information on the nonlinear
features of the system.
One type of peeling experiment that yields dynamical information is carried
out with an adhesive tape mounted on a roller subjected to a constant pull
velocity MB ; CGVB04 . Peeling experiments have also been performed under
constant load conditions BC97 ; CGVB04 . At low pull velocities, the velocity
of the contact point $v$ keeps pace with the imposed velocity $V$. The same is
true at high velocities as well. However, there is an intermediate regime of
traction velocities where the peeling is intermittent. Peeling in this regime
is accompanied by a characteristic audible noise MB ; BC97 ; CGVB04 . It must
be stressed that these two stable dissipative branches refer to stationary
branches. Even so, the stick-slip dynamics observed in the intermediate region
of pull velocities has been attempted by assuming an unstable branch
connecting the two stable branches. The strain energy release rate shows a
power law for low velocities with an exponent around 0.3. The high velocity
branch also shows a power law but with a much higher exponent value of about
5.5 MB . The low velocity branch is known to arise from viscous dissipation
and that at high velocity corresponds to fracture. These studies report a
range of wave forms starting from saw tooth, sinusoidal or even irregular wave
form that has been termed ’chaotic’ MB . More recently, the dynamics of the
peel point has been imaged as well Cortet .
Stick-slip processes are usually observed in systems subjected to a constant
response where-in the force developed in the system is measured by dynamically
coupling the system to a measuring device. The phenomenon is experienced
routinely, for example, while writing with chalk piece on a black board,
playing violin or walking down a staircase with the hand placed on the hand-
rail. A large number of studies on stick-slip dynamics have been reported in
systems ranging from atomic length scales, for instance, stick-slip observed
using atomic force microscope soc04 to geological length scales like the
stick-slip of tectonic plates causing earthquakes BK ; BKC . A few well known
laboratory scale systems are - sliding friction Heslot ; Urbakh04 ; Persson
and the Portevin-Le Chatelier (PLC) effect PLC ; GA07 , a kind of plastic
instability observed during tensile deformation of dilute alloys GA07 ; Anan04
, to name only two. Most stick-slip processes are characterized by the system
spending a large part of the time in the stuck state and a short time in the
slip state. This feature is observed both in experiment and in models. See for
instance Heslot ; soc04 ; BK . A counter example where the time spent in the
stuck is less than that in the slip state (observed at high applied strain
rates) is the PLC effect PLC . These studies show that while the physical
mechanisms that operate in different situations can be quite varied Persson ,
in general, stick-slip results from a competition among the inherent internal
relaxational time scales GA07 ; Anan04 and the applied time scale. In the
case of peeling, one identifiable internal relaxation time scale is the
viscoelastic time scale of the adhesive. Other relevant time scales that may
be operative need to be included for a proper description of the dynamics. All
stick-slip systems are governed by deterministic nonlinear dynamics.
Models that attempt to explain the dynamical features of stick-slip systems
use the macroscopic phenomenological negative force response (NFR) feature as
an input although the unstable region is not accessible. This is also true for
models dealing with the dynamics of the adhesive tape including the present
work. In this context, it must be stated that there is no microscopic theory
that predicts the negative force-velocity relation in most stick-slip
situations except in the case of the PLC effect where we have provided a
dynamical interpretation of the negative strain rate sensitivity of the flow
stress Anan82 ; Rajesh (see below). There are several theoretical attempts to
model the stick-slip process observed during peeling of an adhesive tape.
Maugis and Barquin MB , were the first to write down a model set of equations
suitable for the experimental situation and to carry out approximate dynamical
analysis. These equations were later modified and a dynamical analysis of
these equations was reported HY1 . However, the stick-slip oscillations were
not obtained as a natural consequence of the equations of motion HY2 ; CGB .
Indeed, these equations are singularRumi04 . Subsequently, we devised a
special algorithm to solve these differential algebraic equations (DAE) HLR ;
Rumi04 . This algorithm allows for dynamical jumps across the two stable
branches of the peel force function. This was followed by converting the DAE
into a set of nonlinear ordinary differential equations (ODE) by including the
missing kinetic energy of the stretched tape thereby lifting the singular
nature of the DAE Rumi04 ; Rumi05 . Apart from supporting dynamical jumps, the
ODE model exhibits rich dynamical features. However, all these studies discuss
only contact point dynamics while the tape has a finite width. The ODE model
has been extended to include the spatial degrees of freedom that is crucial
for describing the dynamics of the peel front as also for understanding the
origin of acoustic emission Rumiprl ; Jagdish08 .
Acoustic emission is commonly observed in an unusually large number of systems
such as seismologically relevant fracture studies of rock samples Scholz68a ;
Lockner96 ; Sam92 , martensite transformation vives ; rajeevprl ; kalaprl ,
micro-fracturing process Petri94 , volcanic activity Diodati , collective
dislocation motion Miguel ; Weiss etc. The general mechanism attributed to AE
is the abrupt release of the stored potential energy although the underlying
mechanisms triggering AE are system specific. The nondestructive nature of the
AE technique has been useful in tracking the microstructural changes during
the course of deformation by monitoring the AE signals. For instance, it is
used in fracture studies of rock samples Lockner96 and more recently, a
similar approach has been used in understanding collective behavior of
dislocations Weiss . In both these cases, multiple transducers are used to
locate the hypocenters through an inversion process of arrival times Lockner96
; Weiss . In the latter case, by analysing the dislocations sources generating
AE signals, the study establishes the fractal nature of the collective motion
of dislocations. (In contrast to these dynamical studies, most studies on AE
Petri94 ; Diodati ; Miguel are limited to compiling the statistics of the AE
signals in an effort to find experimental realizations of self-organized
criticality Bak .) However, in the case of peeling, using multiple transducers
is far from easy and only a single transducer is used leading to scalar time
series. In such situations, dynamical information is traditionally recovered
using nonlinear time series analysis GP ; HKS ; KS . However, a major
difficulty arises in the present case due to a high degree of noise present
and the associated difficulties involved in curing the noise content.
Despite large number of experimental investigations and to a lesser extent
model studies, several issues related to intermittent peeling and the
associated acoustic emission remain ill understood. For instance, there are no
models (even in the general area of stick-slip) which show that the duration
of the stick phase can be equal to or even less than that of the slip phase
Cortet , a feature which is quite unlike conventional stick-slip dynamics.
From a dynamical point of view, this is also suggestive of the existence of at
least three time scales. The model represents the acoustic energy in terms of
the Rayleigh dissipation functional that depends on the local strain rate of
the peel front and thus is sensitive to the nature of the peel front dynamics.
While preliminary results of the model Rumiprl ; Jagdish08 based on a small
domain of parameters were encouraging, no systematic study of the influence of
all the relevant time scales on the dynamics of the peel front was carried
out. In particular, while the nature of experimental AE signals changes with
the traction velocity, the study of the influence of pull speed on internal
relaxational mechanisms, the consequent peel front dynamics and its
relationship with the acoustic energy was not studied either.
The principal objective of the present study is to understand the various
contributing mechanisms to the intermittent peel process and its connection to
acoustic emission. The objective is accomplished by carrying out a systematic
study of the influence of the three internal relaxational time scales namely
the two inertial time scales of the tape mass and the roller inertia, and
dissipative time scale of the peel front. In particular, we report the
influence of the experimentally relevant pull velocity (covering the entire
range) on the peel front dynamics. These studies show that the model exhibits
rich spatiotemporal peel front patterns (including the stuck-peeled
configurations that mimic fibrillar patterns seen in experiments) arising due
to the interplay of the three time scales. Consequently, varied patterns of
model acoustic signals are seen. Another consequence of the inclusion of the
three time scales is that it explains the recent observation that the duration
of the slip phase can be larger than that of the stick-phase Cortet .
Interestingly, the model studies show that it is possible to establish a
correspondence between the various types of model acoustic energy profiles
with certain peel front patterns. More importantly, the study shows that even
as the acoustic energy dissipated is the spatial average of the local strain
rate, it can be noisy suggesting the possible deterministic origin of the
experimental acoustic signals. Here, we report a detailed analysis of the
statistical and dynamical analysis of the experimental AE signals. The study
shows that while the intermittent peeling is controlled by the peel force
function, acoustic emission is controlled by the dynamics of the peel front
patterns that determine the local strain rate. This coupled with a comparative
study of a comprehensive nonlinear time series analysis (TSA) of the
experimental AE signals for a wide range of traction velocities supplemented
by a similar study on the model acoustic energy time series provides
additional insights into the connection between AE signals and stick-slip
dynamics. In particular, the model displays the recently observed experimental
feature that the duration of the slip phase can be more than that of the stick
phase with increase in the pull velocity. Finally, the model studies together
with the dynamical analysis of the model acoustic signal provide a dynamical
explanation for the changes in the nature of the experimental AE signal in
terms of the changes in the peel front patterns.
Figure 1: (a) (Color online) A schematic representation of the experimental
setup. (b) Plot of $\phi(v^{s})$ as a function of $v^{s}$.
## II The model
A typical experimental set up consists of an adhesive tape mounted on a
roller. The tape is pulled at a constant pull velocity using a motor. A
schematic representation of the set up is shown in Fig. 1(a). The axis of the
roller passes through the point O into the plane of the paper. The drive motor
is positioned at O′. Let the distance between $O$ and $O^{\prime}$ be denoted
by $l$. $P$ is the contact point on the peel front $PQ$. Let the peeled length
of the tape $PO^{\prime}$ be denoted by $L$. Several geometrical features can
be discussed using a projection on to the plane of the paper. Let the angle
between the tangent to the contact point $P$ and $PO^{\prime}$ be denoted by
$\theta$ and the angle $\angle{POO^{\prime}}$ by $\alpha$. Then, from the
geometry of the Fig. 1(a), we get $L\ {cos}\,\theta=-l\ {sin}\,\alpha$ and $L\
{sin}\,\theta=l\ {cos}\,\alpha-R$ where $R$ is the diameter of the roller
tape. Let the local velocity of the peel point be denoted by $v$ and the
displacement (from a uniform stuck state) of the peel front by $u$. Then, the
pull velocity has to satisfy
$V=v+\dot{u}+R\ {\rm cos}\ \theta\ \dot{\alpha}.$ (1)
As the peel front has a finite width, we define the corresponding quantities
along the peel front coordinate $y$ (i.e., along the contact line) by
$v(y),\theta(y)$ and $\alpha(y)$. Then as the entire tape width is pulled a
constant velocity, the above constraint generalizes to
$\displaystyle{1\over b}\int^{b}_{0}\big{[}V-v(y)-\dot{u}(y)-R\ \
\dot{\alpha}(y)\ \ {\rm cos}\ \theta(y)\big{]}dy=0,$ (2)
where $b$ is the width of the tape. However, we are interested in the
deformation of the peel front of the adhesive, which is a soft visco-elastic
material. For the purpose of modeling, while we shall ignore the viscoelastic
nature of the adhesive, we recognize its low modulus, i.e., we assume an
effective spring constant $k_{g}$ (along the contact line) whose value is much
smaller than the spring constant of the tape material $k_{t}$. This also
implies that the force along $PO^{\prime}$ equilibrates fast and therefore the
integrand in Eq. (2) can be assumed to vanish for all $y$. Thus, the above
equation reduces to Eq. (1).
The present model is an extension of the ODE model for the contact point
dynamics Rumi05 . The ODE model already contains information on the inertial
time scale of the tape mass that allows for dynamical jumps across the two
branches of the peel force function. The extension involves introducing the
Rayleigh dissipation functional to deal with acoustic emission apart from
introducing the spatial degrees of freedom. The equations of motion for the
contact line dynamics are derived by writing down the relevant energy terms
consisting of the kinetic energy, potential energy and the energy dissipated
during the peel process. The total kinetic energy $U_{k}$ is the sum of the
rotational kinetic energy of the roller tape and the kinetic energy of the
stretched part of the tape. This is given by
$U_{K}={1\over 2}\int^{b}_{0}\xi\big{[}\dot{\alpha}(y)+{v(y)\over
R}\big{]}^{2}dy+{1\over 2}\int^{b}_{0}\rho\big{[}\dot{u}(y)\big{]}^{2}dy.$ (3)
Here, $\xi$ is the moment of inertia per unit width of the roller tape and
$\rho$ the mass per unit width of the tape. The total potential energy $U_{p}$
consists of the contribution from the displacement of the peel front due to
stretching of the peeled tape and possible inhomogeneous nature of the peel
front. This is given by
$U_{P}={1\over 2}\int^{b}_{0}{k_{t}\over b}\Big{[}u(y)\Big{]}^{2}dy+{1\over
2}\int^{b}_{0}{k_{g}b}\Big{[}{\partial u(y)\over\partial y}\Big{]}^{2}dy.$ (4)
The peel process always involves dissipation. Indeed, the peel force function
with the two stable branches, one corresponding to low velocities and another
at high velocity arises from two different dissipative mechanisms. Apart from
this, there is an additional dissipation that arises from the rapid rupture of
the peel front which in turn results in the accelerated motion of local
regions of the peel front. We consider this accelerated motion of the local
slip as the source responsible for the generation of acoustic signals Rumiprl
. Any rapid movement also prevents the system from attaining a quasistatic
equilibrium which in turn generates dissipative forces that resist the motion
of the slip. Such dissipative forces are modeled by the Rayleigh dissipation
functional that depends on the gradient of the local displacement rate Land .
Indeed, such a dissipative term has proved useful in explaining the power law
statistics of the AE signals during martensitic transformation vives ;
rajeevprl ; kalaprl as also in explaining certain AE features in fracture
studies of rock sample Rumiepl . Then, the total dissipation can be written as
the sum of these two contributions
${\cal R}={1\over b}\int^{b}_{0}\int f(v(y))dvdy+{1\over
2}\int^{b}_{0}{\Gamma_{u}\over b}\Big{[}{\partial\dot{u}(y)\over\partial
y}\Big{]}^{2}dy,$ (5)
where $f(v)$ physically represents the peel force function assumed to be
derivable from a potential function $\Phi(v)=\int f(v)dv$ (see Ref. Rumi05 ).
We denote the second term in Eq. (5) by ${\cal R}_{ae}$ which is identified
with the energy dissipated in the form of AE. In the context to plastic
deformation, the acoustic energy arising from the abrupt motion of
dislocations is given by ${\cal R}_{ae}\propto\dot{\epsilon}^{2}(r)$, where
$\dot{\epsilon}(r)$ is the local plastic strain rate Rumiepl . Following this,
we interpret ${\cal R}_{ae}$ as the energy dissipated in the form of AE
signals. Note that $\frac{\partial\dot{u}}{\partial y}$ is the local strain
rate of the peel front. As for the first term in Eq. (5), the form of the peel
force function we use is given by
$f(v)=402v^{0.34}+171v^{0.16}+68e^{(v/7.7)}-369.65v^{0.5}-2.$ (6)
We stress here that as we are interested in the generic properties of the
peeling process, the exact form of the peel force function used here is not
important as long as major experimental features like the magnitude of the
jump in the velocity across the two branches, the range of values of the
measured peel force function, in particular the values at the maximum and
minimum, are captured.
As can be seen from Eq. (3), there are two time scales; one corresponding to
the inertia of the tape mass and the other due to the roller inertia. In
addition, there is a third time scale, namely the dissipative time scale in
Eq. (5) (second term). Thus, there are three internal relaxational time scales
in the model. Apart from this, there is also a time scale due to the pull
speed. Then the nature of the dynamics is determined by an interplay among all
these time scales.
It is more convenient to deal with scaled quantities. Consider introducing
basic length and time scales which will be used to rewrite all the energy
terms in scaled form. A natural choice for a time like variable is
$\tau=\omega_{u}t$ with $\omega_{u}^{2}={k_{t}/(b\ \rho)}$. In a similar way,
we introduce a basic length scale defined by $d=f_{max}/k_{t}$, where
$f_{max}$ is the value of $f(v)$ at $v_{max}$ on the left stable branch. We
define scaled variables by $u=Xd=X(f_{max}/k_{t})$, $l=l^{s}d$, $L=L^{s}d$ and
$R=R^{s}d$. The peel force function $f$ can be written as
$\phi(v^{s})=f(v(v^{s}))/f_{max}$. Here $v^{s}=v/v_{c}\omega_{u}d$ and
$V^{s}=V/v_{c}\omega_{u}d$ are the dimensionless peel and pull velocities
respectively with $v_{c}=v_{max}/\omega_{u}d$ representing the dimensionless
critical velocity at which the unstable branch starts. Using this we can
define a few relevant scaled parameters $C_{f}=(f_{max}/k_{t})^{2}(\rho/\xi)$,
$k_{0}=k_{g}b^{2}/(k_{t}a^{2})$,
$\gamma_{u}=\Gamma_{u}\omega_{u}/(k_{t}a^{2})$, and $y=ar$, where $a$ is a
unit length variable along the peel front. The parameter $C_{f}$ is a measure
of the relative strengths of the inertial time scale of the stretched tape to
that of the roller, $k_{0}$ the relative strengths of the effective elastic
constant of the adhesive to that of the tape material and $\gamma_{u}$ the
strength of the dissipation coefficient. Then, the scaled local form of Eq.
(1) takes the form
$\dot{X}=(V^{s}-v^{s})v_{c}+R^{s}\ {l^{s}\over L^{s}}\ ({sin}\ \alpha)\
\dot{\alpha}.$ (7)
In terms of the scaled variables, the scaled kinetic energy $U^{s}_{K}$ and
scaled potential energy $U^{s}_{P}$ can be respectively written as
$\displaystyle U^{s}_{K}$ $\displaystyle=$ $\displaystyle{1\over
2C_{f}}\int^{b/a}_{0}\Big{[}\Big{(}\dot{\alpha}(r)+{v_{c}v^{s}(r)\over
R^{s}}\Big{)}^{2}+C_{f}{\dot{X}}^{2}(r)\Big{]}dr,$ (8) $\displaystyle
U^{s}_{P}$ $\displaystyle=$ $\displaystyle{1\over
2}\int^{b/a}_{0}\Big{[}X^{2}(r)+k_{0}\Big{(}{\partial X(r)\over\partial
r}\Big{)}^{2}\Big{]}dr.$ (9)
The total dissipation in the scaled form is
${\cal R}^{s}=R_{f}^{s}+R_{ae}={1\over
b}\int^{b/a}_{0}\Big{[}\int\phi(v^{s}(r))dv^{s}+{\gamma_{u}\over
2}\Big{(}{\partial\dot{X}(r)\over\partial r}\Big{)}^{2}\Big{]}dr.$ (10)
The first term on the right hand side is the frictional dissipation arising
from the peel force function. The scaled peel force function, $\phi(v^{s})$,
can be obtained by using the scaled velocities in Eq. (6). The nature of
$\phi(v^{s})$ is shown in Fig. 1(b). Note that the maximum occurs at
$v^{s}=1$. We shall refer to the left branch AB as the ‘stuck state’ and the
high velocity branch CD as the ’peeled state’. The second term on the right
hand side denotes the scaled form of the acoustic energy dissipated.
The Lagrange equations of motion in terms of the generalized coordinates
$\alpha(r),\dot{\alpha}(r),X(r)$ and $\dot{X}(r)$ are
$\displaystyle{d\over d\tau}\left({\partial{\cal
L}\over{\partial\dot{\alpha}(r)}}\right)-{\partial{\cal
L}\over{\partial\alpha(r)}}+{\partial{\cal
R}^{s}\over{\partial\dot{\alpha}(r)}}$ $\displaystyle=0$ , (11)
$\displaystyle{d\over d\tau}\left({\partial{\cal
L}\over{\partial\dot{X}(r)}}\right)-{\partial{\cal L}\over{\partial
X}(r)}+{\partial{\cal R}^{s}\over{\partial\dot{X}(r)}}$ $\displaystyle=$
$\displaystyle 0.$ (12)
Using this, we get the equations of motion as
$\displaystyle\ddot{\alpha}$ $\displaystyle=$
$\displaystyle-{v_{c}\dot{v}^{s}\over R^{s}}-C_{f}R^{s}{{l^{s}\over
L^{s}}\,{sin}\,\alpha\over(1+{l^{s}\over L^{s}}\,{sin}\,\alpha)}\phi(v^{s}),$
(13) $\displaystyle\ddot{X}$ $\displaystyle=$
$\displaystyle-X+k_{0}{\partial^{2}X\over\partial
r^{2}}+{\phi(v^{s})\over(1+{l^{s}\over
L^{s}}\,{sin}\,\alpha)}+\gamma_{u}{\partial^{2}\dot{X}\over\partial r^{2}}.$
(14)
However, Eqs. (13, 14) should satisfy the constraint Eq. (7). This consistency
can be imposed by using the theory of mechanical systems with constraints ECG
. This leads to an equation for the acceleration variable $\dot{v}^{s}(r)$
obtained by differentiating Eq. (7) and using Eqs. (14),
$\displaystyle\dot{v}^{s}$ $\displaystyle=$
$\displaystyle\big{[}-{\ddot{X}}+{R^{s}l^{s}\over
L^{s}}\big{(}\dot{\alpha}^{2}(cos\alpha-{R^{s}l^{s}}({sin\alpha\over
L^{s}})^{2}\big{)}$ (15) $\displaystyle+$ $\displaystyle
sin\alpha{\ddot{\alpha}}\big{)}\big{]}/v_{c}.$
These Eqs. (7, 13) and (15) constitute a set of nonlinear partial differential
equations that determine the dynamics of the peel front. They have been solved
by discretizing the peel front on a grid of N points and using an adaptive
step size stiff differential equations solver (MATLAB package). We have used
open boundary conditions appropriate for the problem. The initial conditions
were drawn from the stuck configuration (i.e., the values are from the left
branch of $\phi(v^{s})$) with a small spatial inhomogeneity in $X$ such that
they satisfy Eq. (7) approximately. The system is evolved till a steady state
is reached before the data is accumulated.
The nature of the dynamics depends on the pull velocity $V^{s}$, the
dissipation coefficient $\gamma_{u}$ and $C_{f}$. We have carried out detailed
studies of the dynamics of the model over a wide range of values of these
parameters keeping other parameters fixed at $R^{s}=0.35$, $l^{s}=3.5$,
$k_{0}=0.1$ ($k_{t}=1000$ N/m) and $N=50$ (in units of the grid size). Larger
system size $N=100$ is used whenever necessary.
## III Time series analysis of experimental AE signals
One of the objectives is to carry out statistical and nonlinear time series
analysis of experimental AE signals associated with the jerky peel process
with a view to understand the results on the basis of model studies. Acoustic
emission data files were obtained from peel experiments under constant
traction velocity conditions that cover a wide range of values from $0.2$ to
$7.6$ cm/s Ciccotti07 . Signals were recorded at the standard audio sampling
frequency of $44.1$ kHz (having $6$ kHz band width) using a high quality
microphone. They were digitized and stored as $16$ bit signals in raw binary
files. There are 38 data files each containing approximately $1.2\times
10^{6}$ points. The AE signals are noisy as in most experiments on AE.
Two characteristic features of low dimensional chaos are the existence of a
strange attractor with self similar properties quantified by a fractal
dimension (or equivalently the correlation dimension) and sensitivity to
initial conditions quantified by the existence of a positive Lyapunov
exponent. Given the equations of motion, these quantities can be directly
calculated. However, when a scalar time series is suspected to be a projection
from a higher dimensional dynamics, they are traditionally analyzed by using
embedding methods that attempt to recover the underlying dynamics. The basic
idea is to unfold the dynamics through a phase space reconstruction of the
attractor by embedding the time series in a higher dimensional space using a
suitable time delayPackard ; GP . Consider a scalar time series measured in
units of sampling time $\Delta t$ defined by $[x(k),k=1,2,3,\cdots,N]$. Then,
we can construct $d-$dimensional vectors defined by
$\vec{\xi}_{k}=[x(k),x(k+\tau),\cdots,x(k+(d-1)\tau)];\,\,k=1,\cdots,[N-(d-1)\tau]$.
The delay time $\tau$ suitable for the purpose is either obtained from the
autocorrelation function or from mutual information HKS . Once the
reconstructed attractor is obtained, the existence of converged values of
correlation dimension and a positive exponent is taken to be a signature of
the underlying chaotic dynamics. In real systems, most experimental signals
contain noise which in this case is high. There are several methods designed
to cure the noise component HKS ; KS ; KS1 ; GHKSS ; CH . Usually, the cured
data sets are then subjected to further analysis.
The correlation integral defined as the fraction of pairs of points
$\vec{\xi}_{i}$ and $\vec{\xi}_{j}$ whose distance is less than $r$, is given
by
$C(r)=\frac{1}{N_{p}}\sum_{i,j}\Theta(r-|\vec{\xi}_{i}-\vec{\xi}_{j}|),$ (16)
where $\Theta(\cdots)$ is the step function and $N_{p}$ the number of vector
pairs summed. A window is imposed to exclude temporally correlated points HKS
. The method provides equivalence between the reconstructed attractor and the
original attractor. It has been shown that a proper equivalence is possible if
the time series is noise free and long Ding . For a self similar attractor
$C(r)\sim r^{\nu}$, where $\nu$ is the correlation dimension GP . Then, as $d$
is increased, one expects to find a convergence of the slope $dlnC(r)/dlnr$ to
a finite value in the limit of small $r$. However, in practice, the scaling
regime is found at intermediate length scales due to the presence of noise.
The existence of a positive Lyapunov exponent is considered as an unambiguous
quantifier of chaotic dynamics. However, the presence of superposed noise
component, which in the present case is high, poses problems. In principal the
noise component can be cured and then the Lyapunov exponent calculated HKS ;
KS ; KS1 ; GHKSS ; CH . Here, we use an algorithm that does not require
preprocessing of the data; it is designed to average out the influence of
superposed noise. The algorithm, which is an extension of Eckmann’s algorithm,
has been shown to work well for reasonably high levels of noise in model
systems as well as for short time series. The method has been used to analyze
experimental time series as well (for details, see Ref. Anan99 ; Noro01 ).
In the conventional Eckmann’s algorithm Eckmann , a sequence of tangent
matrices are constructed that connect the initial small difference vector
$\vec{\xi}_{i}-\vec{\xi}_{j}$ to evolved difference vectors
$\vec{\xi}_{i+k}-\vec{\xi}_{j+k}$, where $k$ is the propagation time. In the
algorithm, the number of neighbors used is small typically min$[2d,d+4]$
contained in a spherical shell of size $\epsilon_{s}$. A simple modification
of this is to use those neighbors falling between an inner and outer radii
$\epsilon_{i}$ and $\epsilon_{0}$ respectively. Then, the inner shell
$\epsilon_{i}$ is expected to act as a noise filter. However, so few neighbors
will not be adequate to average out the noise component superposed on the
signal. Thus, the modification we effect is to allow more number of neighbors
so that the noise statistics is sampled properly. (See for details Anan99 ;
Noro01 .) As the sum of the exponents should be negative for a dissipative
system, we impose this as a constraint. In addition, we also demand the
existence of stable positive and zero exponents (a necessary requirement for
continuous time systems like AE) over a finite range of shell sizes
$\epsilon_{s}$. As a cross check, we have also calculated the correlation
integral and Lyapunov spectrum using the TISEAN package as well HKS .
## IV Dynamics of the peel front
A systematic study of the dynamics of the model is essential to understand the
influence of the various parameters on the spatiotemporal dynamics of the peel
front, its connection to intermittent peeling and to the accompanying acoustic
emission. From Eq. (10), it is clear that the acoustic energy $R_{ae}$ is the
spatial average of the local strain rate. As the peel front patterns determine
the nature of acoustic energy, a detailed study of the dependence of the
patterns on the relevant parameters and on the pull velocity should help us to
get insight into AE generation process during peeling.
### IV.1 General considerations on time scales and parameter values
We begin by making some general observation about the various parameters and
their influences. The dynamics of the model is sensitive to the three time
scales (reduced from four due to scaling) determined by the parameters
$C_{f}$, $\gamma_{u}$ and $V^{s}$. $C_{f}$ is related to the ratio of inertial
time of the tape mass to that of roller inertia (see below). The dissipation
parameter $\gamma_{u}$ reflects the rate at which the local strain rate
relaxes. The pull velocity $V^{s}$ determines the duration over which all the
internal relaxations are allowed to occur. The range of $C_{f}$ is determined
by the allowed values of the tape mass $m$ and the roller inertia $I$.
Following our earlier studies, we vary $I$ from $10^{-5}$ to $10^{-2}$ and $m$
from 0.001 to 0.1. Thus, $C_{f}$ can be varied over a few orders of magnitude
keeping one of them fixed. For model calculations, the dissipation parameter
is varied from 0.001 to 1. (However, an order of magnitude estimate shows that
$\gamma_{u}<<1$, see below.) The range of $V^{s}$ of interest is determined by
the instability domain which is from 1 to $\sim 12$ as shown in Fig. 1(b).
To appreciate the influence of inertial time scale of tape mass parameterized
by $C_{f}$, consider the low mass limit of the ODE model Rumi05 which has
been shown to lead to the DAE model equations Rumi04 . In this limit, the
velocity jumps across the two branches of the peel force function are abrupt
with infinite acceleration. However, finite tape mass introduces an additional
time scale that leads to jumps in $v^{s}$ to occur over a finite time scale
which in turn the magnitude of the velocity jumps. Indeed, the phase space
trajectory need not jump to the high velocity branch of $\phi(v^{s})$, as we
shall see. This can be better appreciated by considering the ODE model (that
ignores the spatial degrees of freedom). Consider the relevant ODE model
equations Rumi05 (in unscaled form).
$\displaystyle{\ddot{\alpha}}$ $\displaystyle=$
$\displaystyle-\frac{\dot{v}}{R}+\frac{R}{I}\frac{cos\,\theta}{(1-cos\,\theta)}f(v),$
(17) $\displaystyle m{\ddot{u}}$ $\displaystyle=$
$\displaystyle\frac{1}{(1-cos\,\theta)}[f(v)-ku(1-cos\,\theta)],$ (18)
where $\alpha$ is shown in Fig. 1(a) and $u$ the displacement of the contact
point. $m$ is mass of the tape and $k$ the spring constant of the tape. From,
Eqs. ( 17, 18), two inertial time scales can be identified, one corresponding
to the roller inertia $\Omega_{\alpha}=(Rf/I)^{1/2}$ and another to that of
the tape mass $\Omega_{u}=(k/m)^{1/2}$. (Note that $k$ in Eq. (18) of the ODE
model corresponds to $k_{t}$ in the present model.) Thus, $C_{f}$ in present
model is directly related to the ratio of these two inertial time scales.
Differentiating Eq. (1), we get
${\dot{v}}+{\ddot{u}}+R\,{\ddot{\alpha}}\,cos\,\theta=R\,{\dot{\alpha}}\,{\dot{\theta}}\,sin\theta.$
(19)
Eq. (18) is the force balance equation. In the limit $m\rightarrow 0$, we have
the algebraic constraint $f(v)=F(t)(1-cos\,\theta(t))$. Differentiating this
equation shows that $\dot{v}$ diverges at points of maximum and minimum of the
peel force function $f(v)$. This demonstrates that in the low mass limit, the
orbits jump to the high velocity branch abruptly. Now consider Eq. (19) that
relates the acceleration of the peel point ($\dot{v}$), acceleration of the
displacement $u$ i.e., $\ddot{u}$ and $\ddot{\alpha}$. This again is basically
a force balance equation as can be seen by multiplying the equation by the
tape mass $m$. As the right hand side is small, any increase in one of these
acceleration variables implies a decrease in the other variables. As low mass
limit implies infinite acceleration of the peel front ($\dot{v}$) across the
peel force function, finite mass implies the velocity jumps across the peel
force function is reduced. It is worthwhile to note that the effect of
inertial time scale causing jumps across the unstable branch to occur at a
finite time scale is a general feature. This has been recognized and
demonstrated experimentally in the context of the PLC effect SF85 .
Figure 2: (Color online a, b, e, f) (a, b) Snapshots for $C_{f}=7.88$,
$V^{s}=1.48$, and $\gamma_{u}=1.0$. (a) a smooth peel front with small
amplitude high frequency oscillations due to finite roller inertia and (b) a
smooth peel front. (c) Phase plot for an arbitrary spatial point on the peel
front. Bold line represents $\phi(v^{s})$. (d) Model acoustic energy plot. (e,
f) Snapshots for $C_{f}=7.88$, $V^{s}=1.48$, and $\gamma_{u}=0.1$. (Rugged
peel front and the onset of stuck-peeled configuration.)
Now consider estimating the order of magnitude of the dissipative time scale.
The unscaled dissipation parameter $\Gamma_{u}$ is related to the fluid shear
viscosity $\eta$ Land and thus an order of magnitude estimate can be
obtained. Typical values of $\eta$ for adhesives at low shear rates is $\sim
1000-10000$ Pa.s. As stress is directly related to shear viscosity,
$\Gamma_{u}$ can be estimated using typical dimensions of the peel front. It
has been shown that deformed peel front dimension is about 100 $\mu m$
Dickinson ; Yamazaki , the thickness of the adhesive is $\sim$ 50 $\mu m$ and
the width of the peel front $\sim 20$ mm (width of the tape). It is easy to
show that $\Gamma_{u}\sim 10^{-3}-10^{-2}$ J.s. Thus, the range of
$\gamma_{u}$ is $\sim 10^{-3}-10^{-4}$ taking $\eta\sim 1000$ Pa.s. As some of
the numbers used are material dependent, this is just an order of magnitude
estimate. For model studies, the range of $\gamma_{u}$ is taken to be from 1
to 0.001. However, we will not discuss the results for $\gamma_{u}=0.001$ as
these are similar to 0.01.
Within the scope of the model, the model acoustic energy given by
$R_{ae}(\tau)={1\over 2}\gamma_{u}\sum_{i}(\dot{X}_{i+1}-\dot{X}_{i})^{2}$ (in
the discretized form) depends on the nature of the local displacement rate.
Based on this relation, some general observations can be made on the nature of
$R_{ae}$ and its dependence on the peel front dynamics. From Eq. (14), high
$\gamma_{u}$ implies that the coupling between neighboring sites is strong and
hence the local dynamics at one spatial location has no freedom to deviate
from that of its neighbor. Thus, the displacement rate at a point on the peel
front cannot differ from that of its neighbor. For the same reason, low
$\gamma_{u}$ implies weak coupling between displacement rates on neighboring
points on the peel front which therefore can differ substantially. This
clearly should lead to significantly more inhomogeneous peel velocity profile.
Based on the above arguments, high $\gamma_{u}$ should lead to smooth peel
front and consequently sharp bursts in the model acoustic energy $R_{ae}$ that
occurs during jumps between the two branches of $\phi(v^{s})$. In contrast,
when $\gamma_{u}$ is small, $R_{ae}$ should be high as also spread out in
time. However, as the exact nature of the peel front pattern is sensitive to
the values of $C_{f}$, pull velocity $V^{s}$ and $\gamma_{u}$, the nature of
$R_{ae}$ depends on all the three time scales. Indeed, one should expect that
the more rapidly the peel front patterns change with time, the noisier the
model acoustic energy should be. This is one feature that we hope to compare
with experimental acoustic signals.
### IV.2 Results of the model
We have carried out extensive studies on the nature of the dynamics for a wide
range of values of the parameters stated above. The peel front dynamics is
analyzed by recording the velocity-space-time patterns of the peel front, the
phase plots in the $X^{s}-v^{s}$ plane for an arbitrary spatial point on the
peel front and the model acoustic energy dissipated $R_{ae}$. (Unless
otherwise stated, these plots refer to steady state dynamics after all the
transients have died out.) Here, we present a few representative solutions for
different sets of parameters within the range of interesting dynamics. Our
analysis shows that while the nature of the dynamics results from competing
influences of the three time scales, the dissipation parameter $\gamma_{u}$
appears to have a significant influence on the spatiotemporal dynamics of the
peel front.
#### IV.2.1 Case (i), $C_{f}=7.88$ \- high (low) tape mass, low (high)
roller inertia
Figure 3: (Color online) (a-d) Snapshots during the peeling process for
$C_{f}=7.88,V^{s}=1.48$, and $\gamma_{u}=0.01$ : (a) highly rugged peel front
when the system is on the left branch of $\phi(v^{s})$, (b) the onset of peel
process, (c) stuck-peeled configuration and (d) resulting nearly uniform
peeled state.
Given a value of $C_{f}$ there is a range of values of $(m,I)$. In this case,
the set of values are: $(0.1,10^{-3}),(0.01,10^{-4})$ and $(0.001,10^{-5})$.
The dissipation coefficient is varied from $\gamma_{u}=1$ to 0.01. For high
$\gamma_{u}=1.0$, only smooth peeling is seen independent of the magnitude of
the pull velocity. The peel front switches between the low and high velocity
branches of the peel force function $\phi(v^{s})$. Plots of the smooth nature
of the entire peel front are shown in Figs. 2(a, b) for $V^{s}=1.48$. Figure
2(a) shows the nature of the peel front when the system is on the low velocity
branch of $\phi(v^{s})$, i.e., the local velocities of all spatial elements
follow the AB branch of $\phi(v^{s})$. The small amplitude synchronous high
frequency oscillation of the entire peel front results from the roller
inertia. (Compare the values of $v^{s}$ in the two figures.) A phase plot in
the $X^{s}-v^{s}$ plane for an arbitrary point on the peel front is shown in
Fig. 2(c). The small amplitude oscillation of the peel front shown in Fig.
2(a) corresponds to the velocity oscillations in the phase plot (Fig. 2(c)).
As high $C_{f}$ implies relatively low values of $I$, it can be shown (on
lines similar to Ref. Rumi04 ) that the orbit sticks to the stationary
branches (slow manifold) of the peel force function $\phi(v^{s})$ jumping
between the branches only at the limit of stability typical of relaxation
oscillations. The corresponding model acoustic energy $R_{ae}(\tau)$ shows a
sequence of small amplitude spikes corresponding to the small amplitude
oscillations arising from the roller inertia [Fig. 2(d)] followed by large
bursts that occur at regular intervals. The bursts result from the peel front
jumping from the stuck to the peeled state and back. Note that the duration of
the bursts are short compared to duration between them.
However, as we decrease $\gamma_{u}$ to $0.1$ keeping $V^{s}=1.48$, we observe
rugged and stuck-peeled configurations. The rugged pattern is seen when the
system is on the AB branch of $\phi(v^{s})$. Even so, on reaching the limit of
stability, the entire contact line peels nearly at the same time as shown in
Fig. 2(e). But once it jumps to the high velocity branch CD of $\phi(v^{s})$,
the peel front that has nearly uniform peel velocity commensurate with that of
the right branch of $\phi(v^{s})$ becomes unstable and breaks up into stuck
and peeled segments as shown in Fig. 2(f). The width of these segments
increases in time with a concomitant decrease in the magnitude of the velocity
jumps of peeled segments, eventually the entire peel front goes into a stuck
state. Then, the cycle restarts with the peel front switching between the
rugged and stuck-peeled (SP) states. The phase plot is similar to that for
$\gamma_{u}=1$ again sticking to the slow manifold. The model acoustic energy
dissipated $R_{ae}$ is also similar to that for $\gamma_{u}=1$ except that the
large bursts are comparatively broader as should be expected due to presence
of stuck-peeled configurations that contribute to large changes in the local
velocity.
Figure 4: (Color online a,e) (a, b) Snapshot of a stuck-peeled configuration
and model acoustic energy respectively for $C_{f}=7.88$, $\gamma_{u}=0.01$ and
$V^{s}=2.48$. (c, d) Phase plot and model acoustic energy plot respectively
for $C_{f}=7.88$, $V^{s}=4.48$, and $\gamma_{u}=0.01$. (e) Snapshot of long
lived stuck-peeled configuration for $C_{f}=7.88$, $V^{s}=5.48$, and
$\gamma_{u}=0.01$ and (f) the associated model acoustic energy.
As we decrease $\gamma_{u}$ to 0.01, the observed patterns are similar to
those for $\gamma_{u}=0.1$ but the sequence of the peel front patterns is
different. Starting with a low velocity configuration that is even more rugged
compared to that for $\gamma_{u}=0.1$ as shown in Fig. 3(a), the peel process
starts with a small stuck segment getting peeled [Fig. 3(b)]. There after,
several stuck segments peel out leading to a stuck-peeled pattern as shown in
Fig. 3(c), eventually, the entire peel front peels-out leaving a nearly
uniform peeled state as shown in Fig. 3(d) (with a velocity commensurate with
the high velocity branch of $\phi(v^{s})$). This is again destabilized with
some segments of the peel front getting stuck as in the case of
$\gamma_{u}=0.1$ (similar to Fig. 2(f)). The number of such stuck segments
increases with time, eventually the whole peel front goes into a stuck state.
The cycle restarts. The phase plot is similar to $\gamma_{u}=0.1$ and 1.
Indeed, for a given $C_{f}$, independent of $\gamma_{u}$ the phase plot
changes only when $V^{s}$ is increased. But $R_{ae}$ shows broader bursts
compared to $\gamma_{u}=0.1$ as the corresponding stuck-peeled configurations
last longer. Even so, the duration of the SP configurations in a cycle is
short, i.e., the duration of the bursts is short compared to the duration
between them.
Now we consider the influence of increasing the pull velocity (keeping $C_{f}$
fixed at 7.88) which in turn should leave less time for internal relaxational
mechanisms to operate. Intuitively one should expect that some patterns
observed for low $V^{s}$ may not be seen for higher values of $V^{s}$.
$\gamma_{u}=1$ case is uninteresting for the reasons stated above. But,
reducing $\gamma_{u}$ to 0.1 does provide some degree of freedom for the local
dynamics to operate at each point. Even so, for $V^{s}=2.48$, the peel front
switches between a SP configuration with most segments momentarily in the
stuck state (similar to Fig. 3(b)) and a configuration that has several stuck-
peeled segments (similar to Fig. 3(c)). The corresponding $X^{s}-v^{s}$ phase
plot shows that the orbit jumps moves slightly beyond the upper value of
$\phi(v^{s})$ and jumps back from the right branch even before reaching the
minimum of $\phi(v^{s})$ ( not shown). As we decrease $\gamma_{u}$ to 0.01,
the rugged configuration seen for $V^{s}=1.48$ is no longer seen and only SP
configurations are observed as shown in Fig. 4(a). The SP configurations are
dynamic in the sense, segments that are in stuck state at one time become
unstuck at a later time and vice versa. For this case
($C_{f}=7.88,V^{s}=2.48,\gamma_{u}=0.01$), these rapid changes occur over a
short time scale. Consequently, the model acoustic energy is quite noisy as
shown in Fig. 4(b) but has a noticeable periodic component. The points of
minima correspond to configurations that have fewer peeled segments compared
to those near the peak of $R_{ae}$. The phase plot in the $X^{s}-v^{s}$ plane
is limited to the upper part of $\phi(v^{s})$. Even as the phase plots for any
two spatial points look similar, there is a phase difference. For instance, at
any given time, the phase point of stuck segment will be on the left branch
while that for peeled point will be on the right branch.
As we increase $V^{s}$ to $4.48$ (keeping $C_{f}$ at $7.88$), there is even
lesser time for peel front inhomogeneities to relax and thus, we observe a
smooth peeling for $\gamma_{u}=1$ and as also for $0.1$. As we decrease
$\gamma_{u}$ to $0.01$, we see only SP patterns (not shown but similar to Fig.
4(a)). The corresponding $X^{s}-v^{s}$ phase plot for an arbitrary point on
the peel front shown in Fig. 4(c) is confined to the top of $\phi(v^{s})$. The
corresponding $R_{ae}$ is noisy and irregular as shown in Fig. 4(d). However,
when we increase $V^{s}$ to $5.48$, initially, one does observe the patterns
switching between rugged and SP configurations. If we wait long enough, we
observe only SP configurations that are different from those for lower
$V^{s}$. In this case, the stuck and peeled segments are long lived. A top
view of the SP pattern is shown in Fig. 4(e). The phase space orbit in the
$X^{s}-v^{s}$ plot is pushed beyond the upper limit of $\phi(v^{s})$. The
energy dissipated is quite regular (but aperiodic) unlike that for lower pull
velocities as shown in Fig. 4(f). This regularity is clearly due to the long
lived nature of these SP configurations. The long lived nature of the SP
configurations for high pull velocity is a general feature, i.e., the duration
over which the stuck segments remain stuck (peeled segments remain peeled)
increases as we increase the pull velocity. The dynamics is no longer
interesting beyond $V^{s}=6.48$ as only smooth peeling is seen.
Figure 5: (Color online a, b) (a, b) Snapshots of stuck-peeled configurations
for $C_{f}=0.788,V^{s}=1.48$, and $\gamma_{u}=1.0$. Note that (a) has fewer
peeled segments compared to (b). (c) Phase plot for an arbitrary point on the
peel front, and (d) model acoustic energy plot.
Figure 6: (Color online a, c) (a, b) Snapshot of a stuck-peeled configuration
for $C_{f}=0.788,V^{s}=1.48$, and $\gamma_{u}=0.1$ and model acoustic energy
plot respectively. (c, d) Plot of ‘an edge of peeling’ configuration for
$C_{f}=0.788,V^{s}=1.48$, and $\gamma_{u}=0.01$ and the corresponding model
acoustic energy.
Figure 7: (Color online a, b) (a, b) Snapshots during peel process for
$C_{f}=0.788,V^{s}=2.48$, and $\gamma_{u}=1.0$. (c) Model acoustic energy
plot. (d) Phase plot for an arbitrary spatial point on the peel front. Bold
line represents $\phi(v^{s})$.
#### IV.2.2 Case (ii), $C_{f}=0.788$ \- high (and low) tape mass, high (and
intermediate) roller inertia
For this value of $C_{f}$, the allowed set of values of $(m,I)$ are
$(0.1,10^{-2}),(0.01,10^{-3})$ and $(0.001,10^{-4})$. The dynamics is more
interesting for this case as there is a scope for competition among the three
time scales.
We first study the dynamics keeping $V^{s}=1.48$ and varying the dissipation
parameter. For $\gamma_{u}=$ 1.0, the uniform nature of the peel front seen
for $C_{f}=7.88$ disappears and even for short times, stuck-peeled
configurations are seen. The peel front patterns stabilize to stuck-peeled
configurations as shown in Figs. 5(a, b). As can be seen these SP patterns
have only a few stuck or peeled segments with moderate velocity jumps and
smooth variation along the peel front unlike the SP configurations discussed
earlier. (Note that the SP configuration in Fig. 5(b) has more stuck segments
compared to Fig. 5(a).) The moderate velocity jumps can be understood by
noting that the phase space orbit never visits the high velocity branch of
$\phi(v^{s})$ as can be seen from Fig. 5(c). It is interesting to note that
the trajectory stays close to the unstable branch of $\phi(v^{s})$ even after
attempting to jump from the low velocity branch. Such orbits are reminiscent
of canard type solutions Canard . The trajectory is irregular and is
suggestive of spatiotemporal chaotic nature of the peel front. The energy
dissipated $R_{ae}$ shown in Fig. 5(d) is continuous and irregular due to the
dynamic SP pattern as should be expected, but there is a noticeable periodic
component. The rough periodicity of $R_{ae}$ can be traced to fact that the
peel front configurations switch between patterns with more stuck segments and
less stuck segments. (From the number shown on $x$-axis, Fig. 5(a) can be
identified with minimum and Fig. 5(b) with the peak of $R_{ae}$ in Fig. 5(d).
See the marked arrows as well.)
As we decrease $\gamma_{u}$ to 0.1, the SP configurations observed have more
stuck and peeled segments compared to $\gamma_{u}=1$ (compare Fig. 6(a) with
Fig. 5 (a)). However, the magnitude of the velocity jumps remains moderate as
in the previous case. This is again due to the fact that the orbit never
visits the high velocity branch of $\phi(v^{s})$. (Recall that given a value
of $C_{f}$, the phase plot remains the same for different $\gamma_{u}$ values
as long as $V^{s}$ is fixed). Indeed, for this value of $C_{f}=0.788$, the
orbit never jumping to the high velocity branch is a consequence of finite
inertia of the tape mass compared to that of the roller inertia as discussed
earlier. For this case, the model acoustic energy $R_{ae}$ is also irregular
and continuous as shown in Fig. 6(b) with a noticeable periodic component.
Now, if we decrease $\gamma_{u}$ further to 0.01, the peel front pattern
displays increased number of stuck and peeled segments with each stuck segment
having only a few contiguous stuck points as can be seen from Fig. 6(c). Note
also that there is a large dispersion in the magnitudes of the velocity jumps
of the peeled segments even as the largest one is significantly smaller than
the value of CD branch of $\phi(v^{s})$. As can be seen from the Fig. 6(c),
even though the pattern is dynamic, the segments that are stuck are barely so.
Thus, the configuration shown in Fig. 6(c) gives the feeling of a critically
poised state. The corresponding $X^{s}-v^{s}$ phase plot (similar to that
shown in Fig. 5(c)) is irregular and possibly suggestive of spatiotemporal
chaotic nature of the peel front. The acoustic energy $R_{ae}$ is very
irregular without any trace of periodicity as shown in Fig. 6(d).
Figure 8: (Color online a, b, d, e) (a, b) Snapshots during peel process
$C_{f}=0.788,V^{s}=4.48$, and $\gamma_{u}=1.0$. (c) Phase plot for an
arbitrary spatial point on the peel front. Bold line represents $\phi(v^{s})$.
(d, e) Snapshots during peel process $C_{f}=0.788,V^{s}=4.48$, and
$\gamma_{u}=0.01$.
We now consider the influence of increasing the pull velocity $V^{s}$. As we
increase $V^{s}$ to 2.48, the spatiotemporal patterns seen for
$\gamma_{u}=1.0$, 0.1 and 0.01 are slightly different from those for
$V^{s}=1.48$. For $\gamma_{u}=1.0$, the peel process goes through a cycle of
configurations shown in Figs. 7(a, b). It is clear that Fig. 7(a) has more
segments in the stuck state while Fig. 7(b) is the usual kind of SP
configuration except that the stuck and peel segments are fewer. For this
case, the stuck and peeled segments last longer than those for $V^{s}=1.48$.
The corresponding $R_{ae}$ for each $\gamma_{u}$ exhibits noisy bursts
overriding a periodic component. A typical plot for $\gamma_{u}=1$ is shown in
Fig. 7(c). From the time labels as also the arrows shown, the minima and
maxima in $R_{ae}$ can be identified with Figs. 7(a, b) respectively. The
orbit in the $X^{s}-v^{s}$ plane moves into regions much beyond the values
allowed by $\phi(v^{s})$ as is clear from Fig. 7(d). The phase plots for
$\gamma_{u}=0.1$ and 0.01 are similar to this case.
For $\gamma_{u}=0.1$ also, the peel front pattern goes through a cycle of
stuck-peeled configurations (with more stuck and peeled segments than for
$\gamma_{u}=1.0$) and stuck segments (similar to Fig. 7(a)). Yet, the energy
dissipated $R_{ae}$ is similar to Fig. 7(c) for $\gamma_{u}=1.0$ which is
surprising considering that there are more stuck and peeled segments compared
to $\gamma_{u}=1$ case. This can be traced long lived of the stuck or peeled
configurations that hardly change over a cycle (as in the case of
$C_{f}=7.88,V^{s}=5.48,\gamma_{u}=0.01$, see Fig. 4(e)). The peel process is
similar even for $\gamma_{u}=0.01$.
As we increase the peel velocity to 4.48, the influence of this time scale on
the peel front pattern is discernable even for $\gamma_{u}=1.0$. The
spatiotemporal patterns of the peel front switches sequentially from nowhere
stuck configuration shown in Fig. 8(a) to stuck-peeled configuration with few
stuck and peeled segments shown in Fig. 8 (b). Note that there are very few
stuck and peeled segments. The corresponding $R_{ae}$ exhibits noisy periodic
pattern similar to Fig. 7(c) for $V^{s}=2.48$. The $X^{s}-v^{s}$ phase plot in
Fig. 8(c) shows that the orbit can move much beyond the values allowed by
$\phi(v^{s})$. As we decrease $\gamma_{u}$ to 0.1, the nowhere stuck
configuration [Fig. 8(a)] is replaced by a partly stuck, partly peeled
configuration and a SP configuration. For $\gamma_{u}=1.0$ case ( Fig. 8(c)),
the $X^{s}-v^{s}$ phase plot is slightly different as the orbit makes several
loops before it jumps to low velocity branch without visiting the high
velocity branch of $\phi(v^{s})$. The nature of $R_{ae}$ is still noisy and
periodic similar to Fig. 7(c). As we decrease $\gamma_{u}$ to 0.01, the peel
process goes through SP configurations shown in Figures. 8(d, e). Note that
Fig. 8(e) has large dispersion in the magnitude of velocity jumps of the
peeled segments compared to that in Fig. 8(d). It is worth emphasizing that
the increase in the number of stuck and peeled segments with decrease in
$\gamma_{u}$ is a general feature. Despite the higher number of stuck and
peeled segments, $R_{ae}$ for $\gamma=0.01$ is similar to that for
$\gamma_{u}=1.0$ as these peel front configurations are long lived which again
is a general feature observed at high pull velocities. Finally, it should be
stated that for $C_{f}=0.788$, in general the velocity variation along the
peel front is much more smooth compared to other values of $C_{f}$. The
dynamics is uninteresting beyond $V^{s}=7.48$ as only smooth peeling is seen.
Figure 9: (Color online b, c) (a) $R_{ae}$ as a function for
$C_{f}=0.00788,V^{s}=1.48$, and $\gamma_{u}=1$. (b) a stuck-peeled
configuration with more stuck segments compared to (c) where nearly equal
number of stuck and peeled segments for $C_{f}=0.00788,V^{s}=1.48$, and
$\gamma_{u}=0.01$. (d) The corresponding phase plot for an arbitrary spatial
point along the peeling front. Bold line represents $\phi(v^{s})$. (e)
Corresponding $R_{ae}(\tau)$ as a functions of time and (f) substructure of
(e).
#### IV.2.3 Case (iii), $C_{f}=0.00788$, low tape mass and high roller
inertia
For this value of $C_{f}$, there is just one set of values of tape mass and
roller inertia, namely, $m=0.001$ and $I=0.01$. As the tape mass is low, this
also corresponds to the DAE type of solutions for each spatial point. Thus,
the velocity jumps between the two branches of the peel force function will
always be abrupt with the roller inertia playing a major role in allowing the
orbits to jump between the branches of the peel force function as demonstrated
earlier Rumi04 .
Figure 10: (Color online a, b) (a, b) Snapshots during peel process for
$C_{f}=0.00788,V^{s}=2.48$, and $\gamma_{u}=0.01$. (c) Corresponding model
acoustic energy plot.
Figure 11: (a) (Color online) Snapshot during peel process
$C_{f}=0.00788,V^{s}=4.48$, and $\gamma_{u}=0.01$. (b) Model acoustic energy
plot. (c) Phase plot for an arbitrary spatial point on the peel front for
$C_{f}=0.00788$, $V^{s}=4.48$, and $\gamma_{u}=0.01$. Bold line represents
$\phi(v^{s})$
Consider the influence of the dissipation parameter $\gamma_{u}$ keeping
$V^{s}=1.48$. For $\gamma_{u}=1$, peeling is uniform and thus the whole peel
front switches between the two branches of the peel force function. The
acoustic energy shows a bunch of seven double spikes that appear at regular
interval as shown in Fig. 9(a). (The number of spikes is correlated with the
number of cascading loops seen in the $X^{s}-v^{s}$ phase plot, see below.) As
we decrease $\gamma_{u}$ to 0.01, the peel front goes through a cycle of
patterns with only few peeled segments and those with large number of stuck-
peeled segments as shown in Figs. 9(b) and (c) respectively. The phase plot in
the $X^{s}-v^{s}$ plane of an arbitrary point on the peel front jumps between
the $AB$ to $CD$ branches of $\phi(v^{s})$. As shown in Fig. 9(d), in a cycle,
the trajectory starting at the highest value of $\phi(v^{s})$ stays on $CD$
for a significantly shorter time compared to that on the left branch. The
orbit then cascades down through a series of back and forth jumps between the
two branches of $\phi(v^{s})$. (For $V^{s}=1.48$, independent of $\gamma_{u}$
value, the nature of the phase plot is the same with seven loops.)
The corresponding model acoustic energy consists of rapidly fluctuating time
series with an overall convex envelope of bursts separated by a quiescent
state as shown in Fig. 9(e). (Contrast this with Fig. 9(a) for
$\gamma_{u}=1$.) From the time labels in Figures. 9(b, c), both configurations
belong to the region within the bursts [Fig. 9(e)]. To understand this complex
pattern of bursts in $R_{ae}$ we have looked at the fine structure of each of
these bursts along with the evolution of the associated configurations. One
such plot is shown in Fig. 9(f) which shows that fine structure consists of
seven bursts within each convex envelope. These seven bursts can be correlated
with the seven loops in the phase plot shown in Fig. 9(d). The time interval
marked LM in the phase plot corresponds largely to stuck configuration (not
shown) and hence can be easily identified with the quiescent region in
$R_{ae}$. Following the peel front patterns continuously, it is possible to
identify the sequence of configurations that leads to the substructure shown
in $R_{ae}$ [Fig. 9(f)]. For instance, the loop marked PQRST in the
$X^{s}-v^{s}$ plot corresponds to the burst between P and T in Fig. 9(f).
During this period, the configuration at P is largely in the stuck state (as
in Fig. 9(b)) which gradually evolves with more and more segments peeling out
[Fig. 9(c)] as the trajectory moves from $P\rightarrow Q\rightarrow
R\rightarrow S$. As the number of stuck and peeled segments reaches a maximum,
$R_{ae}$ reaches the peek region. Then, during the interval corresponding to S
to T the number of peeled segments decreases abruptly. Thereafter, the next
cycle of configurations (corresponding to the next loop in the phases plot)
ensues.
As we increase the pull velocity, the peel front is smooth for
$\gamma_{u}=1.0$ as also for 0.1 for the entire range of pull speeds. However,
for $\gamma_{u}=0.01$, as we increase $V^{s}$ to 2.48, the peel process goes
through a cycle of SP configurations shown in Figs. 10(a, b). Note that there
is a large dispersion in the jump velocities as is clear from Fig. 10(a). The
corresponding $R_{ae}$ shows rapidly fluctuating triangular envelope of bursts
with no quiescent region seen for $V^{s}=1.48$ case. This is shown in Fig.
10(c). The corresponding phase plot is similar to Fig. 9(d) but has twelve
loops. In addition, the value of the upper loop extends far beyond that
allowed by $\phi(v^{s})$. We also see a fine structure similar to that in Fig.
9(e). As in the previous case, it is possible to identify configurations that
correspond to minima and near the maxima of $R_{ae}$. As we increase $V^{s}$
further to 4.48, only SP configurations are seen. The energy dissipated
$R_{ae}$ shows continuous bursts overriding a sawtooth form as shown in Fig.
11(b). The $X^{s}-v^{s}$ phase plot shows large excursions way beyond the peel
force function values as shown in Fig. 11(c).
A general comment may be relevant regarding large excursions of the trajectory
in the phase plot $X^{s}-v^{s}$ as we increase the pull velocity. This is
easily explained for the low $C_{f}$ (low tape mass, high roller inertia). It
is clear from Eq. (18) that $m\rightarrow 0$ we have
$F(t)\sim\frac{f(v)}{(1+sin\,\alpha(t))}$. As $\alpha(t)$ can takes on
positive and negative values, one can see $F_{max}$ and $F_{min}$ are
determined by minimum (negative) and maximum values of $\sin{\alpha}$ as
argued in Rumi04 . It is possible to extend this argument to finite tape mass
case. Finally it must be stated that the dynamics is no longer interesting
beyond $V^{s}=7.48$.
Figure 12: Largest Lyapunov exponents of the model for $C_{f}=0.788$ and $\gamma_{u}=0.01$: (a) $V^{s}=1.48$ , (b) $V^{s}=2.48$ and (c) $V^{s}=4.48$. $C_{f}$ | $V^{s}$ | $\gamma_{u}$ | $LLE$
---|---|---|---
$7.88$ | $2.48$ | $0.01$ | $0.110$
| $4.48$ | $0.01$ | $0.102$
$0.788$ | $1.48$ | $1.00$ | $0.095$
| | $0.10$ | $0.120$
| | $0.01$ | $0.148$
$0.788$ | $2.48$ | $1.00$ | $0.068$
| | $0.10$ | $0.090$
| | $0.01$ | $0.139$
$0.788$ | $4.48$ | $1.00$ | $0.028$
| | $0.10$ | $0.030$
| | $0.01$ | $0.035$
$0.00788$ | $1.48$ | $0.01$ | $0.105$
| $2.48$ | $0.01$ | $0.180$
| $4.48$ | $0.01$ | $0.224$
Table 1: Largest Lyapunov exponent for the model for various parameter values.
For all $C_{f}$, the LLE reaches a near value zero for $V^{s}=5.48$.
#### IV.2.4 Spatiotemporal Chaotic Dynamics
As discussed above, there are several sets of parameter values for which the
phase plots are irregular which may suggest the possibility of spatiotemporal
chaotic dynamics. To verify this, we have calculated the largest Lyapunov
exponent (LLE) for all the cases using the model equations. (The transient
solutions for the first 2000 time units have been ignored for calculating the
LLE.) Figure 12 shows a plot of the largest Lyapunov exponent for
$C_{f}=0.788$ and $\gamma_{u}=0.01$ for various pull speeds. The value of LLE
for $V^{s}=5.48$ is close to zero (not displayed in the figure). As can be
seen, the LLE is positive being largest for $V^{s}=1.48$ decreasing to near
zero value for $V^{s}=5.48$. Table 1 shows the values of LLE for various
parameter values for which the spatiotemporal dynamics has been detected. From
this we conclude that the dynamics of the peel front is spatiotemporally
chaotic for a range of parameters values.
## V Analysis of AE signals
### V.1 Statistical Analysis of AE signals
$V$ cm/s | $m_{A1}$ | $m_{A2}$ | $\nu$ | $\lambda_{1}$ | $D_{ky}$
---|---|---|---|---|---
$1.0$ | $...$ | $2.00$ | NC | NC | NC
$1.6$ | $...$ | $2.15$ | NC | NC | NC
$3.0$ | $0.31$ | $2.26$ | NC | NC | NC
$3.8$ | $0.30$ | $2.75$ | $2.80$ | $1.70$ | $2.94$
$5.0$ | $0.32$ | $3.00$ | $2.70$ | $1.73$ | $2.96$
$6.2$ | $0.27$ | $3.00$ | $2.55$ | $1.54$ | $2.84$
$7.4$ | $0.30$ | $2.99$ | NC | NC | NC
Table 2: Statistical and dynamical invariants for the experimental AE signals
for typical traction velocities. The second and third columns show power law
exponents $m_{A1}$ and $m_{A2}$ corresponding to small and large amplitudes.
When only a single power law is seen, $m_{A2}$ is the exponent value. Fourth
to sixth columns list the values of the correlation dimension $\nu$, the
largest exponent $\lambda_{1}$ and Kaplan-Yorke dimension $D_{ky}$ obtained
from dynamical analysis of the AE signals. NC corresponds to nonchaotic
dynamics where we did not find any convergence of the correlation dimension.
The acoustic emission data obtained from experiments are fluctuating and noisy
only within the domain where the peel process is intermittent form $0.2$ to
$7.6$ cm/s. As known from early experiments MB , force wave forms change as
the traction velocity is increased. Correspondingly, the nature of the AE
signals also change with the traction velocity. At low traction velocities,
the AE signals have a burst like character appearing at nearly regular
intervals separated by oscillatory decay of the amplitudes. These bursts are
correlated with the stick-slip events. With increasing traction velocity, the
bursts become increasingly irregular and continuous. Examples of burst and
continuous type of AE time series are shown in Figures. 13(a, b).
As shown in the previous section, the nature of the model AE signal depends on
parameter values. In general $R_{ae}$ can be of noisy burst type overriding a
periodic component, continuous and irregular, rapidly fluctuating triangular
envelope of bursts or simply a set of spikes. Clearly, interesting cases for
comparison with the experimental AE signals are those where $R_{ae}$ is
continuous and noisy. Simplest quantity to compare is the nature of the model
acoustic signal with the energy of the experimental AE signal (i.e., square
the amplitude). Figures 13(c, d) show a comparison between model acoustic
signal for $C_{f}=7.88,\gamma_{u}=0.01$ and $V^{s}=1.48$ and energy of the
experimental signal for $V=0.4$ cm/s. Both show burst type emission. As
another example Figs. 13(e, f) show respectively the continuous model signal
(for $C_{f}=7.88,\gamma_{u}=0.01$ and $V^{s}=4.48$) and experimental acoustic
energy for $V=6.4$ cm/s.
Figure 13: (a, b) Raw AE signal for $V=1.6$ cm/s and 7.6 cm/s respectively.
(c) Burst like model acoustic energy plot for $C_{f}=7.88,V^{s}=1.48$ and
$\gamma_{u}=0.01$. (d) Burst like experimental acoustic energy for $V=0.4$
cm/s. (e) Continuous model acoustic energy plot for $C_{f}=7.88,V^{s}=4.48$
and $\gamma_{u}=0.01$. (f) Continuous experimental acoustic energy for $V=6.4$
cm/s.
Given an experimental time series, the simplest statistical quantity to
compute is the statistics of events. The definition of events depends on the
physical situation, which in the case of the AE signal may be the time
interval between the bursts of AE, the amplitude of bursts etc. Indeed the
former has been computed CGVB04 . Here we compute the distribution of the
amplitudes of the AE signals. The difference between the maximum and next
minimum, denoted by $\Delta A$ can be taken to be a measure of the amplitude
of the AE signal. (In experiments, it is measured by setting a cut-off and
measuring all amplitudes larger than the cut-off.) We have computed the
distribution of the amplitudes $D(\Delta A)$ for all the 38 data files for
pull velocities starting from $0.2$ to $7.6$ cm/s. Surprisingly, we find only
power law distributions for all the data files, i.e., $D(\Delta A)\sim\Delta
A^{-m_{A}}$; we do not find peaked distributions. For small traction
velocities, we find a single power law crossing over to a two stage power law
for high traction velocities. A typical single power law distribution for
$V=1.6$ cm/s is shown in Fig. 14(a) with an exponent $m_{A}=2.15$. Figure
14(b) shows a two stage power law for high velocity $V=5.0$ cm/s. The exponent
values are $m_{A}=0.32$ and $3.0$ respectively for small and large amplitude
regimes. The transition from a single to two stage power law distribution
occurs with the deviation for small values seen in Fig. 14(b) becoming more
dominant with increase in the pull velocity. A two stage power law (over one
order of magnitude range) is first observed for $V=3.0$ cm/s. The exponent
values are functions of the pull velocity. Table 2 shows the exponent values
for a selected set of pull velocities. Even though statistical features are
easy to calculate, they are sufficiently discriminating. The analysis will be
useful while comparing the cured data files as also with the statistics of
model acoustic energy signals.
These results may be compared with the statistics of the amplitude of the
model energy bursts $R_{ae}$, i.e., from the maximum to the next minimum.
Denoting $\Delta R_{ae}$ to be the amplitude of $R_{ae}(\tau)$, let $D(\Delta
R_{ae})$ be the distribution of the amplitude of $R_{ae}$. For
$C_{f}=0.00788$, only $\gamma_{u}=0.01$ case is interesting (see Fig. 10(d)
corresponding to $V^{s}=2.48$). To determine $D(\Delta R_{ae})$, we use long
time series (typically $\sim 10^{5}$ points in units of the integration step).
For this case, we find a two stage power law as shown in Fig. 14(d) i. e.,
$D(\Delta R_{ae})\sim\Delta R_{ae}^{-m_{E}}$ with the exponents $m_{E}\sim
0.60$ and 2.0 for the small and large amplitude regimes respectively. Figure
14(c) shows an example of a single power law for $C_{f}=0.788,\gamma_{u}=0.1$
and $V^{s}=1.48$. The exponent value is $m_{E}=0.7$. To compare, we note that
the experimental time series refers to the amplitude of the AE signals while
the model signal $R_{ae}$ is the energy. Thus, the two exponents are related
through $m_{E}=(m_{A}+1)/2$. Using the value of $m_{E}$ shown in Fig. 14(d),
we obtain $m_{A1}=0.2$ for the exponent corresponding to small amplitude
regime (of the AE signal) and $m_{A2}=3.0$ for large amplitudes. Clearly, the
values are in reasonable agreement with the exponent values for the
experimental signals [Fig. 14(b)]. Unlike the experimental signal where
scaling regime is good for all values of the pull velocity, for the model
acoustic energy $R_{ae}$, the distributions for $\Delta R_{ae}$ show a power
law statistics (of at least one order of scaling regime) only for a certain
sets of parameter values. Table 3 lists the exponent values wherever the power
law distribution is seen. For high and low values of $C_{f}$ we find two stage
power law distributions. However, for the intermediate $C_{f}$( 0.788), only a
single stage power law is found. It is interesting to note that the power law
generated here is purely of dynamical origin.
Figure 14: (a) Plot of a single stage distribution of the amplitudes of the uncured data for V = 1.6 cm/s. (b) Plot of a two stage power law distribution of the amplitudes of the uncured data for V = 5.0 cm/s. (c) A plot of a single stage power law distribution for the magnitudes of acoustic energy dissipated for $C_{f}=0.788,\gamma_{u}=0.01$ and $V^{s}=1.48$. (d) A plot of a two stage power law distribution for the magnitudes of model acoustic energy for $C_{f}=0.00788,\gamma_{u}=0.01$ and $V^{s}=2.48$. $C_{f}$ | $V^{s}$ | $\gamma_{u}$ | $m_{E1}$ | $m_{E2}$ | $\nu$ | $\lambda_{1}$ | $D_{ky}$
---|---|---|---|---|---|---|---
$7.88$ | $2.48$ | $0.01$ | $0.45$ | $2.10$ | $2.40$ | $1.05$ | $2.90$
| $4.48$ | $0.01$ | $0.65$ | $1.97$ | $2.35$ | $0.46$ | $2.74$
$0.788$ | $1.48$ | $1.00$ | $0.55$ | $...$ | $2.45$ | $1.53$ | $2.74$
| | $0.10$ | $0.70$ | $...$ | $2.49$ | $1.80$ | $2.77$
| | $0.01$ | $0.65$ | $...$ | $2.15$ | $1.85$ | $2.45$
$0.788$ | $2.48$ | $1.00$ | $1.00$ | … | 2.45 | 1.50 | 2.48
| | $0.10$ | $0.90$ | … | 2.54 | 1.78 | 2.59
| | $0.01$ | $0.67$ | … | $2.45$ | $1.59$ | $2.86$
$0.788$ | $4.48$ | $1.00$ | $0.70$ | … | 2.55 | 1.48 | 2.86
| | $0.10$ | $0.60$ | … | 2.35 | 1.57 | 2.53
| | $0.01$ | $0.72$ | … | $2.50$ | $1.50$ | $2.52$
$0.00788$ | $1.48$ | $0.01$ | $0.74$ | $2.0$ | NC | NC | NC
| $2.48$ | $0.01$ | $0.60$ | $2.0$ | $2.20$ | $0.32$ | $2.40$
| $4.48$ | $0.01$ | $0.75$ | $2.0$ | $2.70$ | $0.13$ | $2.76$
Table 3: Statistical and dynamical quantities for the model acoustic signal.
Columns 4 and 5 show the power law exponents. Columns 6 to 8 list the
correlation dimension, positive exponent and Lyapunov dimension respectively.
### V.2 Dynamical Analysis of AE time series
In Section IV, we showed that the peel front patterns for several sets of
parameters are spatiotemporally chaotic. More importantly, the model acoustic
energy is quite irregular even as it is of dynamical origin. This suggests the
possibility that the experimental AE signals could be chaotic. However, often
time series have undesirable systematic component, which needs to be removed
from the original data. For instance, in the PLC effect, the stress-strain
time series has an overall increasing stress arising from the work hardening
component of the stress Anan99 which needs to be subtracted. In the present
case, the experimental data for high pull velocities does show a background
variation. A simple way of eliminating this background component is to use a
window averaging and subtract this component from the raw data. Moreover, as
stated in the introduction, the experimental AE data are quite noisy and
therefore it is necessary to cure the data (using standard noise reduction
techniques HKS ) before subjecting them to further analysis. Simple visual
checks for the existence of chaos such as phase plots, power spectrum etc.
have been carried out. We have also used singular value decomposition, false
neighbor search etc. Figures 15(a, b) show the raw and cured data respectively
for $V=5.0$ cm/s. Clearly, the dominant features of the time series are
retained except that small amplitude fluctuations are reduced or washed out
HKS . Statistical features like the distribution function for the amplitude of
the AE signals, power spectrum etc. are not altered. For instance, the two
stage power law distribution for the amplitude of AE signals for the raw data
(shown in Fig. 14(b)) is retained except that the exponent value for the small
amplitude regime is reduced from $0.32$ to $0.24$ without altering the
exponent corresponding to large amplitudes. This reduction is understandable
as small amplitude fluctuations are affected during curing.
The cured data are used to calculate the correlation dimension for all the
data files. However, for calculating the Lyapunov spectrum using our
algorithm, raw data is adequate as our algorithm is designed to process noisy
data. (In contrast, calculating the Lyapunov spectrum using the TISEAN package
requires the cured data). To optimize the computational time, all our
calculations are carried out using one fifth of each data set as each file
contains large number of points $\sim 10^{6}$ points, and there are 38 data
sets.
Figure 15: (a, b) Raw and cured AE signal respectively for $V=5.0$ cm/s. (c)
Correlation integral for pull velocity $3.8$ cm/s for $d=9$ to $13$. Dashed
lines are guide to eye. (d) Lyapunov spectrum for the same data file. (e)
Correlation integral for pull velocity $5.0$ cm/s from $d=7$ to $10$. Dashed
lines are guide to eye. (f) Lyapunov spectrum of the AE signals for traction
velocity $5.0$ cm/s.
Figure 16: (a) Correlation integral of $R_{ae}(\tau)$ for
$C_{f}=0.00788,\gamma_{u}=0.01$ and $V^{s}=2.48$ with $\nu=2.2$ ($d=5$ to
$8$). Dashed lines are guide to eye.(b) The Corresponding Lyapunov spectrum
for $R_{ae}$.
Typical autocorrelation time is about four units in sampling time. However,
using a smaller value of $\tau=1$, we have calculated the correlation integral
$C(r)$ for all the data files. Converged values of correlation dimension are
seen only in the region of pull velocities in the subinterval 3.8 to 6.2 cm/s.
A log-log plot of $C(r)$ for the pull velocity $3.8$ cm/s is shown in Fig.
15(c) for $d=9$ to $13$. A scaling regime of more than three orders of
magnitude is seen with $\nu\sim 2.80\pm 0.05$. This is at the beginning of the
chaotic window.
We have calculated the Lyapunov spectrum using our algorithm. The Lyapunov
spectrum for $V=3.8$ cm/s is shown in Fig. 15(d). (The outer shell radius is
kept at $\epsilon_{o}=0.065$.) Note that the second exponent is close to zero
as should be expected of continuous flow systems. Using the spectrum, we have
calculated the Kaplan-Yorke dimension (also called Lyapunov dimension)
$D_{ky}$ using the relation
$D_{ky}=j+\frac{\sum_{i=1}^{j}\lambda_{i}}{|\lambda_{j+1}|};\sum_{i=1}^{j}\lambda_{i}>0;\sum_{i=1}^{j+1}\lambda_{i}<0$.
The value so obtained in each case should be consistent with that obtained
from the correlation integral. For the case shown in Fig. 15(d) we get
$D_{ky}=2+1.70/1.81=2.94$ consistent with $\nu=2.80$. (Typical error bars on
the first three Lyapunov exponents are $\pm 0.01,\pm 0.005$ and $\pm 0.05$.
Thus the errors in $D_{ky}$ values are $\pm 0.05$.)
As an example of converged value of correlation dimension near the upper end
of the chaotic domain, a log-log plot of $C(r)$ for $V^{s}=5.0$ cm/s is shown
in Fig. 15(e) with $\nu=2.73\pm 0.05$ for $d=7$ to 10. Again, the scaling
regime is seen to be nearly three orders of magnitude. The Lyapunov spectrum
for the data file is shown in Fig. 15(f). The calculated Lyapunov dimension
from the spectrum is $D_{ky}=2+1.73/1.80=2.96$ which is again consistent with
$\nu=2.73$.
The values of $\nu$ for all the files are found to be in the range $2.55$ to
$2.85\pm 0.05$ as can be seen from Table 2. We have calculated the Lyapunov
spectrum for the full range of traction velocities and we find (stable)
positive and zero exponents only in the region 3.8 to 6.2 cm/s, consistent
with the range of converged values of $\nu$ as can be seen from Table 2. The
corresponding values of $D_{ky}$ are in the range of $2.7$ to $3.0$. We have
also calculated the Lyapunov spectrum using the TISEAN package using cured
files. The $D_{ky}$ values obtained from the TISEAN package are uniformly
closer to the $\nu$ values, typically $D_{ky}=\nu+0.1$. Finally, we note that
the positive exponent decreases toward the end of the chaotic domain (6.2
cm/s). These results (see Table 2) show unambiguously that the underlying
dynamics responsible for AE during peeling is chaotic in a mid range of pull
speeds.
In order to compare the low dimensional chaotic nature of the experimental AE
signals with the model acoustic signal, we have analyzed the low dimensional
dynamics of $R_{ae}(\tau)$ using the embedding procedure after subtracting the
periodic component.
We have computed the correlation dimension and Lyapunov spectrum for the
entire instability domain. A log-log plot of the $C(r)$ is shown in Fig. 16(a)
for $d=5$ to 8. The convergence over more than three orders of magnitude is
clear. The value of $\nu=2.20\pm 0.05$. For this file, we find stable positive
and zero exponents for a range of $\epsilon_{o}$ values. A plot of the
spectrum for $C_{f}=0.00788,\gamma_{u}=0.01$ and $V^{s}=2.48$
($\epsilon_{o}=0.08$) is shown in Fig. 16(b). Using this we get
$D_{ky}=2+0.32/0.77=2.4$ which is again consistent with $\nu=2.2\pm 0.02$.
We have calculated both correlation dimension and Lyapunov spectrum of
$R_{ae}$ for a range of values of the parameters. For each $C_{f}$, we find
converged values of $\nu$ and $D_{ky}$ within a window of pull speeds.
Generally, the range of $\nu$ is between 2.15 to 2.70 while $D_{ky}$ is in the
range 2.4 to 2.90. Table 3 shows the values of correlation dimension and
$D_{ky}$ for various sets of parameter values. It is interesting to note that
the magnitude of the largest exponent for the model AE signal also decreases
as we increase the pull velocity, a feature displayed by the experimental time
series as well.
## VI Summary and Conclusions
In summary, the present investigation is an attempt to understand the origin
of the intermittent peeling of an adhesive tape and its connection to acoustic
emission. At the conceptual level, we have established a relationship between
stick-slip dynamics and the acoustic energy, the latter depends on the local
strain rate Land which in turn is controlled by the roughness of the peel
front. As the model is fully dynamical, one basic result that emerges is that
the model acoustic energy is controlled by the nature of spatiotemporal
dynamics of the peel front. Further, even as the model acoustic emission is a
dynamical quantity, the nature of $R_{ae}$ turns out to be quite noisy
depending on the possible interplay of different time scales in the model.
Thus, the highly noisy nature of the experimental signals need not necessarily
imply stochastic origin of AE signal; instead, they could be of deterministic
origin. This motivated us to carry out a detailed analysis of statistical and
dynamical features of the experimental AE signals. Despite the high noise
content, we have been able to demonstrate the existence of finite correlation
dimension and positive Lyapunov exponent for a window of pull speeds. The
Kaplan-Yorke dimension (for various traction velocities) calculated from the
Lyapunov spectrum is consistent with the value obtained from the correlation
integral. Thus, the analysis establishes unambiguously the deterministic
chaotic nature of the experimental AE signals. Interestingly, the largest
Lyapunov exponent shows a decreasing trend toward the end of the chaotic
window, a feature displayed by the model acoustic signal as well. The work
also addresses the general problem of extracting dynamical information from
noisy AE signals. A similar analysis of the model acoustic energy shows that
$R_{ae}$ is chaotic for a range of parameter values. More importantly, several
qualitative features of the experimental AE signals such as the statistics of
the signals and the change from burst to continuous type with increase in the
pull velocity are also displayed by $R_{ae}$. The observed two stage power law
distribution for the experimental AE signals [Fig. 14(b)] is reproduced by the
model [Fig. 14(d)]. It must be emphasized that this power law distribution for
the amplitudes is completely of dynamical origin. This result should be of
general interest in the context of dynamical systems as there are very few
models that generate power laws purely from dynamics. The only other example
known to the authors is that of the PLC effect where the amplitude of the
stress drops shows a power law distribution within the context of the
Ananthakrishna model Anan04 .
The spatiotemporal patterns of the peel front are indeed rich and depend on
the interplay of the three time scales. Although, the nature of spatiotemporal
patterns is quite varied, they can be classified as smooth synchronous,
rugged, stuck-peeled and even nowhere stuck patterns. As expected on general
consideration of dynamics, rich patterns are observed for the case when all
the three time scales are of similar magnitude (illustrated for
$C_{f}=0.788$). All spatiotemporal patterns, except the smooth synchronous
peel front are interesting. As a function of time, the nature of the peel
front can go through a specific sequence of these patterns (depending on the
parameter values). The most interesting pattern is the stuck-peeled
configuration which is reminiscent of fibrils observed in experiments
Dickinson ; Urahama ; Yamazaki . Even among the SP configurations, there are
variations, for example, rapidly changing, long lived, edge of peeling etc.
Despite the varied range of patterns, a few general trends of the influence of
the parameters on the peel front patterns are worth noting. First, in general
the number of stuck and peeled segments increases as $\gamma_{u}$ is
decreased. Second, as the pull velocity is increased, the rapidly varying
stuck-peeled configurations observed at low pull velocities become long lived.
The dynamical signature of these two parameters are reflected in the nature of
the phase space orbit. For instance, given a value of $C_{f}$ and
$\gamma_{u}$, the nature of the phase space orbit changes only when $V^{s}$ is
increased which allows the orbit to move way beyond the values of
$\phi(v^{s})$.
The study of the model shows that while the intermittent peeling is controlled
by the peel force function, the dynamics of the peel front is influenced by
all the three time scales. This together with the dynamical analysis of the
experimental acoustic emission signals establishes that deterministic dynamics
is responsible for AE during peeling. The various sequences of peel front
patterns and their time dependences lead to quite varied model acoustic
signals. These can be classified as bunch of spikes, isolated bursts occurring
at near regular intervals, continuous bursts with an overall envelope
separated by quiescent state, continuous bursts overriding a near periodic
triangular form, irregular waveform overriding a periodic component, and
continuous irregular type. Interestingly, our studies show that there is a
definite correspondence between the model acoustic energy and the nature of
peel front patterns even though $R_{ae}(\tau)$ is the spatial average of the
local strain rate. Despite this, two distinguishable time scales in
$R_{ae}(\tau)$ can be detected, one corresponding to short term fluctuations
and another corresponding to overall periodic component. The short term
fluctuations can be readily identified when the model acoustic signal is
fluctuating without any background component (see for example Fig. 6(d)).
These rapid changes in $R_{ae}$ arise due fast dynamic changes in the SP
configurations. The minimum in $R_{ae}$ corresponds to the situation where the
average velocity jumps of the SP configurations are smaller compared to that
at the preceding maximum. In contrast, the overall periodicity in
$R_{ae}(\tau)$ (for instance see Fig. 7(c) among many other cases) can be
identified with the changes in the peel front patterns that occur over a cycle
in the phase plot. The minima in the $R_{ae}(\tau)$ corresponds to the peel
patterns where more segments are in stuck state than in the peeled state while
the maxima corresponds to more stuck and peeled segments (see Figs. 7(a),
(b)). The corresponding phase plot usually goes through a cycle of visits
between the low and high velocity branches.
Often, however, the nature of the model acoustic energy signal can be
complicated as in the case of low tape mass ($C_{f}=0.00788$). Even in such
cases, some insight is possible. This is aided by the analysis of the
corresponding phase plot. For example for the low $C_{f}$ case where the
roller inertia plays an important role in the dynamics, the rapidly
fluctuating acoustic energy has an overall triangular envelope [Fig. 9(e)]. On
an expanded scale, the convex envelope consists of seven local peaks [Fig.
9(f)]. Each of these is generated when the various peel front segments make
abrupt jumps between the two branches of the peel force function. Note that
the phase space orbit has seven loops in this case [Fig. 9(d)]. The general
identification of the minima in $R_{ae}$ with patterns that have more stuck
segments than peeled segments still holds. Similarly, the maxima in $R_{ae}$
usually correspond to the presence of large number of stuck-peeled
configurations.
The above correspondence between the model acoustic energy and the peel front
patterns provides insight into the transition from burst to continuous type of
AE seen in experiments as a similar transition from burst type to continuous
type is also seen in the model acoustic energy (for large $C_{f}=7.88$ ). At
low pull velocities, the peel front goes through a cycle of patterns where
most segments of the peel front (or the entire peel front) spends substantial
time in the stuck state switching to stuck-peeled configuration. As the
duration of the SP configuration is short and velocity bursts are large,
$R_{ae}(\tau)$ is of burst type [Fig. 13(c)]. With increasing pull velocity,
only dynamic stuck-peeled configurations are seen which in turn leads to
continuous AE signals [Figures. 4(a, b)]. This coupled with time series
analysis of the model acoustic signal shows that the associated positive
Lyapunov exponent decreases with increase in the traction velocity. This is
precisely the trend observed for experimental signals as well. Thus, the
decreasing trend of the largest Lyapunov exponent can be attributed to the
peel front breaking up into large number of small segments providing insight
into stick-slip dynamics and its connection to the AE process.
The present study has relevance to the general area of stick-slip dynamics. As
mentioned earlier, models for stick-slip dynamics use negative force-drive
rate relation. In such models, the phase space orbit generally sticks to the
slow manifold (stable branches) of the force-drive rate function. This leads
to clearly identifiable stick and slip phases, the former lasting much longer
than the latter. However, recent work on imaging the peel point dynamics
Cortet shows that the ratio of the stick phase to the slip phase, is about
two or even less than unity for high peel velocities. While all the known
models of the peel process predict that the duration of the stick phase is
longer than that of the slip phase, our model displays the experimentally
observed feature. This feature emerges in the model due to the interplay of
the three time scales aided by incomplete relaxation of the relevant modes.
Our studies show that only for low pull velocity and high $C_{f}$ do we
observe the stick phase lasting much longer than the slip phase. As the pull
velocity is increased, and for all other parameter values, we find that the
duration of the slip phase (peel velocity being larger than unity) is nearly
the same as or less than that of the stick phase (peel velocity less than
unity). Further, the present model provides an example of the richness of
spatiotemporal dynamics arising when more than two time scales are involved.
In this context, we emphasize that the introduction of the Rayleigh
dissipation functional to model the acoustic energy is crucial for the
richness of the spatiotemporal peel front patterns. It is important to note
that this kind of dissipative term is specific to spatially extended systems
as it represents relaxation of neighboring points on the peel front.
The present study has relevance to time dependent issues of adhesion. For
instance, apart from the fact that the time series analysis addresses the
general problem of extracting dynamical information from noisy AE signals, it
may have relevance to failure of adhesive joints and composites that are
subject to fluctuating loads. The failure time can be estimated by calculating
the Lyapunov spectrum for the AE signals. If the largest Lyapunov exponent is
positive, the inverse of the exponent should give an estimate of the time
scale over which the failure can occur and hence could prove to be useful in
predicting failure of joints. The present study should also help to optimize
production schedules in peeling tapes.
Finally, several features of the present study are common to the PLC effect
even though the underlying mechanism is very different. In this case the
repeated occurrence of stress drops during constant strain rate deformation
PLC ; GA07 ; GA07a , are associated with the formation and possible
propagation of dislocation bands that are visible to the naked eye. The
phenomenon occurs only in a window of applied strain rates. The instability is
attributed to the pinning and unpinning of dislocations from solute
atmosphere, yet, the dominant feature underlying the instability is the
negative strain rate sensitivity of the flow stress that has two stable
branches separated by an unstable branch. Clearly, these features are similar
to the occurrence of the peel instability within a window of pull velocities
and the existence of unstable branch in the peel force function. Further, the
Ananthakrishna (AK) model for the PLC instability predicts that the stress
drops should be chaotic in a subinterval of the instability domain Anan82 .
This prediction has been verified subsequently through the analysis of
experimental stress-strain curves obtained from single and polycrystals Anan93
; Anan95 ; Anan99 ; Bhaprl . This feature is again similar to the existence of
chaotic dynamics observed in a mid range of pull velocities in the peeling
problem, both in experiment and in the model. In the case of the PLC effect,
one finds that the positive Lyapunov exponent characterizing the stress-time
series decreases toward the end of chaotic window, both in experiments and in
the AK model. Again this feature is also seen in the present peel model as
also in experimental AE signals. Dynamically, in the case of the AK model for
the PLC effect the decreasing trend of the positive Lyapunov exponent has been
shown to be a result of a forward Hopf bifurcation (HB) followed by a reverse
HB Anan04 . In the case of peeling problem as well, the instability begins
with a forward HB followed by a reverse HB. Finally, in the PLC effect (both
in experiments and the AK model), as in the peeling problem, the duration of
the slip phase can be longer than that of the stick phase with increasing
drive rate. As many of these features are common to two different systems, it
is likely that these are general features in other stick-slip situations with
multiple time scales that are limited to a window of drive rates with multiple
participating time scales.
A few comments may be in order about the model, in particular about the
parameters that are crucial for the dynamics. While the agreement of several
statistical and dynamical features of the model ( for several sets of
parameter values) with the experimental AE series is encouraging, it would be
interesting to verify model results for other sets of parameters. For
instance, it is clear that the roller inertia and the inertia of the tape mass
are experimentally assessable parameters. Thus, the influence of these two
inertial time scales can in principal be studied in experiments. However,
conventional experiments have been performed keeping these parameters fixed,
presumably, as there has been no suggestion that the dynamics can be sensitive
to these variables. It would be interesting to verify the predicted dynamical
changes in the AE signals as a function of these two parameters. As for the
influence of the dissipation parameter $\gamma_{u}$, the range of physically
reasonable values of $\gamma_{u}$ is expected to be small ($10^{-4}$ to
$10^{-3}$) as argued. Interestingly, the region of low $\gamma_{u}$ is indeed
the region where both statistics and dynamical features compare well with that
of the experiments. However, within the scope of the model, the visco-elastic
properties of the adhesive have been modeled using an effective spring
constant. (This kind of assumption is common to studies in adhesion and
tackiness etc Gay .) However, it is possible to include this feature as well.
Finally, it must be stated that features that critically depend on thickness
of the film and its visco-elastic properties such the shape of peel front are
beyond the scope of the present model.
Acknowledgment: The authors wish to thank Professor M. Ciccotti for providing
the AE data. GA acknowledges support from BRNS Grant No.
$2005/37/16/BRNS,2007/36/62-BRNS/2564$, and Raja Ramanna Fellowship.
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|
arxiv-papers
| 2008-12-01T06:52:41 |
2024-09-04T02:48:59.066397
|
{
"license": "Public Domain",
"authors": "Jagadish Kumar, Rumi De, G. Ananthakrishna",
"submitter": "G. Ananthakrishna",
"url": "https://arxiv.org/abs/0812.0202"
}
|
0812.0231
|
# Measurement of the quadratic Zeeman shift of 85Rb hyperfine sublevels using
stimulated Raman transitions
Run-Bing Li Lin Zhou State Key Laboratory of Magnetic Resonance and Atomic
and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese
Academy of Sciences, Wuhan 430071, China Center for Cold Atom Physics,
Chinese Academy of Sciences, Wuhan 430071, China Graduate School, Chinese
Academy of Sciences, Beijing 100080, China Jin Wang wangjin@wipm.ac.cn Ming-
Sheng Zhan State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy
of Sciences, Wuhan 430071, China Center for Cold Atom Physics, Chinese
Academy of Sciences, Wuhan 430071, China
###### Abstract
Abstract
We demonstrate a technique for directly measuring the quadratic Zeeman shift
using stimulated Raman transitions. The quadratic Zeeman shift has been
measured yielding $\Delta\nu=1296.8$ $\pm 3.3$ Hz/G2 for magnetically
insensitive sublevels ($5S_{1/2},F=2,m_{F}=0\rightarrow$
$5S_{1/2},F=3,m_{F}=0$) of 85Rb by compensating the magnetic field and
cancelling the ac Stark shift. We also measured the cancellation ratio of the
differential ac Stark shift due to the imbalanced Raman beams by using two
pairs of Raman beams ($\sigma^{+}$, $\sigma^{+})$ and it is $1$:$3.67$ when
the one-photon detuning is $1.5$ GHz in the experiment.
###### pacs:
32.80.Qk,03.75.Dg, 37.25.+k
1\. Introduction
Since the atom interferometer was demonstrated in 1991Kasevich1991a , it has
been applied to rotation measurement, such as inertial navigation and even the
rotation rate of the earth Gustavson2000a ; Canuel2006a . Recently, an atom-
interferometer gyroscope of high sensitivity and long-term stability was
reported Durfee2006a . In order to improve the accuracy of the rotation rate
measurement by using an atom-interferometer gyroscope, the potential
systematic errors should be considered and controlled as well as possible. The
quadratic Zeeman shift is considered as a factor that influences the accuracy
of the rotation rate measurement in the atom-interferometer gyroscope.
The atom gyroscope generally uses two counter-propagating cold-atom clouds
launched in strongly curved parabolic trajectories Canuel2006a . The two cold
atom clouds should be overlapped completely in order to cancel common noise
and gravity acceleration, and cold collisions occur between atoms along
similar trajectories. For a dual atom-interferometer gyroscope, Rubidium is a
suitable candidate because it has a smaller collision frequency shift than
Cesium Santos2002a ; Kokkelmans1997a ; Gibble1995a ; Sortais2000a . In our
previous work Li2008a ; Wang2007a , we have experimentally investigated the
stimulated Raman transitions in the cold atom interferometer. Both the
accuracy and the fringe contrast of an atom-interferometer gyroscope can be
improved by studying the magnetic field dependence of the coherent population
transfer. A homogenous magnetic field must be applied along the Raman beams to
keep the quantization axis consistent and resolve degenerate magnetic
sublevels. This magnetic field will cause Zeeman shifts. The quadratic Zeeman
shift induces a relative frequency shift of the two coherent states, which
influences the accuracy of the rotation rate measurement. It is therefore
important to measure accurately and understand the quadratic Zeeman shift of
85Rb in the cold atom interferometer. Similarly, the quadratic Zeeman shift is
important in other applications such as microwave frequency standards
Ramsey1963a ; Thomas1982a ; Hemmer1986a , optical frequency standards
Kajita2005a ; Boyd2007a and coherent population trapping clock Vanier2005a .
The quadratic Zeeman shift can be usually obtained from the Breit-Rabi formula
after the magnetic field is measured by the linear Zeeman effect Bize1999a .
We study this from the field-insensitive clock transitions whose linear Zeeman
shift is zero, thus the magnetic field is calibrated from other $u_{F}\neq 0$
states. We have also studied this quadratic Zeeman shift in the presence of
the ac Stark shift of the Raman pulses.
In this paper, we analyze the hyperfine sublevels of the ground states in the
magnetic field by using second-order perturbation theory, and demonstrate
experimentally the coherent population transfer of the different Zeeman
sublevels by stimulated Raman transitions. The quadratic Zeeman shift of the
ground state of 85Rb was measured by the two-photon resonance of the
stimulated Raman transition after the ac Stark shift was cancelled and the
residual magnetic field was compensated. The value of the magnetic field is
calibrated by the linear Zeeman shift. Our analysis shows that the quadratic
Zeeman shift can be measured to Hz level for magnetically insensitive states
($5S_{1/2},F=2,m_{F}=0\rightarrow$ $5S_{1/2},F=3,m_{F}=0$) in our experiment.
We also measured the cancellation ratio of the differential ac Stark shift due
to the imbalanced Raman beams by using two pairs of Raman beams. This study
provides useful data for higher precision measurement of the quadratic Zeeman
shift of 85Rb, even for improving the accuracy of the rotation rate
measurement of the atom-interferometer gyroscope.
2\. Quadratic Zeeman shift
Including the hyperfine interaction, the ground state energy levels will split
and shift in the magnetic field. The interaction Hamiltonian operator
Sobelman1996a ; Itano2000a within the subspace of hyperfine sublevels
associated with the electronic levels is given by
$H^{{}^{\prime}}=hA_{S}I\cdot J+g_{J}\mu_{B}J\cdot B+g_{I}\mu_{B}I\cdot B$ (1)
where, $h$ is the Plank constant, $A_{S}$ is the hyperfine constant, $I$ and
$J$ are the nuclear spin operators and orbital angular momentum respectively,
$g_{J}$ and $g_{I}$ are the electronic $g$-factor and nuclear $g$-factor
respectively, $\mu_{B}$ is Bohr magneton. Second-order perturbation theory is
valid for low magnetic-field intensity, and the energies of the hyperfine
Zeeman sublevels for the ground states can be derived as following
For F=2
$E(\frac{1}{2},2,0,B)=E(\frac{1}{2})-\frac{7}{4}hA_{S}-\mathstrut\vskip 3.0pt
plus 1.0pt minus 1.0pt\frac{(g_{J}-g_{I})^{2}}{12hA_{S}}\mu_{B}^{2}B^{2}$ (2)
$\displaystyle E(\frac{1}{2},2,\pm 1,B)$
$\displaystyle=E(\frac{1}{2})-\frac{7}{4}hA_{S}\mp\frac{g_{J}-7g_{I}}{6hA_{S}}\mu_{B}B$
(3) $\displaystyle-\frac{2(g_{J}-g_{I})^{2}}{27hA_{S}}\mu_{B}^{2}B^{2}$
$\displaystyle E(\frac{1}{2},2,\pm 2,B)$
$\displaystyle=E(\frac{1}{2})-\frac{7}{4}hA_{S}\mp\frac{g_{J}-7g_{I}}{3hA_{S}}\mu_{B}B$
(4) $\displaystyle-\frac{5(g_{J}-g_{I})^{2}}{108hA_{S}}\mu_{B}^{2}B^{2}$
For F=3
$E(\frac{1}{2},3,0,B)=E(\frac{1}{2})+\frac{5}{4}hA_{S}+\frac{(g_{J}-g_{I})^{2}}{12hA_{S}}\mu_{B}^{2}B^{2}$
(5)
$\displaystyle E(\frac{1}{2},3,\pm 1,B)$
$\displaystyle=E(\frac{1}{2})+\frac{5}{4}hA_{S}\pm\frac{g_{J}+5g_{I}}{6hA_{S}}\mu_{B}B$
(6) $\displaystyle+\frac{2(g_{J}-g_{I})^{2}}{27hA_{S}}\mu_{B}^{2}B^{2}$
$\displaystyle E(\frac{1}{2},3,\pm 2,B)$
$\displaystyle=E(\frac{1}{2})+\frac{5}{4}hA_{S}\pm\frac{g_{J}+5g_{I}}{3hA_{S}}\mu_{B}B$
(7) $\displaystyle+\frac{5(g_{J}-g_{I})^{2}}{108hA_{S}}\mu_{B}^{2}B^{2}$
$E(\frac{1}{2},3,\pm
3,B)=E(\frac{1}{2})-\frac{7}{4}hA_{S}\pm\frac{g_{J}+5g_{I}}{2hA_{S}}\mu_{B}B$
(8)
Here, $E(J,F,m_{F},B)$ denotes the energy of the hyperfine sublevels,
including the effect of the hyperfine interaction and magnetic field
splitting. From eqs.(2) and (5), the quadratic Zeeman shift for the transition
$5S_{1/2},F=2,m_{F}=0\rightarrow 5S_{1/2},F=3,m_{F}=0$ is
$\Delta\nu=(g_{J}-g_{I})^{2}\mu_{B}^{2}B^{2}/6hA_{S}$, which is consistent
with the reference Steck that is obtained from the Breit-Rabi formula when it
is extended to second order in the field strength.
3\. Experimental configuration
Figure 1: Experimental scheme: cold 85Rb atoms fly horizontally from the MOT
to the probe region. Three crossed pairs of Helmholtz coils are applied to
compensate the residual magnetic field in the stimulated Raman interaction
area. The combined Raman beams ($R_{1}$,$R_{2}$) and
($R_{1}^{{}^{\prime}}$,$R_{2}^{{}^{\prime}}$) are parallel to the magnetic
field $B$ and $B_{0}$ respectively. The laser-induced fluorescence signal is
detected by a PMT.
The experimental arrangement is shown in Fig.1, which is similar to our
previous work Li2008a ; Wang2007a . Briefly, the cold atoms are trapped in a
nonmagnetic stainless steel chamber with $14$ windows, where the trapping and
repumping beams are provided by a tapered amplifier diode laser (TOPTICA
TA100) and an external-cavity diode laser (TOPTIC DL100) respectively, whose
frequencies are stabilized using saturated absorption spectroscopy Wang2000a .
After the polarization gradient cooling (PGC) process, the atoms are guided by
a near-resonance laser pulse and fly transversely from the trapping region to
the probe region at a velocity of $24$ m/s Jiang2005a . Then, they are
completely pumped to the ground state $5S_{1/2},F=2$ as the initial state by a
perpendicular linearly polarized laser beam which is near resonance with the
transition $5S_{1/2},F=3\rightarrow 5P_{3/2},F=2$. Three crossed pairs of
Helmholtz coils are used to provide the magnetic field in the Raman
interaction area, where the current of the coils along the Raman beams (R1,
R2) is controlled by the DC power supply (MPS-901) and measured by the digital
multimeters (Flucke 8846A). The magnetic field intensity is calibrated by the
first-order Zeeman shift, whose uncertainty is less than one part in one
thousand. The combined Raman beams (R1, R2) and (R${}_{1}^{{}^{\prime}}$,
R${}_{2}^{{}^{\prime}}$) are applied along the magnetic fields $B$ and $B_{0}$
respectively in the stimulated Raman interaction region. The Raman beams (R1,
R2) are used to measure the frequency shift induced by the external fields
such as the Raman beams (R${}_{1}^{{}^{\prime}}$, R${}_{2}^{{}^{\prime}}$) and
the magnetic field $B$. The Raman beams (R1, R2) and (R${}_{1}^{{}^{\prime}}$,
R${}_{2}^{{}^{\prime}}$) are supplied from the same Raman laser. This
configuration has the benefit for the accurate measurement of the ac stark
shift because two pairs of Raman beams always have the same one-photon
detuning. The detailed description of the Raman laser arrangement is similar
to our previous work Wang2007a . The atoms are transferred to the state
$5S_{1/2},F=3$ from $5S_{1/2},F=2$ when they pass through a Raman $\pi$-pulse.
After coherent population transfer via a simulated Raman transition, the
population of the state is detected by a laser induced fluorescence (LIF)
signal, and we use a photo multiplier tube (PMT) to collect the LIF.
Figure 2: Population transfer dependence on two-photon detuning. The peaks
(-2,-2),(-1,-1),(0,0),(1,1),(2,2) are the resonance transition of the
hyperfine Zeeman sublevels when the Raman beams ($R_{1}$,$R_{2}$) are applied
along the magnetic field $B=220$ mG, where the Raman beams
($R_{1}^{{}^{\prime}}$,$R_{2}^{{}^{\prime}}$) and $B_{0}$ are not used.
4\. Results and analysis
Figure 3: The resonance frequency of the hyperfine Zeeman sublevels (-2,-2)
and (2,2) depends on the current of the Helmholtz coils. It is measured by
Raman beams (R1, R2) under R${}_{1}^{{}^{\prime}}$=$0$,
R${}_{2}^{{}^{\prime}}$=$0$, B0=$0$. The magnetic field can be scaled by the
first Zeeman shift after the linear fit, whose uncertainty is below one per
thousand.
The hyperfine level $F=2$ is split into five sublevels and $F=3$ into seven
sublevels, whose energies are expressed as in eqs.(2-8) when there exists a
magnetic field. After the magnetic field is compensated completely according
to our previous work Li2008a , all sublevels are degenerate. Coherent
population transfer can occur for the transition of the combined hyperfine
Zeeman sublevels $(-2,-2),(-1,-1),(0,0),(1,1),(2,2)$ when the Raman beams (R1,
R2) with ($\sigma^{+}$,$\sigma^{+}$) propagate along the magnetic field $B$ in
the stimulated Raman interaction region as shown in Fig.1 Li2008a ; Peters ;
Gustavson ; Petelsk , where the Raman beams (R${}_{1}^{{}^{\prime}}$,
R${}_{2}^{{}^{\prime}}$) and the magnetic field $B_{0}$ are not used. The
maximum population transfer is achieved when two-photon resonance is satisfied
with the transition selection rules shown in Fig.2. A perfect symmetric Raman
spectrum are achieved when the atoms are interacted with Raman beams
($\sigma^{+}$,$\sigma^{+}$). In Fig.2, the transition probability can be
explained using the oscillator strength of two-photon transition for the
different hyperfine Zeeman sublevels $(-2,-2),(-1,-1),(0,0),(1,1),(2,2)$
respectively. The energy separation of the different sublevels is well
explained by eqs.(2-8) when the bias magnetic field $B=220$ mG is applied. The
magnetic field is calibrated by the linear Zeeman shift of the hyperfine
Zeeman sublevels $(-2,-2),(2,2)$. For different magnetic field, we measured
the resonance frequency for the transitions $(-2,-2)$ and $(2,2)$, as shown in
Fig.3. After a linear fit, the slope is the magnetic field intensity
controlled by the current of the coils in the Raman interaction area. The
scaled method is similar to that of the quadratic Zeeman shift measurement
introduced in our paper. The scaled parameters come from earlier references
Steck ; Bender1958a ; Penselin1962a ; Arimondo1977a . The scale factor of the
magnetic field is $1576.9\pm 1.3$ mG/A after the averaged measurements.
Figure 4: The differential ac Stark shift caused by imbalanced Raman beams
versus the Raman light intensity. The dots are the frequency shift induced by
$R_{1}^{{}^{\prime}}$ while the squares are the frequency shift induced by
$R_{2}^{{}^{\prime}}$, where they are fitted linearly and the slopes are
$3.66$ kHz/(mW/cm2) and $-0.99$ kHz/(mW/cm2) respectively. The ac Stark shift
can be cancelled by adjusting the intensity ratio to $1:3.67$ for the one-
photon detuning $\Delta=1.5$GHz.
The differential ac Stark shift caused by the imbalanced Raman beams will
induce a measurement noise in the determination of the quadratic Zeeman shift.
The difference between the ac Stark shifts of two hyperfine sublevels,
$\delta^{AC}=\Omega_{F=3,m_{F}=0}^{AC}-\Omega_{F=2,m_{F}=0}^{AC}$, can be
cancelled by optimizing the ratio of two Raman beams weiss1994a . We measure
the frequency shift that is induced by one of the Raman beams separately. In
the experiment, we use two pairs of Raman beams (R1, R2) and
(R${}_{1}^{{}^{\prime}}$, R${}_{2}^{{}^{\prime}}$) along the magnetic field
$B$ and $B_{0}$, where $B$ and $B_{0}$ are $250$ mG and $100$ mG respectively.
The Raman beams (R1, R2) are used to measure the ac Stark shift induced by the
other Raman beams (R${}_{1}^{{}^{\prime}}$, R${}_{2}^{{}^{\prime}}$). We
carefully optimize the intensities of the Raman beams ($R_{1}$, $R_{2}$) along
the magnetic field $B$ to obtain a $\pi$-pulse. We scan the frequency
difference of the Raman beams (R1, R2), and the resonant frequency of the
hyperfine Zeeman sublevels ($0,0$) can be obtained by a Gaussian fit for the
different Raman light intensities (R${}_{1}^{{}^{\prime}}$, or
R${}_{2}^{{}^{\prime}}$). The detailed proceedure is similar to test of
quadratic Zeeman shift measurement in the paper. In the experiment, the Raman
beams (R1, R2) and (R${}_{1}^{{}^{\prime}}$, R${}_{2}^{{}^{\prime}}$) are
guided using single mode polarization maintained fiber. The intensity
instability is below one part in one thousand for each of the Raman beams. The
dots are the frequency shift that is induced by R${}_{1}^{{}^{\prime}}$, while
the squares are the frequency shift that is induced by R${}_{2}^{{}^{\prime}}$
in Fig.4, where they are fitted linearly. The slopes are $3.66$ kHz/(mW/cm2)
and $-0.99$ kHz/(mW/cm2) for R${}_{1}^{{}^{\prime}}$, and
R${}_{2}^{{}^{\prime}}$ respectively. The frequency shifts, induced by the
different Raman beams (R${}_{1}^{{}^{\prime}}$, or R${}_{2}^{{}^{\prime}}$),
are referenced to the separation of hyperfine sublevels ($3$ $035$ $732$
$436$) Penselin1962a . The non null values are mainly caused by the quadratic
Zeeman shift when the Raman beams (R${}_{1}^{{}^{\prime}}$,
R${}_{2}^{{}^{\prime}}$) are not applied in Fig.4. The ratio ($1:3.67$) of the
two slopes determines the cancellation of the ac Stark shift when the one-
photon detuning is $1.5$ GHz in our experiment. Therefore, we can cancel the
ac Stark shift by adjusting the ratio of two Raman beam intensities.
Figure 5: Frequency dependence of transition amplitude for a square Raman
$\pi$-pulse (R1, R2). The frequency is referenced to the hyperfine separation
of the two ground states. The dots are the experimental data with $B=600$ mG,
B0=$0$, R${}_{1}^{{}^{\prime}}$=$0$, R${}_{2}^{{}^{\prime}}$=$0$. while the
solid line is the Gaussian fit.
After the magnetic field compensation and the cancellation of the ac Stark
shift, their influence is considerably decreased in the measurement of the
quadratic Zeeman shift. The Raman beams are generated by an acousto-optical
modulator(Brimrose, $1.5$ GHz) driven by microwave generator (Agilent $8257$C)
which is locked by a H-maser. The arrangement of the Raman laser is similar to
our previous work Wang2007a . We carefully optimize the intensities of the
Raman beams ($R_{1}$, $R_{2}$) along the magnetic field $B$ to obtain a
$\pi$-pulse, where $B_{0}$, $R_{1}^{{}^{\prime}}$ and $R_{2}^{{}^{\prime}}$
are not used. The instability of the ratio of the Raman beams
($R_{1}:R_{2}=1:3.67$) is below $10^{-5}$ in the experiment. We scan the
frequency difference of the Raman beams (R1, R2), and observe a typical
stimulated Raman transition which shows the population versus frequency
difference between the two Raman beams in Fig.5 at a magnetic field $B=600$
mG, where the frequency is referenced to the separation between the two ground
states ($3$ $035$ $732$ $436$ Hz) Penselin1962a . In our experiment, the
intensity profile of the Raman beams is a Gaussian distribution and the line
width is mainly limited by the transition time because the spontaneous can be
ignored in large one-photon detuning. In such case, the population dependence
on the two-photon detuning is a Gaussian profile Demtroder2003a . The central
frequency is obtained from a Gaussian fit. We have made a series of such
curves for different magnetic fields, and the dependence of the frequency
shift on the magnetic field is shown in Fig.6. The frequency shift depends on
the magnetic field and it is fitted by a polynomial function (The maximum
power is $2$), while the quadratic dependence is for the quadratic Zeeman
shift. We measured a series of values as shown in table $1$, and the average
frequency shift induced by the quadratic Zeeman effect for the hyperfine
Zeeman sublevels ($5S_{1/2},F=2,m_{F}=0\rightarrow 5S_{1/2},F=3,m_{F}=0$) is
$1296.8$ Hz/G2. The measurement uncertainty comes mainly from the calibrated
magnetic field and the fitted error. As shown in table $1$, the averaged
uncertainty of the quadratic Zeeman shift is $2.1$ Hz/G2 and $2.5$ Hz/G2 for
the scaled magnetic field and the fitted error respectively. The final result
for the quadratic Zeeman shift is $1296.8\pm 3.3$ Hz/G2 by using an
independent error source model, which is in good agreement with the
calculation result Steck within our measurement precision. The result shows
that the second perturbation theory is sufficient when the magnetic field is
less than $1$ mT Itano2000a . The ac Stark shifts induce a systematic shift of
the ground-state hyperfine splitting. This does not influence the value of the
quadratic Zeeman shift when a quadratic dependence term of the polynomial
function is chosen as shown in Fig.6. The fitted error, which is induced by
the instability of the Raman beams, is decreased when the cancellation ratio
of the Raman beams ($1:3.67$) is applied in the experiment.
Figure 6: Dependence of resonance frequency on the magnetic field intensity.
The dots are the frequency shift in the different magnetic field that is
obtained from Fig. 5, while the line is the experimental fit by a polynomial
function.
In the atom interferometer, the bias magnetic field is applied through the
interference area. Although the atoms are always kept in magnetically
insensitive states with $m_{F}=0$, these states still show a quadratic Zeeman
shift that induces a relative frequency shift of two ground states. This
effect is big enough to require well controlled magnetic fields and extensive
magnetic field shielding to achieve the millihertz frequency stability
necessary for gravity measurements at the $1\mu$G level Peters . For the
rotation rate measurement, the quadratic Zeeman shift should be known
accurately when considering the accuracy necessary to determine the rotation
rate of the earth. The sensitivity of the rotation signal to the various bias
magnetic field was determined in detailly performed in the dual atomic
interferometer gyroscope, and the bias magnetic field caused a phase shift
$2\times 10^{-6}\Omega_{E}$/mG for the rotation measurement in the system
Gustavson , which is mainly induced by the quadratic Zeeman shift. In our
experiment, the precision of the quadratic Zeeman shift is mainly limited by
the measurement time, and it can be measured even more accurately by
decreasing the atomic flight velocity and increasing the Raman beam diameter,
and by using the separated oscillation field method in a weak magnetic field
Bize1999a . However, our result provides helpful data for higher precision
measurement of the quadratic Zeeman shift of 85Rb, even for the accuracy of
the rotation rate measurement of the atom-interferometer gyroscope.
Table $1$ Experimental data for the determination of the quadratic Zeeman
shift of hyperfine sublevels ($5S_{1/2},F=2,m_{F}=0\rightarrow$
$5S_{1/2},F=3,m_{F}=0$) of 85Rb.
$Run$ | $\begin{array}[c]{c}Frequency\\\ shift\\\ \text{(Hz/G}^{2}\text{)}\end{array}$ | $\begin{array}[c]{c}Scaled\\\ error\\\ \text{ (Hz/G}^{2}\text{)}\end{array}$ | $\begin{array}[c]{c}Fitted\\\ error\\\ \text{(Hz/G}^{2}\text{)}\end{array}$
---|---|---|---
$1$ | $1294.2$ | $2.1$ | $2.9$
$2$ | $1294.1$ | $2.1$ | $2.9$
$3$ | $1295.7$ | $2.1$ | $2.3$
$4$ | $1296.1$ | $2.1$ | $2.2$
$5$ | $1298.6$ | $2.1$ | $1.9$
$6$ | $1298.7$ | $2.1$ | $1.9$
$7$ | $1298.7$ | $2.1$ | $1.9$
$8$ | $1298.6$ | $2.1$ | $1.9$
$Average$ | $1296.8$ | $2.1$ | $2.5$
$Total$ | $1296.8\pm 3.3$ (Hz/G2)
5\. Conclusion
In summary, we analyzed the energy of the hyperfine sublevels of two ground
states of 85Rb in the magnetic field. We demonstrated experimentally the
coherent population transfer of the hyperfine sublevels between two ground
states by the stimulated Raman transition. The ac Stark shift was
experimentally studied by measuring the ac Stark frequency shift dependence on
the Raman beam intensity, and it was cancelled by adjusting the ratio of two
Raman beam intensities. We measured the quadratic Zeeman shift of the ground
states using the coherent population transfer by a stimulated Raman
transition. The error analysis shows that the quadratic Zeeman shift was
measured to Hz level for magnetically insensitive states
$5S_{1/2},F=2,m_{F}=0\rightarrow$ $5S_{1/2},F=3,m_{F}=0$ in the experiment.
This result provides helpful data to improve the accuracy of the atom-
interferometer gyroscope in future.
Acknowledgments
We acknowledge the financial support from the National Basic Research Program
of China under Grant Nos. 2005CB724505, 2006CB921203, and from the National
Natural Science Foundation of China under Grant No.10774160. We thank
Professor J. P. Connerade and Professor L. You for useful comment and
discussion.
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|
arxiv-papers
| 2008-12-01T07:01:46 |
2024-09-04T02:48:59.081244
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Runbing Li, Lin Zhou, Jin Wang, Mingsheng Zhan",
"submitter": "Jin Wang",
"url": "https://arxiv.org/abs/0812.0231"
}
|
0812.0411
|
Hyperon Physics from Lattice QCD H.-W. Lin
# Hyperon Physics from Lattice QCD
Huey-Wen Lin Speaker Thomas Jefferson National Accelerator Facility, Newport
News, VA 23606
###### Abstract
I review recent lattice calculations of hyperon physics, including hyperon
spectroscopy, axial coupling constants, form factors and semileptonic decays.
## 1 Introduction
The hyperons are extremely interesting because they provide an ideal system in
which to study SU(3) flavor symmetry breaking by replacement of up or down
quarks in nucleons by strange ones. Hyperon semileptonic decays provide an
additional way of extracting the CKM matrix element $V_{us}$ and offer unique
opportunities to understand baryon structure and decay mechanisms. However,
since hyperons decay in less than a nanosecond under weak interactions, their
experimental study is not as easy as for the nucleons, and thus hyperon
properties are not as well determined. Recent successes of lattice QCD in
computing nucleon structure[1, 2, 3] provide assurance that lattice QCD can
reliably predict the properties of hyperons as well. Knowledge of these
properties can be very valuable in understanding hypernuclear physics, the
physics of neutron stars and the structure of nucleons.
In this proceeding, I review the latest progress on the hyperon spectroscopy,
axial coupling constants, electric charge radii, magnetic moments and
semileptonic decay form factors using a mixed action, in which the sea and
valence fermions use different lattice actions. Lattice calculations of
hyperon scattering lengths are not included in this proceeding; for more
details, please see Ref. [4]. In our case, the sea fermions are 2+1 flavors of
staggered fermions (in configuration ensembles generated by the MILC
collaboration[5]), and the valence fermions are domain-wall fermions (DWF).
The pion mass ranges from 300 to 700 MeV in a lattice box of size 2.6 fm. The
gauge fields are hypercubic-smeared and the source field is Gaussian-smeared
to improve the signal. Details on the configurations can be found in Ref. [6].
For three-point functions, the source-sink separation is fixed at 10 time
units.
## 2 Spectroscopy
On the lattice, continuum SO(3) rotational symmetry is broken to the more
restricted symmetry of the cubic group (also known as the octahedral group).
Hadronic states at rest are classified according to irreducible
representations (irreps) of the cubic group. Since cubic symmetry is respected
by the lattice action, an operator belonging to a particular irrep will not
mix with states in other irreps. The most general baryon spectrum from a
lattice calculation is given by a small number of irreps of the double-cover
of the cubic group: $G_{1g/u}$, $H_{g/u}$ and $G_{2g/u}$, where $g$ (German:
gerade) and $u$ (ungerade) denote positive and negative parity, respectively.
We follow the technique introduced in Ref. [7] to construct all the possible
baryon interpolating operators that can be formed from local or quasi-local
$u/d$ and $s$ quark fields; the $G_{2g/u}$ irreps are excluded, since they
require non-(quasi-)local operators. In this paper, we present the
calculations of the masses of the lowest-lying states in the $G_{1g/u}$ and
$H_{g/u}$ representations.
The non-locality in four dimensions of our valence DWF action manifests in
oscillations of the two-point effective mass close to the source. As a result,
a phenomenological form for fitting such data was proposed in Ref. [8],
employing both an oscillating contribution describing the non-local lattice
artifacts and two positive-definite contributions:
$\displaystyle C(t)$ $\displaystyle=$ $\displaystyle A_{0}e^{-M_{0}(t-t_{\rm
src})}+A_{1}e^{-M_{1}(t-t_{\rm src})}$ (1) $\displaystyle+$ $\displaystyle
A_{\rm osc}(-1)^{t}e^{-M_{\rm osc}(t-t_{\rm src})}.$
The first excited-state mass $M_{1}$ is included to extract a better ground-
state mass $M_{0}$; $M_{\rm osc}$ is a non-physical oscillating term due to
lattice artifacts. We use two-point correlators with the same smearing and
interpolating operators at both source and sink and select fitted results with
varying fit ranges, optimized for quality of fit. A standard jackknife
analysis is employed here.
Here we will focus on orbitally excited hyperon resonances. The ground states
of the octet and decuplet and the SU(3) Gell-Mann–Okubo mass relation can be
found in Ref. [6]. There is in this reference an extensive description of
baryon-mass extrapolation (to the physical pion mass) using continuum and
mixed-action heavy-baryon chiral perturbation theory for two and three
dynamical flavors. However, no chiral perturbation theory results have been
published for orbitally excited hyperon resonances; therefore, we apply naive
linear extrapolations in terms of $M_{\pi}^{2}$. Figure 1 summarizes our
results for the lowest-lying $\Lambda$ and $\Omega$ $G_{1g/u}$
(upward/downward-pointing triangles) and $H_{g/u}$ (diamonds and squares)
(since these have less overlap with the data in in Ref. [6]) and their chiral
extrapolations. The leftmost points are extrapolated masses at the physical
pion mass, and the horizontal bars are the experimental masses (if they are
known).
We summarize the lattice hyperon-mass calculations for the $\Sigma$,
$\Lambda$, $\Xi$ and $\Omega$ in Figure 2, divided into vertical columns
according to their discrete lattice spin-parity irreps ($G_{1g/u}$ and
$H_{g/u}$), along with experimental results by subduction of continuum $J^{P}$
quantum numbers onto lattice irreps. $G_{1}$ ground states only overlap with
spin-$1/2$. The spin identification for $H$ can be a bit trickier, since this
irrep could match either to spin-$3/2$ or $5/2$ ground states. We simply
select the lowest-lying of $3/2$ or $5/2$ indicated in the PDG, so it could be
either depending on which one is the ground state for a particular baryon
flavor. (We note below if the lowest $H$ is not spin-$3/2$.)
Although our naive extrapolation neglects contributions at next-to-leading-
order chiral perturbation theory, several interesting patterns are seen. The
better-known $\Lambda$ (with $H_{g}$ being spin-$5/2$) and $\Sigma$ spectra
match up with our calculations well. $G_{1u}$ $\Xi$ lines up well with the
$\Xi(1690)$, indicating that the spin-parity of this resonance could be
$1/2^{-}$; this agrees with SLAC’s recent spin measurement. The spin
assignments for the $\Omega$ channel are the least known; from our mass
pattern, we predict that $\Omega(2250)$ is likely to be $3/2^{-}$ (although we
cannot rule out the possibility of $5/2^{-}$), and $\Omega(2380)$ and
$\Omega(2470)$ are likely spin $1/2^{-}$ and $1/2^{+}$ respectively.
To successfully extract any reliable radially excited hyperon states, we would
need finer lattice spacing, which would require massive computational
resources to achieve. In Euclidean space the excited signals exponentially
decay faster than the ground state. One possible solution is to use an
anisotropic lattice, where the temporal lattice spacing is made finer than the
spatial ones to reduce overall costs. A full-QCD calculation using anisotropic
lattices is underway, and we would expect new results on these excited states
within the next couple of years.
Figure 1: The squared pion-mass dependence of $\Lambda$ and $\Omega$-flavor
baryons with their extrapolated values. The bar at the left indicates the
experimental values for the corresponding spin. Figure 2: A summary of our
hyperon spectrum compared to experiment. Our data are the short bars with
large labels on the right. The experimental states are long bars with small
labels on the left.
## 3 Axial Coupling Constants
The hyperon axial couplings are important parameters entering the low-energy
effective field theory description of the octet baryons. At the leading order
of SU(3) heavy-baryon chiral perturbation theory, these coupling constants are
linear combinations of the universal coupling constants $D$ and $F$, which
enter the chiral expansion of every baryonic quantity, including masses and
scattering lengths. These coupling constants are needed in the effective field
theory description of both the non-leptonic decays of hyperons, and the
hyperon-nucleon and hyperon-hyperon scattering phase shifts[9]. Hyperon-
nucleon and hyperon-hyperon interactions are essential in understanding the
physics of neutron stars, where hyperon and kaon production may soften the
equation of state of dense hadronic matter.
We have calculated the axial coupling constants for $\Sigma$ and $\Xi$ strange
baryons using lattice QCD for the first time. We have done the calculation
using 2+1-flavor staggered dynamical configurations with pion mass as light as
350 MeV. Figure 4 shows our lattice data as a function of
$(M_{\pi}/f_{\pi})^{2}$ with the corresponding chiral extrapolation; the band
shows the jackknife uncertainty. We conclude that $g_{A}=1.18(4)_{\rm
stat}(6)_{\rm syst}$, $g_{\Sigma\Sigma}=0.450(21)_{\rm stat}(27)_{\rm syst}$
and $g_{\Xi\Xi}=-0.277(15)_{\rm stat}(19)_{\rm syst}$. In addition, the
$SU(3)$ axial coupling constants are estimated to be $D=0.715(6)_{\rm
stat}(29)_{\rm syst}$ and $F=0.453(5)_{\rm stat}(19)_{\rm syst}$. The axial
charge couplings of $\Sigma$ and $\Xi$ baryons are predicted with
significantly smaller errors than estimated in the past.
We also study the SU(3) symmetry breaking in the axial couplings through the
quantity $\delta_{\rm SU(3)}$,
$\delta_{\rm SU(3)}=g_{\rm A}-2.0\times
g_{\Sigma\Sigma}+g_{\Xi\Xi}=\sum_{n}c_{n}x^{n},$ (2)
where $x$ is ${(M_{K}^{2}-M_{\pi}^{2})}/{(4\pi f_{\pi}^{2})}$. Figure 3 shows
$\delta_{\rm SU(3)}$ as a function of $x$. Note that the value increases
monotonically as we go to lighter pion masses. Our lattice data suggest that a
$\delta_{\rm SU(3)}\sim x^{2}$ dependence is strongly preferred, as the plot
of $\delta_{\rm SU(3)}/{x^{2}}$ versus $x$ in Figure 3 also demonstrates. A
quadratic extrapolation to the physical point gives 0.227(38), telling us that
SU(3) breaking is roughly 20% at the physical point, where $x=0.332$ using the
PDG values[10] for $M_{\pi^{+}}$, $M_{K^{+}}$ and $f_{\pi^{+}}$. We compare
the result of heavy-baryon SU(3) chiral perturbation theory[11] for
$\delta_{\rm SU(3)}$ as a function of $x$, and we find that the coefficient of
the linear term in Eq. 2 does not vanish. This implies that an accidental
cancellation of the low-energy constants is responsible for this behavior.
Figure 3: (Top) The SU(3) symmetry breaking measure $\delta_{\rm SU(3)}$. The
circles are the measured values at each pion mass, the square is the
extrapolated value at the physical point, and the shaded region is the
quadratic extrapolation and its error band. (Bottom) $\delta_{\rm
SU(3)}/x^{2}$ plot. Symbols as above, but the band is a constant fit. Figure
4: Lattice data (circles) for $g_{A}$, $g_{\Sigma\Sigma}$ and $g_{\Xi\Xi}$ and
chiral extrapolation (lines and bands). The square is the extrapolated value
at the physical point.
## 4 Form Factors
The study of the hadron electromagnetic form factors reveals information
important to our understanding of hadronic structure. The electromagnetic form
factors of an octet baryon $B$ can be written as
$\langle
B\left|V_{\mu}\right|B\rangle={\overline{u}}_{B}\left[\gamma_{\mu}F_{1}(q^{2})+\sigma_{\mu\nu}q_{\nu}\frac{F_{2}(q^{2})}{2M_{B}}\right]u_{B}$
from Lorentz symmetry and vector-current conservation. $F_{1}$ and $F_{2}$ are
the Dirac and Pauli form factors. Another common form-factor definition,
widely used in experiments, are the Sachs form factors; these can be related
to the Dirac and Pauli form factors through
$\displaystyle G_{E}(q^{2})$ $\displaystyle=$ $\displaystyle
F_{1}(q^{2})-\frac{q^{2}}{4M_{B}^{2}}F_{2}(q^{2})$ (3) $\displaystyle
G_{M}(q^{2})$ $\displaystyle=$ $\displaystyle F_{1}(q^{2})+F_{2}(q^{2}).$ (4)
In this work, we will concentrate on Sachs form factors. Note that here we
only calculate the “connected” diagram, which means the inserted quark current
is contracted with the valence quarks in the baryon interpolating fields.
On the lattice, we calculate the quark-component inserted current,
$V_{\mu}=\overline{q}\gamma_{\mu}q$, with $q=u,d$ for the light-quark current
and $q=s$ for the strange-quark vector current. A single interpolating field
for the nucleon, Sigma and cascade octet baryons has the general form
$\chi^{B}(x)=\epsilon^{abc}[q_{1}^{a\mathrm{T}}(x)C\gamma_{5}q_{2}^{b}(x)]q_{1}^{c}(x),$
(5)
where $C$ is the charge conjugation matrix, and $q_{1}$ and $q_{2}$ are any of
the quarks $\\{u,d,s\\}$. For example, to create a proton, we want $q_{1}=u$
and $q_{2}=d$; for the $\Xi^{-}$, $q_{1}=s$ and $q_{2}=d$. By calculating two-
point and three-point correlators on the lattice with the same baryon
operator, we will be able to extract the form factors from Eq. 3. We solve for
$G_{M,E}$ using singular value decomposition (SVD) at each time slice from
source to sink with data from all momenta with the same $q^{2}$ and all $\mu$.
We constrain the fit form to go asymptotically to $1/Q^{4}$ at large $Q^{2}$
and to have $G_{E}(Q^{2}=0)=1$:
$G_{E}=\frac{AQ^{2}+1}{CQ^{2}+1}\frac{1}{(1+Q^{2}/M_{e}^{2})^{2}}.$ (6)
By trying various combinations of fit constraints on our data with squared
momentum transfer ${}<2$ GeV2, we find $C$ is always consistent with 0;
therefore, we set $C=0$. The mean-squared electric charge radii can be
extracted from the electric form factor $G_{E}$ via
$\langle
r_{E}^{2}\rangle=(-6)\frac{d}{dQ^{2}}\left(\frac{G_{E}(Q^{2})}{G_{E}(0)}\right)\Big{|}_{Q^{2}=0}.$
(7)
In Figure 5 we plot the electric charge radii with the neutron and $\Xi^{0}$
omitted, since the vector conserved current gives
$G_{E,\\{n,\Xi^{0}\\}}(Q^{2})\approx 0$ for these. We see that there is small
SU(3) symmetry breaking between the SU(3) partners $p$ and $\Sigma^{+}$ (or
$\Sigma^{-}$ and $\Xi^{-}$); their charge radii are consistent within
statistical errors. Overall, the SU(3) symmetry breaking in the charge radii
is much smaller than what we observed in our study of the axial coupling
constants; for charge radii, the effect is negligible.
We can take a ratio of the electric radii of the baryons which coincide in the
SU(3) limit; for example, $p$ and $\Sigma^{+}$ and $\Sigma^{-}$ and $\Xi^{-}$.
The dominant meson-loop contribution is suppressed; thus, a naive linear fit
to a ratio could be a better description than fitting individual channels.
Following Sec. 3, we use the SU(3) symmetry measure $x$ to parametrize the
deviation of the ratio from 1 due to symmetry breaking:
$1+\sum_{n=1}^{N}c_{n}x^{n},$ where the next-order corrections contribute at
the order of $x^{N+1}$; taking $n=1$, we expect the remaining effect should be
less than 1% in the expansion. The fit works fairly well for the extrapolation
of $\frac{\langle r_{E}^{2}\rangle_{p}}{\langle
r_{E}^{2}\rangle_{\Sigma^{+}}}$ and $\frac{\langle
r_{E}^{2}\rangle_{\Sigma^{-}}}{\langle r_{E}^{2}\rangle_{\Xi^{-}}}$. By using
the experimental value of $\langle r_{E}^{2}\rangle_{p}$ and $\langle
r_{E}^{2}\rangle_{\Sigma^{-}}$, we can make predictions for $\langle
r_{E}^{2}\rangle_{\Sigma^{+}}$ and $\langle r_{E}^{2}\rangle_{\Xi^{-}}$:
$0.93(3)$ and $0.501(10)~{}\mbox{ fm}^{2}$ respectively. (Using $n=2$ in for
the fit yields $c_{2}$ zero within error and thus gives results consistent
with the extrapolation using $n=1$.)
Figure 5: The electric mean-squared radii in units of $\mbox{fm}^{2}$ as
functions of $M_{\pi}^{2}$ (in GeV2) from each quark contribution
Studying the momentum-transfer dependence of magnetic form factors gives us
the magnetic moment via
$\mu_{B}=G_{M}^{B}(Q^{2}=0)$ (8)
with natural units $\frac{e}{2M_{B}}$, where $M_{B}$ are the baryon
($B\in\\{N,\Sigma,\Xi\\}$) masses. To compare among different baryons, we
convert these natural units into nuclear magneton units
$\mu_{N}=\frac{e}{2M_{N}}$; therefore, we convert the magnetic moments with
factors of $\frac{M_{N}}{M_{B}}$.
We can obtain the magnetic moments and radii from polynomial fitting to the
ratio of magnetic and electric form factors, $G_{M}/G_{E}$. From the
definition of the electric and magnetic radii, we expect that
$G_{M}/G_{E}\approx A+CQ^{2}$, where the magnetic moment is $\mu=AG_{E}(0)$,
and $C$ is proportional to $\langle r_{M}^{2}\rangle-\langle
r_{E}^{2}\rangle$. In the case of $n$ and $\Xi^{0}$, we use $G_{E,p}$ and
$G_{E,\Xi^{-}}$ in the ratio instead of $G_{E,n}$ and $G_{E,\Xi^{0}}$. Figure
6 shows the magnetic moments of each baryon compared with its SU(3) partner:
$\\{p,\Sigma^{+}\\}$, $\\{n,\Xi^{0}\\}$, $\\{\Sigma^{-},\Xi^{-}\\}$. We find
that as seen in experiment, the SU(3) breakings of the magnetic moments are
rather small. As we go to larger pion masses (that is, as the light mass goes
to the strange mass), the discrepancy gradually goes to zero as SU(3) is
restored. But even at our lightest pion mass, around 350 MeV, the effects of
SU(3) symmetry breaking effect can be ignored. The fitted results are
consistent with what we obtained from the dipole extrapolations. We examine
the radii differences from the quark contributions and observe less than 10%
discrepancy. The ratio approach also benefits from cancellation of noise due
to the gauge fields, and thus it has smaller statistical error. Therefore, we
will concentrate on the results from this approach for the rest of this work.
Figure 6: Baryon magnetic moments in units of $\mu_{N}$ as functions of
$M_{\pi}^{2}$ (in GeV2). The leftmost points are the experimental numbers.
We extrapolate the baryon magnetic moments using the SU(3) ratios with the
SU(3) symmetry breaking measure $x$. Again, the ratio has cleaner statistical
signal due to the cancellation of fluctuations within gauge configurations,
and the linear extrapolation is not a bad approximation, since potential log
terms are suppressed. With the help of experimental $\mu_{\Sigma^{+}}$,
$\mu_{\Xi^{-}}$ and $\mu_{\Xi^{0}}$[10], we obtain $\mu_{p}=2.56(7)$,
$\mu_{n}=-1.55(8)$ and $\mu_{\Sigma^{-}}=-1.00(3)$. (The fit using up to
$x^{2}$ terms results in a zero-consistent fit parameter $c_{2}$ and yields
numbers consistent with the above.)
SU(6) symmetry predicts the ratio $\mu_{d^{p}}/\mu_{u^{p}}$ should be around
$-1/2$. Compared with what we obtain in this work, the ratio agrees within
$2\sigma$ for all the pion-mass points. The heaviest two pion points have
roughly the same magnitude as in the quenched calculation[12]. However, at the
lightest two pion masses, they are consistent with the $-1/2$ value. The
difference could be due to sea-quark effects, which become larger as the pion
mass becomes smaller. A naive linear extrapolation through all the points
gives $-0.50(10)$. SU(6) symmetry is preserved in the lattice calculations.
We also check the sum of the magnetic moments of the proton and neutron,
$\mu_{p}+\mu_{n}$, which should be about 1 from isospin symmetry. Again, the
values from different pion masses are consistent with each other within 2
standard deviations and differ from 1 by about the same amount. A naive linear
extrapolation suggests the sum is 0.78(13), which is consistent with
experiment but about $2\sigma$ away from 1. This symmetry is softly broken,
possibly due to finite lattice-spacing effects. Finer lattice-spacing
calculations would be needed to confirm this.
## 5 Semileptonic Decays
The hyperons differ from the nucleons by their strangeness, and although
hyperon decays via the weak interaction have been known for more than half a
century, interest in their study has not decayed with time. They provide an
ideal systems in which to study SU(3) flavor symmetry breaking and offer
unique opportunities to understand baryon structure and decay mechanisms. The
low-energy contribution to the transition matrix elements for hyperon beta
decay, $B_{1}\rightarrow B_{2}e^{-}\overline{\nu}$ can be written in general
form as
$\displaystyle{\cal
M}=\frac{G_{s}}{\sqrt{2}}\overline{u}_{B_{2}}(O_{\alpha}^{\rm
V}+O_{\alpha}^{\rm
A}){u}_{B_{1}}\overline{u}_{e}\gamma^{\alpha}(1+\gamma_{5})v_{\nu}.$ (9)
From Lorentz symmetry, we expect the matrix element composed of any two
spin-$1/2$ nucleon states, $B_{1}$ and $B_{2}$, to have the form
$O_{\alpha}^{V}=f_{1}(q^{2})\gamma^{\alpha}+\frac{f_{2}(q^{2})}{M_{B_{1}}}\sigma_{\alpha\beta}q^{\beta}+\frac{f_{3}(q^{2})}{M_{B_{1}}}q_{\alpha}$
$O_{\alpha}^{A}=\left(g_{1}(q^{2})\gamma^{\alpha}+\frac{g_{2}(q^{2})}{M_{B_{1}}}\sigma_{\alpha\beta}q^{\beta}+\frac{g_{3}(q^{2})}{M_{B_{1}}}q_{\alpha}\right)\gamma_{5}$
with transfer momentum $q=p_{B_{2}}-p_{B_{1}}$ and $V,A$ indicating the vector
and axial currents respectively. $f_{1}$ and $g_{1}$ are the vector and axial
form factors, and $f_{2}$ and $g_{2}$ are the weak magnetic and induced-
pseudoscalar form factors. They are non-zero even with $B_{1}=B_{2}$. SU(3)
flavor symmetry breaking accounts for the non-vanishing induced scalar and
weak electric form factors, $f_{3}$ and $g_{2}$ respectively. Due to the
difficulty in disentangling experimental form factor contributions, they tend
to be set to zero. We will be able to determine these form factors using
theoretical technique and will find them to be non-negligible.
So far, there are only two “quenched” (where the fermion masses in the sea
sector are infinitely heavy) lattice calculations of hyperon beta decay, and
they are in different channels, $\Sigma\rightarrow n$ and
$\Xi^{0}\rightarrow\Sigma^{+}$. Guadagnoli et al.[13] calculated the matrix
element $\Sigma\rightarrow n$ with all of the pion masses larger than 700 MeV.
Sasaki et al.[14] used lighter pion masses in the range 530–650 MeV and DWF to
look at the $\Xi^{0}$ decay channel. They extrapolate the vector form factor
$f_{1}$ using the parameter $\delta=(M_{B_{2}}-M_{B_{1}})/M_{B_{2}}$. In this
work, we calculate both decay channels and remove the quenched approximation,
which often causes notoriously large systematic error. Our fermion sea sector
contains degenerate up and down quarks plus the strange. We use pion masses as
light as 350 MeV, which ameliorates some of the uncertainty in the
extrapolation to the physical pion mass. To condense our work for this
proceeding, we will concentrate on the results from $\Sigma\rightarrow n$.
To obtain $V_{us}$, we need to extrapolate $f_{1}$ to zero momentum-transfer
(we cannot calculate this point directly due to the discrete values of
momentum accessible in a finite volume) and the physical pion and kaon masses.
Fortunately, $f_{1}$ is protected by the Ademollo-Gatto (AG) theorem such that
there is no first-order SU(3) breaking. Therefore, the quantity deviates from
its SU(3) value by the order of the symmetry breaking term of
$O({H^{\prime}}^{2})$, where $H^{\prime}$ is the SU(3) symmetry breaking
Hamiltonian; the natural candidate an observable to track this breaking is the
mass splitting between the kaon and pion. Combining with momentum
extrapolation (using a dipole form in this case), we use a single simultaneous
fit:
$f_{1}(q^{2})=\frac{1+\left(M_{K}^{2}-M_{\pi}^{2}\right)^{2}\left(A_{1}+A_{2}\left(M_{K}^{2}+M_{\pi}^{2}\right)\right)}{\left(1-\frac{q^{2}}{M_{0}+M_{1}\left(M_{K}^{2}+M_{\pi}^{2}\right)}\right)^{2}}.$
(10)
Figure 7 shows the result from simultaneously fitting over all $q^{2}$ and
mass combinations for the $\Sigma^{-}\rightarrow n$ decay. The $z$-direction
indicates $f_{1}$, while the $x$\- and $y$-axes indicate mass and transfer
momentum. The surface is the fit using Eq. 10 with color to indicate different
masses. The columns are the data and the momentum points from different pion
masses line up in bands. Our preliminary result for $f_{1}$ is $-0.95(3)$
(which is consistent with the quenched result[13]: $f_{1}=-0.988(29)_{\rm
stat}$.)
Figure 7: Simultaneous extrapolation in $q^{2}$ and mass
The axial form factor $g_{1}$ is not protected by the AG theorem.
Experimentally, one is interested in its ratio with the vector form factor,
$g_{1}/f_{1}$, at zero momentum transfer. Here we adopt naive linear mass
combinations $(M_{K}^{2}+M_{\pi}^{2})$ and $(M_{K}^{2}-M_{\pi}^{2})$ and
momentum dependence to extrapolate and find $g_{1}(0)/f_{1}(0)=-0.336(52)$,
which is consistent with the experimental value of $-0.340(17)$. Similar
extrapolations are applied to other form factor ratios, such as
$f_{2}(0)/f_{1}(0)=-1.28(19)$, which is consistent with Cabibbo model value of
$-1.297$[15].
Figure 8: The momentum dependence of the form factor ratios $f_{3}/f_{1}$
(top) and $g_{2}/f_{1}$ (bottom) with various pion masses labeled by
triangles, squares, downward triangles and diamonds from light to heavy; the
band indicates the extrapolation at the physical mass
Finally, we turn our discussion toward the SU(3)-vanishing weak-electric
$g_{2}$ and induced-scalar $f_{3}$ form factors. Figure 8 presents the
momentum dependence of the ratios $g_{2}(q^{2})/f_{1}(q^{2})$ and
$f_{3}(q^{2})/f_{1}(q^{2})$ for sea-pion masses ranging 350–700 MeV. The band
indicates our mass extrapolation in terms of a naive linear dependence of
$M_{K}^{2}-M_{\pi}^{2}$ and $M_{K}^{2}+M_{\pi}^{2}$. We find
$f_{3}(0)/f_{1}(0)=-0.17(11)$ and $g_{2}(0)/f_{1}(0)=-0.29(20)$, which are 1.5
standard deviations from zero. Experimentally, only a combination of axial
form factors $\left|g_{1}(0)/f_{1}(0)-0.133g_{2}(0)/f_{1}(0)\right|$ is
determined. They find 0.327(7)(19), which is consistent with our result,
$0.297(60)$.
## 6 Summary and Outlook
The lattice formulation is a powerful method for calculating nonperturbative
quantities used in fundamental tests of QCD. The results presented here
demonstrate clear progress toward the long-term goals of determining the
hyperon spectrum, structure and decays from first principles. Ongoing efforts
from lattice community using improved operators and algorithms (to improve the
noise-to-signal ratios), finer lattice spacings (to reduce systematic error
due to discretization), lighter pion masses (to reduce uncertainty introduced
by chiral extrapolation) on dynamical sea quarks will continue to better
approximate the physical world and provide observables that elude experiments.
## Acknowledgements
HWL thanks collaborators Kostas Orginos, David Richards, Colin Morningstar and
Saul Cohen for useful discussions. These calculations were performed using the
Chroma software suite[16] on clusters at Jefferson Laboratory using time
awarded under the SciDAC Initiative. Authored by Jefferson Science Associates,
LLC under U.S. DOE Contract No. DE-AC05-06OR23177. The U.S. Government retains
a non-exclusive, paid-up, irrevocable, world-wide license to publish or
reproduce this manuscript for U.S. Government purposes.
## References
* [1] K. Orginos PoS LAT2006 (2006) 018.
* [2] P. Hagler PoS LAT2007 (2007) 013.
* [3] J. Zanotti PoS LAT2008 (2008).
* [4] S. R. Beane et. al. Nucl. Phys. A794 (2007) 62–72 [hep-lat/0612026].
* [5] C. W. Bernard et. al. Phys. Rev. D64 (2001) 054506 [hep-lat/0104002].
* [6] A. Walker-Loud et. al. 0806.4549.
* [7] S. Basak et. al. Phys. Rev. D72 (2005) 094506 [hep-lat/0506029].
* [8] S. Syritsyn and J. W. Negele PoS LAT2007 (2007) 078 [0710.0425].
* [9] S. R. Beane et. al. Nucl. Phys. A747 (2005) 55–74 [nucl-th/0311027].
* [10] W. M. Yao et. al. J. Phys. G33 (2006) 1–1232.
* [11] W. Detmold and C. J. D. Lin Phys. Rev. D71 (2005) 054510 [hep-lat/0501007].
* [12] S. Boinepalli et. al. Phys. Rev. D74 (2006) 093005 [hep-lat/0604022].
* [13] D. Guadagnoli, V. Lubicz, M. Papinutto and S. Simula Nucl. Phys. B761 (2007) 63–91 [hep-ph/0606181].
* [14] S. Sasaki and T. Yamazaki hep-lat/0610082.
* [15] N. Cabibbo, E. C. Swallow and R. Winston Ann. Rev. Nucl. Part. Sci. 53 (2003) 39–75 [hep-ph/0307298].
* [16] R. G. Edwards and B. Joo Nucl. Phys. Proc. Suppl. 140 (2005) 832 [hep-lat/0409003].
|
arxiv-papers
| 2008-12-02T01:39:06 |
2024-09-04T02:48:59.090411
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Huey-Wen Lin",
"submitter": "Huey-Wen Lin",
"url": "https://arxiv.org/abs/0812.0411"
}
|
0812.0429
|
# Photonic Feshbach Resonance
D. Z. Xu Institute of Theoretical Physics, The Chinese Academy of
Sciences,100190, P.R. China Department of Modern Physics, University of
Science and Technology of China, Hefei, 230026, P.R. China H. Ian Institute
of Theoretical Physics, The Chinese Academy of Sciences,100190, P.R. China T.
Shi Institute of Theoretical Physics, The Chinese Academy of Sciences,100190,
P.R. China H. Dong dhui@itp.ac.cn Institute of Theoretical Physics, The
Chinese Academy of Sciences,100190, P.R. China C. P. Sun suncp@itp.ac.cn
Institute of Theoretical Physics, The Chinese Academy of Sciences,100190, P.R.
China
Hermann Feshbach predicted fifty years ago feshbach58 that when two atomic
nuclei are scattered within an open entrance channel— the state observable at
infinity, they may enter an intermediate closed channel — the locally bounded
state of the nuclei. If the energy of a bound state of in the closed channel
is fine-tuned to match the relative kinetic energy, then the open channel and
the closed channel “resonate”, so that the scattering length becomes divergent
pethick02 . We find that this so-called Feshbach resonance phenomenon not only
exists during the collisions of massive particles, but also emerges during the
coherent transport of massless particles, that is, photons confined in the
coupled resonator arrays lzhou08 . We implement the open and the closed
channels inside a pair of such arrays, linked by a separated cavity or a
tunable qubit. When a single photon is bounded inside the closed channel by
setting the relevant physical parameters appropriately, the vanishing
transmission appears to display this photonic Feshbach resonance. The general
construction can be implemented through various experimentally feasible solid
state systems, such as the couple defected cavities in photonic crystals. The
numerical simulation based on finite-different time-domain(FDTD) method
confirms our conceive about physical implementation.
The phenomenon of Feshbach resonance has been found in many physical systems
over the years, such as the electron scattering of atoms schulz73a and
diatomic molecules schulz73b . More recently, the development of laser cooling
technologies has enabled the observation of low-energy Feshbach resonance in
ultra-cold atoms courteille98 ; roberts98 ; vuletic99 and Bose-Einstein
condensates (BEC) ketterle98 ; timmermans99 . These experiments have helped
verify the simulation of various theoretical predictions of condensing
phenomena in solid state systems dsjin08rev . The latter in particular
exemplifies the resonance phenomenon as a means for adjusting inter-atomic
coupling in realizing various quantum phases ranging from BEC to BCS d.s.jin02
; d.s.jin08 .
On the other hand, since their first discovery by von Neumann and Wigner
neumann29 , bound states have been studied in a general continuum friedrich85
and the emergence of a bounded energy level has been verified by various
models lee54 ; fano61 ; anderson61 , extracting from the simple coupling of a
discrete level with a continuum. Quasi-bound states have been predicted in
tight-binding fermionic quantum wires nakamura07 for localized fermions and
in the optical coupled resonator arrays lzhou08 ; hdong08 ; zrgong08 for
confined photons. It has also found various applications in many quantum
optical devices fan05 ; fan05_optic ; fan07 , including the single-photon
transistor lukin07 . Under this retrospect, we ask if it is possible to
control exactly when the photons become bounded and unbounded through an
external parameter, _viz._ to implement an optical version of Feshbach
resonance.
The desired resonance between the bound and the unbound states can be found
inside a pair of parallelly placed coupled resonator arrays lzhou08 , a series
of consecutively placed optical microcavities that entrap photons and allow
photon-hopping from one of the cavities to its closest neighbors at left and
right. The two arrays are connected by a central cavity that acts as a quantum
controller and couples separately to one cavity in each of the arrays, as
shown in Fig. 1(a), forming an H-shape system. We designate the upper array as
array A with the Hamiltonian
$H=\omega_{\mathrm{A}}\sum_{j}a_{j}^{\dagger}a_{j}+(J_{\mathrm{A}}\sum_{j}a_{j}^{\dagger}a_{j+1}+g_{\mathrm{A}}a_{0}c^{\dagger}+\mathrm{h.c}),$
(1)
where the second summation describes the tight-binding hopping of photons
between neighboring cavities with hopping coefficient $J_{\mathrm{A}}$ while
the first one accounts for the static photon occupations in the cavities.
$a_{j}$ denotes the annihilation operator of the bosonic mode for the $j$-th
cavity field and we assume that the mode frequency $\omega_{\mathrm{A}}$ of
each cavity field is identical. The last term is the interaction between the
central cavity $C$ of single mode frequency $\Omega$ and the zeroth resonator
of array A with coupling strength $g_{\mathrm{A}}$. $c$ is corresponding
annihilation bosonic operator for central cavity. The lower array is
designated as array B, the Hamiltonian $H_{\mathrm{B}}$ for which is no
different from Eq. (1) except the change of the bosonic mode operator to
$b_{j}$,the hopping coefficient to $J_{\mathrm{B}}$, the mode frequency to
$\omega_{\mathrm{B}}$ and the coupling strength to $g_{\text{B}}$. Then,the
total model Hamiltonian reads $H=H_{\mathrm{A}}+H_{\mathrm{B}}+\Omega
c^{\dagger}c.$
One experimental available system of the model is based on photonic crystal,
which will be described in details below. We have to point out that there is a
drawback in our setup: the inter-cavity coupling cannot be externally
manipulated, once the photonic-crystal based metamaterial is fabricated as an
all-optical chip. However, for single photon transferring, the role of central
cavity as that of a qubit with two levels
$\left|e\right\rangle=\left|1\right\rangle$ and $\left|g\right\rangle=$
$\left|0\right\rangle$, corresponding to the single photon state and vacuum of
the central cavity. This identification motivates us to use a qubit controller
replacing the central cavity equivalently in the single photon case.
Therefore, the phenomenon predicted here can be realized in the hybrid system
of photonic crystal and quantum dots with external controllable parameter. Our
general construction can also be implemented through the circuit QED system
wallraff04 including two coupled superconducting transmission line cavity
linked by charge or flux qbit. The replacement of the central cavity by a
controllable qubit can overcome the drawback mentioned above.
Figure 1: The photonic Feshbach resonance based on the coupled resonator
arrays: (a) schematic of the H-shape system consisting of two coupled
resonator arrays, array A and array B, connected by a qubit controller of
level spacing $\Omega$. Each of the cavity in the two arrays is characterized
by the cavity field mode it confines and hereby indicated by the bosonic
operator $a_{j}$ or $b_{j}$ and indexed by the relative distance $j$ from the
zeroth cavity with which the qubit controller couples; (b) probability
distribution of a single photon in a coupled resonator array for a quasi-bound
state. When the level spacing $\Omega$ of the qubit controller matches certain
values, the probability amplitude would vanish at the two ends of the array;
(c) the energy state distribution for array $\alpha$,
$\alpha\in\left\\{A,B\right\\}$, relative to the common eigenfrequency
$\omega_{\alpha}$ of the cavity fields: a pair of a continuum and a discrete
level above and below $\omega_{\alpha}$.
In the following , we use the Jaynes-Cummings couplings
$g_{\alpha}c^{\dagger}\left|g\right\rangle\left\langle e\right|+\mathrm{h.c}$
($\alpha=A,B$) to modeling the central cavity couplings in the single photon
case. The Hilbert space is spanned by the tensor product
$\left\\{\left|e\right\rangle,\left|g\right\rangle\right\\}\bigotimes_{j}\left|n_{\mathrm{A},j}\right\rangle\bigotimes_{j}\left|n_{\mathrm{B},j}\right\rangle$
where $\left|n_{\alpha,j}\right\rangle$ denotes the state of array $\alpha$
with its $j$-th cavity being occupied with $n_{\alpha,j}$ number of photons.
When separated from the other and studied individually, each coupled resonator
array is described by the subsystem Hamiltonian $H_{\alpha}$ and possesses two
bound states, reminiscent that of the Feshbach resonance. The states are the
particular superposition of eigenstates for the Hamiltonian $H_{\alpha}$,
comprised by a subset of the basis vectors described above. For array A, the
state is namely
$\left|\varphi_{\mathrm{A}}\right\rangle=\sum_{j}u_{\mathrm{A},g}(j)\left|g,1_{j},0\right\rangle+u_{\mathrm{A},e}\left|e,0,0\right\rangle$
where only the excited state of the central cavity and a single-photon
excitation in one of the cavities, as indicated by the $1_{j}$ symbol, are
included. The coefficients $u_{\mathrm{A},g}(j)$ and $u_{\mathrm{A},e}$ in the
equation constitute the spectrum of probability distributions of these states.
That of array B takes a similar form with amplitudes $u_{\mathrm{B},g}(j)$ and
$u_{\mathrm{B},e}$.
To see whether there are bounded single-photon states within their individual
coupled resonator array, we can solve the discrete-coordinate scattering
equation associated with their corresponding eigenvalue $E$ for the
probability spectrum,
$[E-\omega_{\alpha}-V_{\alpha}(E)]u_{\alpha,g}(j)=-J_{\alpha}[u_{\alpha,g}(j+1)+u_{\alpha,g}(j-1)]$
(2)
where the term $V_{\alpha}(E)=g_{\alpha}^{2}\delta_{j0}/(E-\Omega)$ on the
left hand side is contributed by the JC type interaction between the central
cavity and the coupled resonator array. $V_{\alpha}(E)$ is a resonate
potential that depends on the eigenenergy $E$. In the continuous limit of the
coordinate $j\rightarrow x$, this term reduces to a $\delta$ -type potential
$V_{\alpha}(E,x)=\frac{g_{\alpha}^{2}}{E-\Omega}\delta(x).$ (3)
The $\delta$-type potential forms a confining barrier to the transportation of
single photon in the coupled resonator array and informs a bounded single
photon within, similar to those in the models proposed in Refs. hdong08 ;
zrgong08 .
It has a singularity at $E$ being equal to the level spacing $\Omega$, leading
to a quasi-plane-wave type solution lzhou08 to Eq. (2),
$u_{\alpha,g}(j)=C_{\alpha}\exp(-i\kappa_{\alpha}|j|)$ where $C_{\alpha}$
denotes a constant and the wave number is complex,
$\kappa_{\alpha}=\kappa_{\alpha,R}-i\kappa_{\alpha,I}$. The imaginary part
$\kappa_{\alpha,I}$ of the wave number can admit a positive value and for the
non-zero coupling $g_{\alpha}$, resulting in a decay of the probability
distribution of single-photon states over the discrete spatial coordinate $j$.
The vanishing probability amplitude towards the ends of the arrays, i.e. along
with $|j|\rightarrow\infty$, demonstrates the existence of a bound state of a
single photon, as shown in Fig. 1(b). For this system, continuum band has a
bandwidth of $4J_{\alpha}$ and their paired discrete levels, denoted
respectively by $E_{\alpha+}$ and $E_{\alpha-}$, are gapped from either below
or above, as illustrated in Fig. 1(c). Reverted to the conventional language
of atomic scattering, the continua of eigenenergies can be considered open
channels of multiple admissible energy states in the continuous range
$\omega_{\alpha}-2J_{\alpha}<E<\omega_{\alpha}+2J_{\alpha}$. Out of this
range, the energy states can only admit two discrete levels that associate
with a non-real $k_{\alpha}$, representing closed channels or bound states.
These two discrete levels
$E_{\alpha\pm}=\Omega\pm\frac{g_{\alpha}^{2}}{\sqrt{(E_{\alpha\pm}-\omega_{\alpha})^{2}-4J_{\alpha}^{2}}}.$
(4)
are exactly solved from the above discrete-coordinate scattering equation. The
dependence of these two bound state energies on the various system parameters
gives hints to their potential of tunability and controllability. The readers
familiar with the approaches by Lee lee54 , Fano fano61 , and Anderson
anderson61 shall also find our result here familiar.
The next logical step is to study how the resonance phenomenon arises when two
individual arrays are paired to form the H-shape system. The scattering state
$\left|\varphi\right\rangle$ of the H-system for single-photon reads
$\left|\varphi\right\rangle=\sum_{j}\left[u_{\mathrm{A},g}(j)\left|g,1_{j},0\right\rangle+u_{\mathrm{B},g}(j)\left|g,0,1_{j}\right\rangle\right]+u_{e}\left|e,0,0\right\rangle$.
The probability amplitudes $u_{\mathrm{A},g}(j)$ and $u_{\mathrm{B},g}(j)$ of
the photonic occupation among the cavities and $u_{e}$ of the atomic
excitation in the system can still be analyzed through the time-independent
Schrödinger equations, leading to a pair of algebraic scattering equations
similar to Eq. (2), one for the probability amplitudes in each array. The
distinction of the case here lies in the adding
$W_{\alpha}(E)=g_{\alpha}g_{\bar{\alpha}}u_{\bar{\alpha},g}(0))/(E-\Omega)$ in
the right hand side with $\alpha$ indexing either A or B. We have used
$\bar{\alpha}$ to indicate the dual array relative to $\alpha$; namely, when
$\alpha$ indexes A, then $\bar{\alpha}$ indexes B and vice versa. This term
reflects a potential again contributed by the interaction of each array with
the central cavity. However, because of the coupling between the central
cavity and the dual array, additional contribution from the dual array has to
be considered.
The set of solutions to the pair of scattering equations are many. The portion
we are concerned with are those illustrating a simultaneously existent set of
open channel and closed channel in the two coupled resonator arrays. We select
one of the particular cases when a single photon is inserted into an open
channel in array A from the left for this purpose. The distribution of this
single photon in the array is then described by a plane wave,
$u_{\mathrm{A}}(j)=\exp(ikj)+r\exp(-ikj)\;(j<0);\;s\exp(ikj)\;(j>0)$. $s$ and
$r$ denote, respectively, the transmission and the reflection coefficients of
the optical plane wave, indicating the scattering of photon by the effective
potential Eq. (3) at the zeroth resonator in the one-dimensional coordinate
space. Meanwhile, the distribution amplitude for the single photon in array B
can be quasi-plane-wave type,
$u_{\mathrm{B}}(j)=C_{\mathrm{B}}\exp\left\\{-i\kappa|j|\right\\}$, with a
complex wave number $\kappa$, and indicate a closed channel, same as that of
the individually discussed case. These two distributions in the paired arrays,
when combined through the coupled scattering equations, give rise to a unified
dual-channel coupling equation
$(1-s)\sin
k\frac{J_{\mathrm{A}}}{g_{\mathrm{A}}}=C_{\mathrm{B}}\sin\kappa\frac{J_{\mathrm{B}}}{g_{\mathrm{B}}}=\frac{g_{\mathrm{A}}s+g_{\mathrm{B}}C_{\mathrm{B}}}{2i(\Omega-E)}.$
(5)
Therefore, the optical dual-channel resonance occurs when there exists a
solution of real $k$ and complex $\kappa$ to Eq. (5) and the eigenenergy $E$
of the photon in array A matches either of the discrete energy levels
$E_{\mathrm{B}\pm}$ of array B. The process is illustrated in Fig. 2 for two
particular cases with $E$ matching $E_{\mathrm{B}+}$ in Fig. 2 (a) or
$E_{\mathrm{B}-}$ in Fig. 2(b). The two possibilities of channel resonances is
further illustrated in Fig. 2 (c) with the equivalent potential of array A as
a function of the resonator position $j$. The zeroth resonator locates the
position of local minimal energy for both the open channels and the closed
channel, reflecting the potential barrier set up by the central cavity. The
dual-channel resonance occurs as well when the roles of array A and array B
are exchanged.
Figure 2: Diagrams of the energy state distributions of the single photon in
array A and array B, showing the process of Feshbach resonance between an open
channel and a closed channel. Two particular resonance cases exist for an
incident photon inserted into array A: (a) the photon energy level in the
continuum band in array A is resonant with the upper discrete level
$E_{\mathrm{B}+}$ in array B; (b) the photon energy level in the continuum
band in array A is resonant with the lower discrete level $E_{\mathrm{B}-}$ in
array B. (c) A profile plot illustrating the same two cases of channel
resonances relative to the cavity position $j$ the photon occupies.
The criteria of the dual-channel resonance can be met if the transmission
coefficient $s$ in Eq. (5) vanishes. This condition implies the circumstance
where the incident photon in array A is totally reflected or scattered by the
potential barrier set up by the central cavity at position $j=0$. In other
words, the level spacing $\Omega$ of the controller becomes our tuning
parameter for the photonic Feshbach resonance. Written in terms of the other
variables in the dual-channel coupling equation
$s=-2iC_{\mathrm{B}}J_{\mathrm{B}}(E-\Omega)\sin\kappa/(g_{\mathrm{A}}g_{\mathrm{B}})-C_{\mathrm{B}}g_{\mathrm{B}}/g_{\mathrm{A}}$,
the transmission coefficient vanishes when $E=\Omega-
g_{\mathrm{B}}^{2}/(2iJ_{\mathrm{B}}\sin\kappa)$, leading to a complex wave
number $\kappa$ as expected. The complete reflection in the open channel can
be understood as the divergence effect of s-wave scattering length for the
usual Feshbach resonances in three-dimensional space reduced to a version in
one-dimensional space. Eliminating the various variables, the transmission
coefficient can be expressed as a function of the incident energy $E,i.e.,$
$s=\begin{cases}F_{\mathrm{A}}(E)\left[F_{\mathrm{A}}(E)-G_{-}(E)\right]^{-1},&E>\omega_{\mathrm{B}}+2J_{\mathrm{B}}\\\
F_{\mathrm{A}}(E)\left[F_{\mathrm{A}}(E)-G_{+}(E)\right]^{-1},&E<\omega_{\mathrm{B}}+2J_{\mathrm{B}}\end{cases}$
(6)
where we have used the shorthands
$F_{\alpha}(E)=\sqrt{(E-\omega_{\alpha})^{2}-4J_{\alpha}^{2}}$ with
$\alpha\in\left\\{\mathrm{A},\mathrm{B}\right\\}$ and
$G_{\pm}(E)=g_{\mathrm{A}}^{2}F_{\mathrm{B}}(E)/\left((E-\Omega)F_{\mathrm{B}}(E)\pm
g_{\mathrm{B}}^{2}\right)$. The norm-squared reflection coefficient
$|1-s|^{2}$ is plotted against the photon energy in families of varying level
spacing $\Omega$ and coupling constant $g_{A}$ of the qubit controller in Fig.
3. The photon encounters two kinds of characteristic points while propagating
through array A. The first one is an indifferentiable turning point where
$s=1$ or $E=\omega_{\mathrm{B}}+2J_{\mathrm{B}}$. The potential barrier
becomes transparent and the photon is completely transmitted because of the
matching coupling between the qubit controller and the dual array B. The
second one is the maximum point where the photon is fully reflected when the
transmission coefficient is vanishing $s=0$. We hence see the shifting of this
peak while $\Omega$ is varied. The reliance on the coupling coefficient
$g_{\mathrm{A}}$ determines the width of the peaking.
Figure 3: Plots of the norm-squared reflection coefficient $|1-s|^{2}$ against
the eigenenergy $E$ of a propagating photon in the coupled resonator array A
of the process illustrated in Fig. 2(a). Two tuning parameters are varied: (a)
the level spacing $\Omega$ of the qubit controller; and (b) the coupling
coefficient $g_{\mathrm{A}}$ between the qubit controller and the zeroth
cavity in array A. Other parameters are chosen to be: $J_{\mathrm{A}}=1$,
$J_{\mathrm{B}}=0.5$, $\omega_{\mathrm{A}}=2$, $\omega_{\mathrm{B}}=1$ and
$g_{\mathrm{B}}=0.7$. The incident energy ranges from $0$ to $4$ and the
continuum band for the array B is set $\left[0,2\right]$. The corresponding
bound state energies are marked as $E_{b1}$,$E_{b2}$,$E_{b3}$ in (a) and
$E_{b}$ in (b).
Next we numerically examine the feasibility of our theoretical prediction on a
two-dimensional photonic crystal fanapl02 ; joan07 . The crystal is made up of
a square lattice of high-index dielectric rods of radius $0.2a$, $0.1a$ and
$0.05a$, where $a$ is the lattice spacing. The artificial design is made by
two parallel waveguides of coupled defected cavity arrays linked through a
central defected cavity on the two-dimensional photonic crystal, as
illustrated in Fig. 4(a). The two resonator arrays yariv99 is constructed
with different frequencies, inter-resonator tunneling rates, and coupling
strengths with the central cavity.
For this photonic crystal, the material of all the rods is assumed to be
silicon, with a dielectric constant $\epsilon=11.56$, and the background is
filled by air. We make the simulation of the designed structure with the
finite-difference time-domain (FDTD) method FDTD in freely available Meep
code Meep . The steady field vector of the incident wave at frequency
$\omega_{0}=0.3628\times 2\pi c/a$ is plotted in Fig. 4(b), with red showing
positive amplitudes pointing out from the plane and blue negative amplitudes
into the plane. The wave travels horizontally from left to right and hence,
according to the convention for characterizing photonic crystals, carries a
transverse-magnetic (TM) polarization. The notice-worthy region is located at
the center where the highly-saturated colors indicate a localized bounded
photon from the lower waveguide. Moreover, the blank portion in the upper
waveguide indicates a completely reflected wave. Finally, we point out that,
though the numerical simulation based on FDTD is of classical, but the weak
light calculation can also reflect the single photon nature with the intensity
distribution illustrated in Fig. 4(b), which is only relevant to the first
order coherence function.
Figure 4: An experimental protocol based on a photonic crystal made up of
silicon rods of radius $0.2a$. (a) The structure of the design: the upper
waveguide is implemented by removing a row of original rods and substituting
with a set of rods of radius $0.1a$ and spacing $d_{\mathrm{UP}}=4a$. The
lower waveguide, $D=7a$ apart from the upper one, is constructed by removing
three rods out of every five rods, i.e. lattice spacing
$d_{\mathrm{DOWN}}=5a$. The central cavity is created by reducing the radius
of three vertically-placed rods between the two waveguides to $0.05a$. (b)
Plotting the steady electric field vector for an incident wave of frequency
$\omega_{0}=0.3628\times 2\pi c/a$ with TM-polarization.
In conclusion, we have shown the existence of a photonic bound state in a
qubit-controlled coupled resonator array and predicted photonic Feshbach
resonance emerges from a pair of these coupled resonator arrays coupled in an
H-shape fashion. An FDTD simulation of the system implemented on a photonic
crystal has verified the validity of the proposal. The resonance phenomenon
arises from the dual-channel coupling between an unbound state in one array
and a bound state in the other, the occurring moment of which is indicated by
a total reflection of an incident photon in the array. Our analysis for the
resonant scattering process was carried out for the single photon case and did
not rely on the photonic statistics. Our prediction here is thus applicable to
fermionic models, such as the electron transportation along a H-shape array of
quantum dots. For the case where multiple photons are assumed to exist in the
arrays, Bethe-ansatz must be used for the analysis and we shall defer its
discussion in a future work.
###### Acknowledgements.
C.P.S. acknowledges the helpful discussion with S. Yang, Peng Zhang and X. H.
Wang. This work is supported by NSFC No.10474104, No. 60433050, and No.
10704023, NFRPC No. 2006CB921205 and 2005CB724508.
## References
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* (2) Pethick, C. J. & Smith _,_ H. _Bose-Einstein Condensation in Dilute Gases_ (Cambridge University Press, 2002).
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* (4) Schulz, G. J. Resonances in electron impact on atoms. _Rev. Mod. Phys._ 45, 378-422 (1973).
* (5) _ibid_. Resonances in electron impact on diatomic molecules. _Rev. Mod. Phys._ 45, 423-486 (1973).
* (6) Courteille, P., Freeland, R. S., Heinzen, D. J., van Abeelen, F. A. & Verhaar, B. J. Observation of a feshbach resonance in cold atom scattering. _Phys. Rev. Lett._ 81, 69-72 (1998).
* (7) Roberts, J. L., Claussen, N. R., Burke, J. P. Jr., Greene, C., Cornell, E. A. & Wieman C. E. Resonant magnetic field control of elastic scattering in cold 85Rb. _Phys. Rev. Lett._ 81, 5109-5112 (1998).
* (8) Vuletic, V., Kerman, A. J., Chin, C. & Chu, S. Observation of low-field feshbach resonances in collisions of cesium atoms. _Phys. Rev. Lett._ 82, 1406-1409 (1999).
* (9) Inouye, S., Andrews, M. R., Stenger, J., Miesner, H. J., Stamper-Kurn, D. M. & Ketterle, W. Observation of Feshbach resonances in a Bose-Einstein condensate. _Nature_ 392, 151-154 (1998).
* (10) Timmermans, E., Tommasini, P., Hussein, M. & Kerman, A. Feshbach resonances in atomic Bose-Einstein condensates. _Phys. Rep._ 315, 199-230 (1999).
* (11) Jin, D. S. & Regal, C. A. Proceedings of the International school of physics Enrico Fermi, course CLXIV (IOS Press, Amsterdam, 2008).
* (12) Loftus, T., Regal, C. A., Ticknor, C., Bohn, J. L. & Jin, D. S. Resonant control of elastic collisions in an optically trapped fermi gas of atoms. _Phys. Rev. Lett._ 88, 173201 (2002).
* (13) Zirbel, J. J., Ni, K.-K., Ospelkaus, S., D’Incao, J. P., Wieman, C. E., Ye, J. & Jin, D. S. Collisional stability of fermionic feshbach molecules. _Phys. Rev. Lett._ 100, 143201 (2008).
* (14) von Neumann, J. & Wigner, E. Üer merkwürdige diskrete Eigenwerte. _Z. Phys._ 30, 465-467 (1929).
* (15) Friedrich, H. & Wintgen, D. Physical realization of bound states in the continuum. _Phys. Rev. A_ 31, 3964-3966 (1985).
* (16) Lee, T. D. Some special examples in renormalizable field theory. _Phys. Rev._ 95, 1329-1334 (1954).
* (17) Fano, U _._ Effects of configuration interaction on intensities and phase shifts. _Phys. Rev._ 124, 1866-1878 (1961).
* (18) Anderson, P. W. Localized magnetic states in metals. _Phys. Rev._ 124, 41-53 (1961).
* (19) Nakamura, H., Hatano, N., Garmon, S. & Petrosky, T. Quasibound states in the continuum in a two channel quantum wire with an adatom. _Phys. Rev. Lett._ 99, 210404 (2007).
* (20) Gong, Z. R., Ian, H., Zhou, L. & Sun, C. P. Controlling quasibound states in 1D continuum through electromagnetic induced transparency mechanism. arXiv:0805.3042 (2008).
* (21) Dong, H., Gong, Z. R., Ian, H., Zhou, L. & Sun, C. P. Intrinsic cavity QED and emergent quasi-normal modes for single photon. arXiv:0805.3085 (2008).
* (22) Shen, J. T. & Fan, S. Coherent single photon transport in a one-dimensional waveguide coupled with superconducting quantum bits. _Phys. Rev. Lett._ 95, 213001 (2005).
* (23) ibid. Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system. _Phys. Rev. Lett._ 98, 153003 (2007).
* (24) _ibid._ Coherent photon transport from spontaneous emission in one-dimensional waveguides. _Opt. Lett._ 30, 2001-2003 (2005).
* (25) Chang, D. E., Sørensen, A. S., Demler, E. A. & Lukin, M. D. A single-photon transistor using nanoscale surface plasmons. _Nature Phys._ 3, 807-812 (2007).
* (26) Wallraff, A., Schuster, D. I., Blais, A., Frunzio, L., Huang, R.-S., Majer, J., Kumar, S., Girvin, S. M. & Schoelkopf, R. J. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. _Nature_ 431, 162-167 (2004).
* (27) Fan, S., Sharp asymmetric line shapes in side-coupled waveguide-cavity systems. _App. Phys. Lett._ 80, 908-910 (2002).
* (28) Joannopoulus, J. D., Johnson S. G., Winn, J. N., & Meade, R. D., _Photonic Crystal: Molding the flow of light_ (Princeton University Press, 2007).
* (29) Yariv, A., Xu, Y., Lee, R. K. & Scherer, A., Coupled-resonator optical waveguide: a proposal and analysis. _Opt. Lett._ 24, 711-713 (1999).
* (30) Taflove, A., & Hagness, S. C., _Computational Electrodynamics: The Finite-Difference Time-Domain Method_ (Artech: Norwood, MA, 2000).
* (31) Farjadpour, A., Roundy, D., Rodriguez, A., Ibanescu, M., Bermel, P., Joannopoulos, J. D., Johnson, S. G., & Burr, G. Improving accuracy by subpixel smoothing in FDTD. _Optics Letters_ 31, 2972–2974 (2006). http://ab-initio.mit.edu/wiki/index.php/Meep.
|
arxiv-papers
| 2008-12-02T04:54:58 |
2024-09-04T02:48:59.097089
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D.Z. Xu, H. Ian, T. Shi, H. Dong and C.P. Sun",
"submitter": "H. Dong",
"url": "https://arxiv.org/abs/0812.0429"
}
|
0812.0497
|
# Connecting anomaly and tunneling methods for Hawking effect through
chirality
Rabin Banerjee, Bibhas Ranjan Majhi
S. N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata-700098, India
E-mail: rabin@bose.res.inE-mail: bibhas@bose.res.in
###### Abstract
The role of chirality is discussed in unifying the anomaly and the tunneling
formalisms for deriving the Hawking effect. Using the chirality condition and
starting from the familiar form of the trace anomaly, the chiral
(gravitational) anomaly, manifested as a nonconservation of the stress tensor,
near the horizon of a black hole, is derived. Solution of this equation yields
the stress tensor whose asymptotic infinity limit gives the Hawking flux.
Finally, use of the same chirality condition in the tunneling formalism gives
the Hawking temperature that is compatible with the flux obtained by anomaly
method.
Introduction: Ever since Hawking’s original observation [1] that black holes
radiate, there have been several derivations [2, 3, 4, 5, 6, 7, 8] of this
effect. A common feature in these derivations is the universality of the
phenomenon; the Hawking radiation is determined universally by the horizon
properties of the black hole leading to the same answer. This, in the absence
of direct experimental evidence, definitely reinforces Hawking’s original
conclusion. Moreover, it strongly suggests that there is some fundamental
mechanism which could, in some sense, unify the various approaches.
In this paper we show that chirality is the common property which connects the
tunneling formalism [4, 5, 9, 10] and the anomaly method [3, 6, 7, 8, 11, 12,
13] in studying Hawking effect. Apart from being among the most widely used
approaches, interest in both the anomaly and tunneling methods has been
revived recently leading to different variations and refinements in them [7,
8, 9, 10, 11, 12, 13, 14]. The calculation will be performed using a family of
metrics that includes a subset of the stationary, spherically symmetric space-
times which are asymptotically flat. Also, the results are derived using
mostly physical reasoning and do not require any specific technical skill.
Before commencing on our analysis we briefly recapitulate the basic tenets of
the tunneling and anomaly methods. The idea of a tunneling description, quite
akin to what we know in usual quantum mechanics where classically forbidden
processes might be allowed through quantum tunneling, dates back to 1976 [15].
Present day computations generally follow either the null geodesic method [5,
9] or the Hamilton-Jacobi method [4, 9, 10], both of which rely on the
semiclassical WKB approximation yielding equivalent results. The essential
idea is that a particle-antiparticle pair forms close to the event horizon.
The ingoing negative energy mode is trapped inside the horizon while the
outgoing positive energy mode is observed at infinity as the Hawking flux.
Although the notion of an anomaly, which represents the breakdown of some
classical symmetry upon quantisation, is quite old, its implications for
Hawking effect were first studied in [3]. It was based on the conformal
(trace) anomaly but the findings were confined only to two dimensions. However
it is possible to apply this method to general dimensions. Recently a new
method was put forward in [6] where a general (any dimensions) derivation was
given. It was based on the well known fact that the effective theory near the
event horizon is a two dimensional conformal theory. The ingoing modes are
trapped within the horizon and cannot contribute to the effective theory near
the horizon. Thus the near horizon theory becomes a two dimensional chiral
theory. Such a chiral theory suffers from a general coordinate
(diffeomorphism) anomaly manifested by a nonconservation of the stress tensor.
Using this gravitational anomaly and a suitable boundary condition the Hawking
flux was obtained. A simplified version of this method was given in [7]. This
was followed by another, new, anomaly based approach in [8, 11].
The first step in our procedure is to derive the gravitational anomaly using
the notion of chirality. This is a new method of obtaining the gravitational
anomaly. Once this anomaly is obtained, the flux is easily deduced. Exploiting
the same notion of chirality the probability of the outgoing mode in the
tunneling approach will be computed. The Hawking temperature then follows from
this probability. At an intermediate stage of this computation we further show
that the chiral modes obtained in the tunneling formalism reproduce the
gravitational anomaly thereby completing the circle of arguments regarding the
connection of the two approaches.
Metric and null coordinates: Consider a black hole characterised by a
spherically symmetric, static space-time and asymptotically flat metric of the
form,
$\displaystyle ds^{2}=F(r)dt^{2}-\frac{dr^{2}}{F(r)}-r^{2}d\Omega^{2}$ (1)
whose event horizon $r=r_{H}$ is defined by $F(r_{H})=0$. Now it is well known
[16, 6] that near the event horizon the effective theory reduces to a two
dimensional conformal theory whose metric is given by the ($r-t$) sector of
the original metric (1).
It is convenient to express (1) in the null tortoise coordinates which are
defined as,
$\displaystyle u=t-r_{*},\,\,\,v=t+r_{*};\,\,\,\,dr_{*}=\frac{dr}{F(r)}.$ (2)
Under these set of coordinates the relevant ($r-t$)-sector of the metric (1)
takes the form,
$\displaystyle ds^{2}=\frac{F(r)}{2}(du~{}dv+dv~{}du)$ (3)
Chiral conditions, to be discussed in the next section, are most appropriately
described in these coordinates.
Chirality conditions: Consider the Klein-Gordon equation for a massless scalar
particle governed by the metric (3),
$\displaystyle
g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi=\frac{4}{F}\partial_{u}\partial_{v}\phi=0.$
(4)
The general solution of this can be taken as
$\phi(u,v)=\phi^{(R)}(u)+\phi^{(L)}(v)$ where $\phi^{(R)}(u)$ and
$\phi^{(L)}(v)$ are the right (outgoing) and left (ingoing) modes satisfying
$\displaystyle\nabla_{v}\phi^{(R)}=0,\,\,\nabla_{u}\phi^{(R)}\neq 0;\,\,\,\
\nabla_{u}\phi^{(L)}=0,\,\,\,\nabla_{v}\phi^{(L)}\neq 0.$ (5)
These equations are expressed simultaneously as,
$\displaystyle\nabla_{\mu}\phi=\pm\bar{\epsilon}_{\mu\nu}\nabla^{\nu}\phi=\pm\sqrt{-g}\epsilon_{\mu\nu}\nabla^{\nu}\phi$
(6)
where $+(-)$ stand for left (right) mode and $\epsilon_{\mu\nu}$ is the
numerical antisymmetric tensor with $\epsilon_{uv}=\epsilon_{tr}=-1$. This is
the chirality condition 111In analogy with studies in 2d CFT this condition is
usually referred as holomorphy condition.. In fact the condition (6) holds for
any chiral vector $J_{\mu}$ in which case
$J_{\mu}=\pm{\bar{\epsilon}}_{\mu\nu}J^{\nu}$. Likewise, the chirality
condition for the energy-momentum tensor is [17],
$\displaystyle
T_{\mu\nu}=\pm\frac{1}{2}(\bar{\epsilon}_{\mu\sigma}T^{\sigma}_{\nu}+\bar{\epsilon}_{\nu\sigma}T^{\sigma}_{\mu})+\frac{1}{2}g_{\mu\nu}T^{\alpha}_{\alpha}$
(7)
This shows that corresponding to right mode (i.e. ($-$) sign)
$\displaystyle T^{(R)}_{vv}=0,\,\,\,\,T^{(R)}_{uu}\neq 0$ (8)
which is analogous to the first equation of (5). The other sign of (7) gives
the energy-momentum tensor for the left mode. This essentially corresponds to
the interchange $u\leftrightarrow v$ and $L\leftrightarrow R$. In the next
section, using these chirality conditions we will derive the explicit form for
the gravitational anomaly that reproduces the Hawking flux.
Chirality, gravitational anomaly and Hawking flux: It is well known that for a
non-chiral (vector like) theory it is not possible to simultaneously preserve,
at the quantum level, general coordinate invariance as well as conformal
invariance. Since the former invariance is fundamental in general relativity,
conformal invariance is sacrificed leading to a nonvanishing trace of the
stress tensor, called the trace anomaly. Using this trace anomaly and the
chirality condition we will derive an expression for the chiral gravitational
(diffeomophism) anomaly from which the Hawking flux is computed.
The energy-momentum tensor near an evaporating black hole is split into a
traceful and traceless part by [18],
$\displaystyle T_{\mu\nu}=\frac{R}{48\pi}g_{\mu\nu}+\theta_{\mu\nu}$ (9)
where $\theta_{\mu\nu}$ is symmetric (i.e. $\theta_{\mu\nu}=\theta_{\nu\mu}$),
so that it preserves the symmetricity of $T_{\mu\nu}$, and traceless (i.e.
$\theta_{\mu}^{\mu}=0$ so that in $u,v$ coordinates $\theta_{uv}=0$). The
traceful part is contained in the first piece leading to the trace anomaly,
$T^{\mu}_{\mu}=\frac{R}{24\pi}$. Also, since general coordinate invariance is
preserved, $\nabla^{\mu}T_{\mu\nu}=0$, from which it follows that the
solutions of $\theta_{\mu\nu}$ satisfy,
$\displaystyle\nabla^{\mu}\theta_{\mu\nu}=-\frac{1}{48\pi}\nabla_{\nu}R$ (10)
Now the energy-momentum tensor (9) can be regarded as the sum of the
contributions from the right and left moving modes. Symmetry principle tells
that the contribution from one mode is exactly equal to that from the other
mode, only that $u,v$ have to be interchanged. Since $T_{\mu\nu}$ is symmetric
we have $T_{\mu\nu}=T_{\mu\nu}^{(R)}+T_{\mu\nu}^{(L)}$ with
$\displaystyle
T_{\mu\nu}^{(R/L)}=\frac{R}{96\pi}g_{\mu\nu}+\theta_{\mu\nu}^{(R/L)}$ (11)
where $\theta_{\mu\nu}=\theta_{\mu\nu}^{(R)}+\theta_{\mu\nu}^{(L)}$ (in
analogy with $T_{\mu\nu}$). Therefore the chirality condition (8) and the
traceless condition of $\theta_{\mu\nu}$ immediately show
$\displaystyle\theta_{uv}^{(R)}=0,\,\,\,\theta_{vv}^{(R)}=0,\,\,\,\,\theta_{uu}^{(R)}\neq
0;\,\,\,\theta_{uv}^{(L)}=0,\,\,\,\theta_{uu}^{(L)}=0,\,\,\,\,\theta_{vv}^{(L)}\neq
0$ (12)
The trace anomaly for the chiral components follows from (11) and (12),
$\displaystyle
T{{}^{\mu}_{\mu}}{{}^{(R)}}=T{{}^{\mu}_{\mu}}{{}^{(L)}}=\frac{1}{2}T^{\mu}_{\mu}=\frac{R}{48\pi}$
(13)
To find out the diffeomorphism anomaly for the chiral components we will use
(11). Considering only the right mode, for example, we have
$\displaystyle\nabla^{\mu}T_{\mu\nu}^{(R)}=\frac{1}{96\pi}\nabla_{\nu}R+\nabla^{\mu}\theta_{\mu\nu}^{(R)}$
(14)
Next, using (10) and (12) for the right mode we obtain,
$\displaystyle\nabla^{\mu}\theta_{\mu
u}^{(R)}=-\frac{1}{48\pi}\nabla_{u}R;\,\,\,\nabla^{\mu}\theta_{\mu v}^{(R)}=0$
(15)
Substituting these in (14) we get, once for $\nu=u$ and then $\nu=v$,
$\displaystyle\nabla^{\mu}T_{\mu
u}^{(R)}=-\frac{1}{96\pi}\nabla_{u}R;\,\,\,\,\,\nabla^{\mu}T_{\mu
v}^{(R)}=\frac{1}{96\pi}\nabla_{v}R.$ (16)
Therefore, combining both the above results yields
$\displaystyle\nabla^{\mu}T_{\mu\nu}^{(R)}=\frac{1}{96\pi}\bar{\epsilon}_{\nu\lambda}\nabla^{\lambda}R$
(17)
which is the chiral (gravitational) anomaly for the right mode. Similarly the
chiral anomaly for left mode can also be obtained which has a similar form
except for a minus sign on the right side of (17). This anomaly is in
covariant form and so it is also called the covariant gravitational anomaly.
The structure, including the normalization, agrees with that found by using
explicit regularization of the chiral stress tensor [19, 20].
From (17) and (13) a simple relation follows between the gravitational anomaly
(${\cal{A}}_{\nu}$) and the trace anomaly ($T$),
$\displaystyle{\cal{A}}_{\nu}=\frac{1}{2}\bar{\epsilon}_{\nu\lambda}\nabla^{\lambda}T.$
(18)
Such a relation is not totally unexpected since covariant expressions must
involve the Ricci scalar. However (18) should not be interpreted as a Wess-
Zumino consistency condition which involves only ‘consistent’ expressions
[20]. Here, on the contrary, we are dealing with covariant expressions.
The covariant anomaly (17) is now used to obtain the Hawking flux. As was
mentioned earlier the effective two dimensional theory near the horizon
becomes chiral. The chiral theory has the anomaly (17). Taking its $\nu=u$
component we obtain,
$\displaystyle\partial_{r}T_{uu}^{(R)}=\frac{F}{96\pi}\partial_{r}R=\frac{F}{96\pi}\partial_{r}(F^{\prime\prime})=\frac{1}{96\pi}\partial_{r}(FF^{\prime\prime}-\frac{F^{\prime
2}}{2})$ (19)
which yields,
$\displaystyle
T_{uu}^{(R)}=\frac{1}{96\pi}\Big{(}FF^{{}^{\prime\prime}}-\frac{F^{\prime
2}}{2}\Big{)}+C_{uu}$ (20)
where $C_{uu}$ is an integration constant. Imposing the usual boundary
condition which requires $T_{uu}^{(R)}(r\rightarrow r_{H})=0$, implying that a
freely falling observer sees a finite amount of flux at the outer horizon,
leads to $C_{uu}=\frac{F^{\prime 2}(r_{H})}{192\pi}$. This condition on the
outgoing modes is similar to that of the Unruh vacuum [17]. The corresponding
condition on the ingoing mode for the Unruh vacuum is satisfied by default
since, due to chirality, these are absent ($T_{vv}^{(R)}=0$). Note, however,
that the Unruh condition on the ingoing modes
$T_{vv}^{(R)}(r\rightarrow\infty)=0$ is applied at asymptotic infinity where
the theory is non-chiral. This does not affect our interpretation since,
asymptotically, the anomaly (17) vanishes. Hence the results from the chiral
expressions will agree with the non-chiral ones at asymptotic infinity.
Indeed, the Hawking flux, obtained by taking the asymptotic infinity limit
($r\rightarrow\infty$) of (20) [6, 7, 8, 11, 13],
$\displaystyle T_{uu}^{(R)}(r\rightarrow\infty)=C_{uu}=\frac{F^{\prime
2}(r_{H})}{192\pi}=\frac{K^{2}}{48\pi}$ (21)
where $K=\frac{F^{\prime}(r_{H})}{2}$ is the surface gravity of the black
hole, reproduces the known result corresponding to the Hawking temperature
$T_{H}=\frac{K}{2\pi}$. The other terms in (20) drop out due to asymptotic
flatness.
Chirality, quantum tunneling and Hawking temperature: Here, using the
chirality condition (6), we will derive the tunneling probability, which will
eventually yield the Hawking temperature. Under the metric (1) this condition
corresponds to,
$\displaystyle\partial_{t}\phi(r,t)=\pm F(r)\partial_{r}\phi(r,t)$ (22)
As before $+(-)$ stand for left (right) mode. Putting the standard WKB ansatz
$\displaystyle\phi(r,t)=e^{-\frac{i}{\hbar}S(r,t)}$ (23)
in (22), where $S(r,t)$ is the action, we get the familiar semiclassical
Hamilton-Jacobi equation
$\displaystyle\partial_{t}S(r,t)=\pm F(r)\partial_{r}S(r,t)$ (24)
which is the basic equation in the tunneling mechanism for studying Hawking
radiation. This has been derived earlier from the Klein-Gordon equation with
the background metric (1) and the ansatz (23) [4, 10].
Now since the metric (1) is stationary we choose a solution for $S(r,t)$ as
$\displaystyle S(r,t)=\omega t+S(r)$ (25)
where $\omega$ is the energy of the particle. Substituting this in (24) a
solution for $S(r)$ is obtained. Inserting this back in (25) yields,
$\displaystyle S(r,t)=\omega t\pm\omega\int\frac{dr}{F(r)}$ (26)
It is important to note that expressing (26) in the null tortoise coordinates
(see (2)) defined inside and outside of the event horizon and then
substituting in (23) one can obtain the right and left modes for both sectors:
$\displaystyle\Big{(}\phi^{(R)}\Big{)}_{\textrm{in}}=e^{-\frac{i}{\hbar}\omega
u_{\textrm{in}}};\,\,\,\Big{(}\phi^{(L)}\Big{)}_{\textrm{in}}=e^{-\frac{i}{\hbar}\omega
v_{\textrm{in}}}$
$\displaystyle\Big{(}\phi^{(R)}\Big{)}_{\textrm{out}}=e^{-\frac{i}{\hbar}\omega
u_{\textrm{out}}};\,\,\,\Big{(}\phi^{(L)}\Big{)}_{\textrm{out}}=e^{-\frac{i}{\hbar}\omega
v_{\textrm{out}}}$ (27)
which satisfy the condition (5). Precisely these modes were used previously to
find the trace anomaly [18] as well as the chiral (gravitational) anomaly [19]
by the point splitting regularization technique. In our formulation these
modes (27) are a natural consequence of chirality.
Now in the tunneling formalism a virtual pair of particles is produced in the
black hole. One of this pair can quantum mechanically tunnel through the
horizon. This particle is observed at infinity while the other goes towards
the center of the black hole. While crossing the horizon the nature of the
coordinates changes. This can be explained in the following way. The Kruskal
time ($T$) and space ($X$) coordinates inside and outside the horizon are
defined as,
$\displaystyle
T_{\textrm{in}}=e^{K(r_{*})_{\textrm{in}}}~{}{\textrm{cosh}}(Kt_{\textrm{in}});\,\,\,X_{\textrm{in}}=e^{K(r_{*})_{\textrm{in}}}~{}{\textrm{sinh}}(Kt_{\textrm{in}})$
$\displaystyle
T_{\textrm{out}}=e^{K(r_{*})_{\textrm{out}}}~{}{\textrm{sinh}}(Kt_{\textrm{out}});\,\,\,X_{\textrm{out}}=e^{K(r_{*})_{\textrm{out}}}~{}{\textrm{cosh}}(Kt_{\textrm{out}})$
(28)
where, as before, $K=\frac{F^{\prime}(r_{H})}{2}$ is the surface gravity of
the black hole. These two sets of coordinates are connected by the relations,
$\displaystyle
t_{\textrm{in}}=t_{\textrm{out}}-i\frac{\pi}{2K};\,\,\,\,(r_{*})_{\textrm{in}}=(r_{*})_{\textrm{out}}+i\frac{\pi}{2K}$
(29)
Therefore following the definition (2) we obtain the relations connecting the
null coordinates defined inside and outside the horizon,
$\displaystyle
u_{\textrm{in}}=t_{\textrm{in}}-(r_{*})_{\textrm{in}}=u_{\textrm{out}}-i\frac{\pi}{K}$
$\displaystyle
v_{\textrm{in}}=t_{\textrm{in}}+(r_{*})_{\textrm{in}}=v_{\textrm{out}}$ (30)
Since the left moving mode travels towards the center of the black hole, its
probability to go inside is
$\displaystyle P^{(L)}=|\phi^{(L)}_{\textrm{in}}|^{2}=1$ (31)
i.e. the left moving (ingoing) mode is trapped inside the black hole. On the
other hand the right moving mode ($\phi^{(R)}_{\textrm{in}}$) tunnels through
the event horizon. To calculate the tunneling probability as seen by an
external observer it is first necessary to rewrite $u_{\textrm{in}}$ in terms
of $u_{\textrm{out}}$ using(30). We find, from (27),
$\displaystyle
P^{(R)}=|\phi^{(R)}_{\textrm{in}}|^{2}=|e^{-\frac{i}{\hbar}\omega(u_{\textrm{out}}-i\frac{\pi}{K})}|^{2}=e^{-\frac{2\pi\omega}{\hbar
K}}$ (32)
Then using the principle of “detailed balance” [4, 10]
$P^{(R)}=e^{-\frac{\omega}{T_{H}}}P^{(L)}=e^{-\frac{\omega}{T_{H}}}$ yields
the Hawking temperature (in the units of $\hbar=1$) as,
$\displaystyle T_{H}=\frac{K}{2\pi}$ (33)
This is the standard Hawking temperature corresponding to the flux (21).
As we observe the ingoing modes are trapped and do not play any role in the
computation of the Hawking temperature. A similar feature occurs in the
anomaly approach where the ingoing modes are neglected leading to a chiral
theory that eventually yields the flux. These observations provide a physical
picture of chirality connecting the tunneling and anomaly methods.
Conclusions: We have shown that the notion of chirality pervades the anomaly
and tunneling formalisms thereby providing a close connection between them.
This is true both from a physical as well as algebraic perspective. The chiral
restrictions play a pivotal role in the abstraction of the anomaly from which
the flux is computed. The same restrictions, in the tunneling formalism, lead
to the Hawking temperature corresponding to that flux.
A dimensional reduction is known to reduce the theory effectively to a two
dimensional conformal theory near the event horizon. The ingoing (left moving)
modes are lost inside the horizon. They cannot contribute to the near horizon
theory thereby rendering it chiral and, hence, anomalous. Using the
restrictions imposed by chirality we obtained a form for this (gravitational)
anomaly, manifested by a nonconservation of the stress tensor, by starting
from the familiar form of the trace anomaly. From a knowledge of the
gravitational anomaly we were able to obtain the flux.
The chirality constraints were then exploited to obtain the equations for the
ingoing and outgoing modes in the tunneling formalism, following the standard
geometrical (WKB) approximation. We reformulated the tunneling mechanism to
highlight the role of coordinate systems in the chiral framework. A specific
feature of this reformulation is that explicit treatment of the singularity in
(26) is not required since we do not carry out the integration. Only the modes
inside ($\phi_{\textrm{in}}$) and outside ($\phi_{\textrm{out}}$) the horizon,
both of which are well defined, are required. The singularity now manifests in
the complex transformations (29) that connects these modes across the horizon.
The probability for finding the ingoing modes was shown to be unity. These
modes do not play any role in the tunneling approach which is the exact
analogue of omitting them when considering the effective near horizon theory
in the anomaly method.
It is useful to observe that the crucial role of chirality in both approaches
is manifested in the near horizon regime. This reaffirms the universality of
the Hawking effect being governed by the properties of the event horizon.
## References
* [1] S.W.Hawking, Commun. Math. Phys. 43, 199 (1975).
* [2] G.W.Gibbons and S.W.Hawking, Phys. Rev. D 15, 2752 (1977).
* [3] S.M.Christensen and S.A.Fulling, Phys. Rev. D 15, 2088 (1977).
* [4] K.Srinivasan and T.Padmanabhan, Phys. Rev. D 60, 024007 (1999) [arXiv:gr-qc/9812028].
* [5] M.K.Parikh and F.Wilczek, Phys. Rev. Lett. 85, 5042 (2000) [arXiv:hep-th/9907001].
* [6] S.P.Robinson and F.Wilczek, Phys. Rev. Lett. 95, 011303 (2005) [arXiv:gr-qc/0502074].
S.Iso, H.Umestsu and F.Wilczek, Phys. Rev. Lett. 96, 151302 (2006) [arXiv:hep-
th/0602146].
* [7] R.Banerjee and S.Kulkarni, Phys. Rev. D 77, 024018 (2008) [arXiv:0707.2449].
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* [9] M.Arzano, A.J.M.Medved and E.C.Vagenas, JHEP 0509, 037 (2005) [arXiv:hep-th/0505266].
E.T.Akhmedov, V.A.Akhmedova and D.Singleton, Phys. Lett. B 642, 124 (2006)
[arXiv:hep-th/0608098].
M.Angheben, M.Nadalini, L.Vanzo and S.Zerbini, JHEP 0505, 014 (2005)
[arXiv:hep-th/0503081].
R.Kerner and R.B.Mann, Phys. Rev. D 73, 104010 (2006) [arXiv:gr-qc/0603019].
P.Mitra, Phys. Lett. B 648, 240 (2007) [arXiv:hep-th/0611265].
R.Banerjee and B.R.Majhi, Phys. Lett. B 662, 62 (2008) [arXiv:0801.0200].
* [10] R.Banerjee and B.R.Majhi, JHEP 0806, 095 (2008) [arXiv:0805.2220].
* [11] R.Banerjee, [arXiv:0807.4637].
* [12] S.Ghosh, [arXiv:0804.2999].
* [13] S.Iso, T.Morita and H.Umestsu, Phys. Rev. D 77, 045007 (2008) [arXiv:0710.0456].
L.Bonora and M.Cvitan, JHEP 0805, 071 (2008)[arXiv:0804.0198].
L.Bonora, M.Cvitan, S.Pallua and I.Smolic, [arXiv:0808.2360].
* [14] T.Morita, [arXiv:0811.1741].
* [15] T.Damour and R.Ruffini, Phys. Rev. D 14, 332 (1976).
* [16] S.Carlip, Phys. Rev. Lett. 82, 2828 (1999) [arXiv:hep-th/9812013].
* [17] R.Banerjee and S.Kulkarni, [arXiv:0810.5683].
* [18] P.C.W.Davies and S.A.Fulling, Proc. R. Soc. Lond. A 354, 59 (1977).
* [19] S.A.Fulling, General Relativity and Gravitation 18, 609 (1986).
* [20] R.Bertlmann and E.Kohlprath, Ann. Phys. 288, 137 (2001).
|
arxiv-papers
| 2008-12-02T13:06:17 |
2024-09-04T02:48:59.104720
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rabin Banerjee and Bibhas Ranjan Majhi",
"submitter": "Bibhas Majhi Ranjan",
"url": "https://arxiv.org/abs/0812.0497"
}
|
0812.0551
|
# On some consequences of the Snyder–Sidharth deformation of Special
Relativity
Łukasz Andrzej GLINKA111Electronic address: laglinka@gmail.com
_Dipartimento di Matematica e Informatica,_
_Università degli Studi di Udine,_
_Via delle Scienze 206, 33100 Udine, Italia_
###### Abstract
The hypothesis on a minimal scale existence in the Universe leads to
noncommutative geometry of Spacetime and thence to a modification of the
Special Relativity constraint. Sidharth has deduced that this is equivalent to
the Lorentz symmetry violation. This latter consideration was also used by
Glashow, Coleman and other scholars though based on purely phenomenological
models that have been suggested by the observation of Ultra High Energy Cosmic
Rays and Gamma Bursts. On the other hand a parallel development has been the
proposal of a small but nonzero photon mass $m_{\gamma}>0$ by some authors
including Sidharth, such a mass being within experimentally allowable limits.
This too leads to a small violation of the Lorentz symmetry observable in
principle in very high energy gamma rays, as in fact is claimed. In this paper
we study the Snyder–Sidharth Hamiltonian and briefly comment the
Dirac–Sidharth Hamiltonian, that is a possible explanation for observable
violation of the Lorentz symmetry.
## 1 Introduction
In Special Relativity, the Einstein Hamiltonian constraint holds222We use the
units system $\hbar=c=1$.
$E^{2}=m^{2}+p^{2},$ (1)
and is in fact a minimal quadratic form in the momentum $p$ of a relativistic
particle. For this case the Lorentz symmetry is validate. However, recent
observations of Ultra High Energy Cosmic Rays and rays from Gamma Bursts seem
to suggest a small violation of the Lorentz symmetry [1]. Also a number of
scholars like Glashow, Coleman, and others have considered schemes which
depart from the Einstein theory (1). It must be stressed here that these all
schemes are purely _ad hoc_.
From a theoretical physics point of view, the problem seems to have a source
in the fact that the equation (1) is not the only possible Lorentz invariant
quadratic form in momentum $p$. As the example let us consider the natural
possibility of deformation of the energetic constraint (1) by simple adding
linear term in $p$
$E^{2}=m^{2}+p^{2}+\beta_{i}p_{i},$ (2)
where $\beta^{i}$ is deformation parameter. The modification (2) nontrivially
deforms invariant hyperboloid in the energy-momentum space. This quadratic
form can be led by elementary algebraic manipulation to its canonical form
$E^{2}+\dfrac{|\beta|^{2}}{4}=\left(\dfrac{\beta_{i}}{|\beta|}p_{i}+\dfrac{|\beta|}{2}\right)^{2}+m^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ |\beta|^{2}=\beta_{i}\beta_{i}$ (3)
which can be worked out by three different ways of possible interpretation.
1. 1.
First case is the following identification
$\left\\{\begin{array}[]{l}m^{2}=\dfrac{|\beta|^{2}}{4}\\\
E^{2}=\left(\dfrac{\beta_{i}}{|\beta|}p_{i}+\dfrac{|\beta|}{2}\right)^{2}\end{array}\right.,$
(4)
which leads to the physical mass values
$m=\pm\dfrac{|\beta|}{2}.$ (5)
The solution of the second equation determines energy as
$\gamma_{0}E=\gamma_{i}p_{i}+m\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
{\gamma_{0}}^{2}=1\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
\gamma_{i}=\dfrac{\beta_{i}}{2m}\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
\left[\gamma_{i},\gamma_{j}\right]_{+}=2\delta_{ij}.$ (6)
This is the _linear_ Dirac constraint with the classical Clifford
$\gamma$-algebra. After change algebra on four-dimensional
$\gamma_{i}\rightarrow\gamma_{\mu}=\left\\{(-\gamma_{0},\gamma_{i}):\left[\gamma_{\mu},\gamma_{\nu}\right]_{+}=-2g_{\mu\nu}\right\\},$
(7)
and using of the canonical relativistic quantization
$(E,p)\rightarrow i\partial_{\mu}=i(-\partial_{0},\partial_{i}),$ (8)
the Lorentz symmetry is fully valid for this case.
2. 2.
The second possible identification is
$\left\\{\begin{array}[]{l}m^{2}=E^{2}\\\
\dfrac{|\beta|^{2}}{4}=\left(\dfrac{\beta_{i}}{|\beta|}p_{i}+\dfrac{|\beta|}{2}\right)^{2}\end{array}\right..$
(9)
In this situation we have the Einstein-type relation between energy and rest
mass
$E=m,$ (10)
where negative mass was rejected as nonphysical, as well as the following
relation is established
$\dfrac{\beta_{i}}{|\beta|}p_{i}=\left\\{0,-|\beta|\right\\}\longrightarrow
p_{i}=\left\\{0,-\beta_{i}\right\\}$ (11)
It physically means either rest frame or motion under constant momentum, that
is generally an inertial frame. For this case the Lorentz symmetry also holds.
3. 3.
The third possibility is
$\left\\{\begin{array}[]{l}-E^{2}=\dfrac{|\beta|^{2}}{4}\\\
-m^{2}=\left(\dfrac{\beta_{i}}{|\beta|}p_{i}+\dfrac{|\beta|}{2}\right)^{2}\end{array}\right..$
(12)
which leads to the energy values
$\pm i\gamma_{0}E=\dfrac{|\beta|}{2}\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ {\gamma_{0}}^{2}=1.$ (13)
The solution of the second equation again establishes the Dirac–Clifford
classical constraint
$\gamma_{0}E=\gamma_{i}p_{i}+m\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
\gamma_{i}=-\dfrac{\beta_{i}}{2E}\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
\left\\{\gamma_{i},\gamma_{j}\right\\}=2\delta_{ij},$ (14)
for which the Lorentz symmetry is valid after application of the relativistic
quantization procedure (7)-(8).
However, the linear deformation (2) is not the only one which gives the
Hamiltonian constraint that is a quadratic form in momentum. It be more
general Lorentz invariant constraint is naturally
$E^{2}=m^{2}+p^{2}+\left(\beta_{i}+\delta_{i}p^{2}\right)p_{i}+\alpha p^{4},$
(15)
where $(\alpha,\beta_{i},\delta_{i})$ is heptavalent family of deformation
parameters. The modification (15) is a good candidate for new type deductions
in context of so called ”New Physics”. Especially so the mysterious and almost
mystic question as violation of the Lorentz symmetry in high energy physics
and cosmology can be studied by _ad hoc_ application of the deformations of
the Einstein energetic constraint similar to (15).
## 2 The Snyder–Sidharth deformation
Let us consider now the following deformation of the Einstein constraint (1)
$E^{2}=m^{2}+p^{2}+\alpha\ell^{2}p^{4},$ (16)
where $\ell$ is any minimal physical scale, deduced by Sidharth (Refs. [2, 3,
4]) in the astroparticle physics context, and investigated by Snyder [5] in
context of the infrared catastrophe of soft photons in the Compton scattering.
In fact this modification follows from the manipulation in phase space of any
special relativistic particle
$i[p,x]=1+\alpha\ell^{2}p^{2}\quad,\quad[x,y]=O(\ell^{2})\quad,\quad\alpha\sim
1,$ (17)
so that we have to deal with the structure of a nondifferentiable manifold, or
lattice type model. It must be emphasized that (17) is Lorentz invariant
deformation. Sidharth proposed taking into account the Hamiltonian constraint
(16), and studying this deformation in wider sense as some type of
perturbational series in the minimum scale $\ell$, that can be the Planck
scale (or the Compton scale).
As in the case of the linear deformation, the Hamiltonian constraint (16) can
be seen easily lead to the canonical form
$E^{2}+\dfrac{1}{4\alpha\ell^{2}}=\alpha\ell^{2}\left(p^{2}+\dfrac{1}{2\alpha\ell^{2}}\right)^{2}+m^{2},$
(18)
and as previously there are three possible mathematical interpretations of
this equation. However, in the considered deformation (16) we have not linear
or 3rd-order terms, there are only powers of $p^{2}$. According to standard
rules of Quantum Theory [6, 7] this means that in considered situation the
Dirac–Clifford algebraic structure must be absent or hidden.
1. 1.
First, we can interpret the constraint equation (18) as system of two
equations
$\left\\{\begin{array}[]{l}m^{2}=\dfrac{1}{4\alpha\ell^{2}}\\\
E^{2}=\alpha\ell^{2}\left(p^{2}+\dfrac{1}{2\alpha\ell^{2}}\right)^{2}\end{array}\right.$
(19)
The first equation leads to solution that looks like formally as the bosonic
string tension
$m=\dfrac{1}{2\sqrt{{\alpha}}\ell}.$ (20)
Expressing $\alpha\ell^{2}$ by $m$, one can write the solution of the second
equality as follows
$E=m+\dfrac{p^{2}}{2m}.$ (21)
This is the Hamiltonian of a free point particle in semi-classical mechanics;
it is a sum of the Newtonian kinetic energy and the Einstein–Poincare rest
energy correction. Interestingly (20) and (21) are consistent if $m$ is the
Planck mass, and $\ell$ is the Planck length. It must be remembered that the
Planck mass defines a scale where the classical and the quantum meet. For the
Planck mass, as is well known the Schwarzschild radius equals to Compton
length [8]. In comparison with the non deformed case we have not here higher
relativistic corrections. After canonical quantization this is exactly the
Schrödinger Hamiltonian of free quarks in Quantum Chromo-Dynamics (QCD)
because already the quarks are heavy and so non-relativistic [9]. For the case
of vanishing scale and nonzero $\alpha$ as well as for vanishing $\alpha$ and
fixed nonzero scale $\ell$, formally $m\equiv\infty$ and energy is also
infinite, so that this is a nonphysical black-hole type singularity. For the
large scale limit and nonvanishing $\alpha$, there mass spectrum is $m=0$, and
for nonzero momentum, this system has infinite energy, so this too is a
nonphysical situation. In any case this shows that (21) is compatible with
(17).
2. 2.
The second case changes the role of energy and mass
$\left\\{\begin{array}[]{l}-m^{2}=\alpha\ell^{2}\left(p^{2}+\dfrac{1}{2\alpha\ell^{2}}\right)^{2}\\\
-E^{2}=\dfrac{1}{4\alpha\ell^{2}}\end{array}\right.,$ (22)
and gives discrete energy spectrum for fixed scale $\ell$. However, it should
be mentioned that while (16) with $\alpha>0$ is true for fermions, as was
shown by Sidharth [1], $\alpha<0$ for bosons. So, for the case of fermions we
have here
$iE=\dfrac{1}{2\sqrt{{\alpha}}\ell},$ (23)
as well as the mass one
$im=\sqrt{{\alpha}}\ell p^{2}+\dfrac{1}{2\sqrt{{\alpha}}\ell}.$ (24)
(rejecting the negative value from). However, one can eliminate scale by
energy with the result
$m=-\dfrac{p^{2}}{2E}+E.$ (25)
The last equation can be rewritten in the form
$E^{2}=mE+\dfrac{p^{2}}{2},$ (26)
and by using of the deformed constraint (16) it yields
$m^{2}+p^{2}+\alpha\ell^{2}p^{4}=mE+\dfrac{p^{2}}{2}.$ (27)
One can find now the energy (not square of energy!), that is $4th$-power in
momentum
$E=m+\dfrac{p^{2}}{2m}+\dfrac{\alpha\ell^{2}}{m}p^{4}.$ (28)
So, again one can apply the canonical form of a quadratic form
$E+\dfrac{1}{16m\alpha\ell^{2}}=\dfrac{\alpha\ell^{2}}{m}\left(p^{2}+\dfrac{1}{4\alpha\ell^{2}}\right)^{2}+m,$
(29)
and consider three possible cases of identification mass-energy.
1. (a)
The first obvious interpretation yields
$\left\\{\begin{array}[]{l}m=\dfrac{1}{16m\alpha\ell^{2}}\vspace*{5pt}\\\
E=\dfrac{\alpha\ell^{2}}{m}\left(p^{2}+\dfrac{1}{4\alpha\ell^{2}}\right)^{2}\end{array}\right.$
(30)
and again solution of the first equation is easy to extract
$m=\dfrac{1}{4\sqrt{\alpha}\ell},$ (31)
and solution of the second equality can be written in the form of the Pauli
Hamiltonian constraint
$E=m+\dfrac{p^{2}}{2m}+\dfrac{p^{4}}{16m^{3}}=4\alpha^{3/2}\ell^{3}\left(p^{2}+\dfrac{1}{4\alpha\ell^{2}}\right)^{2}.$
(32)
However, one can easily see by the relation (23) that for this case holds
$iE=2m\longrightarrow m^{2}=-\dfrac{E^{2}}{2}<0,$ (33)
and in consequence values of momentum are non hermitian, so that we have to
deal with tachyon.
Moreover, by direct using of the relation (31) together with the formula (24)
one establishes the spectrum of momentum $p$ in dependence on the minimal
scale $\ell$
$p=\mp\dfrac{1}{2\sqrt{\alpha}\ell}\left(\sqrt{{\dfrac{1}{2}\sqrt{{5}}-1}}-i\sqrt{{\dfrac{1}{2}\sqrt{{5}}+1}}\right).$
(34)
2. (b)
The second case is
$\left\\{\begin{array}[]{l}m=E\\\
\dfrac{\alpha\ell^{2}}{m}\left(p^{2}+\dfrac{1}{4\alpha\ell^{2}}\right)^{2}=\dfrac{1}{16m\alpha\ell^{2}}\end{array}\right.$
(35)
Again, solution of the first equation is elementary
$E=m=-i\dfrac{1}{2\sqrt{{\alpha}}\ell}\quad,\quad m^{2}<0,$ (36)
and again the tachyon is obtained – there are particles with momentum spectrum
$p=\left\\{0,\dfrac{1}{\sqrt{{2\alpha}}\ell}\right\\}.$ (37)
This is discrete momenta spectrum for fixed scale $\ell$. For running scale
this is non compact spectrum, but compactification to the point is done for
large scale
$\lim_{\ell\rightarrow\infty}p=0,$ (38)
and it is the rest. For $\alpha=0$ and fixed scale $\ell$ there are two
singular values of the momentum $p$. For all $\ell\neq 0$ and $\alpha\neq 0$,
the case of nonzero $p$ is related to the existence of tachyon.
3. (c)
The third interpretation yields
$\left\\{\begin{array}[]{l}-E=\dfrac{1}{16m\alpha\ell^{2}}\vspace*{5pt}\\\
-m=\dfrac{\alpha\ell^{2}}{m}\left(p^{2}+\dfrac{1}{4\alpha\ell^{2}}\right)^{2}\end{array}\right..$
(39)
Again by using of the relation (23) we obtain from the first equation
$im=\dfrac{1}{8\sqrt{\alpha}\ell},$ (40)
and in consequence momentum spectrum is
$p=\left\\{\pm i\sqrt{{\dfrac{1}{8\alpha}}}\dfrac{1}{\ell},\pm
i\sqrt{{\dfrac{3}{8\alpha}}}\dfrac{1}{\ell}\right\\},$ (41)
so that it is again tachyonic case. By this reason this case - that is the
Pauli Hamiltonian constraint with mass related to minimum scale describes
tachyon, the hypothetical particle with velocity faster then light.
Results for bosons can be obtained by a simple change
$\alpha\longrightarrow-|\alpha|,$ (42)
and are more realistic from a current experimental particle physics point of
view. In this case the Pauli energetic constraint (28) has the following form
$E=m+\dfrac{p^{2}}{2m}-\dfrac{|\alpha|\ell^{2}}{m}p^{4},$ (43)
and tachyon-like states are absent. We have here the solutions
$E=\dfrac{1}{2\sqrt{|\alpha|}\ell}\quad,\quad m=\sqrt{|\alpha|}\ell
p^{2}+\dfrac{1}{2\sqrt{|\alpha|}\ell},$ (44)
and three possible situations, that can be easily deduced from the fermionic
case presented above, with the change (42).
3. 3.
The third possible solution of the Snyder–Sidharth Hamiltonian constraint can
be constructed by the system of equations
$\left\\{\begin{array}[]{l}E^{2}=m^{2}\vspace*{5pt}\\\
\dfrac{1}{4\alpha\ell^{2}}=\alpha\ell^{2}\left(p^{2}+\dfrac{1}{2\alpha\ell^{2}}\right)^{2}\end{array}\right..$
(45)
First equation gives standard Einstein relation
$E=m,$ (46)
and the second equality leads to the discrete momentum spectrum
$p=\left\\{0,\dfrac{1}{\sqrt{{\alpha}}}\dfrac{1}{\ell}\right\\}.$ (47)
For fixed scale $\ell$ this is discrete spectrum. For running scale this is
non compact spectrum, but compactification to the point spectrum is done in
the large scale limit
$\lim_{\ell\rightarrow\infty}p=0,$ (48)
and it is the rest. For $\alpha=0$ and fixed scale $\ell$ there are two
singular values of the momentum $p$. For all $\ell\neq 0$ and $\alpha\neq 0$,
the case of nonzero $p$ is related to the existence of a relativistic
particle.
## 3 Scale-modified Compton effect
Let us consider now the case of the Compton effect with the Sidharth
Hamiltonian constraint (16). In the standard case, in the CM system, wave
vector of outgoing photon $k$ is related to wave vector of incoming photon
$k_{0}$ scattered on the electron with mass $m$ by the relation
$k=\dfrac{mk_{0}}{m+k_{0}(1-\cos\theta)}.$ (49)
The point is the modification of this wave vector according to the idea
$\omega^{2}_{eff}=m^{2}+k^{2}_{eff},$ (50)
where $k_{eff}$ is the corrected wave vector
$k^{2}_{eff}=k^{\prime 2}+\alpha\ell^{2}k^{\prime
4}\quad,\quad\alpha=-|\alpha|.$ (51)
Effectively one can obtain the relation
$k^{\prime}=k+\epsilon,$ (52)
where $\epsilon$ is the correction from non vanishing scale $\ell$
$\epsilon=[Q^{2}+2mQ]^{2}\dfrac{\omega}{\omega_{0}}\dfrac{\alpha\ell^{2}}{2m},$
(53)
$Q$ is the difference
$Q=k-k_{0},$ (54)
and the frequencies were introduced
$\omega_{0}=\dfrac{k_{0}}{m}\quad,\quad\omega=\dfrac{k}{m}=\dfrac{\omega_{0}}{1+\omega_{0}(1-\cos\theta)}.$
(55)
According to energy conservation law one can establish the mass of the photon
as
$m_{\gamma}=k_{eff}-k^{\prime}=k^{\prime}\left(\sqrt{{1+\alpha\ell^{2}k^{\prime
2}}}-1\right),$ (56)
and it is non zeroth for non vanishing scales $\ell\neq 0$. This relation
leads to the formula
$m_{\gamma}+k^{\prime}=k^{\prime}\sqrt{{1+\alpha\ell^{2}k^{\prime 2}}},$ (57)
which can be rewritten in the form of equation for $k^{\prime}$
$\alpha\ell^{2}k^{\prime 4}-2m_{\gamma}k^{\prime}-m_{\gamma}^{2}=0.$ (58)
For finite photon mass $m_{\gamma}>0$ and nonzeroth scale $\ell\neq 0$ this
equation has complex solution
$\omega^{\prime}=\dfrac{k^{\prime}}{m}=\omega_{R}+i\omega_{I},$ (59)
where $\omega_{R}$ and $\omega_{I}$ are real and imaginary parts of
$\omega^{\prime}$
$\displaystyle\omega_{R}$ $\displaystyle=$
$\displaystyle\pm\dfrac{m_{\gamma}}{m}\dfrac{1}{\eta\xi(\eta)},$ (60)
$\displaystyle\omega_{I}$ $\displaystyle=$
$\displaystyle\mp\omega_{R}\sqrt{{1+\dfrac{1}{3}\xi^{3}(\eta)}}.$ (61)
Here we have introduced the function $\xi(\eta)$
$\xi(\eta)=\sqrt{{\dfrac{18^{1/3}\left({1+\mathrm{sgn}(\alpha)\sqrt{{1+\dfrac{4}{9}\eta^{3}}}}\right)^{1/3}}{\left(1+\mathrm{sgn}(\alpha)\sqrt{1+\dfrac{4}{9}\eta^{3}}\right)^{2/3}-12^{1/3}\eta}}},$
(62)
where $\eta$ is the parameter
$\eta^{3}=\dfrac{4}{3}\alpha\ell^{2}m_{\gamma}^{2}.$ (63)
One can easily establish $\epsilon$ by difference of frequencies as
$\dfrac{\epsilon(\ell)}{m}=\omega^{\prime}-\omega,$ (64)
and by similarly one can determine the constant $Q$ directly from (53)
$Q(\ell)=-m\left(1\mp\sqrt{{1+\dfrac{2}{3}\dfrac{m_{\gamma}}{m}\sqrt{{\dfrac{6\omega_{0}}{\eta^{3}}}}\sqrt{{\dfrac{\epsilon(\ell)/m}{\omega}}}}}\right).$
(65)
By using of the relation (54) rewritten in terms of the frequencies
$\dfrac{Q(\ell)}{m}=\omega-\omega_{0},$ (66)
one can finally establish the energy gap $\epsilon$ as
$\dfrac{\epsilon(\ell)}{m}=\omega^{\prime}-\omega_{0}+1\mp\sqrt{{1+\dfrac{2}{3}\sqrt{{\dfrac{6}{\eta^{3}}}}\sqrt{{\dfrac{\omega_{0}}{\omega}}}\dfrac{m_{\gamma}}{m}\sqrt{{\omega^{\prime}-\omega}}}}.$
(67)
Since $\omega^{\prime}$ is a complex number, one can write the gap energy by
employing of the polar representation
$\dfrac{\epsilon(\ell)}{m}=\sqrt{{\epsilon_{R}^{2}(\ell)+\epsilon_{I}^{2}(\ell)}}\exp\left(i\vartheta(\ell)\right)\quad,\quad\vartheta(\ell)\equiv\mathrm{arg}\epsilon(\ell)$
(68)
where $\vartheta$ is a phase
$\vartheta=n\cdot\arctan\dfrac{\omega_{I}\pm\sqrt{{-\dfrac{1+ax}{2}+\dfrac{1}{2}\sqrt{{(1+ax)^{2}+y^{2}}}}}}{\omega_{R}-\omega_{0}+1\pm\sqrt{{\dfrac{1+ax}{2}+\dfrac{1}{2}\sqrt{{(1+ax)^{2}+y^{2}}}}}},$
(69)
where $n$ is any integer, $\epsilon_{R}$ is real part of the gap energy
$\epsilon_{R}=\omega_{R}-\omega_{0}+1\pm\sqrt{{\dfrac{1+ax}{2}+\dfrac{1}{2}\sqrt{{(1+ax)^{2}+y^{2}}}}},$
(70)
and $\epsilon_{I}$ is its imaginary part
$\epsilon_{I}=\omega_{I}\pm\sqrt{{-\dfrac{1+ax}{2}+\dfrac{1}{2}\sqrt{{(1+ax)^{2}+y^{2}}}}}.$
(71)
Here, for shorten notation, we have introduced the abbreviations
$a\equiv\dfrac{2}{3}\sqrt{{\dfrac{6}{\eta^{3}}}}\sqrt{{\dfrac{\omega_{0}}{\omega}}}\dfrac{m_{\gamma}}{m},$
(72) $\displaystyle x$ $\displaystyle\equiv$
$\displaystyle\sqrt{{\dfrac{\omega_{R}-\omega}{2}+\dfrac{1}{2}\sqrt{{\left(\omega_{R}-\omega\right)^{2}+\omega_{I}^{2}}}}},$
(73) $\displaystyle y$ $\displaystyle\equiv$
$\displaystyle\sqrt{{-\dfrac{\omega_{R}-\omega}{2}+\dfrac{1}{2}\sqrt{{\left(\omega_{R}-\omega\right)^{2}+\omega_{I}^{2}}}}},$
(74)
which are consistent for $\omega_{R}\geq\omega$, $\omega_{I}\geq 0$, $1+ax\geq
0$. The existence of the scale-dependent nonzero phase (69) reflects the
property of multiply connected space.
However the fundamental part of multi-energy (68) can be established by the
Bohr–Sommerfeld quantization rule for the phase of energy, _i.e._
$\vartheta=n^{\prime}\cdot 2\pi,$ (75)
where $n^{\prime}$ is any integer. In this case we have simply
$\epsilon_{I}=0$, and the total energy gap is determined as
$\dfrac{\epsilon(\ell)}{m}=\left|\epsilon_{R}(\ell)\right|=\left|\omega_{R}-\omega_{0}+1\pm\sqrt{{\omega_{I}^{2}+ax+1}}\right|.$
(76)
Taking the minimal scale as the Planck scale
$\ell=\ell_{P}=\sqrt{{\dfrac{\hslash G}{c^{3}}}}\approx 1.61625\times
10^{-35}m$ (in SI units) one receives the fundamental energy gap as $1$ eV.
The multiple value of an energy gap can be interpreted as the multiple
connected property of Spacetime [2].
## 4 Conclusion
The supposition about non vanishing photon mass was deduced based on a
background Dark Energy or the Zero Point Field (Cf. ref. [2]) at the Planck
scale. On the other hand employing the Planck scale as the minimum scale, it
is known that spacetime geometry becomes nondifferentiable and noncommutative.
In fact it modifies the usual Lorentz symmetry existing in the Klein–Gordon
and Dirac equations.
The photon mass within the experimental constraints, but also leads to
observable results in the High Energy Gamma Ray spectrum and Gamma Bursts
astrophysics. We are full of hopes that NASA’s GLAST satellite will throw
further light on this. We emphasize that the energy–momentum relation (16)
leads to a Dirac type Hamiltonian, the Dirac–Sidharth Hamiltonian with
interesting consequences in the Ultra High Energy regime [2, 3].
Formally, we have shown that Special Relativity modified by the noncommutative
geometry of Spacetime (16) can be resolved (_i.e._ Hamiltonian can be
established within) by some nontrivial classical ways. It means that the
Snyder–Sidharth deformation of the Einstein theory leads to some nontrivial
quantum theories, that are dependent on relations of energy $E$, spatial
momentum $p$ and mass $m$ of a relativistic particle and the minimal scale
$\ell$.
## Acknowledgements
The author thanks A. De Angelis and F. Freschi for full hospitality at the
University of Udine.
## References
* [1] Sidharth B.G., Found. Phys. 38, 89-95 (2008), and several references therein.
* [2] Sidharth B.G., _The Thermodynamic Universe_. World Scientific, Singapoure 2008.
* [3] Sidharth B.G., _The Universe of Fluctuations. The Architecure of Spacetime and the Universe_. Springer, Dordrecht 2005.
* [4] Sidharth B.G., _Chaotic Universe: From th Planck to the Hubble Scale_. Nova Science, New York 2001.
* [5] Snyder H.S., Phys. Rev. 72, 68-71 (1947),
Snyder H.S., Phys. Rev. 71, 38-41 (1947).
* [6] Greiner W., _Relativistic Quantum Mechanics_. 3rd ed., Springer–Verlag Berlin Heidelberg New York, Berlin 2000.
* [7] Peskin M.E. and Schroeder D.V., _An Introduction to Quantum Field Theory_. Addison–Weseley Publishing Company, New York 1995.
* [8] Kiefer C., _Quantum Gravity_. 2nd ed., Oxford University Press 2007.
* [9] Lee T.D., _Particle Physics and Introduction to Field Theory_. Harwood Academic 1981,
Lee T.D., Phys. Lett. 122B, 217-220 (1983).
|
arxiv-papers
| 2008-12-02T17:18:59 |
2024-09-04T02:48:59.111587
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L. A. Glinka",
"submitter": "Lukasz Andrzej Glinka",
"url": "https://arxiv.org/abs/0812.0551"
}
|
0812.0630
|
# The Uniqueness Problem of Sequence Product on Operator Effect Algebra ${\cal
E}(H)$††thanks: E-mail: wjd@zju.edu.cn
Liu Weihua, Wu Junde
Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China
Abstract. A quantum effect is an operator on a complex Hilbert space $H$ that
satisfies $0\leq A\leq I$. We denote the set of all quantum effects by ${\cal
E}(H)$. In this paper we prove, Theorem 4.3, on the theory of sequential
product on ${\cal E}(H)$ which shows, in fact, that there are sequential
products on ${\cal E}(H)$ which are not of the generalized Lüders form. This
result answers a Gudder’s open problem negatively.
PACS numbers: 02.10-v, 02.30.Tb, 03.65.Ta.
1\. Introduction
If a quantum-mechanical system $\cal S$ is represented in the usual way by a
complex Hilbert space $H$, then a self-adjoint operator $A$ on $H$ such that
$0\leq A\leq I$ is called the quantum effect on $H$ ([1, 2]). Quantum effects
represent yes-no measurements that may be unsharp. The set of quantum effects
on $H$ is denoted by ${\cal E}(H)$. The subset ${\cal P}(H)$ of ${\cal E}(H)$
consisting of orthogonal projections represents sharp yes-no measurements. Let
${\cal T}(H)$ be the set of trace class operators on $H$ and ${\cal S}(H)$ the
set of density operators, i.e., the trace class positive operators on $H$ of
unit trace, which represent the states of quantum system. An operation is a
positive linear mapping $\Phi:{\cal T}(H)\rightarrow{\cal T}(H)$ such that for
each $T\in{\cal S}(H)$, $0\leq tr[\Phi(T)]\leq 1$ ([3-5]). Each operation
$\Phi$ can define a unique quantum effect $B$ such that for each $T\in{\cal
T}(H)$, $tr[\Phi(T)]=tr[TB]$.
Let ${\cal B}(H)$ be the set of bounded linear operators on $H$, the dual
mapping $\Phi^{*}:{\cal B}(H)\rightarrow{\cal B}(H)$ of an operation $\Phi$ is
defined by the relation $tr[T\Phi^{*}(A)]=tr[\Phi(T)A]$, $A\in{\cal
B}(H),T\in{\cal T}(H)$ ([4]). The effect $B$ defined by an operation $\Phi$
satisfies that $B=\Phi^{*}(I)$ ([5]).
For each $P\in{\cal P}(H)$ is associated a so-called Lüders operation
$\Phi_{L}^{P}:T\rightarrow PTP$, its dual is $(\Phi_{L}^{P})^{*}(A)=PAP$ and
the corresponding quantum effect is $(\Phi_{L}^{P})^{*}(I)=P$. These
operations arise in the context of ideal measurements. Moreover, each quantum
effect $B\in{\cal E}(H)$ gives to a general Lüders operation
$\Phi_{L}^{B}:T\rightarrow B^{\frac{1}{2}}TB^{\frac{1}{2}}$ and $B$ is
recovered as $(\Phi_{L}^{B})^{*}(I)=B$ as well.
Let $\Phi_{1},\Phi_{2}$ be two operations. The composition
$\Phi_{2}\circ\Phi_{1}$ is a new operation, called a sequential operation as
it is obtained by performing first $\Phi_{1}$ and then $\Phi_{2}$. In general,
$\Phi_{2}\circ\Phi_{1}\neq\Phi_{1}\circ\Phi_{2}$. Note that for any two
quantum effects $B,C\in{\cal E}(H)$ we have
$(\Phi_{L}^{C}\circ\Phi_{L}^{B})^{*}(I)=B^{\frac{1}{2}}CB^{\frac{1}{2}}$
$([5,P_{26-27}])$. It shows that the new quantum effect
$B^{\frac{1}{2}}CB^{\frac{1}{2}}$ yielded by $B$ and $C$ has important physics
meaning. Professor Gudder called it the sequential product of $B$ and $C$, and
denoted it by $B\circ C$. It represents the quantum effect produced by fist
measuring $A$ then measuring $B$ ([6-8]). This sequential product has also
been generalized to an algebraic structure called a sequential effect algebra
([7]).
Now, we introduce the abstract sequential product on ${\cal E}(H)$ as
following:
Let $\circ$ be a binary operation on ${\cal E}(H)$, i.e., $\circ$: ${\cal
E}(H)\times{\cal E}(H)\rightarrow{\cal E}(H)$, if it satisfies:
(S1). The map $B\rightarrow A\circ B$ is additive for each $A\in{\cal E}(H)$,
that is, if $B+C\leq I$, then
$(A\circ B)+(A\circ C)\leq I$ and $(A\circ B)+(A\circ C)=A\circ(B+C)$.
(S2). $I\circ A=A$ for all $A\in{\cal E}(H)$.
(S3). If $A\circ B=0$, then $A\circ B=B\circ A$.
(S4). If $A\circ B=B\circ A$, then $A\circ(I-B)=(I-B)\circ A$ and
$A\circ(B\circ C)=(A\circ B)\circ C$ for
all $C\in{\cal E}(H)$.
(S5). If $C\circ A=A\circ C$, $C\circ B=B\circ C$, then $C\circ(A\circ
B)=(A\circ B)\circ C$ and
$C\circ(A+B)=(A+B)\circ C$ whenever $A+B\leq I$.
If ${\cal E}(H)$ has a binary operation $\circ$ satisfying conditions
(S1)-(S5), then $({\cal E}(H),0,I,\circ)$ is called a sequential operator
effect algebra. Professor Gudder showed that for any two quantum effects $B$
and $C$, the operation $\circ$ defined by $B\circ
C=B^{\frac{1}{2}}CB^{\frac{1}{2}}$ satisfies conditions (S1)-(S5), and so is a
sequential product of ${\cal E}(H)$, which we call the generalized Lüders
form. In 2005, Professor Gudder presented 25 open problems about the general
sequential effect algebras. The second problem is:
Problem 1.1 ([9]). Is $B\circ C=B^{\frac{1}{2}}CB^{\frac{1}{2}}$ the only
sequential product on ${\cal E}(H)$?
As we see the five properties are base on the measurement logics and the the
uniqueness property has been asked many times in Gudder’s paper. In this
paper, we construct a new sequential product on ${\cal E}(H)$ which differs
from the generalized Lüders form, thus, we answer the open problem negatively.
2\. Sequential Product on ${\cal E}(H)$
In this section, we study some abstract properties of sequential product
$\circ$ on ${\cal E}(H)$. For convenience, we introduce the following
notations: If $A,B\in{\cal E}(H)$, we say that $A\oplus B$ is defined if and
only if $A+B\leq I$ and define $A\oplus B=A+B$; if $A\circ B=B\circ A$, we
denote $A|B$ .
Lemma 2.1. If $A,B\in{\cal E}(H),a\in[0,1]$, then
$A\circ(aB)=a(A\circ B).$
Proof. It is clear that for $a=1$, the conclusion is true. If $a>0$ is a
rational number, i.e., $a=\frac{m}{n}$, where $n,m$ are positive integer, it
follows from $\bigoplus\limits_{i=1}^{n}(A\circ{\frac{1}{n}B})=A\circ B$ that
$A\circ(\frac{1}{n}B)=\frac{1}{n}(A\circ B)$, thus,
$A\circ(\frac{m}{n}B)=\bigoplus\limits_{i=1}^{m}A\circ(\frac{1}{n}B)=\frac{m}{n}(A\circ
B)$. If $a\in[0,1]$ is not a rational number, then for each $q=\frac{m}{n}>a$
we have $q(A\circ B)=A\circ(qB)=A\circ[(q-a)B]+A\circ(aB)\geq A\circ(aB)$, so
$q(A\circ B)\geq A\circ(aB)$. Let $q\rightarrow a$ we have $a(A\circ B)\geq
A\circ(aB)$. Similarly, we can get that $A\circ(aB)\geq a(A\circ B)$ by taking
$q=\frac{m}{n}<a$. So $A\circ(aB)=a(A\circ B)$. Moreover, it follows from the
proof process that for $a=0$ the conclusion is also true.
Lemma 2.2 ([9], Theorem 3.4 (i)). Let $A\in{\cal E}(H)$ and $E\in{\cal P}(H)$.
If $A\leq E$, then $A|E$ and $E\circ A=A$.
Lemma 2.3. If $a\in[0,1]$, $E\in{\cal P}(H)$, then $aI|E$ and $(aI)\circ
E=E\circ(aI)=aE$.
Proof. Since $aE\leq E$, so $aE|E$ and $E\circ E=E$ by Lemma 2.2, it follows
from $E=E\circ I=(E\circ E)\oplus(E\circ(I-E))=E\oplus(E\circ(I-E))$ that
$E\circ(I-E)=0$, note that $E\circ(a(I-E))\leq E\circ(I-E)=0$, so
$E\circ(a(I-E))=0$, thus, it follows from (S3) that $E|a(I-E)$, moreover, by
(S5) we have $E|a(I-E)\oplus aE=aI$, so, it follows from Lemma 2.1 and Lemma
2.2 that $(aI)\circ E=E\circ(aI)=a(E\circ I)=aE$.
Lemma 2.4. If $E,F\in{\cal P}(H),E\leq F$ and $0\leq a\leq 1$, then $E|aF$ and
$E\circ(aF)=aE$.
Proof. It follows from $E\leq F$ that $I-E\geq I-F\geq a(I-F)$, by Lemma 2.2
and Lemma 2.3, we have $I-E|a(I-F)$ and $I-E|(1-a)I$, thus,
$I-E|a(I-F)\oplus(1-a)I=I-aF$, it follows from (S4) that $E|I-aF$ and so by
(S4) again that $E|aF$, moreover, by Lemma 2.1 and Lemma 2.2, we have
$(aF)\circ E=E\circ(aF)=a(E\circ F)=aE$.
Lemma 2.5. If $E\in{\cal P}(H),A\in{\cal E}(H),0\leq a\leq 1$ and $A\leq E$,
then $aE|A$, and $(aE)\circ A=A\circ(aE)=aA$.
Proof. It follows from Lemma 2.2 that $A|E$, so by (S4) we have $A|I-E$. Since
$A\circ E=A=A\circ I=A\circ E\oplus A\circ(I-E)$, so $A\circ(I-E)=0$. Note
that $A\circ(a(I-E))\leq A\circ(I-E)$, we have $A\circ(a(I-E))=0$, so
$A|a(I-E)$.
Let $\\{E_{\lambda}\\}$ be the identity resolution of $A$ and denote
$A_{n}=\sum\limits_{i=0}^{2^{n}-1}\frac{i}{2^{n}}(E_{\frac{i+1}{2^{n}}}-E_{\frac{i}{2^{n}}}),$
$B_{n}=\sum\limits_{i=1}^{2^{n}}\frac{i}{2^{n}}(E_{\frac{i}{2^{n}}}-E_{\frac{i-1}{2^{n}}}).$
Note that $A\in\varepsilon(H)$, so $E_{\lambda}=0$ when $\lambda<0$ and
$E_{\lambda}=I$ when $1\leq\lambda$. Moreover, for each $n\in\mathbb{N}$,
$A_{n}\leq A_{n+1}$, $B_{n+1}\leq B_{n}$, and when $n\rightarrow\infty$,
$\|A_{n}-A\|\rightarrow 0,\|B_{n}-A\|\rightarrow 0$ ([10]).
Let $0\leq b\leq 1$. Then it follows from Lemma 2.1 and Lemma 2.3 that
$(bI)\circ
A_{n}=\sum\limits_{i=1}^{2^{n}-1}(bI)\circ(\frac{i}{2^{n}})(E_{\frac{i+1}{2^{n}}}-E_{\frac{i}{2^{n}}})$
$=\sum\limits_{i=1}^{2^{n}-1}(\frac{ib}{2^{n}})(E_{\frac{i+1}{2^{n}}}-E_{\frac{i}{2^{n}}})=bA_{n}$
and
$(bI)\circ B_{n}=bB_{n}.$
Note that $A\geq A_{n}$, so $(bI)\circ A\geq(bI)\circ A_{n}=bA_{n}$. Let
$n\rightarrow\infty$. Then $(bI)\circ A\geq bA$, do the same
with$\\{B_{n}\\}$, we get $(bI)\circ A\leq bA$, so $(bI)\circ
A=bA=A\circ(bI)$. That is $A|bI$ for each $0\leq b\leq 1$, in particular,
$A|(1-a)I$. Thus, it follows from $A|(1-a)I+a(I-E)$ that $A|I-aE$, by (S4) we
have $A|aE$, Hence, $(aE)\circ A=A\circ(aE)=a(A\circ E)=aA$.
Lemma 2.6. Let $0\leq a\leq 1$ and $A,B\in{\cal E}(H)$. Then
$(aA)\circ B=A\circ(aB)=a(A\circ B).$
Proof. It follows from Lemma 2.5 that $(aA)\circ B=(A\circ(aI))\circ
B=A\circ((aI)\circ B)=A\circ(aB)=a(A\circ B)$.
Lemma 2.6 showed that we can write $a(A\circ B)$ for $(aA)\circ B$ and
$A\circ(aB)$.
In order to obtain our main result in this section, we need to extent
$\circ:{\cal E}(H)\times{\cal E}(H)\rightarrow{\cal E}(H)$ to ${\cal
E}(H)\times{\cal S}(H)\rightarrow{\cal S}(H)$, where ${\cal S}(H)$ is the set
of bounded linear self-adjoint operators on $H$.
Let $B\in{\cal E}(H)$, $A\in{\cal S}^{+}(H)$. Then there exists a number $M>0$
such that $\frac{A}{M}\in{\cal E}(H).$ Now we define
$B\circ A=M(B\circ\frac{A}{M}).$
If there is another positive number $M^{\prime}$ such that
$\frac{A}{M^{\prime}}\in{\cal E}(H)$, without losing generality, we assume
that $M\leq M^{\prime}$, then
$M^{\prime}(B\circ\frac{A}{M^{\prime}})=M^{\prime}(B\circ(\frac{M}{M^{\prime}}\frac{A}{M}))=M^{\prime}(\frac{M}{M^{\prime}}(B\circ\frac{A}{M}))=M(B\circ\frac{A}{M})$,
this showed that $B\circ A$ is well defined for each bounded linear positive
operator $A$ on $H$.
In general, if $A\in{\cal S}(H)$, we can express $A$ as $A_{1}-A_{2}$, where
$A_{1},A_{2}$ are two bounded linear positive operators on $H$ ([10]). Now we
define
$B\circ A=B\circ A_{1}-B\circ A_{2}.$
If $A_{1}^{\prime}-A_{2}^{\prime}$ is another expression of $A$ with the above
properties, then $A_{1}+A_{2}^{\prime}=A_{1}^{\prime}+A_{2}=K$ is a bounded
linear positive operator on $H$. If take positive real number $M$ such that
$\frac{K}{M}\in{\cal E}(H)$, then
$B\circ(A_{1}+A_{2}^{\prime})=M(B\circ(\frac{A_{1}}{M}+\frac{A_{2}^{\prime}}{M}))=M(B\circ\frac{A_{1}}{M})+M(B\circ\frac{A_{2}^{\prime}}{M})=B\circ
A_{1}+B\circ A_{2}^{\prime}$. Similarly, $B\circ(A_{1}^{\prime}+A_{2})=B\circ
A_{1}^{\prime}+B\circ A_{2}$. Thus, it follows from $B\circ
A_{1}^{\prime}+B\circ A_{2}=B\circ A_{1}+B\circ A_{2}^{\prime}$, $B\circ
A_{1}-B\circ A_{2}=B\circ A_{1}^{\prime}-B\circ A_{2}^{\prime}$. This showed
that $\circ$ is well defined on ${\cal E}(H)\times S(H)$.
From the above discussion we can easily prove the following important result:
Theorem 2.7. If $B\in{\cal E}(H)$, $A_{1},A_{2}\in S(H)$ and $a\in\mathbb{R}$,
then we have
$B\circ(A_{1}+A_{2})=B\circ A_{1}+B\circ A_{2},\,\,B\circ(aA_{1})=a(B\circ
A_{1}).$
3\. Sequential Product on ${\cal E}(H)$ with dim$(H)=2$
In this section, we suppose that dim$(H)=2$. Now, we explore the key idea of
constructing our sequential product.
Lemma 3.1. If $E\in{\cal P}(H),B\in{\cal E}(H)$, then $E\circ B=EBE.$
Proof. Since $E$ is a orthogonal projection on ${\cal E}(H)$ with dim$(H)=2$,
so there exists a normal basis $\\{e_{1},e_{2}\\}$ of $H$ such that
$E(e_{i})=\lambda_{i}e_{i}$, where $\lambda_{i}\in\\{0,1\\}$, $i=1,2$. If
$\lambda_{i}=0,i=1,2$, then $E=0$, if $\lambda_{i}=1,i=1,2$, then $E=I$. It is
clear that for $E=0$ or $E=I$, the conclusion is true. Without losing
generality, we now suppose that $\lambda_{1}=1$ and $\lambda_{2}=0$, i.e.,
$(E(e_{1}),E(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}1&0\\\ 0&0\\\
\end{array}\right).$ Let $B\in S(H)$. Then we have
$\begin{array}[]{cc}(B(e_{1}),B(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}x&y\\\
\bar{y}&z\\\ \end{array}\right)\end{array}$, where $x,z\in\mathbb{R}$ ([10]).
Now we define two linear operators $X$ and $Z$ on $H$ satisfy that
$(X(e_{1}),X(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}x&0\\\ 0&0\\\
\end{array}\right)$
and
$(Z(e_{1}),Z(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}0&0\\\ 0&z\\\
\end{array}\right).$
Then $X=xE,Z=z(I-E)\in{\cal E}(H)$ and it follows from (S1) and Lemma 2.2 that
$E\circ X=X$ and $E\circ Z=0$. Denote
$(E\circ B(e_{1}),E\circ
B(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}f(x,y,z)&g(x.y.z)\\\
\overline{g(x.y.z)}&h(x,y,z)\\\ \end{array}\right).$
Since $S(H)$ is a real linear space and by Theorem 2.7 that $B\rightarrow
E\circ B$ is a real linear map of $S(H)\rightarrow S(H)$, so $f,g$ and $h$ are
real linear maps of vector $(x,y,z)$ and $f$ and $g$ are real-valued functions
of $(x,y,z)$, thus, function $f(x,y,z)$ must have the form ([10]):
$f(x,y,z)=kx+lz+n(y+\bar{y})+im(y-\bar{y}),$ where $k,l,m,n\in R$. Let $B=X$
and $B=Z$, respectively, it follows from $E\circ X=X$ and $E\circ Z=0$ that
$l=0,k=1$, so $f(x,y,z)=x+n(y+\bar{y})+mi(y-\bar{y})$. Note that when
$B\in{\cal S}^{+}(H)$, $E\circ B$ should be a positive operator, so when
$x,z\geq 0$ and $xz-|y|^{2}\geq 0$, we have $f(x,y,z)\geq 0$. Take $y\in R$,
then $f(x,y,z)=x+2ny$. Thus, when $x,z\geq 0$, $y\in R$ and $xz-y^{2}\geq 0$,
$f(x,y,z)=x+2ny\geq 0$. If $n\neq 0$, take $y=-\frac{1}{n}$, $x=1$,
$z=\frac{1}{n^{2}}$, then we have $f<0$, this is a contradiction and so $n=0$.
Similarly, if $m\neq 0$, take $y=-\frac{i}{m}$, $x=1$, $z=\frac{1}{m^{2}}$, we
will get $f<0$, this is also a contradiction and so $m=0$. Thus, we have
$f(x,y,z)=x$.
Moreover, note that $E\circ((I-E)\circ B)=(E\circ(I-E))\circ B=0\circ
B=0=((I-E)\circ E))\circ B=(I-E)\circ(E\circ B)$, as above, we may prove that
$((I-E)\circ(E\circ B)(e_{1}),(I-E)\circ(E\circ
B)(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}0&0\\\ 0&h(x,y,z)\\\
\end{array}\right)=(e_{1},e_{2})\left(\begin{array}[]{cc}0&0\\\ 0&0\\\
\end{array}\right)$, thus $h(x,y,z)=0$. For each $y\in\mathbb{C}$, take $x=1$,
$z=|y|^{2}$, then $B$ is a positive operator, so $E\circ B$ is also a positive
operator, thus we have $fh-|g|^{2}\geq 0$. It follows from $h=0$ that $g=0$,
so $E\circ B=X=EBE$.
Corollary 3.2. Let $E\in{\cal P}(H),a\in[0,1]$ and $A=aE$. Then for each
$B\in{\cal E}(H)$,
$A\circ B=(aE)\circ B=a(E\circ
B)=a(EBE)=a^{\frac{1}{2}}EBa^{\frac{1}{2}}E=A^{\frac{1}{2}}BA^{\frac{1}{2}}.$
Now, we prove the following important result:
Theorem 3.2. Let $H$ be a complex Hilbert space with dim$(H)=2$, $A,B\in{\cal
E}(H)$. If $\\{e_{1},e_{2}\\}$ is a normal basis of $H$ such that
$(A(e_{1}),A(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}a^{2}&0\\\
0&b^{2}\\\ \end{array}\right)$ and
$(B(e_{1}),B(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}x&y\\\ \bar{y}&z\\\
\end{array}\right)$, then there exists a $\theta\in\mathbb{R}$ such that
$(A\circ B(e_{1}),A\circ
B(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}a^{2}x&abe^{i\theta}y\\\
abe^{-i\theta}\bar{y}&b^{2}z\\\ \end{array}\right).$
Proof. Let $\\{e_{1},e_{2}\\}$ be a normal basis of $H$ such that
$(A(e_{1}),A(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}a^{2}&0\\\
0&b^{2}\\\ \end{array}\right)$ and
$(B(e_{1}),B(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}x&y\\\ \bar{y}&z\\\
\end{array}\right)$, where $0\leq a,b\leq 1$, $0\leq x,0\leq z,0\leq
xz-|y|^{2}$. Now we define a linear operator $E$ on $H$ such that
$(E(e_{1}),E(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}1&0\\\ 0&0\\\
\end{array}\right)$, then $E\in{\cal P}(H)$. By Corollary 3.2, we can suppose
$a,b\in(0,1]$ and $a\neq b$. Thus, $A=a^{2}E+b^{2}(I-E)$. Denote $(A\circ
B(e_{1}),A\circ
B(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}f(x,y,z)&g(x,y,z)\\\
\overline{g(x,y,z)}&h(x,y,z)\\\ \end{array}\right)$, where $f,g,h$ are real
linear functions with respect to
$(x,y,z)\in\mathbb{R}\times\mathbb{C}\times\mathbb{R}$ and $f,h$ take values
in $\mathbb{R}$. Since $E\circ(A\circ B)=(E\circ A)\circ
B)=(E\circ(a^{2}E+b^{2}(I-E)))\circ B=a^{2}(E\circ B)$, we have
$f(x,y,z)=a^{2}x$. Similarly, we have also $h(x,y,z)=b^{2}z$. Moreover, since
$E|E,E|(I-E)$, by (S5), we have $E|A$, so by (S4), we have $(I-E)|A$, thus,
$A\circ(xE)=xa^{2}E$, $A\circ z(I-E)=zb^{2}(I-E)$, this showed that $g$ is
independent of $x$ and $z$, so $g(x,y,z)=\alpha y$, where $\alpha\in C$. On
the other hand, if $B\in{\cal S}(H)$ is a positive operator, then $A\circ B$
is also a positive operator, so for each positive number $x$ and $z$, and each
complex number $y$, when $xz-|y|^{2}\geq 0$, we have $a^{2}b^{2}xz-|\alpha
y|^{2}\geq 0$. Let $x=1$, $z=|y|^{2}$. Then we get that
$a^{2}b^{2}-|\alpha|^{2}\geq 0.$ $None$
Let $B,C$ be two positive operators. We show that if both $B\leq C$ and $C\leq
B$ are not true, then both $A\circ B\leq A\circ C$ and $A\circ C\leq A\circ B$
are also not true. In fact, let $D=b^{2}E+a^{2}(I-E)$. Then
$A|b^{2}E+a^{2}(I-E)=D$ and $A\circ D=A\circ(b^{2}E+a^{2}(I-E))=a^{2}b^{2}I$.
So if $A\circ B\leq A\circ C$, then $D\circ(A\circ B)\leq D\circ(A\circ C)$.
But $D\circ(A\circ B)=(D\circ A)\circ B=a^{2}b^{2}I\circ B=a^{2}b^{2}B\leq
D\circ(A\circ C)=a^{2}b^{2}C$, thus we will have $B\leq C$, this is a
contradiction. So $A\circ B\leq A\circ C$ is not true. Similarly, we have
$A\circ C\leq A\circ B$ is also not true.
Let $y\in\mathbb{C}$, $y\neq 0$, $\epsilon$ be a positive number satisfy that
$a^{2}|y|-\epsilon>0$. If we define
$(B(e_{1}),B(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}|y|&y\\\
\bar{y}&|y|\\\ \end{array}\right)$ and
$(C(e_{1},C(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}\epsilon&0\\\ 0&0\\\
\end{array}\right)$, then $B,C\in{\cal E}(H)$, $B\leq C$ and $C\leq B$ are
both not true. Thus we have both $A\circ B\leq A\circ C$ and $A\circ B\leq
A\circ C$ are also not true, i.e., the self-adjoint operator $A\circ B-A\circ
C$ is not positive operator. Note that $((A\circ B-A\circ C)(e_{1}),(A\circ
B-A\circ
C)(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}a^{2}|y|-\epsilon&\alpha y\\\
\overline{\alpha y}&b^{2}|y|\\\ \end{array}\right)$, and
$a^{2}|y|-\epsilon>0$, $b^{2}|y|>0$, so we have
$b^{2}(a^{2}|y|-\epsilon)|y|-|\alpha y|^{2}<0$. Let $\epsilon\rightarrow 0$,
we get that $|\alpha y|^{2}\geq b^{2}a^{2}|y|^{2}$. Thus, we have
$|\alpha|^{2}\geq b^{2}a^{2}.$ $None$
It follows from (1) and (2) that $|\alpha|^{2}=a^{2}b^{2}.$ So $|\alpha|=ab$
and $\alpha=abe^{i\theta}$.
4\. A New Sequential Product on ${\cal E}(H)$
Theorem 3.2 motivated us to construct the new sequential product on ${\cal
E}(H)$. First, we need the following:
For each $A\in{\cal E}(H)$, denote $R(A)=\\{Ax,x\in H\\}$, $N(A)=\\{x,x\in
H,Ax=0\\}$, $P_{0}$ and $P_{1}$ be the orthogonal projections on
$\overline{R(A)}$ and $N(A)$, respectively. It follows from $A\in{\cal E}(H)$
that $N(A)=N(A^{1/2})$, so $R(A)=R(A^{1/2})$. Moreover, $P_{0}(H)\bot
P_{1}(H)$ and $H=P_{0}(H)\oplus P_{1}(H)$ ([10]).
Denote $f_{z}(u)$ be the complex-valued Borel function defined on $[0,1]$,
where $f_{z}(u)=\exp z(\ln u)$ if $u\in(0,1]$ and $f_{z}(0)=0$. Now, we define
$A^{i}=f_{i}(A),\,\,A^{-i}=f_{-i}(A).$
It is easily to show that $||A^{i}||\leq 1$, $||A^{-i}||\leq 1$ and
$(A^{i})^{*}=A^{-i},A^{i}A^{-i}=A^{-i}A^{i}=P_{0}.$
Theorem 4.1. Let $H$ be a complex Hilbert space and $A,B\in{\cal E}(H)$. If we
define $A\circ B=A^{1/2}A^{i}BA^{-i}A^{1/2}$, then $\circ$ satisfies the
conditions (S1)-(S3).
Proof. If $A,B\in{\cal E}(H)$, note that $||A^{i}||\leq 1$ and $||A^{-i}||\leq
1$, we have
$\|A\circ
B\|=\|A^{1/2}A^{i}BA^{-i}A^{1/2}\|\leq\|A^{1/2}\|\|A^{i}\|\|B\|\|A^{-i}\|\|A^{1/2}\|\leq
1$
and
$<A^{1/2}A^{i}BA^{-i}A^{1/2}x,x>=\|B^{1/2}A^{-i}A^{1/2}x\|\geq 0$
for all $x\in H$, so $A\circ B=A^{1/2}A^{i}BA^{-i}A^{1/2}$ is a binary
operation on ${\cal E}(H)$. Moreover, it is clear that the map $B\rightarrow
A\circ B$ is additive for each $A\in{\cal E}(H)$, so the operation $\circ$
satisfies (S1).
It follows from $I\circ A=I^{1/2}I^{i}AI^{-i}I^{1/2}=A$ that $\circ$ satisfies
(S2).
If $A\circ B=A^{1/2}A^{i}BA^{-i}A^{1/2}=0$, now, we represent $A$ and $B$ on
$H=P_{0}(H)\oplus P_{1}(H)$ by $\left(\begin{array}[]{cc}A_{1}&0\\\ 0&0\\\
\end{array}\right)$ and $\left(\begin{array}[]{cc}B_{1}&B_{2}\\\
B_{3}&B_{4}\\\ \end{array}\right)$, then
$A\circ
B=\left(\begin{array}[]{cc}A_{1}^{1/2}A_{1}^{i}B_{1}A_{1}^{-i}A_{1}^{1/2}&0\\\
0&0\\\ \end{array}\right)=0,$
so we have $A_{1}^{1/2}A_{1}^{i}B_{1}A_{1}^{-i}A_{1}^{1/2}=0$ on $P_{0}(H)$,
i.e., $(A_{1}^{1/2}A_{1}^{i}B_{1}A_{1}^{-i}A_{1}^{1/2}x,x)=0$ for each $x\in
P_{0}(H)$. Note that $R(A)=R(A^{1/2})$ and $A^{i}$ is a unitary operator on
$P_{0}(H)$, so $R(A^{1/2})$ is dense in $P_{0}(H)$, thus for each $y\in
P_{0}(H)$, there is a sequence $\\{z_{n}\\}\subseteq R(A^{1/2})$ such that
$z_{n}\rightarrow A^{i}y$, so there is a sequence $\\{x_{n}\\}\subseteq H$
such that $A^{1/2}x_{n}=z_{n}\rightarrow A^{i}y$. Let $x_{n}=y_{n}+u_{n}$,
where $y_{n}\in P_{0}(H),u_{n}\in P_{1}(H)$. Then $A^{1/2}x_{n}=A^{1/2}y_{n}$.
Thus, there is a sequence $\\{y_{n}\\}$ in $P_{0}(H)$ such that
$A^{1/2}y_{n}=z_{n}\rightarrow A^{i}y$. Note that $A^{i}$ is a unitary
operator on $P_{0}(H)$, so we have $A^{-i}A^{1/2}y_{n}\rightarrow y$. But,
$\|B_{1}^{1/2}A_{1}^{-i}A_{1}^{1/2}y_{n}\|=(A_{1}^{1/2}A_{1}^{i}B_{1}A_{1}^{-i}A_{1}^{1/2}y_{n},y_{n})=0,$
so $B_{1}^{1/2}y=0$ for each $y\in P_{0}(H)$, that is, $B_{1}^{1/2}=0$. Since
$B\in{\cal E}(H)$, so $B_{2}=0,B_{3}=0$, thus we have
$B=\left(\begin{array}[]{cc}0&0\\\ 0&B_{4}\\\ \end{array}\right)$, so $B\circ
A=B^{1/2}B^{i}AB^{-i}B^{1/2}=0=A\circ B$. This showed that $\circ$ satisfies
(S3).
Theorem 4.2. Let $H$ be a complex Hilbert space with dim$(H)<\infty$,
$A,B\in{\cal E}(H)$. If we define $A\circ B=A^{1/2}A^{i}BA^{-i}A^{1/2}$, then
$A\circ B=A^{1/2}A^{i}BA^{-i}A^{1/2}=B\circ A=B^{1/2}B^{i}AB^{-i}B^{1/2}$ if
and only if $AB=BA$.
Proof. Firstly, it is obvious that if $AB=BA$, then $A\circ
B=A^{1/2}A^{i}BA^{-i}A^{1/2}=B\circ A=B^{1/2}B^{i}AB^{-i}B^{1/2}$. Now, if
$A\circ B=A^{1/2}A^{i}BA^{-i}A^{1/2}=B\circ A=B^{1/2}B^{i}AB^{-i}B^{1/2}$, we
show that $AB=BA$. Note that $A\in{\cal E}(H)$ and dim$(H)<\infty$, so $A$ has
the form $\sum\limits_{i=1}^{n}a_{i}E_{i}$, where
$\sum\limits_{k=1}^{n}E_{k}=I$, $a_{k}\geq 0$, $E_{k}\in{\cal P}(H)$,
$a_{k}\neq a_{l}$, $E_{k}E_{l}=0$ for all $k,l=1,2,\cdots,n,k\neq l$. Without
losing generality, we suppose that $0\leq a_{1}<\cdots<a_{n}$, then
$0\leq|a_{1}^{1/2}f_{i}(a_{1})|<\cdots<|a_{n}^{1/2}f_{i}(a_{n})|$ since
$a_{k}^{1/2}=|a_{k}^{1/2}f_{i}(a_{k})|$. It follows from the operator theory
that $A^{1/2}=\sum\limits_{k=1}^{n}a_{k}^{1/2}E_{k}$ and
$f_{i}(A)=A^{i}=\sum\limits_{k=1}^{n}f_{i}(a_{k})E_{k}$,
$f_{-i}(A)=A^{-i}=\sum\limits_{k=1}^{n}f_{-i}(a_{k})E_{k}$ ([10]). Note that
$A^{1/2}A^{i}BA^{-i}A^{1/2}=B^{1/2}B^{i}AB^{-i}B^{1/2}$, so for each $x\in H$,
$(A^{1/2}A^{i}BA^{-i}A^{1/2}x,x)=(B^{1/2}B^{i}AB^{-i}B^{1/2}x,x)$, thus we
have
$\|B^{1/2}A^{-i}A^{1/2}x\|=\|A^{1/2}B^{-i}B^{1/2}x\|.$ $None$
Take $x\in E_{n}(H)$, then
$A^{1/2}A^{-i}x=A^{-i}A^{1/2}x=a_{n}^{1/2}f_{-i}(a_{n})x$, note that
$|a_{n}f_{-i}(a_{n})|=|a_{n}f_{i}(a_{n})|=|a_{n}|$,
$\overline{R(B)}=\overline{R(B^{1/2})}$ and $B^{-i}$ is a unitary operator on
$\overline{R(B)}$ and $B^{-i}B^{1/2}=B^{1/2}B^{-i}$, we have
$\|A^{1/2}B^{1/2}B^{-i}x\|^{2}=\|\sum\limits_{k=1}^{n}a_{k}^{1/2}E_{k}B^{1/2}B^{-i}x\|^{2}=$
$\sum\limits_{k=1}^{n}a_{k}\|E_{k}B^{1/2}B^{-i}x\|^{2}\leq\sum\limits_{k=1}^{n}a_{n}\|E_{k}B^{1/2}B^{-i}x\|^{2}=$
$a_{n}||B^{1/2}B^{-i}x\|^{2}=||a_{n}^{1/2}B^{-i}B^{1/2}x\|^{2}=$
$||a_{n}^{1/2}B^{1/2}x\|^{2}=||B^{1/2}A^{1/2}A^{-i}x\|^{2}.$
Thus, it follows from equation (3), $B^{-i}B^{1/2}=B^{1/2}B^{-i}$,
$A^{-i}A^{1/2}=A^{1/2}A^{-i}$ and $0\leq a_{1}<\cdots<a_{n}$ that for each
$k<n$, we have $E_{k}B^{1/2}B^{-i}x=0$, so $B^{1/2}B^{-i}x\in E_{n}(H)$. Thus
we have $E_{n}B^{1/2}B^{-i}E_{n}=B^{1/2-i}E_{n}$. This showed that
$B^{1/2}B^{-i}$ has the matrix form $\left(\begin{array}[]{cc}C&D\\\ 0&K\\\
\end{array}\right)$ on $H=E_{n}(H)\oplus(I-E_{n})(H)$, where $C\in{\cal
B}(E_{n}(H),E_{n}(H))$, $D\in{\cal B}((I-E_{n})(H),E_{n}(H)),K\in{\cal
B}((I-E_{n})(H),(I-E_{n})(H))$. Note that $B\in{\cal E}(H)$, $B$ has the form
$\sum\limits_{k=1}^{m}b_{k}F_{k}$, and
$B^{1/2}B^{-i}=\sum\limits_{k=1}^{m}b^{1/2}f_{-i}(b_{k})F_{k}$, where
$\sum\limits_{k=1}^{m}F_{k}=I$, $b_{k}\geq 0$, $F_{k}\in{\cal P}(H)$,
$b_{k}\neq b_{l}$, $F_{k}F_{l}=0$ for all $k,l=1,2,\cdots,m,k\neq l$. Now we
define a polynomial
$G_{k}(z)=\prod\limits_{j\neq
k}(z-b_{j}^{1/2}f_{-i}(b_{j}))/\prod\limits_{j\neq
k}(b_{k}^{1/2}f_{-i}(b_{j})-b_{j}^{1/2}f_{-i}(b_{j}))$
on $\mathbb{C}$. It is easily to show that for each $1\leq k\leq m$,
$G_{k}(B^{1/2}B^{-i})=F_{k}$. Note that $B^{1/2}B^{-i}$ has the up-triangulate
form, so $G_{k}(B^{1/2}B^{-i})$ has also the up-triangulate form. But $F_{k}$
is a self-adjoint operator, so $F_{k}$ has the diagonal matrix form on
$E_{n}(H)\oplus(I-E_{n})(H)$. This implies that $F_{k}$ commutes with $E_{n}$
for each $k$, so $B$ commutes with $E_{n}$. Denote $A_{0}=A-a_{n}E_{n}$, then
we still have $A_{0}\circ B=B\circ A_{0}$ as discussed before, thus we get
that $B$ commutes with $E_{n-1}$. Continuously, we will have that $B$ commutes
with all $E_{k}$ and so with A. In this case we have $A\circ B=AB$.
Our main result is:
Theorem 4.3. Let $H$ be a complex Hilbert space with $dim(H)<\infty$ and
$A,B\in{\cal E}(H)$. If we define $A\circ B=A^{1/2}A^{i}BA^{-i}A^{1/2}$, then
$\circ$ is a sequential product on ${\cal E}(H)$.
Proof. By Theorem 4.1, we only need to prove that $\circ$ satisfies (S4) and
(S5). In fact, if $A|B$, i.e., $A\circ B=A^{1/2}A^{i}BA^{-i}A^{1/2}=B\circ
A=B^{1/2}B^{i}AB^{-i}B^{1/2}$, then it follows from Theorem 4.2 that $A$
commutes with $B$ and of course $I-B$, so $A|I-B$. If $C\in{\cal E}(H)$, we
have
$A\circ(B\circ
C)=A^{\frac{1}{2}}A^{i}B^{\frac{1}{2}}B^{i}CB^{-i}B^{\frac{1}{2}}A^{-i}A^{\frac{1}{2}}$
$=A^{\frac{1}{2}}B^{\frac{1}{2}}A^{i}B^{i}CA^{-i}B^{-i}A^{\frac{1}{2}}B^{\frac{1}{2}}$
$=(AB)^{\frac{1}{2}}(AB)^{i}C(AB)^{-i}(AB)^{\frac{1}{2}}$ $=(AB)\circ
C=(A\circ B)\circ C.$
So (S4) is satisfied.
Moreover, if $C|B$ and $C|A$, then $C(AB)=ACB=(AB)C$, $C(A\oplus B)=(B+A)C$,
so it is easily to prove that $C(A\circ B)=(A\circ B)C$, thus, by Theorem 4.2,
we have $C|A\circ B$ and $C|(A\oplus B)$ whenever $A\oplus B$ is defined, this
showed that (S5) is hold.
By using Theorem 4.3 we can prove the following corollary:
Corollary 4.4. Let $H$ be a complex Hilbert space with dim$(H)=2$,
$A,B\in{\cal E}(H)$. Take a normal basis $\\{e_{1},e_{2}\\}$ of $H$ such that
$(A(e_{1}),A(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}a^{2}&0\\\
0&b^{2}\\\ \end{array}\right)$ and
$(B(e_{1}),B(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}x&y\\\ \bar{y}&z\\\
\end{array}\right)$. If when $a,b>0$, define
$((A\circ B)(e_{1}),(A\circ
B)(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}a^{2}x&abe^{i\theta}y\\\
abe^{-i\theta}\bar{y}&b^{2}z\\\ \end{array}\right),$
where $\theta=\ln a^{2}-\ln b^{2}$; when $a>0,b=0$, define
$((A\circ B)(e_{1}),(A\circ
B)(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}a^{2}x&0\\\ 0&0\\\
\end{array}\right),$
when $a=0,b>0$, define
$((A\circ B)(e_{1}),(A\circ
B)(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}0&0\\\ 0&b^{2}z\\\
\end{array}\right),$
then $\circ$ is a sequential product of ${\cal E}(H)$.
Remark 1. In conclusion, we construct a new sequential product $A\circ
B=A^{\frac{1}{2}}A^{i}BA^{-i}A^{\frac{1}{2}}$ on $\varepsilon(H)$ with
dim$(H)<\infty$, which is different from the generalized Lüders form
$A^{\frac{1}{2}}BA^{\frac{1}{2}}$. In this proof we can also get a more
general one $A\circ B=A^{\frac{1}{2}}A^{ti}BA^{-ti}A^{\frac{1}{2}}$ for $t\in
R$. It indicates that with the measurement rule (S1)-(S5), there can be a time
parameter $t$ to describe the phase change. In particular, if dim$(H)=2$,
$A\in{\cal E}(H)$ and $\\{e_{1},e_{2}\\}$ is a normal basis of $H$ such that
$(A(e_{1}),A(e_{2}))=(e_{1},e_{2})\left(\begin{array}[]{cc}a^{2}&0\\\
0&b^{2}\\\ \end{array}\right)$, then when $a>0,b>0$ and $a\neq b$, Corollary
4.4 showed that $\theta=(\ln a^{2}-\ln b^{2})t$ can be used to describe the
phase-changed phenomena of quantum effect $A\circ B$. As the proof showed, it
is the only form that the sequential product can be. This is much more
important in physics.
Remark 2. As we knew, in the quantum computation and quantum information
theory, if $(A_{i})^{n}_{i=1}\subseteq{\cal B}(H)$ satisfying
$\sum\limits_{i=1}^{n}A_{i}A_{i}^{*}=I$, then the operators
$(A_{i})^{n}_{i=1}$ are called the operational elements of the quantum
operation $U:{\cal T}(H)\rightarrow{\cal T}(H)$ defined by
$U(\rho)=\sum\limits_{i=1}^{n}A_{i}\rho A_{i}^{*},$
where ${\cal T}(H)$ is the set of trace class operators. Any trace preserving,
normal, completely positive map has the above form. This is very important in
describing dynamics, measurements, quantum channels, quantum interactions, and
quantum error, correcting codes, etc. If $(A_{i})^{n}_{i=1}$ is a set of
quantum effects with $\sum\limits_{i=1}^{n}A_{i}=I$, then the transformation
$U^{\prime}(\rho)=\sum\limits_{j=1}^{n}A_{j}^{\frac{1}{2}}A_{j}^{ti}\rho
A_{j}^{-ti}A_{j}^{\frac{1}{2}}$ is a well defined quantum operation since
$\sum\limits_{j=1}^{n}A_{j}^{\frac{1}{2}}A_{j}^{ti}A_{j}^{-ti}A_{j}^{\frac{1}{2}}=\sum\limits_{i=1}^{n}A_{i}=I$.
So this new sequential product yields a natural and interesting quantum
operation.
Remark 3. Theorem 4.3 indicates that the conditions (S1)-(S5) of sequential
product of ${\cal E}(H)$ are not sufficient to characterize the generalized
Lüders form $A^{\frac{1}{2}}BA^{\frac{1}{2}}$ of $A$ and $B$. Recently,
Professor Gudder presented a characterization of the sequential product of
${\cal E}(H)$ is the generalized Lüders form ([11]).
Acknowledgement. The authors wish to express their thanks to the referees for
their valuable comments and suggestions. In particular, their comments
motivated the authors to prove Theorem 4.3 for any finite dimensional Hilbert
spaces. This project is supported by Natural Science Foundations of China
(10771191 and 10471124).
References
[1] Ludwig G 1983 Foundations of Quantum Mechanics (I-II) (Springer, New York)
[2] Ludwig G 1986 An Axiomatic Basis for Quantum Mechanics (II) (Springer, New
York)
[3] Kraus K 1983 Effects and Operations (Springer-Verlag, Beilin)
[4] Davies E B 1976 Quantum Theory of Open Systems (Academic Press, London)
[5] Busch P, Grabowski M and Lahti P J 1999 Operational Quantum Physics
(Springer-Verlag, Beijing Word Publishing Corporation)
[6] Gudder S, Nagy G 2001 J. Math. Phys. 42 5212
[7] Gudder S, Greechie R 2002 Rep Math. Phys. 49 87
[8] Gheondea A, Gudder S 2004 Proc. Am. Math. Soc. 132 503
[9] Gudder S 2005 Inter. J. Theory. Physi. 44 2199
[10] Kadison R V, Ringrose J R 1983 Fundamentals of the Theory of Operator
algebra (Springer, New York)
[11] Gudder S, Latremoliere F 2008 J. Math. Phys. 49 052106
|
arxiv-papers
| 2008-12-03T01:28:15 |
2024-09-04T02:48:59.118904
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Liu Weihua and Wu Junde",
"submitter": "Junde Wu",
"url": "https://arxiv.org/abs/0812.0630"
}
|
0812.0656
|
# Black Boxes
Saharon Shelah
###### Abstract.
We shall deal comprehensively with Black Boxes, the intention being that
provably in ZFC we have a sequence of guesses of extra structure on small
subsets, the guesses are pairwise with quite little interaction, are far but
together are ”dense”. We first deal with the simplest case, were the existence
comes from winning a game by just writing down the opponent’s moves. We show
how it help when instead orders we have trees with boundedly many levels,
having freedom in the last. After this we quite systematically look at
existence of black boxes, and make connection to non-saturation of natural
ideals and diamonds on them.
Publication 309; Was supposed to be Chapter IV to the book “Non-structure” and
probably will be if it materialize
## 0\. Introduction
The non-structure theorems we have discussed in [Sh:E59] rests usually on some
freedom on finite sequences and on a kind of order. When our freedom is
related to infinite sequences, and to trees, our work is sometimes harder. In
particular, we may consider, for $\lambda\geq\chi$, $\chi$ regular, and
$\varphi=\varphi(\bar{x}_{0},\ldots,\bar{x}_{\alpha},\ldots)_{\alpha<\chi}$ in
a vocabulary $\tau$:
$(\ast)$:
For any $I\subseteq{}^{\chi\geq}\lambda$ we have a $\tau$–model $M_{I}$ and
sequences $\bar{a}_{\eta}$ (for $\eta\in{}^{\chi>}\lambda$), where
$[\eta\triangleleft\nu\ \Rightarrow\
{\bar{a}}_{\eta}\neq{\bar{a}}_{\nu}],\qquad\mathop{\rm\ell
g}({\bar{a}}_{\eta})=\mathop{\rm\ell g}({\bar{x}}_{\mathop{\rm\ell
g}(\eta)}),$
such that for $\eta\in{}^{\chi}\lambda$ we have:
$M_{I}\models\varphi(\ldots,{\bar{a}}_{\eta\restriction\alpha},\ldots)_{\alpha<\chi}\qquad\mbox{
if and only if }\qquad\eta\in I.$
(Usually, $M_{I}$ is to some extend “simply defined” from $I$). Of course, if
we do not ask more from $M_{I}$, we can get nowhere: we certainly restrict its
cardinality and/or usually demand it is $\varphi$–representable (see
Definition [Sh:E59, 2.4] clauses (c),(d)) in (a variant of)
${\mathscr{M}}_{\mu,\kappa}(I)$ (for suitable $\mu,\kappa$). Certainly for $T$
unsuperstable we have such a formula $\varphi$:
$\varphi(\ldots,\bar{a}_{\eta\restriction
n},\ldots)=(\exists{\bar{x}})\bigwedge_{n}\varphi_{n}({\bar{x}},{\bar{a}}_{\eta\restriction
n}).$
There are many natural examples.
Formulated in terms of the existence of $I$ for which our favorite “anti-
isomorphism” player has a winning strategy, we prove this in 1969/70 (in
proofs of lower bounds of ${\dot{\mathbb{I}}}(\lambda,T_{1},T)$, $T$
unsuperstable), but it was shortly superseded. However, eventually the method
was used in one of the cases in [Sh:a, VIII,§2]: for strong limit singular
[Sh:a, VIII 2.6]. It was developed in [Sh 172], [Sh 227] for constructing
Abelian groups with prescribed endomorphism groups. See further a
representation of one of the results here in Eklof-Mekler [EM], [EM02] a
version which was developed for a proof of the existence of Abelian (torsion
free $\aleph_{1}$–free) group $G$ with
$G^{\ast\ast\ast}=G^{\ast}\oplus A\qquad(G^{\ast}=:{\rm Hom}(G,{\bf Z}))$
in a work by Mekler and Shelah. A preliminary version of this paper appeared
in [Sh 300, III,§4,§5] but §3 here was just almost ready, and §4 on partitions
of stationary sets and $\diamondsuit_{D}$ was written up as a letter to
Foreman in the late nineties answering his question on what I know on this.
## 1\. The Easy Black Box and an Easy Application
In this section we do not try to get the strongest results, just provide some
examples (i.e., we do not present the results when $\lambda=\lambda^{\chi}$ is
replaced by $\lambda=\lambda^{<\chi})$. By the proof of [Sh:a, VIII 2.5] (see
later for a complete proof):
###### Theorem 1.1.
Suppose that
1. (*)
1. (a)
$\lambda=\lambda^{\chi}$
2. (b)
$\tau$ a vocabulary
$\varphi=\varphi(\bar{x}_{0},\bar{x}_{1},\ldots,\bar{x}_{\alpha}\ldots)_{\alpha<\chi}$
is a formula in ${\mathscr{L}}(\tau)$ for some logic ${\mathscr{L}}$.
3. (c)
τ,φ For any $I\subseteq{}^{\chi\geq}\lambda$ we have a $\tau$–model $M_{I}$
and sequences $\bar{a}_{\eta}$ (for $\eta\in{}^{\chi>}\lambda$), where
$[\eta\triangleleft\nu\ \Rightarrow\
{\bar{a}}_{\eta}\neq{\bar{a}}_{\nu}],\qquad\mathop{\rm\ell
g}({\bar{a}}_{\eta})=\mathop{\rm\ell g}({\bar{x}}_{\mathop{\rm\ell
g}(\eta)}),$
such that for $\eta\in{}^{\chi}\lambda$ we have:
$M_{I}\models\varphi(\ldots,{\bar{a}}_{\eta\restriction\alpha},\ldots)_{\alpha<\chi}\qquad\mbox{
if and only if }\qquad\eta\in I.$
4. (d)
$\|M_{I}\|=\lambda$ for every $I$ satisfying
${}^{\chi>}\lambda\subseteq I\subseteq{}^{\chi\leq}\lambda,$
and $\mathop{\rm\ell g}({\bar{a}}_{\eta})\leq\chi$ or just
$\lambda^{\mathop{\rm\ell g}({\bar{a}}_{\eta})}=\lambda$.
Then (using ${}^{\chi>}\lambda\subseteq I\subseteq{}^{\chi\geq}\lambda)$:
1. (1)
There is no model $M$ of cardinality $\lambda$ into which every $M_{I}$ can be
$(\pm\varphi)$–embedded (i.e., by a function preserving $\varphi$ and
$\neg\varphi)$.
2. (2)
For any $M_{i}$ (for $i<\lambda$), $\|M_{i}\|\leq\lambda$, for some $I$
satisfying ${}^{\chi>}\lambda\subseteq I\subseteq{}^{\chi\geq}\lambda$, the
model $M_{I}$ cannot be $(\pm\varphi)$–embedded into any $M_{i}$.
###### Example 1.2.
Consider the class of Boolean algebras and the formula
$\varphi(\ldots,x_{n},\ldots)=:(\bigcup\limits_{n}x_{n})=1$
(i.e., there is no $x\neq 0$ such that $x\cap x_{n}=0$ for each $n$).
For ${}^{\omega>}\lambda\subseteq I\subseteq{}^{\omega\geq}\lambda$, let
$M_{I}$ be the Boolean algebra generated freely by $x_{\eta}$ (for $\eta\in
I$) except the relations:
$\mbox{ for }\eta\in I,\quad\mbox{ if }n<\mathop{\rm\ell g}(\eta)=\omega\mbox{
then }x_{\eta}\cap x_{\eta\restriction n}=0.$
So $\|M_{I}\|=|I|\in[\lambda,\lambda^{\aleph_{0}}]$ and in $M_{I}$ for
$\eta\in{}^{\omega}\lambda$ we have:
$M_{I}\models(\bigcup\limits_{n}x_{\eta\restriction n})=1$ if and only if
$\eta\notin I$ (work a little in Boolean algebras).
So
###### Conclusion 1.3.
If $\lambda=\lambda^{\aleph_{0}}$, then there is no Boolean algebra
${\mathbf{B}}$ of cardinality $\lambda$ universal under $\sigma$–embeddings
(i.e., ones preserving countable unions).
###### Remark 1.4.
This is from [Sh:a, VIII,Ex.2.2]).
Proof of the Theorem 1.1. First we note 1.5, 1.6 below:
###### Fact 1.5.
There are functions $f_{\eta}$ (for $\eta\in{}^{\chi}\lambda$) such that:
(i):
$\mathop{\mathrm{Dom}}(f_{\eta})=\\{\eta\restriction\alpha:\alpha<\chi\\}$,
(ii):
$\mathop{\mathrm{Rang}}(f_{\eta})\subseteq\lambda$,
(iii):
if $f:{}^{\chi>}\lambda\longrightarrow\lambda$, then for some
$\eta\in{}^{\chi}\lambda$ we have $f_{\eta}\subseteq f$.
Proof: For $\eta\in{}^{\chi}\lambda$ let $f_{\eta}$ be the function (with
domain $\\{\eta\restriction\alpha:\alpha<\chi\\})$ such that:
$f_{\eta}(\eta\restriction\alpha)=\eta(\alpha).$
So $\langle f_{\eta}:\eta\in{}^{\chi}\lambda\rangle$ is well defined.
Properties (i), (ii) are straightforward, so let us prove (iii). Let
$f:{}^{\chi>}\lambda\longrightarrow\lambda$. We define
$\eta_{\alpha}=\langle\beta_{i}:i<\alpha\rangle$ by induction on $\alpha$.
For $\alpha=0$ or $\alpha$ limit — no problem.
For $\alpha+1$: let $\beta_{\alpha}$ be the ordinal such that
$\beta_{\alpha}=f(\eta_{\alpha})$.
So $\eta=:\langle\beta_{i}:i<\chi\rangle$ is as required. $\hskip
5.0pt\hbox{\hskip 5.0pt\vrule width=4.0pt,height=6.0pt,depth=1.5pt\hskip
1.0pt}_{\ref{4.5A}}$
###### Fact 1.6.
In 1.5:
(a):
we can replace the range of $f$, $f_{\eta}$ by any fixed set of cardinality
$\lambda$,
(b):
we can replace the domains of $f$, $f_{\eta}$ by
$\\{{\bar{a}}_{\eta}:\eta\in{}^{\chi>}\lambda\\}$,
$\\{{\bar{a}}_{\eta\restriction\alpha}:\alpha<\chi\\}$, respectively, as long
as
$\alpha<\beta<\chi\ \wedge\
\eta\in{}^{\chi}\lambda\quad\Rightarrow\quad{\bar{a}}_{\eta\restriction\alpha}\neq{\bar{a}}_{\eta\restriction\beta}.$
###### Remark 1.7.
We can present it as a game. (As in the book [Sh:a, VIII 2.5]).
Continuation of the Proof of Theorem 1.1. It suffices to prove 1.1(2). Without
loss of generality $\langle|M_{i}|:i<\lambda\rangle$ are pairwise disjoint.
Now we use 1.6; for the domain we use
$\langle{\bar{a}}_{\eta}:\eta\in{}^{\chi>}\lambda\rangle$ from the assumption
of 1.1, and for the range: $\bigcup\limits_{i<\lambda}{}^{\chi\geq}|M_{i}|$
(it has cardinality $\leq\lambda$ as $\|M_{i}\|\leq\lambda=\lambda^{\chi}$).
We define
$\begin{array}[]{ll}I=({}^{\chi>}\lambda)\cup\big{\\{}\eta\in{}^{\chi}\lambda:&\mbox{for
some }i<\lambda,\ \mathop{\mathrm{Rang}}(f_{\eta})\mbox{ is a set of
sequences}\\\ &\mbox{from }|M_{i}|\mbox{ and
}M_{i}\models\neg\varphi(\ldots,f_{\eta}({\bar{a}}_{\eta\restriction\alpha}),\ldots)_{\alpha<\chi}\big{\\}}.\end{array}$
Look at $M_{I}$. It suffices to show:
$(\otimes)$:
for $i<\lambda$ there is no $(\pm\varphi)$–embedding of $M_{I}$ into $M_{i}$.
Why does $(\otimes)$ hold?
If $f:M_{I}\longrightarrow M_{i}$ is a $(\pm\varphi)$–embedding, then by Fact
1.6, for some $\eta\in{}^{\chi}\lambda$ we have
$f\restriction\\{{\bar{a}}_{\eta\restriction\alpha}:\alpha<\kappa\\}=f_{\eta}.$
By the choice of $f$,
$M_{I}\models\varphi\left[\ldots,{\bar{a}}_{\eta\restriction\alpha},\ldots\right]_{\alpha<\chi}\iff
M_{i}\models\varphi\left[\ldots,f({\bar{a}}_{\eta\restriction\alpha}),\ldots\right]_{\alpha<\chi},$
but by the choice of $I$ and $M_{I}$ we have
$M_{I}\models\varphi\left[\ldots,{\bar{a}}_{\eta\restriction\alpha},\ldots\right]_{\alpha<\chi}\iff
M_{i}\models\neg\varphi\left[\ldots,f_{\eta}({\bar{a}}_{\eta\restriction\alpha}),\ldots\right]_{\alpha<\chi}.$
A contradiction, as by the choice of $\eta$,
$\bigwedge\limits_{\alpha<\chi}f({\bar{a}}_{\eta\restriction\alpha})=f_{\eta}(\bar{a}_{\eta\restriction\alpha}).$
$\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{4.2}}$
###### Discussion 1.8.
We may be interested whether in 1.1, when $\lambda^{+}<2^{\lambda}$, we may
(a):
allow in (1) $\|M\|=\lambda^{+}$, and/or
(b):
get $\geq\lambda^{++}$ non-isomorphic models of the form $M_{I}$, assuming
$2^{\lambda}>\lambda^{+}$
The following lemma shows that we cannot prove those better statements in ZFC,
though (see 1.10) in some universes of set theory we can. So this require
(elementary) knowledge of forcing, but is not used later . It is here just to
justify the limitations of what we can prove and the reader can skip it.
###### Lemma 1.9.
Suppose that in the universe ${\bf V}$ we have
$\kappa<\lambda=\mathop{\mathrm{cf}}(\lambda)=\lambda^{<\lambda},\quad\mbox{
and
}\quad(\forall\lambda_{1}<\lambda)[\lambda^{\kappa}_{1}<\lambda]\quad\mbox{
and }\quad\lambda<\mu=\mu^{\lambda}.$
Then for some notion forcing ${\mathbb{P}}$:
(a):
${\mathbb{P}}$ is $\lambda$–complete and satisfies the $\lambda^{+}$–c.c., and
$|{\mathbb{P}}|=\mu$, $\Vdash_{{\mathbb{P}}}$“ $2^{\lambda}=\mu$ ” (so forcing
with ${\mathbb{P}}$ collapses no cardinals, changes no cofinalities, adds no
new sequences of ordinals of length $<\lambda$, and $\Vdash_{{\mathbb{P}}}$“
$\lambda^{<\lambda}=\lambda$ ”).
(b):
We can find $\varphi$, $M_{I}$ (for ${}^{\kappa>}\lambda\subseteq
I\subseteq{}^{\kappa\geq}\lambda$) as in $(\ast)$ of ‣ 0\. Introduction, so
with $\|M_{I}\|=\lambda$ ($\tau$–models with $|\tau|=\kappa$ for simplicity)
such that:
$(\oplus)$:
there are up to isomorphism exactly $\lambda^{+}$ models of the form $M_{I}$
(${}^{\kappa>}\lambda\subseteq I\subseteq{}^{\lambda\geq}\lambda$).
(c):
In (b), there is a model $M$ such that $\|M\|=\lambda^{+}$ and every model
$M_{I}$ can be $(\pm\varphi)$–embedded into $M$.
Remark:
1. (1)
Essentially $M_{I}$ is $(I^{+},\vartriangleleft)$, the addition of level
predicates is immaterial, where $I^{+}$ extends $I$ “nicely” so that we can
let $a_{\eta}=\eta$ for $\eta\in I$.
2. (2)
Clearly clause (c) also shows that weakening $\|M\|=\lambda$, even when
$\lambda^{+}<2^{\lambda}$ may make 1.1 false.
Proof: Let $\tau=\\{R_{\zeta}:\zeta\leq\kappa\\}\cup\\{<\\}$ with $R_{\zeta}$
being a monadic predicate, and $<$ being a binary predicate. For a set $I$,
${}^{\kappa>}\lambda\subseteq I\subseteq{}^{\kappa\geq}\lambda$ let $N_{I}$ be
the $\tau$-model:
$|N_{I}|=I,\quad
R^{N_{I}}_{\zeta}=I\cap{}^{\zeta}\lambda,\quad<^{N_{I}}=\\{(\eta,\nu):\eta\in
I,\nu\in I,\eta\vartriangleleft\nu\\},$
and
$\varphi(\ldots,x_{\zeta},\ldots)_{\zeta<\kappa}=\bigwedge\limits_{\zeta<\xi<\kappa}\left(x_{\zeta}<x_{\xi}\
\&\ R_{\zeta}(x_{\zeta})\right)\wedge(\exists y)[R_{\kappa}(y)\ \&\
\bigwedge\limits_{\zeta<\kappa}x_{\zeta}<y].$
Now we define the forcing notion ${\mathbb{P}}$. It is
${\mathbb{P}}_{\lambda^{+}}$, where
$\langle{\mathbb{P}}_{i},\mathchoice{\oalign{$\displaystyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{j}:i\leq\lambda^{+},\ j<\lambda^{+}\rangle$
is an iteration with support $<\lambda$, of $\lambda$–complete forcing
notions, where $\mathchoice{\oalign{$\displaystyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{j}$ is defined as follows.
For $j=0$ we add $\mu$ many Cohen subsets to $\lambda$:
${\mathbb{Q}}_{0}=\\{f:f\mbox{ is a partial function from }\mu\mbox{ to
}\\{0,1\\},\ |\mathop{\mathrm{Dom}}(f)|<\lambda\\},$
the order is the inclusion.
For $j>0$, we define
$\mathchoice{\oalign{$\displaystyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{j}$ in ${\bf V}^{{\mathbb{P}}_{j}}$. Let $\langle
I(j,\alpha):\alpha<\alpha(j)\rangle$ list all sets $I\in{\bf
V}^{{\mathbb{P}}_{j}}$, ${}^{\kappa>}\lambda\subseteq
I\subseteq{}^{\kappa\geq}\lambda$ (note that the interpretation of
${}^{\kappa\geq}\lambda$ does not change from ${\bf V}$ to ${\bf
V}^{{\mathbb{P}}_{j}}$ as $\kappa<\lambda$ but the family of such $I$-s
increases). Now
$\begin{array}[]{ll}\mathchoice{\oalign{$\displaystyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{j}=\Big{\\{}\bar{f}:&f=\langle
f_{\alpha}:\alpha<\alpha(j)\rangle,\ f_{\alpha}\mbox{ is a partial
isomorphism}\\\ &\mbox{from }N_{I(j,\alpha)}\mbox{ into
}N_{({}^{\kappa\geq}\lambda)},\\\
&w({\bar{f}})=:\\{\alpha:f_{\alpha}\neq\emptyset\\}\mbox{ has cardinality
}<\lambda,\\\ &\mathop{\mathrm{Dom}}(f_{\alpha})\mbox{ has the form
}\bigcup\limits_{\beta<\gamma}{}^{\kappa\geq}\beta\cap N_{I(j,\alpha)}\mbox{
for some }\gamma<\lambda;\\\ &\mbox{and if
}\alpha_{1},\alpha_{2}<\alpha(j)\mbox{ and
}\eta_{1},\eta_{2}\in{}^{\kappa}\lambda,\mbox{ and for every }\zeta<\kappa,\\\
&f_{\alpha_{1}}(\eta_{1}\restriction\zeta),f_{\alpha_{2}}(\eta_{2}\restriction\zeta)\mbox{
are defined and equal, then}\\\ &\eta_{1}\in I(j,\alpha_{1})\iff\eta_{2}\in
I(j,\alpha_{2})\Big{\\}}.\end{array}$
The order is:
$\begin{array}[]{lcl}{\bar{f}}^{1}\leq{\bar{f}}^{2}&\mbox{ {\text@underline{if
and only if}}}&(\forall\alpha<\alpha(j))(f^{1}_{\alpha}\subseteq
f^{2}_{\alpha})\mbox{ and}\\\ &&\mbox{for all }\alpha<\beta<\alpha(j),\quad
f^{1}_{\alpha}\neq\emptyset\wedge f^{1}_{\beta}\neq\emptyset\mbox{ implies}\\\
&&\mathop{\mathrm{Rang}}(f^{2}_{\alpha})\cap\mathop{\mathrm{Rang}}(f^{2}_{\beta})=\mathop{\mathrm{Rang}}(f^{1}_{\alpha})\cap\mathop{\mathrm{Rang}}(f^{1}_{\beta}).\end{array}$
Then, $\mathchoice{\oalign{$\displaystyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{j}$ is $\lambda$–complete and it satisfies the version of
$\lambda^{+}$–c.c. from [Sh 80] (see more [Sh 546]), hence each
${\mathbb{P}}_{j}$ satisfies the $\lambda^{+}$–c.c. (by [Sh 80]).
Now the ${\mathbb{P}}_{j+1}$-name $\mathchoice{\oalign{$\displaystyle
I$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle I$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle I$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle I$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{j}$, (interpreting it in ${\bf V}^{{\mathbb{P}}_{j+1}}$ we
get $I^{\ast}_{j}$) is:
$\begin{array}[]{r}I^{\ast}_{j}={}^{\kappa>}\lambda\cup\Big{\\{}\eta\in{}^{\kappa}\lambda:\mbox{
for some }{\bar{f}}\in\mathchoice{\oalign{$\displaystyle G$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle G$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle G$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle G$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{{\mathbb{Q}}_{j}},\ \alpha<\alpha(j)\mbox{ and }\nu\in
N_{I(j,\alpha)},\\\ \mathop{\rm\ell g}(\nu)=\kappa\mbox{ and
}f_{\alpha}(\nu)=\eta\Big{\\}}.\end{array}$
This defines also $f^{j}_{\alpha}:I(j,\alpha)\longrightarrow I^{\ast}_{j}$,
which is forced to be a $(\pm\varphi)$–embedding and also just an embedding.
So now we shall define for every $I$, ${}^{\kappa>}\lambda\subseteq
I\subseteq{}^{\kappa\geq}\lambda$, a $\tau$–model $M_{I}$: clearly $I$ belongs
to some ${\bf V}^{{\mathbb{P}}_{j}}$. Let $j=j(I)$ be the first such $j$, and
let $\alpha=\alpha(I)$ be such that $I=I(j,\alpha)$. Let
$M_{I(j,\alpha)}=N_{I^{\ast}_{j}}\qquad(\mbox{ and }\quad
a_{\rho}=f^{j}_{\alpha}(\rho)\mbox{ for}\rho\in I(j,\alpha)).$
We leave the details to the reader. $\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{4.6}}$
On the other hand, consistently we may easily have a better result.
###### Lemma 1.10.
Suppose that, in the universe ${\bf V}$,
$\lambda=\mathop{\mathrm{cf}}(\lambda)=\lambda^{\kappa}=\lambda^{<\lambda},\quad\lambda<\mu=\mu^{\lambda}.$
For some forcing notion ${\mathbb{P}}$:
(a):
as in 1.9.
(b):
In ${\bf V}^{{\mathbb{P}}}$, assume that $\varphi$ and the function
$I\mapsto(M_{I},\langle{\bar{a}}^{I}_{\eta}:\eta\in{}^{\kappa>}\lambda\rangle)$
are as required in clauses (a),(b),(c) of $(\ast)$ of 1.1), $\zeta(\ast)<\mu$,
and $N_{\zeta}$ (for $\zeta<\zeta(\ast)$) is a model in the relevant
vocabulary, $\sum\limits_{\zeta<\zeta(\ast)}\|N_{\zeta}\|^{\kappa}<\mu$ (if
the vocabulary is of cardinality $<\lambda$ and each predicate or relation
symbol has finite arity, then requiring just
$\sum\limits\\{|N_{\zeta}\|:\zeta<\zeta(*)\\}<\mu$ suffices). Then for some
$I$, the model $M_{I}$ cannot be $(\pm\varphi)$–embedded into any $N_{\zeta}$.
(c):
Assume $\mu_{1}=\mathop{\mathrm{cf}}(\mu_{1})$, $\lambda<\mu_{1}\leq\mu$ and
${\bf V}\models(\forall\chi<\mu_{1})[\chi^{\lambda}<\mu_{1}]$. Then, in ${\bf
V}^{{\mathbb{P}}}$, if $\langle M_{I_{i}}:i<\mu_{1}\rangle$ are pairwise non-
isomorphic, ${}^{\kappa>}\lambda\subseteq
I_{i}\subseteq{}^{\kappa\geq}\lambda$, and $M_{I_{i}}$, ${\bar{a}}_{\eta}^{i}$
($\eta\in I_{i}$) are as in $(\ast)$ of ‣ 0\. Introduction), then for some
$i\neq j$, $M_{I_{i}}$ is not embeddable into $M_{I_{j}}$.
(d):
In ${\bf V}^{\mathbb{P}}$ we can find a sequence $\langle
I_{\zeta}:\zeta<\mu\rangle$ (so ${}^{\kappa>}\lambda\subseteq
I_{\zeta}\subseteq{}^{\kappa\geq}\lambda$) such that the $M_{I_{\zeta}}$’s
satisfy that no one is $(\pm\varphi)$–embeddable into another.
Proof: ${\mathbb{P}}$ is ${\mathbb{Q}}_{0}$ from the proof of 1.9. Let
${\mathbf{f}}$ be the generic function that is
$\cup\\{f:f\in\mathchoice{\oalign{$\displaystyle G$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle G$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle G$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle G$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{{\mathbb{Q}}_{0}}\\}$, clearly it is a function from $\mu$
to $\\{0,1\\}$. Now clause (a) is trivial. Next, concerning clause (b), we are
given $\langle N_{\zeta}:\zeta<\zeta(*)\rangle$. Clearly for some $A\in{\bf
V}$ of size smaller than $\mu$, $A\subseteq\mu$, to compute the isomorphism
types of $N_{\zeta}$ (for $\zeta<\zeta(\ast))$ it is enough to know
${\mathbf{f}}\restriction A$. We can force by
$\\{f\in{\mathbb{Q}}_{0}:\mathop{\mathrm{Dom}}(f)\subseteq A\\}$, then ${\bf
f}\upharpoonright B$ for any $B\subseteq\lambda\setminus A$ of cardinality
$\lambda$, (from ${\bf V}$) gives us an $I$ as required.
To prove clause (c) use $\Delta$–system argument for the names of various
$M_{I}$’s.
The proof of (d) is like that of (c). $\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{4.7}}$
## 2\. An Application for many models in $\lambda$
###### Discussion 2.1.
Next we consider the following:
Assume $\lambda$ is regular, $(\forall\mu<\lambda)[\mu^{<\chi}<\lambda]$. Let
${\mathscr{U}}_{\alpha}\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\chi\\}$
for $\alpha<\lambda$ be pairwise disjoint stationary sets. For
$A\subseteq\lambda$, let
${\mathscr{U}}_{A}=\bigcup\limits_{i\in A}{\mathscr{U}}_{i}.$
We want to define $I_{A}$ such that ${}^{\chi>}\lambda\subseteq
I_{A}\subseteq{}^{\chi\geq}\lambda$ and
$A\not\subseteq B\quad\Rightarrow\quad M_{I_{A}}\not\cong M_{I_{B}}.$
We choose $\langle\langle
M_{I_{A}}^{i}:i<\lambda\rangle:A\subseteq\lambda\rangle$ with
$M_{I_{A}}=\bigcup\limits_{i<\lambda}M^{i}_{I_{A}}$,
$\|M^{i}_{I_{A}}\|<\lambda$, $M^{i}_{I_{A}}$ increasing continuous.
Of course, we have to strengthen the restrictions on $M_{I}$. For $\eta\in
I_{A}\cap{}^{\chi}\lambda$, let $\delta(\eta)=:\bigcup\\{\eta(i)+1:i<\chi\\}$,
we are specially interested in $\eta$ such that $\eta$ is strictly increasing
converging to some $\delta(\eta)\in{\mathscr{U}}_{A}$; we shall put only such
$\eta$’s in $I_{A}$. The decision whether $\eta\in I_{A}$ will be done by
induction on $\delta(\eta)$ for all sets $A$. Arriving to $\eta$, we assume we
know quite a lot on the isomorphism $f:M_{I_{A}}\rightarrow M_{I_{B}}$,
specially we know
$f\restriction\bigcup\limits_{{\alpha}<\chi}{\bar{a}}_{\eta\restriction\alpha},$
which we are trying to “kill”,and we can assume
$\delta(\eta)\notin{\mathscr{U}}_{B}$ and $\delta$ belongs to a thin enough
club of $\lambda$ and using all this information we can “compute” what to do.
Note: though this is the typical case, we do not always follow it.
###### Notation 2.2.
1. (1)
For an ordinal $\alpha$ and a regular $\theta\geq\aleph_{0}$, let
${\mathscr{H}}_{<\theta}(\alpha)$ be the smallest set $Y$ such that:
(i):
$i\in Y$ for $i<\alpha$,
(ii):
$x\in Y$ for $x\subseteq Y$ of cardinality $<\theta$.
2. (2)
We can agree that ${\mathscr{M}}_{\lambda,\theta}(\alpha)$ from [Sh:E59, §2]
is interpretable in $({\mathscr{H}}_{<\theta}(\alpha),\in)$ when
$\alpha\geq\lambda$, and in particular its universe is a definable subset of
${\mathscr{H}}_{<\theta}(\alpha)$, and also $R$ is, where:
$\begin{array}[]{r}R=\Big{\\{}(\sigma^{\ast},\langle
t_{i}:i<\gamma_{x}\rangle,x):x\in{\mathscr{M}}_{\lambda,\theta}({}^{\theta>}\alpha),\sigma^{*}\mbox{
is a }\tau_{\lambda,\kappa}-\mbox{ term }\mbox{ and
}\theta\leq\lambda\leq\alpha,\\\ x=\sigma^{\ast}(\langle
t_{i}:i<\gamma_{x}\rangle)\Big{\\}}.\end{array}$
Similarly ${\mathscr{M}}_{\lambda,\theta}(I)$, where
$I\subseteq{}^{\kappa>}\lambda$ is interpretable in
$({\mathscr{H}}_{<\chi}(\lambda^{\ast}),\in)$ if $\lambda\leq\lambda^{\ast}$,
$\theta\leq\chi$, $\kappa\leq\chi$.
The main theorem of this section is:
###### Theorem 2.3.
${\dot{I}\dot{E}}_{\pm\varphi}(\lambda,K)=2^{\lambda}$, provided that:
(a):
$\lambda=\lambda^{\chi}$,
(b):
$\varphi=\varphi(\ldots,{\bar{x}}_{\alpha},\ldots)_{\alpha<\chi}$ is a formula
in the vocabulary $\tau_{K}$,
(c):
for every $I$ such that ${}^{\chi>}\lambda\subseteq
I\subseteq{}^{\chi\geq}\lambda$ we have a model $M_{I}\in K_{\lambda}$ and a
function $f_{I}$, and ${\bar{a}}_{\eta}\in{}^{\chi\geq}|M_{I}|$ for
$\eta\in{}^{\chi>}\lambda$ with $\mathop{\rm\ell
g}({\bar{a}}_{\eta})=\mathop{\rm\ell g}({\bar{x}}_{\mathop{\rm\ell g}(\eta)})$
such that:
$(\alpha)$:
for $\eta\in{}^{\chi}\lambda$ we have
$M_{I}\models\varphi(\ldots,{\bar{a}}_{\eta\restriction\alpha},\ldots)$ if and
only if $\eta\in I$,
$(\beta)$:
$f_{I}:M_{I}\longrightarrow{\mathscr{M}}_{\mu,\kappa}(I)$, where
$\mu\leq\lambda$, $\kappa=\chi^{+}$, and:
(d):
for $I$, ${}^{\chi>}\lambda\subseteq I\subseteq{}^{\chi\geq}\lambda$ and
${\bar{b}}_{\alpha}\in M_{I}$, $\mathop{\rm\ell
g}({\bar{x}}_{\alpha})=\mathop{\rm\ell g}({\bar{b}}_{\alpha})$ for
$\alpha<\chi$,
$f_{I}({\bar{b}}_{\alpha})={\bar{\sigma}}_{\alpha}({\bar{t}}_{\alpha})$ we
have: the truth value of
$M_{I}\models\varphi[\ldots,{\bar{b}}_{\alpha},\ldots]_{\alpha<\chi}$ can be
computed from $\langle{\bar{\sigma}}_{\alpha}:\alpha<\chi\rangle$,
$\langle{\bar{t}}_{\alpha}:\alpha<\chi\rangle$ (not just its q.f. type in $I$)
and the truth values of statements of the form
$(\exists\nu\in
I\cap{}^{\chi}\lambda)[\bigwedge\limits_{i<\chi}\nu\restriction\epsilon_{i}={\bar{t}}_{\beta_{i}}(\gamma_{i})\restriction\epsilon_{i}]\qquad\mbox{
for }\alpha_{i},\beta_{i},\gamma_{i},\epsilon_{i}<\chi$
(i.e., in a way not depending on $I,f_{I})$ [we can weaken this].
We shall first prove 2.3 under stronger assumptions.
###### Fact 2.4.
Suppose
$(\ast)$:
$\lambda=\lambda^{2^{\chi}}$, (so $\mathop{\mathrm{cf}}(\lambda)>\chi$) and
$\chi\geq\kappa$.
Then there are $\\{(M^{\alpha},\eta^{\alpha}):\alpha<\alpha(\ast)\\}$ such
that:
(i):
for every model $M$ with universe ${\mathscr{H}}_{<\chi^{+}}(\lambda)$ such
that $|\tau(M)|\leq\chi$ (and, e.g.,
$\tau\subseteq{\mathscr{H}}_{<\chi^{+}}(\lambda)$), for some $\alpha$ we have
$M^{\alpha}\prec M$,
(ii):
$\eta^{\alpha}\in{}^{\chi}\lambda$, $(\forall
i<\chi)[\eta^{\alpha}\restriction i\in M^{\alpha}]$, $\eta^{\alpha}\notin
M^{\alpha}$, and $\alpha\neq\beta\ \Rightarrow\
\eta^{\alpha}\neq\eta^{\beta}$,
(iii):
for every $\beta<\alpha(\ast)$ we have: $\\{\eta^{\alpha}\restriction
i:i<\chi\\}\not\subseteq M^{\beta}$,
(iv):
for $\beta<\alpha$ if $\\{\eta^{\beta}\restriction i:i<\chi\\}\subseteq
M^{\alpha}$, then $|M^{\beta}|\subseteq|M^{\alpha}|$,
(v):
$\|M^{\alpha}\|=\chi$.
Proof of 2.4: By 3.18+3.19 below with $\lambda,2^{\chi},\chi$ here standing
for $\lambda,\chi(\ast),\theta$ there.
Proof of 2.3 from the Conclusion of 2.4. Without loss of generality the
universe of $M_{I}$ is $\lambda$ in 2.3.
We shall define for every $A\subseteq\lambda$ a set $I[A]$ satisfying
${}^{\chi>}\lambda\subseteq I[A]\subseteq{}^{\chi\geq}\lambda$, moreover
$I[A]\setminus{}^{\chi>}\lambda\subseteq\\{\eta^{\alpha}:\alpha<\alpha(\ast)\\}.$
For $\alpha<\alpha(\ast)$, let
${\mathscr{U}}_{\alpha}=\\{\eta\in{}^{\chi}\lambda:\\{\eta\restriction
i:i<\chi\\}\subseteq M^{\alpha}\\}$. We shall define by induction on $\alpha$,
for every $A\subseteq\lambda$ the set $I[A]\cap{\mathscr{U}}_{\alpha}$ so that
on the one hand those restrictions are compatible (so that we can define I[A]
in the end, for each $A\subseteq\lambda$), and on the other hand they
guarantee the non $(\pm\varphi)$–embeddability.
For each $\alpha$: (essentially we decide whether $\eta^{\alpha}\in I[A]$
assuming $M^{\alpha}$ “guesses” rightly a function $g:M_{I_{1}}\longrightarrow
M_{I_{2}}$ ($I_{\ell}=I[A_{\ell}]$), and $A_{\ell}\cap M^{\alpha}$ for
$\ell=1,2$, and we make our decision to prevent this)
Case I: there are distinct subsets $A_{1},A_{2}$ of $\lambda$ and
$I_{1},I_{2}$ satisfying ${}^{\chi>}\lambda\subseteq
I_{\ell}\subseteq{}^{\chi\geq}\lambda$, and a $(\pm\varphi)$–embedding $g$ of
$M_{I_{1}}$ into $M_{I_{2}}$ and
$M^{\alpha}\prec\left({\mathscr{H}}_{<\chi^{+}}(\lambda),\in,R,A_{1},A_{2},I_{1},I_{2},M_{I_{1}},M_{I_{2}},f_{I_{1}},f_{I_{2}},g\right),$
where
$\begin{array}[]{r}R=\Big{\\{}\\{(0,\sigma_{x},x),(1+i,t^{x}_{i},x)\\}:i<i_{x}\mbox{
and }x\mbox{ has the form }\\\ \sigma_{x}(\langle
t^{x}_{i}:i<i_{x}\rangle)\Big{\\}}\end{array}$
(we choose for each $x$ a unique such term $\sigma$), and
$I_{2}\cap{\mathscr{U}}_{\alpha}\subseteq
I_{2}\cap(\bigcup\limits_{\beta<\alpha}{\mathscr{U}}_{\beta})$, and
$I_{1},I_{2}$ satisfy the restrictions we already have imposed on $I[A_{1}]$,
$I[A_{2}]$, respectively for each $\beta<\alpha$. Computing according to
clause (d) of 2.3 the truth value for
$M_{I_{2}}\models\varphi[\ldots,f({\bar{a}}_{\eta^{\alpha}\restriction
i}),\ldots]_{i<\chi}$ we get ${\bf t}^{\alpha}$.
Then we restrict:
(i):
if $B\subseteq\lambda$, $B\cap|M^{\alpha}|=A_{2}\cap|M^{\alpha}|$, then
$I[B]\cap({\mathscr{U}}^{\alpha}\setminus\bigcup\limits_{\beta<\alpha}{\mathscr{U}}^{\beta})=\emptyset$,
(ii):
if $B\subseteq\lambda$, $B\cap|M^{\alpha}|=A_{1}\cap|M^{\alpha}|$ and ${\bf
t}^{\alpha}$ is true,, then
$I[B]\cap({\mathscr{U}}^{\alpha}\setminus\bigcup\limits_{\beta<\alpha}{\mathscr{U}}^{\beta})=\emptyset,$
or just
$\eta^{\alpha}\notin I[B]$
(iii):
if $B\subseteq\lambda$, $B\cap|M^{\alpha}|=A_{1}\cap|M^{\alpha}|$ and ${\bf
t}^{\alpha}$ is false, then
$I[B]\cap({\mathscr{U}}^{\alpha}\setminus\bigcup\limits_{\beta<\alpha}{\mathscr{U}}^{\beta})=\\{\eta^{\alpha}\\}$
or just
$\eta^{\alpha}\in I[B]$
Case II: Not I.
No restriction is imposed.
The point is the two facts below which should be clear.
###### Fact 2.5.
The choice of $A_{1},A_{2},I_{1},I_{2},g$ is immaterial (any two candidates
lead to the same decision).
Proof: Use clause (d) of 2.3.
###### Fact 2.6.
$M_{I[A]}$ (for $A\subseteq\lambda$) are pairwise non-isomorphic. Moreover,
for $A\neq B$ (subsets of $\lambda$) there is no $(\pm\varphi)$–embedding of
$M_{I[A]}$ into $M_{I[B]}$.
Proof: By the choice of the I[A]’s and (i) of 2.4. $\hskip 5.0pt\hbox{\hskip
5.0pt\vrule width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{5.2}}$
$*\qquad*\qquad*$
Still the assumption of 2.4 is too strong: it do not cover all the desirable
cases, though it cover many of them. However, a statement weaker than the
conclusion of 2.4 holds under weaker cardinality restrictions and the proof of
2.3 above works using it, thus we will finish the proof of 2.3.
###### Fact 2.7.
Suppose $\lambda=\lambda^{\chi}$.
Then there are
$\\{(M^{\alpha},A^{\alpha}_{1},A^{\alpha}_{2},\eta^{\alpha}):\alpha<\alpha(\ast)\\}$
such that:
$(\ast)$:
(i):
for every model $M$ with universe ${\mathscr{H}}_{<\chi^{+}}(\lambda)$ such
that $|\tau(M)|\leq\chi$ and
$\tau(M)\subseteq{\mathscr{H}}_{<\chi^{*}}(\lambda)$ (arity of relations and
functions finite) and sets $A_{1}\neq A_{2}\subseteq\lambda$, for some
$\alpha<\alpha(\ast)$ we have
$(M^{\alpha},A^{\alpha}_{1},A^{\alpha}_{2})\prec(M,A_{1},A_{2})$,
(ii):
$\eta^{\alpha}\in{}^{\chi}\lambda$, $\\{\eta^{\alpha}\restriction
i:i<\chi\\}\subseteq|M^{\alpha}|$, $\eta^{\alpha}\notin M^{\alpha}$, and
$\alpha\neq\beta\ \Rightarrow\ \eta^{\alpha}\neq\eta^{\beta}$,
(iii):
for every $\beta<\alpha(\ast)$, if $\\{\eta^{\alpha}\restriction
i:i<\chi\\}\subseteq M^{\beta}$, then $\alpha<\beta+2^{\chi}$, and
$\alpha+2^{\chi}=\beta+2^{\chi}$ implies $A^{\alpha}_{1}\cap|M^{\alpha}|\neq
A^{\beta}_{2}\cap|M^{\alpha}|$,
(iv):
for every $\beta<\alpha$ if $\\{\eta^{\beta}\restriction i:i<\chi\\}\subseteq
M^{\alpha}$, then $|M^{\beta}|\subseteq|M^{\alpha}|$,
(v):
$\|M^{\alpha}\|=\chi$.
Proof: See 3.41.
Proof of 2.3: Should be clear, We act as in the proof of 2.3 from the
conclusion of 2.4 but now we have to use the ”or just” version in (ii),(iii)
there, $\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{5.1}}$
###### Conclusion 2.8.
1. (1)
If $T\subseteq T_{1}$ are complete first order theories, $T$ in the vocabulary
$\tau$, $\kappa=\mathop{\mathrm{cf}}(\kappa)<\kappa(T)$, hence $T$
unsuperstable and $\lambda=\lambda^{\aleph_{0}}\geq|T_{1}|$, then
${\dot{\mathbb{I}}}_{\tau}(\lambda,T_{1})=2^{\lambda}$
(${\dot{\mathbb{I}}}_{\tau}$ — see Definition [Sh:E59, 1.2(2)]).
2. (2)
Assume $\kappa=\mathop{\mathrm{cf}}(\kappa)$, $\Phi$ is proper and almost nice
for $K^{\kappa}_{{\mathord{\mathrm{tr}}}}$ , see [Sh:E59, 1.7],
${\bar{\sigma}}^{i}$ ($i\leq\kappa$) finite sequence of terms,
$\tau\subseteq\tau_{\Phi}$, $\varphi_{i}({\bar{x}},{\bar{y}})$ first order in
${\mathscr{L}}[\tau]$ and for $\nu\in{}^{i}\lambda$,
$\eta\in{}^{\kappa}\lambda$, $\nu\vartriangleleft\eta$ we have
$EM({}^{\kappa}\lambda,\Phi)\models\varphi_{i}({\bar{\sigma}}^{\kappa}_{i}(x_{\eta}),{\bar{\sigma}}^{i+1}(x_{\eta{}^{\frown}\\!\langle\alpha\rangle}))\
\mbox{ holds\quad{\rm{\text@underline{if and only if}}}\quad}\alpha=\eta(i).$
Then
$2^{\lambda}=|\\{EM_{\tau}(S,\Phi)/\cong:{}^{\kappa>}\lambda\subseteq
S\subseteq{}^{\kappa\geq}\lambda\\}|.$
Proof: (1) By [Sh:E59, 1.10] there is a template $\Phi$ proper for
$K^{\kappa}_{{\mathord{\mathrm{tr}}}}$, as required in part (2).
(2) By 2.3. $\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{5.6}}$
###### Discussion 2.9.
What about Theorem 2.3 in the case we assume only $\lambda=\lambda^{<\chi}$?
There is some information in [Sh:a, Ch.VIII, §2]. Of course, concerning
unsuperstable $T$, that is 2.8, more is done there: the assumption is just
$\lambda>|T|$.
###### Claim 2.10.
In 2.3, we can restrict ourselves to $I$ such that
$I^{0}_{\lambda,\chi}\subseteq I\subseteq{}^{\chi\geq}\lambda$, where
$I^{0}_{\lambda,\chi}={}^{\chi>}\lambda\cup\big{\\{}\eta\in{}^{\chi}\lambda:\eta(i)=0\mbox{
for every }i<\chi\mbox{ large enough}\big{\\}}.$
Proof: By renaming. $\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{2.12}}$
## 3\. Black Boxes
We try to give comprehensive treatment of black boxes, not few of them are
useful in some contexts and some parts are redone here, as explained in §0,
§1.
Note that “omitting countable types” is a very useful device for building
models of cardinality $\aleph_{0}$ and $\aleph_{1}$. The generalization to
models of higher cardinality, $\lambda$ or $\lambda^{+}$, usually requires us
to increase the cardinality of the types to $\lambda$, and even so we may
encounter problems (see [Sh:E60] and background there). Note we do not look
mainly at the omitting type theorem per se, but its applications.
Jensen defined square and proved existence in ${\bf L}$: in Facts 3.1 — 3.8,
we deal with related just weaker principles which can be proved in ZFC. E.g.,
for $\lambda$ regular $>\aleph_{1}$,
$\\{\delta<\lambda^{+}:\mathop{\mathrm{cf}}(\delta)<\lambda\\}$ is the union
of $\lambda$ sets, each has square (as defined there). You can skip them in
first reading, particularly 3.1 (and later take references on belief).
Then we deal with black boxes. In 3.11 we give the simplest case: $\lambda$
regular $>\aleph_{0}$, $\lambda=\lambda^{<\chi(\ast)}$; really
$\lambda^{<\theta}=\lambda^{<\chi(\ast)}$ is almost the same. In 3.11 we also
assume “$S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\theta\\}$
is a good stationary set”. In 3.15 we weaken this demand such that enough sets
$S$ as required exists (provably in ZFC!). The strength of the cardinality
hypothesis ($\lambda=\lambda^{<\chi(\ast)}$,
$\lambda^{<\theta}=\lambda^{<\chi(\ast)}$,
$\lambda^{\theta}=\lambda^{<\chi(\ast)}$) vary the conclusion. In 3.13 – 3.16
we prepare the ground for replacing “$\lambda$ regular” by
“$\mathop{\mathrm{cf}}(\lambda)\geq\chi(\ast)$”, which is done in 3.17.
As we noted in section 2, it is much nicer to deal with
$({\bar{M}}^{\beta},\eta^{\beta})$, this is the first time we deal with
$\eta^{\beta}$, i.e., for no $\alpha<\beta$,
$\\{\eta^{\beta}\restriction
i:i<\theta\\}\subseteq\bigcup_{i<\theta}M^{\alpha}_{i}.$
In 3.18, 3.19 (parallel to 3.11, 3.17, respectively) we guarantee this, at the
price of strengthening $\lambda^{<\theta}=\lambda^{<\chi(\ast)}$ to
$\lambda^{<\theta}=\lambda^{\chi(1)},\quad\chi(1)=\chi(\ast)+(<\chi(\ast))^{\theta}.$
Later, in 3.41, we draw the conclusion necessary for section 2 (in its proof
the function $h$, which may look redundant, plays the major role). This (as
well as 3.18, 3.19) exemplifies how those principles are self propagating —
better ones follow from the old variant (possibly with other parameters).
In 3.20 — 3.25 we deal with the black boxes when $\theta$ (the length of the
game) is $\aleph_{0}$. We use a generalization of the $\Delta$–system lemma
for trees and partition theorems on trees (see Rubin-Shelah [RuSh 117, §4],
[Sh:b, Ch.XI]=[Sh:f, Ch.XI] [Sh:E62, 1.16], [Sh:E62, 1.19] and here the proof
of 3.22; see history there, and 3.6). We get several versions of the black box
— as the cardinality restriction becomes more severe, we get a stronger
principle.
It would be better if we can use for a strong limit
$\kappa>\aleph_{0}=\mathop{\mathrm{cf}}(\kappa)$,
$\begin{array}[]{r}\kappa^{\aleph_{0}}=\mathop{\mathrm{sup}}\big{\\{}\lambda:\mbox{
for some }\kappa_{n}<\kappa\mbox{ and uniform ultrafilter }D\mbox{ on
}\omega,\\\
\mathop{\mathrm{cf}}\big{(}\prod\limits_{n<\omega}\kappa_{n}/D\big{)}=\lambda\big{\\}}.\end{array}$
We know this for the uncountable cofinality case (see [Sh 111] or [Sh:g]), but
then there are other obstacles. Now [Sh 355] gives a partial remedy, but
lately by [Sh 400] there are many such cardinals.
In 3.36, 3.37 we deal with the case $\mathop{\mathrm{cf}}(\lambda)\leq\theta$.
Note that $\mathop{\mathrm{cf}}(\lambda^{<\chi(\ast)})\geq\chi(\ast)$ is
always true, so you may wonder why wouldn’t we replace $\lambda$ by
$\lambda^{<\chi(\ast)}$? This is true in quite many applications, but is not
true, for example, when we want to construct structures with density character
$\lambda$.
Several times, we use results quoted from [Sh 331, §2], but no vicious circle.
Also, several times we quote results on pcf quoting [Sh:E62, §3] . We end with
various remarks and exercises.
###### Fact 3.1.
1. (1)
If $\mu^{\chi}=\mu<\lambda\leq 2^{\mu}$, $\chi$ and $\lambda$ are regular
uncountable cardinals, and
$S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\chi\\}$ is a
stationary set, then there are a stationary set $W\subseteq\chi$ and functions
$h_{a},h_{b}:\lambda\longrightarrow\mu$ and $\langle
S_{\zeta}:0<\zeta<\lambda\rangle$ such that:
(a):
$S_{\zeta}\subseteq S$ is stationary,
(b):
$\xi\neq\zeta\quad\Rightarrow\quad S_{\xi}\cap S_{\zeta}=\emptyset$,
(c):
if $\delta\in S_{\xi}$, then for some increasing continuous sequence
$\langle\alpha_{i}:i<\chi\rangle$ we have
$\delta=\bigcup\limits_{i<\chi}\alpha_{i},\quad h_{b}(\alpha_{i})=i,\quad
h_{a}(\alpha_{i})\in\\{\xi,0\\},$
and the set $\\{i<\chi:h_{a}(\alpha_{i})=\xi\\}$ is stationary, in fact is
$W$.
2. (2)
If in (1), a sequence $\langle
C_{\delta}:\delta<\lambda,\mathop{\mathrm{cf}}(\delta)\leq\chi\rangle$
satisfying
$(\forall\alpha\in C_{\delta})[\alpha\mbox{ limit }\ \Rightarrow\
\alpha=\mathop{\mathrm{sup}}(\alpha\cap C_{\delta})]$
is given, $C_{\delta}$ is closed unbounded subset of $\delta$ of order type
$\mathop{\mathrm{cf}}(\delta)$, then in the conclusion we can get also
$S^{\ast}$, $\langle C^{\ast}_{\delta}:\delta\in S^{\ast}\rangle$ such that
(a), (b), (c) hold, and
(c)′:
in (c) we add $C_{\delta}=\\{\alpha_{i}:i<\chi\\}$,
(d):
$\bigcup\limits_{0<\xi<\lambda}S_{\xi}\subseteq
S^{\ast}\subseteq\bigcup\limits_{0<\xi<\lambda}S_{\xi}\cup\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)<\chi\\}$,
(e):
$W$ is a $(>\aleph_{0})$-closed, stationary in cofinality $\aleph_{0}$, subset
of $\chi$, which means:
(i):
if $i<\chi$ is a limit ordinal, $i=\mathop{\mathrm{sup}}(i\cap W)$ has
cofinality $>\aleph_{0}$ then $i\in W$,
(ii):
$\\{i\in W:\mathop{\mathrm{cf}}(i)=\aleph_{0}\\}$ is a stationary111we can ask
$\notin I$ if $I$ is any normal ideal on
$\\{i<\chi:\mathop{\mathrm{cf}}(i)=\aleph_{0}\\}$ subset of $\chi$,
(f):
for $\delta\in\bigcup\limits_{0<\xi<\lambda}S_{\xi}$ we have
$C^{\ast}_{\delta}=\\{\alpha\in C_{\delta}:\mathop{\mathrm{otp}}(\alpha\cap
C_{\delta})=\mathop{\mathrm{sup}}(W\cap\mathop{\mathrm{otp}}(\alpha\cap
C_{\delta})),\\}$
(g):
$C^{\ast}_{\delta}$ is a club of $\delta$ included in $C_{\delta}$ for
$\delta\in S^{\ast}$, and if $\delta(1)\in C^{\ast}_{\delta}$, $\delta\in
S^{\ast}$, $\delta\in\bigcup\limits_{0<\zeta<\lambda}S_{\zeta}$,
$\delta(1)=\mathop{\mathrm{sup}}(\delta(1)\cap C^{\ast}_{\delta})$ and
$\mathop{\mathrm{cf}}(\delta(1))>\aleph_{0}$
then $C^{\ast}_{\delta(1)}\subseteq C^{\ast}_{\delta}$,
(h):
if $C$ is a closed unbounded subset of $\lambda$, and $0<\xi<\lambda$ then the
set
$\\{\delta\in S_{\xi}:C^{\ast}_{\delta}\subseteq C\\}$
is stationary.
Proof: (1) We can find $\\{\langle h^{1}_{\xi},h^{2}_{\xi}\rangle:\xi<\mu\\}$
such that:
(a):
for every $\xi$ we have $h^{1}_{\xi}:\lambda\longrightarrow\mu$ and
$h^{2}_{\xi}:\lambda\longrightarrow\mu$,
(b):
if $A\subseteq\lambda$, $|A|\leq\chi$, and $h^{1},h^{2}:A\longrightarrow\mu$,
then for some $\xi$, $h^{1}_{\xi}\restriction A=h^{1}$, and
$h^{2}_{\xi}\restriction A=h^{2}$.
This holds by Engelking-Karlowicz [EK] (see for example [Sh:c, AP]).
For $\delta<\lambda$ let $C_{\delta}$ be a closed unbounded subset of $\delta$
of order type $\mathop{\mathrm{cf}}(\delta)$. Now for each $\xi<\mu$ and a
stationary $a\subseteq\chi$ ask whether for every $i<\lambda$ for some
$j<\lambda$ we have
$(\ast)^{\xi,a}_{i,j}$:
the following subset of $\lambda$ is stationary:
$\begin{array}[]{ll}S^{\xi,a}_{i,j}=\\{\delta\in S:&\mbox{(i) if }\alpha\in
C_{\delta},\ \mathop{\mathrm{otp}}(\alpha\cap C_{\delta})\notin a\mbox{ then
}h^{1}_{\xi}(\alpha)=0,\\\ &\mbox{(ii) if }\alpha\in C_{\delta},\
\mathop{\mathrm{otp}}(\alpha\cap C_{\delta})\in a\mbox{ then the
$h^{1}_{\xi}(\alpha)$--th}\\\ &\quad\mbox{ member of $C_{\alpha}$ belongs to
$[i,j)$,}\\\ &\mbox{(iii) if }\alpha\in C_{\delta}\mbox{ then
}h^{2}_{\xi}(\alpha)=\mathop{\mathrm{otp}}(\alpha\cap C_{\delta})\ \
\\}\end{array}$
###### Subfact 3.2.
For some $\xi<\mu$ and a stationary set $a\subseteq\chi$, for every
$i<\lambda$ for some $j\in(i,\lambda)$, the statement $(\ast)^{\xi,a}_{i,j}$
holds.
Proof: If not, then for every $\xi<\mu$ and a stationary $a\subseteq\chi$, for
some $i=i(\xi,a)<\lambda$, for every $j<\lambda$, $j>i(\xi,a)$, there is a
closed unbounded subset $C(\xi,a,i,j)$ of $\lambda$ disjoint from
$S^{\xi,a}_{i,j}$.
Let
$i(\ast)=\bigcup\\{i(\xi,a)+\omega:\xi<\mu\mbox{ and }a\subseteq\chi\mbox{ is
stationary }\\}.$
Clearly $i(\ast)<\lambda$.
For $i(\ast)\leq j<\lambda$ let
$C(j)=\bigcap\\{C(\xi,a,i(\xi,a),j):\ a\subseteq\chi\mbox{ is stationary and
}\xi<\mu\\}\cap(i(\ast)+\omega,\lambda),$
clearly it is a closed unbounded subset of $\lambda$. Let
$C^{\ast}=\\{\delta<\lambda:\delta>i(\ast)\mbox{ and }(\forall
j<\delta)[\delta\in C(j)]\\}.$
So $C^{\ast}$ is a closed unbounded subset of $\lambda$, too. Let $C^{+}$ be
the set of accumulation points of $C^{\ast}$. Choose $\delta(\ast)\in
C^{+}\cap S$, and we shall define
$h^{1}:C_{\delta(\ast)}\longrightarrow\mu,\quad
h^{2}:C_{\delta(\ast)}\longrightarrow\mu.$
For $\alpha\in C_{\delta(\ast)}$ let $h^{0}(\alpha)$ be:
$\mathop{\mathrm{min}}\\{\gamma<\chi:\gamma>0\mbox{ and the }\gamma\mbox{--th
member of }C_{\alpha}\mbox{ is }>i(\ast)\\}$
if $\alpha=\mathop{\mathrm{sup}}(C_{\delta(*)}\cap\alpha)>i(\ast)$, and zero
otherwise. Clearly the set
$\\{\alpha\in C_{\delta(\ast)}:\ h^{0}(\alpha)=0\\}$
is not stationary. Now we can define
$g:C_{\delta(\ast)}\longrightarrow\delta(\ast)$ by:
$g(\alpha)\mbox{ is the }h^{0}(\alpha)\mbox{--th member of }C_{\alpha}.$
Note that $g$ is pressing down and $\\{\alpha\in
C_{\delta(\ast)}:g(\alpha)\leq i(\ast)\\}$ is not stationary. So (by the
variant of Fodor’s Lemma speaking on an ordinal of uncountable cofinality) for
some $j<\mathop{\mathrm{sup}}(C_{\delta(\ast)})=\delta(\ast)$ the set
$a=:\\{\alpha\in C_{\delta(\ast)}\cap C^{\ast}:i(\ast)<g(\alpha)<j\\}$
is a stationary subset of $\delta(\ast)$, and let
$h^{1}:C_{\delta(\ast)}\longrightarrow\mu$ be
$h^{1}(\alpha)=\left\\{\begin{array}[]{ll}0&\mbox{ if
}\mathop{\mathrm{otp}}(\alpha\cap C^{\ast}_{\delta})\notin a,\\\
h^{0}(\alpha)&\mbox{ if }\mathop{\mathrm{otp}}(\alpha\cap
C^{\ast}_{\delta})\in a.\end{array}\right\\}.$
Let $h^{2}:C_{\delta(\ast)}\rightarrow\mu$ be
$h^{2}(\alpha)=\mathop{\mathrm{otp}}(\alpha\cap C_{\delta(\ast)})$. By the
choice of $\langle(h^{1}_{\xi},h^{2}_{\xi}):\xi<\mu\rangle$, for some $\xi$,
we have $h^{1}_{\xi}\restriction C_{\delta(\ast)}=h^{1}$ and
$h^{2}_{\xi}\restriction C_{\delta(\ast)}=h^{2}$. Easily, $\delta(\ast)\in
S^{\xi,a}_{i,j}$ which is disjoint to $C(\xi,a,i(\ast),j)$, a contradiction to
$\delta(\ast)\in C^{\ast}$ by the definition of $C(j)$ and $C^{*}$.
So we have proved the subfact 3.2. $\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{6.1A}}$
Having chosen $\xi$, $a$ we define by induction on $\zeta<\lambda$ an ordinal
$i(\zeta)<\lambda$ such that $\langle i(\zeta):\zeta<\lambda\rangle$ is
increasing continuous, $i(0)=0$, and $(\ast)^{\xi,a}_{i(\zeta),i(\zeta+1)}$
holds.
Now, for $\alpha<\lambda$ we define $h_{a}(\alpha)$ as follows: it is $\zeta$
if $h^{1}_{\xi}(\alpha)>0$ and the $h^{1}_{\xi}(\alpha)$-th member of
$C_{\alpha}$ belongs to $[i(1+\zeta),i(1+\zeta+1))$, and it is zero otherwise.
Lastly, let $h_{b}(\alpha)=:h^{2}_{\xi}(\alpha)$ and $W=a$ and
$\begin{array}[]{ll}S_{\zeta}=:\big{\\{}\delta\in S:&\mbox{(i)\ \ \ for
}\alpha\in C_{\delta},\ \mathop{\mathrm{otp}}(\alpha\cap
C_{\delta})=h_{b}(\alpha),\\\ &\mbox{(ii)\ \ for }\alpha\in C_{\delta},\
h_{b}(i)\in a\ \Rightarrow\ h_{a}(\alpha)=\zeta,\\\ &\mbox{(iii)\ for
}\alpha\in C_{\delta},\ h_{b}(i)\notin a\ \Rightarrow
h_{a}(i)=0\quad\big{\\}}.\end{array}$
Now, it is easy to check that $a$, $h_{a}$, $h_{b}$, and $\langle
S_{\zeta}:0<\zeta<\lambda\rangle$ are as required.
(2) In the proof of 3.1(1) we shall now consider only sets $a\subseteq\chi$
which satisfy the demand in clause $(e)$ of 3.1(2) on $W$ [i.e., in the
definition of $C(j)$ during the proof of Subfact 3.2 this makes a difference].
Also in $(\ast)^{\xi,a}_{i,j}$ in the definition of $S^{\xi,a}_{i,j}$ we
change (iii) to
(iii)′:
if $\alpha\in C_{\delta}$, $h^{2}_{\xi}(\alpha)$ codes the isomorphism type
of, for example,
$\left(C_{\delta}\cup\bigcup\limits_{\beta\in{C_{\delta}}}C_{\beta},<,\alpha,C_{\delta},\\{\langle
i,\beta\rangle:i\in C_{\beta}\\}\right).$
In the end, having chosen $\xi$, $a$ we can define $C^{\ast}_{\delta}$ and
$S^{\ast}$ in the natural way. $\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{6.1}}$
###### Fact 3.3.
1. (1)
If $\lambda$ is regular $>2^{\kappa}$, $\kappa$ regular,
$S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\kappa\\}$ is
stationary and for $\delta\in S$, $C^{0}_{\delta}$, is a club of $\delta$ of
order type $\kappa$ ($=\mathop{\mathrm{cf}}(\delta)$),
then we can find a club $c^{\ast}$ of $\kappa$ (see 3.4(1)) such that letting
for $\delta\in S$, $C_{\delta}=C_{\delta}[c^{\ast}]=:\\{\alpha\in
C^{0}_{\delta}:\mathop{\mathrm{otp}}(C^{0}_{\delta}\cap\alpha)\in
c^{\ast}\\}$, it is a club of $\delta$ and
$(\ast)$:
for every club $C\subseteq\lambda$ we have:
(a):
if $\kappa>\aleph_{0}$, $\\{\delta\in S:C_{\delta}\subseteq C\\}$ is
stationary,
(b):
if $\kappa=\aleph_{0}$, then the set
$\\{\delta\in S:(\forall\alpha,\beta)[\alpha<\beta\wedge\alpha\in
C_{\delta}\wedge\beta\in C_{\delta}\ \Rightarrow\ (\alpha,\beta)\cap
C\neq\emptyset]\\}$
is stationary.
2. (2)
If $\lambda$ is a regular cardinal $>2^{\kappa}$, then we can find
$\langle\langle C^{\zeta}_{\delta}:\delta\in
S_{\zeta}\rangle:\zeta<2^{\kappa}\rangle$ such that:
(a):
$\bigcup\\{S_{\zeta}:\zeta<2^{\kappa}\\}=\\{\delta<\lambda:\aleph_{0}<\mathop{\mathrm{cf}}(\delta)\leq\kappa\\}$,
(b):
$C^{\zeta}_{\delta}$ is a club of $\delta$ of order type
$\mathop{\mathrm{cf}}(\delta)$,
(c):
if $\alpha\in S_{\zeta}$, $\mathop{\mathrm{cf}}(\alpha)>\theta>\aleph_{0}$,
then
$\\{\beta\in C^{\zeta}_{\alpha}:\mathop{\mathrm{cf}}(\beta)=\theta,\ \beta\in
S_{\zeta}\mbox{ and }C^{\zeta}_{\beta}\subseteq C^{\zeta}_{\alpha}\\}$
is a stationary subset of $\alpha$.
3. (3)
If $\lambda$ is regular, $2^{\mu}\geq\lambda>\mu^{\kappa}$, then we can find
$\langle\langle C^{\zeta}_{\delta}:\delta\in
S_{\zeta}\rangle:\zeta<\mu\rangle$ such that:
(a):
$\bigcup\\{S_{\zeta}:\zeta<2^{\kappa}\\}=\\{\delta<\lambda:\aleph_{0}<\mathop{\mathrm{cf}}(\delta)\leq\kappa\\}$,
(b):
$C^{\zeta}_{\delta}$ is a club of $\delta$ of order type
$\mathop{\mathrm{cf}}(\delta)$,
(c):
if $\alpha\in S_{\zeta}$, $\beta\in C^{\zeta}_{\alpha}$,
$\mathop{\mathrm{cf}}(\beta)>\aleph_{0}$, then $\beta\in S_{\zeta}$ and
$C^{\zeta}_{\beta}\subseteq C^{\zeta}_{\alpha}$,
(d):
moreover, if $\alpha$, $\beta\in S_{\zeta}$, $\beta\in C^{\zeta}_{\alpha}$,
then
$\\{(\mathop{\mathrm{otp}}(\gamma\cap
C^{\zeta}_{\beta}),\mathop{\mathrm{otp}}(\gamma\cap
C^{\zeta}_{\alpha})):\gamma\in C_{\beta}\\}$
depends on $(\mathop{\mathrm{otp}}(\beta\cap
C_{\alpha}),\mathop{\mathrm{otp}}(C_{\alpha}))$ only.
4. (4)
We can replace in (1)(a) and (b) of $(\ast)$ “stationary” by “$\notin I$” for
any normal ideal I on $\lambda$.
###### Remark 3.4.
1. (1)
A club $C$ of $\delta$ where $\mathop{\mathrm{cf}}(\delta)=\aleph_{0}$ means
here just an unbounded subset of $\delta$.
2. (2)
In 3.3(1) instead of $2^{\kappa}$, the cardinal
$\mathop{\mathrm{min}}\\{|{\mathscr{F}}|:{\mathscr{F}}\subseteq{}^{\kappa}\kappa\
\&\ (\forall g\in{}^{\kappa}\kappa)(\exists f\in
F)(\forall\alpha<\kappa)[g(\alpha)<f(\alpha)]\\}$
suffices.
3. (3)
In (b) above, it is equivalent to ask
$\\{\delta\in S:(\forall\alpha,\beta)[\alpha<\beta\wedge\alpha\in
C_{\delta}\wedge\beta\in C_{\delta}\ \Rightarrow\
\mathop{\mathrm{otp}}((\alpha,\beta)\cap C)>\alpha]\\}$
is stationary.
Proof: (1) If 3.3(1) fails, for each club $c^{\ast}$ of $\kappa$ there is a
club $C[c^{\ast}]$ of $\lambda$ exemplifying its failure. So
$C^{+}=\bigcap\\{C[c^{\ast}]:c^{\ast}\subseteq\kappa$ a club$\\}$ is a club of
$\lambda$. Choose $\delta\in S$ which is an accumulation point of $C^{+}$ and
get contradiction easily.
(2) Let $\lambda=\mathop{\mathrm{cf}}(\lambda)>2^{\kappa}$, $C_{\alpha}$ be a
club of $\alpha$ of order type $\mathop{\mathrm{cf}}(\alpha)$, for each limit
$\alpha<\lambda$. Without loss of generality
$\beta\in C_{\alpha}\ \&\ \beta>\mathop{\mathrm{sup}}(\beta\cap
C_{\alpha})\quad\Rightarrow\quad\beta\mbox{ is a successor ordinal.}$
For any sequence ${\bar{c}}=\langle
c_{\theta}:\aleph_{0}<\theta=\mathop{\mathrm{cf}}(\theta)\leq\kappa\rangle$
such that each $c_{\theta}$ is a club of $\theta$, for $\delta\in
S^{\ast}=\\{\alpha<\lambda:\aleph_{0}<\mathop{\mathrm{cf}}(\alpha)\leq\kappa\\}$
we let:
$C^{\bar{c}}_{\delta}=\left\\{\alpha\in
C_{\delta}:\mathop{\mathrm{otp}}(C_{\delta}\cap\alpha)\in
c_{\mathop{\mathrm{cf}}(\delta)}\right\\}.$
Now we define $S_{\bar{c}}$, by defining by induction on $\delta<\lambda$, the
set $S_{\bar{c}}\cap\delta$; the only problem is to define whether $\alpha\in
S_{\bar{c}}$ knowing $S_{\bar{c}}\cap\delta$; now
$\begin{array}[]{lcl}\alpha\in S_{\bar{c}}&\mbox{ {\text@underline{if and only
if}}}&\mbox{(i) }\quad\aleph_{0}<\mathop{\mathrm{cf}}(\alpha)\leq\kappa,\\\
&&\mbox{(ii)}\quad\mbox{if
}\aleph_{0}<\theta=\mathop{\mathrm{cf}}(\theta)<\mathop{\mathrm{cf}}(\alpha)\\\
&&\mbox{\ \ \ \ \quad then the set }\\{\beta\in
C^{\bar{c}}_{\alpha}:\mathop{\mathrm{cf}}(\beta)=\theta,\ \beta\in
S_{\bar{c}}\cap\alpha\\}\\\ &&\mbox{\ \ \ \ \quad is stationary in
}\alpha.\end{array}$
Let $\langle{\bar{c}}^{\zeta}:\zeta<2^{\kappa}\rangle$ list the possible
sequences ${\bar{c}}$, and let $S_{\zeta}=S_{{\bar{c}}^{\zeta}}$ and
$C^{\zeta}_{\delta}=C^{{\bar{c}}^{\zeta}}_{\delta}$. To finish, note that for
each $\delta<\lambda$ satisfying
$\aleph_{0}<\mathop{\mathrm{cf}}(\delta)\leq\kappa$, for some $\zeta$,
$\delta\in S_{\zeta}$.
(3) Combine the proof of (2) and of 3.1.
(4) Similarly. $\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{6.2}}$
We may remark
###### Fact 3.5.
Suppose that $\lambda$ is a regular cardinal $>2^{\kappa}$,
$\kappa=\mathop{\mathrm{cf}}(\kappa)>\aleph_{0}$, a set
$S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\kappa\\}$
is stationary, and $I$ is a normal ideal on $\lambda$ and $S\notin I$. If $I$
is $\lambda^{+}$–saturated (i.e., in the Boolean algebra
${\mathscr{P}}(\lambda)/I$, there is no family of $\lambda^{+}$ pairwise
disjoint elements), then we can find $\langle C_{\delta}:\delta\in S\rangle$,
$C_{\delta}$ a club of $\delta$ of order type $\mathop{\mathrm{cf}}(\delta)$,
such that:
$(\ast)$:
for every club $C$ of $\lambda$ we have $\\{\delta\in S:C_{\delta}\setminus
C\mbox{ is unbounded in }\delta\\}\in I$.
Proof: For $\delta\in S$, let $C^{\prime}_{\delta}$ be a club of $\delta$ of
order type $\mathop{\mathrm{cf}}(\delta)$. Call ${\bar{C}}=\langle
C_{\delta}:\delta\in S^{\ast}\rangle$ (where $S^{\ast}\subseteq\lambda$
stationary, $S^{\ast}\notin I$, $C_{\delta}$ a club of $\delta$) $I$-large if:
for every club $C$ of $\lambda$ the set
$\\{\delta<\lambda:\delta\in S^{\ast}\mbox{ and }C_{\delta}\setminus C\mbox{
is bounded in }\delta\\}$
does not belong to $I$.
We call $\bar{C}$ $I$-full if above $\\{\delta\in S^{\ast}:C_{\delta}\setminus
C$ unbounded in $\delta\\}\in I$.
3.3(4) for every stationary $S^{\prime}\subseteq S$, $S^{\prime}\notin I$
there is a club $c^{\ast}$ of $\kappa$ such that $\langle
C^{\prime}_{\delta}[c^{\ast}]:\delta\in S^{\prime}\rangle$ is $I$–large. Now
note:
$(\ast)$:
if $\langle C_{\delta}:\delta\in S^{\prime}\rangle$ is $I$-large,
$S^{\prime}\subseteq S$, then for some $S^{\prime\prime}\subseteq S^{\prime}$,
$S^{\prime\prime}\notin I$, $\langle C_{\delta}:\delta\in
S^{\prime\prime}\rangle$ is $I$–full (hence $S^{\prime\prime}\notin I)$.
[Proof of $(\ast)$: Choose by induction on $\alpha<\lambda^{+}$, a club
$C^{\alpha}$ of $\lambda$ such that:
(a):
for $\beta<\alpha$, $C^{\alpha}\setminus C^{\beta}$ is bounded in $\lambda$,
(b):
if $\beta=\alpha+1$ then $A_{\beta}\setminus A_{\alpha}\in I^{+}$, where
$A_{\gamma}=:\\{\delta\in S^{\prime}:C_{\delta}\setminus C^{\gamma}\mbox{ is
unbounded in }\delta\\}.$
As clearly
$\beta<\alpha\quad\Rightarrow\quad A_{\beta}\setminus A_{\alpha}\mbox{ is
bounded in }\lambda$
(by (a) and the definition of $A_{\alpha},A_{\beta}$) and as $I$ is
$\lambda^{+}$–saturated, clearly for some $\alpha$ we cannot define
$C^{\alpha}$. This cannot be true for $\alpha=0$ or a limit $\alpha$, so
necessarily $\alpha=\beta+1$. Now $S^{\prime}\setminus A_{\beta}$ is not in
$I$ as ${\bar{C}}$ was assumed to be $I$–large. Check that
$S^{\prime\prime}=:S^{\prime}\setminus A_{\beta}$ is as required.]
Using repeatedly 3.3(4) and $(\ast)$ we get the conclusion $\hskip
5.0pt\hbox{\hskip 5.0pt\vrule width=4.0pt,height=6.0pt,depth=1.5pt\hskip
1.0pt}_{\ref{6.2B}}$
###### Claim 3.6.
Suppose $\lambda=\mu^{+}$, $\mu=\mu^{\chi}$, $\chi$ is a regular cardinal and
$S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\chi\\}$ is
stationary. Then we can find $S^{\ast}$, $\langle C_{\delta}:\delta\in
S^{\ast}\rangle$ and $\langle S_{\xi}:\xi<\lambda\rangle$ such that:
(a):
$\bigcup\limits_{\zeta<\mu}S_{\zeta}\subseteq S^{\ast}\subseteq
S\cup\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)<\chi\\}$,
(b):
$S_{\zeta}\cap S$ is a stationary subset of $\lambda$ for each $\zeta<\mu$,
(c):
for $\alpha\in S^{\ast}$, $C_{\alpha}$ is a closed subset of $\alpha$ of order
type $\leq\chi$, if $\alpha\in S^{\ast}$ is a limit then $C_{\alpha}$ is
unbounded in $\alpha$ (so is a club of $\alpha$),
(d):
$\langle C_{\alpha}:\alpha\in S_{\zeta}\rangle$ is a square on $S_{\zeta}$,
i.e., $(S_{\zeta}$ is stationary in $\mathop{\mathrm{sup}}(S_{\zeta})$ and):
(i):
$C_{\alpha}$ is a closed subset of $\alpha$, unbounded if $\alpha$ is limit,
(ii):
if $\alpha\in S_{\zeta}$, $\alpha(1)\in C_{\alpha}$ then $\alpha(1)\in
S_{\zeta}$ and $C_{\alpha(1)}=C_{\alpha}\cap\alpha(1)$,
(e):
for each club $C$ of $\lambda$ and $\zeta<\mu$, for some $\delta\in
S_{\zeta}$, $C_{\delta}\subseteq C$.
Proof: Similar to the proof of 3.1 (or see [Sh 237e]).
We shall use in 3.25
###### Claim 3.7.
Suppose $\lambda=\mu^{+}$, $\gamma$ a limit ordinal of cofinality $\chi$,
$h:\gamma\longrightarrow\\{\theta:\theta=1\mbox{ or
}\theta=\mathop{\mathrm{cf}}(\theta)\leq\mu\\},$
$\mu=\mu^{|\gamma|}$, and
$S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\chi\\}$ is
stationary. Then we can find $S^{\ast}$, $\langle C_{\delta}:\delta\in
S^{\ast}\rangle$ and $\langle S_{\zeta}:\zeta<\lambda\rangle$ such that:
(a):
$\bigcup\limits_{\zeta<\lambda}S_{\zeta}\subseteq
S^{\ast}\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)\leq\chi\\}$,
(b):
$S_{\zeta}\cap S$ is stationary for each $\zeta<\lambda$,
(c):
for $\delta\in S^{\ast}$,
(i):
$C_{\delta}$ is a club of $\delta$ of order type $\leq\gamma$ and
(ii):
$\mathop{\mathrm{otp}}(C_{\delta})=\gamma\iff\delta\in S\cap S^{\ast}$,
(iii):
$\alpha\in
C_{\delta}\wedge\mathop{\mathrm{sup}}(C_{\delta}\cap\alpha)<\alpha\quad\Rightarrow\quad\alpha$
has cofinality $h[\mathop{\mathrm{otp}}(C_{\delta}\cap\alpha)]$,
(d):
if $\delta\in S_{\zeta}$, $\delta(1)$ a limit ordinal $\in C_{\delta}$ then
$\delta(1)\in S_{\zeta}$ and $C_{\delta(1)}=C_{\delta}\cap\delta(1)$,
(e):
for each club $C$ of $\lambda$ and $\zeta<\lambda$ for some $\delta\in
S_{\zeta}$, $C_{\delta}\subseteq C$.
Proof: Like 3.6.
###### Claim 3.8.
(1):
Suppose $\lambda$ is regular $>\aleph_{1}$, then
$\\{\delta<\lambda^{+}:\mathop{\mathrm{cf}}(\delta)<\lambda\\}$ is a good
stationary subset of $\lambda^{+}$ (i.e., it is in ${\check{I}^{\rm
gd}}[\lambda^{+}]$, see [Sh:E62, 3.3] or [Sh 88r, 0.6,0.7] or 3.9(2) below).
(2):
Suppose $\lambda$ is regular $>\aleph_{1}$. Then we can find $\langle
S_{\zeta}:\zeta<\lambda\rangle$ such that:
(a):
$\bigcup\limits_{\zeta<\lambda}S_{\zeta}=\\{\alpha<\lambda^{+}:\mathop{\mathrm{cf}}(\alpha)<\lambda\\}$,
(b):
on each $S_{\zeta}$ there is a square (see 3.6 clause (d)), say it is $\langle
C^{\zeta}_{\alpha}:\alpha\in S_{\zeta}\rangle$ with
$|C^{\zeta}_{\delta}|<\lambda$,
(c):
if $\delta(\ast)<\lambda$, and $\kappa=\mathop{\mathrm{cf}}(\kappa)<\lambda$,
then: for some $\zeta<\lambda$ for every club $C$ of $\lambda^{+}$, for some
accumulation point $\delta$ of $C$, $\mathop{\mathrm{cf}}(\delta)=\kappa$ and
$\mathop{\mathrm{otp}}(C^{\zeta}_{\delta}\cap C)$ is divisible by
$\delta(\ast)$,
(d):
if $\mathop{\mathrm{cf}}(\delta(\ast))=\kappa$, we can add in (c)’s
conclusion:
$C^{\zeta}_{\delta}\subseteq C\ \mbox{ and }\
\mathop{\mathrm{otp}}(C^{\zeta}_{\delta})=\delta(\ast).$
###### Remark 3.9.
1. (1)
For $\lambda=\aleph_{1}$ the conclusion of 3.8(1), (2)(a), (b) becomes totally
trivial; but for $\delta<\omega_{1}$, it means something if we add:
$\\{\alpha\in S_{\zeta}:\mathop{\mathrm{otp}}(C^{\zeta}_{\alpha})=\delta\\}\ \
\mbox{ is stationary and}$
for every club $C$ of $\lambda$ the set
$\\{\alpha\in
S_{\delta}:\mathop{\mathrm{otp}}(C^{\zeta}_{\alpha})=\delta,C^{\zeta}_{\alpha}\subseteq
C\\}\ \ \mbox{ is stationary.}$
So 3.8(2)(c,d) are not so trivial, but still true. Their proofs are similar so
we leave them to the reader (used only in [Sh 331, 2.7]).
2. (2)
Recall that for a regular uncountable cardinal $\mu$, the family
${\check{I}^{\rm gd}}[\mu]$ of good subsets of $\mu$ is the family of
$S\subseteq\mu$ such that there are a sequence $\bar{a}=\langle
a_{\alpha}:\alpha<\lambda\rangle$ and a club $C\subseteq\mu$ satisfying:
$a_{\alpha}\subseteq\alpha$ of order type $<\alpha$ when $\lambda$ is a
successor cardinal , $\beta\in a_{\alpha}\ \Rightarrow\
a_{\beta}=a_{\alpha}\cap\beta$ and
$(\forall\delta\in S\cap C)(\mbox{{\rm sup}}(a_{\delta})=\delta\ \&\
\mathop{\mathrm{otp}}(a_{\alpha})=\mathop{\mathrm{cf}}(\delta)).$
We may say that the sequence $\bar{a}$ as above exemplifies that $S$ is good;
if $C=\mu$ we say “explicitly exemplifies”.
Proof: Appears also in detail in [Sh 351] (originally proved for this work but
as its appearance was delayed we put it there, too). Of course,
(1) follows from (2).
(2) Let $S=\\{\alpha<\lambda^{+}:\mathop{\mathrm{cf}}(\alpha)<\lambda\\}$. For
each $\alpha\in S$ choose $\bar{A}^{\alpha}$ such that:
$(\alpha)$:
$\bar{A}^{\alpha}=\langle A^{\alpha}_{i}:i<\lambda\rangle$ is an increasing
continuous sequence of subsets of $\alpha$ of cardinality $<\lambda$, such
that $\bigcup\limits_{i<\lambda}A^{\alpha}_{i}=\alpha\cap S$,
$(\beta)$:
if $\beta\in A^{\alpha}_{i}\cup\\{\alpha\\}$ , $\beta$ is a limit ordinal and
$\mathop{\mathrm{cf}}(\beta)<\lambda$
(the last actually follows), then
$\beta=\mathop{\mathrm{sup}}(A^{\alpha}_{i}\cap\beta)$,
$(\gamma$):
if $\beta\in A^{\alpha}_{i}\cup\\{\alpha\\}$ is limit and
$\aleph_{0}<\mathop{\mathrm{cf}}(\beta)<\lambda$
then $A^{\alpha}_{i}$ contains a club of $\beta$,
$(\delta)$:
$0\in A^{\alpha}_{i}$ and $\beta\in S\ \&\ \beta+1\in
A^{\alpha}_{i}\cup\\{\alpha\\}\quad\Rightarrow\quad\beta\in A^{\alpha}_{i}$,
$(\varepsilon)$:
the closure of $A^{\alpha}_{i}$ in $\alpha$ (in the order topology) is
included in $A^{\alpha}_{i+1}$.
There are no problems with choosing $\bar{A}^{\alpha}$ as required.
We define $B^{\alpha}_{i}$ (for $i<\lambda$, $\alpha\in S$) by induction on
$\alpha$ as follows:
$B^{\alpha}_{i}=\left\\{\begin{array}[]{ll}{\rm
closure}(A^{\alpha}_{i})\cap\alpha&\mbox{ if
}\mathop{\mathrm{cf}}(\alpha)\neq\aleph_{1},\\\
\bigcap\\{\bigcup\limits_{\beta\in C}B^{\beta}_{i}:C\mbox{ a club of
}\alpha\\}&\mbox{ if
}\mathop{\mathrm{cf}}(\alpha)=\aleph_{1}.\end{array}\right.$
For $\zeta<\lambda$ we let:
$\begin{array}[]{lll}S_{\zeta}=\left\\{\alpha\in S:\right.&\alpha\mbox{
satisfies}&\mbox{(i) }B^{\alpha}_{\zeta}\mbox{ is a closed subset of
}\alpha,\\\ &&\mbox{(ii) if }\beta\in B^{\alpha}_{\zeta},\mbox{ then
}B^{\beta}_{\zeta}=B^{\alpha}_{\zeta}\cap\beta\mbox{ and}\\\ &&\mbox{(iii) if
}\alpha\mbox{ is limit, then
}\alpha=\mathop{\mathrm{sup}}(B^{\alpha}_{\zeta})\left.\right\\}\end{array}$
and for $\alpha\in S_{\zeta}$ let $C^{\zeta}_{\alpha}=B^{\alpha}_{\zeta}$.
Now, demand (b) holds by the choice of $S_{\zeta}$. To prove clause (a) we
shall show that for any $\alpha\in S$, for some $\zeta<\lambda$, $\alpha\in
S_{\zeta}$; moreover we shall prove
$(*)^{0}_{\alpha}$:
$E_{\alpha}:=\\{\zeta<\lambda:\mbox{ if
}\mathop{\mathrm{cf}}(\zeta)=\aleph_{1}\mbox{ then }\alpha\in
S_{\zeta}\\}\mbox{ contains a club of }\lambda$
For $\alpha\in S$ define $E^{0}_{\alpha}=\\{\zeta<\lambda:\mbox{ if
}\mathop{\mathrm{cf}}(\zeta)={\aleph_{1}}\mbox{ then }B^{\alpha}_{\zeta}={\rm
closure}(A^{\alpha}_{\zeta})\cap\alpha\\}$. We prove by induction on
$\alpha\in S$ that $E_{\alpha}\cap E^{0}_{\alpha}$ contains a club of
$\lambda$ and we then choose such a club $E^{1}_{\alpha}$. Arriving to
$\alpha$, let
$E=\\{\zeta<\lambda:\mbox{ if }\beta\in A^{\alpha}_{\zeta}\mbox{ then
}\zeta\in E^{1}_{\beta}\mbox{ and
}A^{\beta}_{\zeta}=A^{\alpha}_{\zeta}\cap\beta\\}.$
Clearly $E$ is a club of $\lambda$. Let $\zeta\in E$,
$\mathop{\mathrm{cf}}(\zeta)=\aleph_{1}$, and we shall prove that $\alpha\in
S_{\zeta}\cap E_{\alpha}\cap E^{0}_{\alpha}$, this clearly suffices. By the
choice of $\zeta$ (and the definition of $E$) we have: if $\beta$ belongs to
$A^{\alpha}_{\zeta}$ then $A^{\beta}_{\zeta}=A^{\alpha}_{\zeta}\cap A$ and
$B^{\beta}_{\zeta}={\rm closure}(A^{\beta}_{\zeta})\cap\beta$, so
$(*)_{1}$:
$\beta\in A^{\alpha}_{\zeta}\ \Rightarrow\ B^{\beta}_{\zeta}={\rm
closure}(A^{\alpha}_{\zeta})\cap\beta$.
Let us check the three conditions for “$\alpha\in S_{\zeta}$” this will
suffice for clause (a) of the claim.
Clause (i): $B^{\alpha}_{\zeta}$ is a closed subset of $\alpha$.
If $\mathop{\mathrm{cf}}(\alpha)\neq\aleph_{1}$ then $B^{\alpha}_{\zeta}={\rm
closure}(A^{\alpha}_{\zeta})\cap\alpha$, hence necessarily it is a closed
subset of $\alpha$.
If $\mathop{\mathrm{cf}}(\alpha)=\aleph_{1}$ then
$B^{\alpha}_{\zeta}=\bigcap\\{\bigcup\limits_{\beta\in
C}B^{\beta}_{\zeta}:C\mbox{ is a club of }\beta\\}$. Now, for any club $C$ of
$\beta$, $C\cap A^{\alpha}_{\zeta}$ is a club of $\alpha$ (see clause
$(\gamma)$ above). By $(*)_{1}$ above,
$\bigcup_{\beta\in C}B^{\beta}_{\zeta}\supseteq\bigcup_{\beta\in C\cap
A^{\alpha}_{\zeta}}B^{\beta}_{\zeta}={\rm
closure}(A^{\alpha}_{\zeta})\cap\beta.$
Note that we have gotten
$(*)_{2}$:
$\alpha\in E^{0}_{\zeta}$.
[Why? If $\mathop{\mathrm{cf}}(\alpha)=\aleph_{1}$ see above, if
$\mathop{\mathrm{cf}}(\alpha)\neq\aleph_{1}$ this is trivial.]
Clause (ii): If $\beta\in B^{\alpha}_{\zeta}$ then
$B^{\beta}_{\zeta}=B^{\alpha}_{\zeta}\cap\beta$.
We know that $B^{\alpha}_{\zeta}={\rm closure}(A^{\alpha}_{\zeta})\cap\alpha$,
by $(*)_{2}$ above. If $\beta\in A^{\alpha}_{\zeta}$ then (by $(*)_{1}$) we
have $B^{\beta}_{\zeta}={\rm closure}(A^{\alpha}_{\zeta})\cap\beta$, so we are
done. So assume $\beta\notin A^{\alpha}_{\zeta}$. Then, by clause $(\epsilon)$
necessarily
$\varepsilon<\zeta\quad\Rightarrow\quad\beta>\mathop{\mathrm{sup}}(A^{\alpha}_{\varepsilon}\cap\beta)\
\mbox{ and }\ \mathop{\mathrm{sup}}(A^{\alpha}_{\varepsilon}\cap\beta)\in
A^{\alpha}_{\varepsilon+1}\subseteq A^{\alpha}_{\zeta}.$
But $\beta\in B^{\alpha}_{\zeta}={\rm closure}(A^{\alpha}_{\zeta})$ by
$(*)_{2}$, hence together $A^{\alpha}_{\zeta}$ contains a club of $\beta$ and
$\mathop{\mathrm{cf}}(\beta)=\mathop{\mathrm{cf}}(\zeta)$, but
$\mathop{\mathrm{cf}}(\zeta)=\aleph_{1}$, so
$\mathop{\mathrm{cf}}(\beta)=\aleph_{1}$. Now, as in the proof of clause (i),
we get $B^{\beta}_{\zeta}=\bigcup\\{B^{\gamma}_{\zeta}:\gamma\in
A^{\alpha}_{\zeta}\cap\beta\\}$, so by the induction hypothesis we are done.
Clause (iii): If $\alpha$ is limit then
$\alpha=\mathop{\mathrm{sup}}(A^{\alpha}_{i})$.
By clause $(\gamma)$ we know $A^{\alpha}_{\zeta}$ is unbounded in $\alpha$,
but $A^{\alpha}_{\zeta}\subseteq B^{\alpha}_{\zeta}$ (by $(*)_{2}$) and we are
done.
So we have finished proving $(*)^{0}_{\alpha}$ by induction on $\alpha$ hence
clause (a) of the claim.
For proving (c) of 3.8(2), note that above, if $\alpha$ is limit, $C$ is a
club of $\alpha$, $C\subseteq S$, and $|C|<\lambda$, then for every $i$ large
enough, $C\subseteq A^{\alpha}_{i}$, and even $C\subseteq B^{\alpha}_{i}$.
Now assume that the conclusion of (c) fails (for fixed $\delta(\ast)$ and
$\kappa$). Then for each $\zeta<\lambda$ we have a club $E^{0}_{\zeta}$
exemplifying it. Now, $E^{0}=:\bigcap\limits_{\zeta<\lambda}E^{0}_{\zeta}$ is
a club of $\lambda^{+}$, hence for some $\delta\in E^{0}$,
$\mathop{\mathrm{otp}}(E^{0}\cap\delta)$ is divisible by $\delta(\ast)$ and
$\mathop{\mathrm{cf}}(\delta)=\kappa$. Choose an unbounded in $\delta$ set
$e\subseteq E^{0}\cap\delta$ of order type divisible by $\delta(*)$. Then, for
a final segment of $\zeta<\lambda$ we have $e\cap\delta\subseteq
C^{\zeta}_{\delta}$.
Note that for any set $C_{1}$ of ordinals, $\mathop{\mathrm{otp}}(C_{1})$ is
divisible by $\delta(\ast)$ if $C_{1}$ has an unbounded subset of order type
divisible by $\delta(\ast)$, so we get a contradiction because by
$(*)^{0}_{\delta(\ast))}$ for some $\zeta\in E_{\delta(\ast)}$ ( so
$\delta(\ast)\in S_{\zeta}$) by $E^{0}_{\zeta}\cap C^{\zeta}_{\delta}\supseteq
E^{0}\cap\delta\supseteq e$, $\mathop{\mathrm{sup}}(e)=\delta$ and $e$ has
order type divisible by $\delta(\ast)$.
We are left with clause (d) of 3.8(2). Fix $\kappa$ , $\delta(\ast)$ and
$\zeta$ as above, we may add $\leq\lambda$ new sequences of the form $\langle
C_{\alpha}:\alpha\in S_{\zeta}\rangle$ as long as each is a square. First
assume that for every $\gamma$, $\beta<\lambda$, such that
$\mathop{\mathrm{cf}}(\beta)=\kappa=\mathop{\mathrm{cf}}(\gamma)$, $\gamma$
divisible by $\delta(\ast)$ we have
$(\ast)^{3}_{\beta,\gamma}$:
there is a club $E_{\beta,\gamma}$ of $\lambda^{+}$ such that for no
$\delta\in S_{\zeta}$ do we have
$\mathop{\mathrm{otp}}(C^{\zeta}_{\delta})=\beta$ and
$\mathop{\mathrm{otp}}(C^{\zeta}_{\delta}\cap E_{\beta,\gamma})=\gamma$,
then let
$E=:\bigcap\\{E_{\beta,\gamma}:\gamma<\lambda,\ \beta<\lambda,\
\mathop{\mathrm{cf}}(\beta)=\kappa=\mathop{\mathrm{cf}}(\gamma),\ \gamma\mbox{
divisible by }\delta(\ast)\\}.$
Applying part (c) we get a contradiction.
So for some $\gamma$, $\beta<\lambda$,
$\mathop{\mathrm{cf}}(\beta)=\kappa=\mathop{\mathrm{cf}}(\gamma)$, $\gamma$
divisible by $\delta(\ast)$ and $(\ast)^{3}_{\beta,\gamma}$ fails. Also there
is a club $E^{\ast}$ of $\lambda^{+}$ such that for every club $E\subseteq
E^{\ast}$ for some $\delta\in S_{\zeta}$,
$\mathop{\mathrm{otp}}(C^{\zeta}_{\delta})=\beta$,
$\mathop{\mathrm{otp}}(C^{\zeta}_{\delta}\cap E)=\gamma$ and
$C^{\zeta}_{\delta}\cap E=C^{\zeta}_{\gamma}\cap E^{\ast}$ (by 3.10 below).
Let $e\subseteq\gamma=\mathop{\mathrm{sup}}(e)$ be closed and such that
$\mathop{\mathrm{otp}}(e)=\delta(\ast)$ and
$\epsilon\in e\mbox{ is limit
}\quad\Rightarrow\quad\epsilon=\mathop{\mathrm{sup}}(e\cap\epsilon).$
We define ${}^{*}C^{\zeta}_{\delta}$ (for $\delta\in S_{\zeta}$) as follows:
if $\delta\notin E^{\ast}$,
${}^{\ast}C^{\zeta}_{\delta}=:C^{\zeta}_{\delta}\setminus(\mathop{\mathrm{max}}(\delta\cap
E^{\ast})+1),$
if $\delta\in E^{\ast}$, $\mathop{\mathrm{otp}}(C^{\zeta}_{\delta}\cap
E^{\ast})\in e\cup\\{\gamma\\}$ then
${}^{\ast}C^{\zeta}_{\delta}=\\{\alpha\in C^{\zeta}_{\delta}\cap
E^{*}:\mathop{\mathrm{otp}}(\alpha\cap C^{\zeta}_{\delta}\cap E^{\ast})\in
e\\},$
and if $\delta\in E^{\ast}$, $\mathop{\mathrm{otp}}(C^{\zeta}_{\delta}\cap
E^{\ast})\notin e\cup\\{\gamma\\}$ let
${}^{\ast}C^{\zeta}_{\delta}=C^{\zeta}_{\delta}\setminus\left(\mathop{\mathrm{max}}\\{\alpha:\mathop{\mathrm{otp}}(C^{\zeta}_{\delta}\cap
E^{\ast}\cap\alpha)\in e\cup\\{\gamma\\}\\}+1\right).$
One easily checks that (d) and square hold for
$\langle{}^{*}C^{\zeta}_{\delta}:\delta\in S_{\zeta}\rangle$. So, we just have
to add $\langle{}^{*}C^{\zeta}_{\delta}:\delta\in S_{\zeta}\rangle$ to
$\\{\langle C^{\zeta}_{\delta}:\delta\in S_{\zeta}\rangle:\zeta<\lambda\\}$
for any $\zeta,\delta(*),\kappa$ (for which we choose $\zeta$ and $E^{*}$).
$\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{6.4}}$
###### Claim 3.10.
(1):
Assume that $\aleph_{0}<\kappa=\mathop{\mathrm{cf}}(\kappa)$,
$\kappa^{+}<\lambda=\mathop{\mathrm{cf}}(\lambda)$,
$S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\kappa\\}$ is
stationary, $C_{\delta}$ is a club of $\delta$ (for $\delta\in S$), and
$(\forall\delta\in S)(|C_{\delta}|=\kappa)$, or at least
$\mathop{\mathrm{sup}}\limits_{\delta\in S}|C_{\delta}|^{+}<\lambda$. Then for
some club $E^{\ast}\subseteq\lambda$, for every club $E\subseteq E^{\ast}$,
the set $\\{\delta\in S^{\ast}:C_{\delta}\cap E^{\ast}\subseteq E\\}$ is
stationary, where
$S^{\ast}=:\\{\delta\in S:\delta\in\mathop{\mathrm{acc}}(E^{\ast})\\}.$
(2):
Assume that
$\kappa=\mathop{\mathrm{cf}}(\kappa),\quad\kappa^{+}<\lambda=\mathop{\mathrm{cf}}(\lambda),\quad
S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\kappa\\}\mbox{ is
stationary,}$
$C_{\delta}$ is a club of $\delta$ (for $\delta\in S$),
$\mathop{\mathrm{sup}}\limits_{\delta\in S}|C_{\delta}|^{+}<\lambda$,
$I_{\delta}$ is an ideal on $C_{\delta}$ including the bounded subsets, and
for every club $E$ of $\lambda$ for stationarily many $\delta\in S$,
$C_{\delta}\cap E\notin I_{\delta}$ (or $C_{\delta}\setminus E\in
I_{\delta}$).
Then for some club $E^{\ast}$ of $\lambda$, for every club $E\subseteq
E^{\ast}$ of $\lambda$
$\mbox{ the set }\\{\delta\in S^{\ast}:C_{\delta}\cap E^{\ast}\subseteq
E\\}\mbox{ is stationary,}$
where
$\begin{array}[]{lr}S^{\ast}=:\left\\{\delta\in
S:\right.&\delta\in\mathop{\mathrm{acc}}(E^{\ast}),\
\delta=\mathop{\mathrm{sup}}(C_{\delta}\cap E^{\ast})\mbox{ and }\\\
&C_{\delta}\cap E^{\ast}\notin I_{\delta}\mbox{ (or }C_{\delta}\setminus
E^{\ast}\in I_{\delta})\left.\right\\}.\end{array}$
Remark: This also was written in [Sh 365].
Proof: (1) If not, choose by induction on
$i<\mu=:\mathop{\mathrm{sup}}\limits_{\delta\in S}(|C_{\delta}|^{+})$ a club
$E^{\ast}_{i}\subseteq\lambda$, decreasing with $i$, $E^{\ast}_{i+1}$
exemplifies that $E^{\ast}_{i}$ is not as required, i.e.,
$\\{\delta\in S^{*}(E^{*}_{i}):C_{\delta}\cap E^{*}_{i}\subseteq
E^{*}_{i+1}\\}=\emptyset.$
Now, $\mathop{\mathrm{acc}}\big{(}\bigcap\limits_{i<\mu}E^{\ast}_{i}\big{)}$
is a club of $\lambda$, so there is $\delta\in
S\cap\mathop{\mathrm{acc}}\big{(}\bigcap\limits_{i<\mu}E^{\ast}_{i}\big{)}$.
The sequence $\langle C_{\delta}\cap E^{\ast}_{i}:i<\mu\rangle$ is necessarily
strictly decreasing, and we get an easy contradiction.
(2) Similarly
$\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{6.4B}}$
$*\qquad*\qquad*$
Now we turn to the main issue: black boxes.
###### Lemma 3.11.
Suppose that $\lambda,\theta$ and $\chi(\ast)$ are regular cardinals and
$\lambda^{\theta}=\lambda^{<\chi(\ast)}$, $\theta<\chi(\ast)\leq\lambda$, and
a set $S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\lambda)=\theta\\}$ is
stationary and in ${\check{I}^{\rm gd}}[\lambda]$ (if $\theta=\aleph_{0}$ this
holds trivially; see [Sh:E62, 3.3] or [Sh 88r, 0.6,0.7] or just 3.9(2)). Then
we can find
${\mathbf{W}}=\\{(\bar{M}^{\alpha},\eta^{\alpha}):\alpha<\alpha(\ast)\\}$
(pedantically, ${\mathbf{W}}$ is a sequence) and functions
${\dot{\zeta}}:\alpha(\ast)\longrightarrow S$, and
$h:\alpha(\ast)\longrightarrow\lambda$ such that:
(a0):
$h(\alpha)$ depends on ${\dot{\zeta}}(\alpha)$ only, and ${\dot{\zeta}}$ is
non-decreasing function (but not necessarily strictly increasing).
(a1):
We have
($\alpha$):
$\bar{M}^{\alpha}=\langle M^{\alpha}_{i}:i\leq\theta\rangle$ is an increasing
continuous chain, ($\tau(M^{\alpha}_{i})$, the vocabulary, may be increasing),
($\beta$):
each $M^{\alpha}_{i}$ is an expansion of a submodel of
$({\mathscr{H}}_{<\chi(*)}(\lambda),\in,<)$ belonging to
${\mathscr{H}}_{<\chi(\ast)}(\lambda)$ [so necessarily has cardinality
$<\chi(\ast)$, of course the order mean the order on the ordinals and, for
transparency, the vocabulary belongs to
${\mathscr{H}}_{<\chi(\ast)}(\chi(\ast))$],
($\gamma$):
$M^{\alpha}_{i}\cap\chi(\ast)$ is an ordinal, $[\chi(\ast)=\chi^{+}\
\Rightarrow\ \chi+1\subseteq M^{\alpha}_{i}]$, and
$M^{\alpha}_{i}\in{\mathscr{H}}_{<\chi(\ast)}(\eta^{\alpha}(i))$,
($\delta$):
$M^{\alpha}_{i}\cap\lambda\subseteq\eta^{\alpha}(i)$,
($\epsilon$):
$\langle M^{\alpha}_{j}:j\leq i\rangle\in M^{\alpha}_{i+1}$,
$(\zeta)$:
$\eta^{\alpha}\in{}^{\theta}\lambda$ is increasing with limit
${\dot{\zeta}}(\alpha)\in S$, $\eta^{\alpha}\restriction(i+1)\in
M^{\alpha}_{i+1}$.
(a2):
In the following game, ${\Game}(\theta,\lambda,\chi(\ast),{\mathbf{W}},h)$,
player I has no winning strategy. A play lasts $\theta$ moves, in the $i$-th
move player I chooses a model $M_{i}\in{\mathscr{H}}_{<\chi(\ast)}(\lambda)$,
and then player II chooses $\gamma_{i}<\lambda$. In the first move player I
also chooses $\beta<\lambda$. In the end player II wins the play if
$(\alpha)\Rightarrow(\beta)$ where
$(\alpha)$:
the pair $(\langle M_{i}:i<\theta\rangle,\langle\gamma_{i}:i<\theta\rangle)$
satisfies the relevant demands on the pair 222so $\langle M_{j}:j\leq
i\rangle$ is an increasing continuous chain, $M_{i}\cap\chi(\ast)$ an ordinal,
$\chi(\ast)=\chi^{+}\ \Rightarrow\ \chi+1\subseteq M_{i}$, $\langle
M_{\epsilon}:\epsilon\leq j\rangle\in M_{j+1}$ and
$\langle\gamma_{\epsilon}:\epsilon\leq j\rangle\in M_{j+1}$ for $j<i$,
$M_{i}\in{\mathscr{H}}_{<\chi(\ast)}(\gamma_{i})$ and $\langle\gamma_{i}:j\leq
i\rangle\in M_{i+1}$ and $M_{i}$ expand a submodel of
$({\mathscr{H}}_{<\chi(\ast)}(\lambda),\in,<)$
$(\bar{M}^{i}\upharpoonright\theta,\eta^{\alpha})$ in clause (a1).
$(\beta)$:
for some $\alpha<\alpha(\ast)$,
$\eta^{\alpha}=\langle\gamma_{i}:i<\theta\rangle$, $M_{i}=M^{\alpha}_{i}$ (for
$i<\theta)$ and $h(\alpha)=\beta$.
(b0):
$\eta^{\alpha}\neq\eta^{\beta}$ for $\alpha\neq\beta$,
(b1):
if $\\{\eta^{\alpha}\restriction i:i<\theta\\}\subseteq M^{\beta}_{\theta}$
then $\alpha<\beta+(<\chi(\ast))^{\theta}$, see below, and
${\dot{\zeta}}(\alpha)\leq{\dot{\zeta}}(\beta)$,
(b2):
if also $\lambda^{<\theta}=\lambda^{<\chi(\ast)}$, then for every
$\alpha<\alpha(\ast)$ and $i<\theta$, there is $j<\theta$ such that:
$\eta^{\alpha}\restriction j\in M^{\beta}_{\theta}$ implies $M^{\alpha}_{i}\in
M^{\beta}_{\theta}$ (hence $M^{\alpha}_{i}\subseteq M^{\beta}_{\theta}$),
(b3):
if $\lambda=\lambda^{<\chi(\ast)}$ and $\eta^{\alpha}\restriction(i+1)\in
M^{\beta}_{j}$ then $M^{\alpha}_{i}\in M^{\beta}_{j}$ (and hence $x\in
M^{\alpha}_{i}\ \Rightarrow\ x\in M^{\beta}_{j}$) and
$[\eta^{\alpha}\restriction i\neq\eta^{\beta}\restriction
i\quad\Rightarrow\quad\eta^{\alpha}(i)\neq\eta^{\beta}(i)].$
###### Remark 3.12.
(1):
If ${\mathbf{W}}$ (with ${\dot{\zeta}}$, $h$, $\lambda$, $\theta$,
$\chi(\ast)$) satisfies (a0), (a1), (a2), (b0), (b1) we call it a barrier.
(2):
Remember, $(<\chi)^{\theta}=:\sum\limits_{\mu<\chi}\mu^{\theta}$.
(3):
The existence of a good stationary set
$S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\theta\\}$ follows,
for example, from $\lambda=\lambda^{<\theta}$ (see [Sh:E62, 3.3] or [Sh 88r,
0.6,0.7]) and from “$\lambda$ is the successor of a regular cardinal and
$\lambda>{\theta^{+}}$”. But see 3.15(1),(2),(3).
(4):
Compare the proof below with [Sh 227] Lemma 1.13, p.49 and [Sh 140].
Proof: First assume $\lambda=\lambda^{<\chi(\ast)}$.
Let $\langle S_{\gamma}:\gamma<\lambda\rangle$ be a sequence of pairwise
disjoint stationary subsets of $S$,
$S=\bigcup\limits_{\gamma<\lambda}S_{\gamma}$ and without loss of generality
$\gamma<\mathop{\mathrm{min}}(S_{\gamma})$. We define
$h^{\ast}:S\longrightarrow\lambda$ by $h^{\ast}(\alpha)=$ “the unique $\gamma$
such that $\alpha\in S_{\gamma}$”, and below we shall let
$h(\alpha)=:h^{\ast}({\dot{\zeta}}(\alpha))$.
Let ${\mathord{\mathrm{cd}}}={\mathord{\mathrm{cd}}}_{\lambda,\chi(\ast)}$ be
a one-to-one function from ${\mathscr{H}}_{<\chi(\ast)}(\lambda)$ onto
$\lambda$ such that: ${\mathord{\mathrm{cd}}}(\langle\alpha,\beta\rangle)$ is
an ordinal $>\alpha,\beta$, but $<|\alpha+\beta|^{+}$ or $<\omega$, and
$x\in{\mathscr{H}}_{<\chi(\ast)}({\mathord{\mathrm{cd}}}(x))$ for every
relevant $x$. For $\xi\in S$ let:
$\begin{array}[]{lr}{\mathbf{W}}^{0}_{\xi}=:\left\\{(\bar{M},\eta):\mbox{ the
pair }\right.&(\bar{M},\ \eta)\mbox{ satisfies (a1) of \ref{6.5},
}\mathop{\mathrm{sup}}\\{\eta(i):i<\theta\\}=\xi,\\\ &\mbox{and for every
}i<\theta\mbox{ for some }y\in{\mathscr{H}}_{<\chi(\ast)}(\lambda),\\\
&\eta(i)={\mathord{\mathrm{cd}}}(\langle\bar{M}\restriction i,\eta\restriction
i,y\rangle)\left.\right\\}.\end{array}$
So (a0), (a1), (b0), (b3) (hence (b2)) should be clear.
We can choose
$\langle(\bar{M}^{\alpha},\eta^{\alpha}):\alpha<\alpha(\ast)\rangle$ an
enumeration of $\bigcup\limits_{\xi\in S}{\mathbf{W}}^{0}_{\xi}$ to satisfy
(b1) (and
${\dot{\zeta}}(\alpha)=\mathop{\mathrm{sup}}\mathop{\mathrm{Rang}}(\eta^{\alpha})$,
of course) because:
$(\ast)$:
if
$(\bar{M}^{\ast},\eta^{\ast})\in\bigcup\limits_{\xi}{\mathbf{W}}^{0}_{\xi}$,
then
$|\\{\eta\in{}^{\theta}\lambda:\\{\eta\restriction i:i<\theta\\}\subseteq
M^{\ast}_{\theta}\\}|\leq\|M^{\ast}_{\theta}\|^{\theta}\leq(<\chi(\ast))^{\theta}.$
This, in fact, defines the function ${{\dot{\zeta}}}$ as follows: we have
${\dot{\zeta}}(\alpha)=\xi$ if and only if
$(\bar{M}^{\alpha},\eta^{\alpha})\in{\mathbf{W}}^{0}_{\xi}$.
We are left with proving (a2). Let $G$ be a strategy for player I.
Let $\langle C_{\delta}:\delta<\lambda\rangle$ exemplify “$S$ is a good
stationary subset of $\lambda$”, see 3.9(2), and let $R=\\{(i,\alpha):i\in
C_{\alpha},\alpha<\lambda\\}$.
Let $\langle{\mathscr{A}}_{i}:i<\lambda\rangle$ be a representation of the
model
${\mathscr{A}}=\left({\mathscr{H}}_{<\chi(\ast)}(\lambda),\in,G,R,{\mathord{\mathrm{cd}}}\right),$
i.e. it is increasing continuous, $\|{\mathscr{A}}_{i}\|<\lambda$, and
$\bigcup\limits_{i}{\mathscr{A}}_{i}={\mathscr{A}}$; without loss of
generality ${\mathscr{A}}_{i}\prec{\mathscr{A}}$ and
$|{\mathscr{A}}_{i}|\cap\lambda$ is an ordinal for $i<\lambda$. Let $G$ “tell”
player I to choose $\beta^{\ast}<\lambda$ in his first move. So there is
$\delta\in S_{\beta^{\ast}}$ (hence $\delta>\beta^{\ast})$ such that
$|{\mathscr{A}}_{\delta}|\cap\lambda=\delta$. Now, necessarily
$C_{\delta}\cap\alpha\in{\mathscr{A}}_{\delta}$ for $\alpha<\delta$. Let
$\\{\alpha_{i}:i<\mathop{\mathrm{cf}}(\delta)\\}$ list $C_{\delta}$ in
increasing order.
Lastly, by induction on $i$, we choose $M_{i}$, $\eta(i)$ as follows:
$\eta(i)={\mathord{\mathrm{cd}}}(\langle\langle M_{j}:j\leq
i\rangle,\langle\eta(j):j<i\rangle,\langle\alpha_{j}:j<i\rangle\rangle),$
and $M_{i}$ is what the strategy $G$ “tells” player I to choose in his $i$-th
move if player II have chosen $\langle\eta(j):j<i\rangle$ so far.
Now, for each $i<\theta$ the sequences $\langle M_{j}:j\leq i\rangle$,
$\langle\eta(j):j<i\rangle$ are definable in ${\mathscr{A}}_{\delta}$ with
$\langle\alpha_{j}:j\leq i\rangle$ as the only parameter, hence they belong to
${\mathscr{A}}_{\delta}$. So
$\mathop{\mathrm{sup}}\\{\eta(j):j<\theta\\}\leq\delta$; however, by the
choice of $\eta(i)$ (and ${\mathord{\mathrm{cd}}}$),
$\eta(i)\geq\mathop{\mathrm{sup}}\\{\alpha_{j}:j<i\\}$ and hence
$\mathop{\mathrm{sup}}\\{\eta(j):j<\theta\\}$ is necessarily $\delta$. Now
check.
The case $\lambda<\lambda^{<\theta}=\lambda^{<\chi(\ast)}$ is similar. For a
set $A\subseteq\theta$ of cardinality $\theta$ we let
${\mathord{\mathrm{cd}}}^{A}={\mathord{\mathrm{cd}}}^{A}_{\lambda,\chi(\ast)}$
be a one-to-one function from ${\mathscr{H}}_{<\chi(\ast)}(\lambda)$ onto
$A_{\lambda}$ where:
$\begin{array}[]{ll}A_{\lambda}=\left\\{h:\right.&h\mbox{ is a function from
$A$ to $\lambda$}\left.\right\\}.\end{array}$
We strengthen (b2) to
(b2)′:
let $A_{i}:=\\{{\mathord{\mathrm{cd}}}(i,j):j<\theta\\}$ for $i\in[1,\theta)$
and $A_{0}:=\theta\setminus\cup\\{A_{1+i}:i<\theta\\}$ so $\langle
A_{i}:i<\theta\rangle$ is a sequence of pairwise disjoint subsets of $\theta$
each of cardinality $\theta$ with $\mathop{\mathrm{min}}({A}_{i})\geq i$ and
we have
$(\ast)$:
$\eta^{\alpha}\restriction
A_{i}={\mathord{\mathrm{cd}}}^{A_{i}}((\bar{M}^{\alpha}{\restriction}i,\eta^{\alpha}{\restriction}i)$.
$\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{6.5}}$
* * *
What can we do when $S$ is not good? As we say in 3.12(3), in many cases a
good $S$ exists, note that for singular $\lambda$ we will not have one.
The following rectifies the situation in the other cases (but is interesting
mainly for $\lambda$ singular). We shall, for a regular cardinal $\lambda$,
remove this assumption in 3.15(1)–(3), while 3.16 helps for singular
$\lambda$. (This is carried in 3.17).
###### Definition 3.13.
Let $\partial$ be an ordinal and for $\alpha<\partial$ let $\kappa_{\alpha}$
be a regular uncountable cardinal,
$S_{\alpha}\subseteq\\{\delta<\kappa_{\alpha}:\mathop{\mathrm{cf}}(\delta)=\theta)\\}$
be a stationary set. Assume $\theta$, $\chi$ are regular cardinals such that
for every $\alpha<\partial$ we have $\theta<\chi\leq\kappa_{\alpha}$. Let
$\bar{S}=\langle S_{\alpha}:\alpha<\partial\rangle$,
$\bar{\kappa}=\langle\kappa_{\alpha}:\alpha<\partial\rangle$.
If $\partial=1$ we may write $S_{0},\kappa_{0}$.
We say that $\bar{S}$ is good for $(\bar{\kappa},\theta,\chi)$ when: for every
large enough $\mu$ and model ${\mathscr{A}}$ expanding
$({\mathscr{H}}_{<\chi}(\mu),\in)$, $|\tau({\mathscr{A}})|\leq\aleph_{0}$,
there are $M_{i}$ for $i<\theta$ such that:
* •
$M_{i}\prec{\mathscr{A}}$ and $\bar{S}\in M_{i}$
* •
$\langle M_{j}:j\leq i\rangle\in M_{i+1}$, $\|M_{i}\|<\chi$,
$M_{i}\cap\chi\in\chi$, $\chi=\chi^{+}_{1}\ \Rightarrow\ \chi_{1}+1\subseteq
M_{i}$, and
* •
$\alpha<\partial$, $\alpha\in\bigcup\limits_{j<\theta}M_{j}$ implies that
$\mathop{\mathrm{sup}}[\kappa_{\alpha}\cap(\bigcup\limits_{j<\theta}M_{j})]$
belongs to $S_{\alpha}$.
If $\partial=1$, we may write $S_{0},\kappa_{0}$ instead
$\bar{S},\bar{\kappa}$. If $\partial<\chi$ then we can demand
$\partial\subseteq M_{0}$.
###### Definition 3.14.
For regular uncountable cardinal $\lambda$ and regular $\theta<\lambda$ let
${\check{J}}_{\theta}[\lambda]$ be the family of subsets $S$ of $\lambda$ such
that $(\\{\delta\in S:\mathop{\mathrm{cf}}(\delta)=\theta\\}$ is not good for
$(\lambda,\lambda,\theta)$.
###### Claim 3.15.
Assume
$\theta=\mathop{\mathrm{cf}}(\theta)<\chi=\mathop{\mathrm{cf}}(\chi)\leq\kappa=\mathop{\mathrm{cf}}(\kappa)$.
Then
(1):
$\\{\delta<\kappa:\mathop{\mathrm{cf}}(\delta)=\theta\\}$ is good for
$(\kappa,\theta,\chi)$, i.e. is not in ${\check{J}}_{\theta}[\lambda]$,
(2):
Any $S\subseteq\kappa$ good for $(\kappa,\theta,\chi)$ is the union of
$\kappa$ pairwise disjoint such sets,
(3):
In 3.11 it suffices to assume that $S$ is good for $(\lambda,\theta,\chi)$,
(4):
${\check{J}}_{\theta}[\lambda]$ is a normal ideal on $\lambda$ and there is no
stationary
$S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\theta\\}$ which
belongs to ${\check{J}}_{\theta}[\lambda]\cap{\check{I}^{\rm gd}}[\lambda]$,
(5):
In Definition 3.13, any $\mu>\lambda^{<\chi}$ is O.K.; and we can preassign
$x\in{\mathscr{H}}_{<\chi}(\mu)$ and demand $x\in M_{i}$.
(6):
In 3.11 we can replace the assumption
“$S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\theta\\}$ is
stationary and in ${\check{I}^{\rm gd}}[\lambda]$” by
“$S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\theta\\}$ is
stationary not in ${\check{J}}_{\theta}[\lambda]$” (which holds for
$S=\\{\delta<\kappa:\mathop{\mathrm{cf}}(\delta)=\theta\\}$).
Proof:
(1):
Straightforward (play the game).
(2):
Similar to the proof of 3.1.
(3):
Obvious.
(4):
Easy.
(5):
Easy.
(6):
Follows.
$\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{6.7}$
###### Claim 3.16.
Assume that $\bar{\kappa},\theta,\chi$ are as in 3.13 with
$|\partial|\leq\chi$. Then
(1):
the sequence
$\langle\\{\delta<\kappa_{i}:\mathop{\mathrm{cf}}(\delta)=\theta\\}:i<\partial\rangle$
is good for $(\bar{\kappa},\theta,\chi)$.
(2):
If $\partial_{1}<\partial$ and $\langle S_{i}:i<\partial_{1}\rangle$ is good
for $(\bar{\kappa}\restriction\partial_{1},\theta,\chi)$ then
$\langle
S_{i}:i<\partial_{1}\rangle{}^{\frown}\\!\langle\\{\delta<\kappa_{i}:\mathop{\mathrm{cf}}(\delta)=\theta\\}:\partial_{1}\leq
i<\partial\rangle$
is good for $(\bar{\kappa},\theta,\chi)$.
(3):
If $\langle S_{i}:i<\partial_{1}\rangle$ is good for
$(\bar{\kappa},\theta,\chi)$ and $i(\ast)<\partial$, then we can partition
$S_{i(\ast)}$ to pairwise disjoint sets $\langle
S_{i(\ast),\epsilon}:\epsilon<\kappa_{i}\rangle$ such that for each
$\epsilon<\kappa_{i}$, the sequence
$\langle S_{i}:i<i(\ast)\rangle{}^{\frown}\\!\langle
S_{i(\ast),\epsilon}\rangle{}^{\frown}\\!\langle\\{\delta:\delta<\kappa_{i},\mathop{\mathrm{cf}}(\delta)=\theta\\}:i(\ast)<i<\partial\rangle$
is good for $(\bar{\kappa},\theta,\chi)$,
(4):
$\bar{S}$ good for $(\bar{\kappa},\theta,\chi)$ implies that $S_{i}$ is a
stationary subset of $\kappa_{i}$ for each $i<\mathop{\rm\ell
g}(\bar{\kappa})$.
Proof: Like 3.15 [in 3.16(3) we choose for $\delta\in S_{i(\ast)}$, a club
$C_{\delta}$ of $\delta$ of order type $\mathop{\mathrm{cf}}(\delta)$; for
$j<\theta$, $\epsilon<\kappa_{i(\alpha)}$, let
$S^{j}_{i(\ast),\epsilon}=\\{\delta\in S_{i(\ast)}:\epsilon\mbox{ is the
}j\mbox{th member of }C_{\delta}\\};$
for some $j$ and unbounded $A\subseteq\kappa_{i(\ast)}$, $\langle
S^{j}_{i(\ast),\epsilon}:\epsilon\in A\rangle$ are as required]. $\hskip
5.0pt\hbox{\hskip 5.0pt\vrule width=4.0pt,height=6.0pt,depth=1.5pt\hskip
1.0pt}_{\ref{6.7A}}$
Now we remove from 3.11 (and subsequently 3.18) the hypothesis “$\lambda$ is
regular” when $\mathop{\mathrm{cf}}(\lambda)\geq\chi(\ast)$.
###### Lemma 3.17.
Suppose $\lambda^{\theta}=\lambda^{<\chi(\ast)}$, $\lambda$ is singular,
$\theta$ and $\chi(\ast)$ are regular, $\theta<\chi(\ast)$ and
$\mathop{\mathrm{cf}}(\lambda)\geq\chi(\ast)$. Suppose further that
$\lambda=\sum\limits_{i<\mathop{\mathrm{cf}}(\lambda)}\mu_{i}$, each $\mu_{i}$
is regular $>\chi(\ast)+\theta^{+}$. Then we can find
${\mathbf{W}}=\\{(\bar{M}^{\alpha},\eta^{\alpha}):\alpha<\alpha(\ast)\\}$
and functions
${\dot{\zeta}}:\alpha(\ast)\longrightarrow\mathop{\mathrm{cf}}(\lambda),\quad{{\dot{\xi}}}:\alpha(\ast)\longrightarrow\lambda,\
\mbox{ and }\ h:\alpha(\ast)\longrightarrow\lambda,$
and $\\{\mu^{\prime}_{i}:i<\mathop{\mathrm{cf}}(\lambda)\\}$ such that
($\\{\mu^{\prime}_{i}:i<\mathop{\mathrm{cf}}(\lambda)\\}=\\{\mu_{i}:i<\mathop{\mathrm{cf}}(\lambda)\\}$
and):
(a0):
$h(\alpha)$ depends only on
$\langle{{\dot{\zeta}}}(\alpha),{{\dot{\xi}}}(\alpha)\rangle$,
$[\alpha<\beta\ \Rightarrow\
{{\dot{\zeta}}}(\alpha)\leq{{\dot{\zeta}}}(\beta)],\quad[\alpha<\beta\wedge{{\dot{\zeta}}}(\alpha)={{\dot{\zeta}}}(\beta)\
\Rightarrow\ {{\dot{\xi}}}(\alpha)\leq{{\dot{\xi}}}(\beta)],$
and ${\dot{\xi}}(\alpha)<\mu^{\prime}_{{\dot{\zeta}}(\alpha)}$
(a1):
as in 3.11 except that: $\langle\eta^{\alpha}(3i):i<\theta\rangle$ is strictly
increasing with limit ${{\dot{\zeta}}}(\alpha)$ and
$\langle\eta^{\alpha}(3i+1):i<\theta\rangle$ is strictly increasing with limit
${{\dot{\xi}}}(\alpha)$ for $i<\theta$,
$\mathop{\mathrm{sup}}(|M^{\alpha}_{i}|\cap\mu^{\prime}_{\zeta(\alpha)})<{{\dot{\xi}}}(\alpha)=\mathop{\mathrm{sup}}(|M^{\alpha}_{\theta}|\cap\mu^{\prime}_{{\dot{\zeta}}(\alpha)})$
and for every $i<\theta$,
$\mathop{\mathrm{sup}}(|M^{\alpha}_{i}|\cap\mathop{\mathrm{cf}}(\lambda))<{{\dot{\zeta}}}(\alpha)=\mathop{\mathrm{sup}}(|M^{\alpha}_{\theta}|\cap\mathop{\mathrm{cf}}(\lambda)),$
(a2):
as in 3.11.
(b0):
, (b1), (b2) as in 3.11 but in clause (b3) we demand $i=2\mod 3$.
Remark: To make it similar to 3.11, we can fix $S^{a}$, $S^{a}_{i}$,
$S^{b}_{i}$, $S^{b}_{i,a}$, $\mu^{\prime}_{i}$ as in the first paragraph of
the proof below.
Proof: First, by 3.15 [(1) + (2)], we can find pairwise disjoint
$S^{a}_{i}\subseteq\mathop{\mathrm{cf}}(\lambda)$ for
$i<\mathop{\mathrm{cf}}(\lambda)$, each good for
$(\mathop{\mathrm{cf}}(\lambda),\theta,\chi(\ast))$ (and $\alpha\in S^{a}_{i}\
\Rightarrow\ \alpha>i\ \&\ \mathop{\mathrm{cf}}(\alpha)=\theta$), and let
$S^{a}=\bigcup\limits_{i<\mathop{\mathrm{cf}}(\lambda)}S^{a}_{i}$. We define
$\mu^{\prime}_{i}\in\\{\mu_{j}:j<i\\}$ such that for each
$i<\mathop{\mathrm{cf}}(\lambda):[j\in
S^{a}_{i}\Rightarrow\mu^{\prime}_{j}=\mu_{i}]$.
Then for each $i$, by 3.16 parts (2) (3) (with
$1,2,S_{0},\kappa_{0},\kappa_{1}$ standing for $\sigma_{1}$, $\sigma$,
$S^{a}_{i}$, $\mathop{\mathrm{cf}}(\lambda)$, $\mu^{\prime}_{i})$, we can find
pairwise disjoint subsets $\langle
S^{b}_{i,\alpha}:\alpha<\mu^{\prime}_{i}\rangle$ of
$\\{\delta<\mu^{\prime}_{i}:\mathop{\mathrm{cf}}(\delta)=\theta\\}$ such that
for each $\alpha<\mu^{\prime}_{\alpha}$, $(S^{a}_{i},S^{b}_{i,\alpha})$ is
good for
$(\langle\mathop{\mathrm{cf}}(\lambda),\mu^{\prime}_{i}\rangle,\theta,\chi)$.
Let $S^{b}_{i}=\bigcup\\{S^{b}_{i,\alpha}:\alpha<\mu^{\prime}_{i}\\}$.
Let ${\mathord{\mathrm{cd}}}$ be as in 3.11’s proof coding only for ordinals
$i=2\mod 3$, and for $\zeta\in S^{a}_{i}$, $\xi\in S^{a}_{i,j}$ let
$\begin{array}[]{ll}{\mathbf{W}}^{0}_{\zeta,\xi}=\left\\{(\bar{M},\eta):\right.&\bar{M}\mbox{
satisfies {\bf(a1)}, }\zeta=\mathop{\mathrm{sup}}\\{\eta(3i):i<\theta\\},\\\
&\xi=\mathop{\mathrm{sup}}\\{\eta(3i+1):i<\theta\\}\mbox{ and}\\\ &\mbox{for
each $i<\theta$, for some $y\in{\mathscr{H}}_{<\chi(\ast)}(\lambda)$,}\\\
&\eta(3i+2)={\mathord{\mathrm{cd}}}(\langle M_{j}:j\leq
3i+1\rangle,\eta\restriction(3i+1),y)\left.\right\\}.\end{array}$
The rest is as in 3.11’s proof. $\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{6.9}}$
* * *
The following Lemma improves 3.11 when $\lambda$ satisfies a stronger
requirement making the distinct $(\bar{M}^{\alpha},\eta^{\alpha})$ interact
less. Lemmas 3.18 \+ 3.17 were used in the proof of 2.4 (and 2.3).
###### Lemma 3.18.
In 3.11, if $\lambda=\lambda^{\chi(\ast)}$, $\chi(\ast)^{\theta}=\chi(\ast)$,
then we can strengthen clause (b1) to
(b1)+:
if $\alpha\neq\beta$ and $\\{\eta^{\alpha}\restriction i:i<\theta\\}\subseteq
M^{\beta}$ then $\alpha<\beta$ and
$x\in M^{\alpha}_{\theta}\quad\Rightarrow\quad x\in M^{\beta}_{\theta}.$
Proof: Apply 3.11 (actually, its proof) but using $\lambda$, $\chi(\ast)^{+}$,
$\theta$, instead of $\lambda$, $\chi(\ast)$, $\theta$; and get
${\mathbf{W}}=\\{(\bar{M}^{\alpha},\eta^{\alpha}):\alpha<\alpha(\ast))\\}$,
and the functions ${\dot{\zeta}},h$.
Let ${\mathord{\mathrm{cd}}}$ be as in the proof of 3.11. Let $<^{\ast}$ be
some well ordering of ${\mathscr{H}}_{<\chi(\ast)}(\lambda)$, and let
${\mathscr{U}}$ be the set of ordinals $\alpha<\alpha(\ast)$ such that for
$i<\theta$, $M^{\alpha}_{i}$ has the form
$(N^{\alpha}_{i},\in^{\alpha}_{i},<^{\alpha})$ and
$(|N^{\alpha}_{i}|,\in^{\alpha}_{i},<^{\alpha})\prec({\mathscr{H}}_{<\chi(\ast)}(\lambda),\in,<^{\ast})$.
Let $\alpha\in{\mathscr{U}}$, by induction on $\epsilon<\chi(\ast)$ we define
$M^{\epsilon,\alpha}_{i}$, $\eta^{\epsilon,\alpha}$ as follows:
(A):
$\eta^{\epsilon,\alpha}(i)$ is
${\mathord{\mathrm{cd}}}(\langle\eta^{\alpha}(i),\epsilon\rangle)$, (which is
an ordinal $<\lambda$ but $>\eta^{\alpha}(i)$ and $>\epsilon)$.
(B):
$M^{\epsilon,\alpha}_{i}\prec N^{\alpha}_{i}$ is the Skolem Hull of
$\\{\eta^{\epsilon,\alpha}\restriction(j+1):j<i\\}$ inside $N^{\alpha}_{i}$,
using as Skolem functions the choice of the $<^{\ast}$–first element and
making $M^{\epsilon,\alpha}_{i}\cap\chi(\ast)$ an ordinal [if we want we can
use $\eta^{\epsilon,\alpha}$ such that it fits the definition in the proof of
3.11].
Note that $\chi(\ast)=\chi^{+}\ \Rightarrow\ \chi+1\subseteq M^{\alpha}_{i}$
and $M^{\epsilon,\alpha}_{i}$ is definable in $M^{\epsilon,\alpha}_{i+1}$ as
$M^{\epsilon,\alpha}_{i}\in M^{\epsilon,\alpha}_{i+1}$ (by the definition of
${{\mathbf{W}}}^{0}_{\xi}$ in the proof of 3.11). Similarly, $\langle
M^{\epsilon,\alpha}_{j}:j\leq i\rangle$ is definable in $M^{\alpha}_{i+1}$. It
is easy to check that the pair
$(\bar{M}^{\epsilon,\alpha},\eta^{\epsilon,\alpha})$ satisfies condition (a1)
of 3.11.
Next we choose by induction on $\alpha\in{\mathscr{U}}$,
$\epsilon(\alpha)<\chi(\ast)$ as follows:
(C):
$\epsilon(\alpha)$ is the first $\epsilon<\chi(\ast)$ such that: if
$\beta<\alpha$ but $\beta+\chi(\ast)>\alpha$ then:
$(\ast)$:
$\\{\eta^{\alpha,\epsilon}\restriction j:j<\theta\\}\not\subseteq
M^{\beta,\epsilon(\beta)}_{\theta}$.
This is possible and easy, as for $(\ast)$ it suffices to have for each
suitable $\beta$, $\epsilon\notin M^{\beta,\epsilon(\beta)}_{\theta}$, so each
$\beta$ “disqualifies” $<\chi(\ast)$ ordinals as candidates for
$\epsilon(\alpha)$, and there are $<\chi(\ast)$ such $\beta$’s, and
$\chi(\ast)$ is by the assumptions (see 3.11) regular.
Now
${\mathbf{W}}^{\prime}=\\{(\bar{N}^{\alpha,\epsilon(\alpha)},\eta^{\alpha,\epsilon(\alpha)}):\alpha\in{\mathscr{U}}\\},\quad{\dot{\zeta}}\restriction{\mathscr{U}},\quad
h\restriction{\mathscr{U}}$
are as required except that we should replace ${\mathscr{U}}$ by an ordinal
(and adjust $\zeta,h$ accordingly). In the end replace $N^{\alpha}_{i}$ by
$N^{\alpha}_{i}\cap{\mathscr{H}}_{<\chi(*)}(\lambda)$. ${}_{\hskip
5.0pt\hbox{\hskip 5.0pt\vrule width=4.0pt,height=6.0pt,depth=1.5pt\hskip
1.0pt}_{\ref{6.6}}}$
###### Claim 3.19.
If in 3.17 we add “$\lambda=\lambda^{\chi(\ast)^{\theta}}$” (or the condition
from 3.18) ??$\theta$ can replace (b1) by
(b1)+:
if $\\{\eta^{\alpha}\restriction i:i<\theta\\}\subseteq M^{\beta}_{\theta}$
then $\alpha\leq\beta$.
Proof: The same as the proof of 3.18 combined with the proof of 3.17. $\hskip
5.0pt\hbox{\hskip 5.0pt\vrule width=4.0pt,height=6.0pt,depth=1.5pt\hskip
1.0pt}_{\ref{6.10}}$
* * *
Next we turn to the case (of black boxes with) $\theta=\aleph_{0}$. We shall
deal with several cases.
###### Lemma 3.20.
Suppose that
(*):
$\lambda$ is a regular cardinal, $\theta=\aleph_{0}$,
$\mu=\mu^{<\chi(\ast)}<\lambda\leq 2^{\mu}$,
$S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\aleph_{0}\\}$ is
stationary and $\aleph_{0}<\chi(\ast)=\mathop{\mathrm{cf}}(\chi(\ast))$.
Then we can find
${\mathbf{W}}=\\{(\bar{M}^{\alpha},\eta^{\alpha}):\alpha<\alpha(\ast)\\}$
and functions
${\dot{\zeta}}:\alpha(\ast)\longrightarrow S\ \mbox{ and }\
h:\alpha(\ast)\longrightarrow\lambda$
such that:
(a0)-(a2):
as in 3.11,
(b0)-(b2):
as in 3.11, and even
(b1)∗:
$\alpha\neq\beta$, $\\{\eta^{\alpha}\restriction n:n<\omega\\}\subseteq
M^{\beta}_{\omega}$ implies $\alpha<\beta$ and even
${{\dot{\zeta}}}(\alpha)<{\dot{\zeta}}(\beta)$,
(c1):
if ${\dot{\zeta}}(\alpha)={\dot{\zeta}}(\beta)$ then
$|M^{\alpha}_{\omega}|\cap\mu=|M^{\beta}_{\omega}|\cap\mu$ and there is an
isomorphism $h_{\alpha,\beta}$ from $M^{\alpha}_{\omega}$ onto
$M^{\beta}_{\omega}$, mapping $\eta^{\alpha}(n)$ to $\eta^{\beta}(n)$, and
$M^{\alpha}_{n}$ to $M^{\beta}_{n}$ for $n<\omega$, and
$h_{\alpha,\beta}\restriction(|M^{\alpha}_{\omega}|\cap|M^{\beta}_{\omega}|)$
is the identity,
(c2):
there is $\bar{C}=\langle C_{\delta}:\delta\in S\rangle$, $C_{\delta}$ an
$\omega$–sequence converging to $\delta$, $0\notin C_{\delta}$, and letting
$\langle\gamma^{\delta}_{n}:n<\omega\rangle$ enumerate $\\{0\\}\cup
C_{\delta}$ we have, when ${\dot{\zeta}}(\alpha)=\delta$:
(i):
$\lambda\cap|M^{\alpha}_{n}|\subseteq\gamma^{\delta}_{n+1}$ but
$\lambda\cap|M^{\alpha}_{n}|$ is not a subset of $\gamma^{\delta}_{n}$, (hence
$M^{\alpha}_{n}\cap[\gamma^{\delta}_{n},\gamma^{\delta}_{n+1})\neq\emptyset$);
(ii):
$C_{\delta}\cap|M^{\alpha}_{\omega}|=\emptyset$;
(iii):
if ${\dot{\zeta}}(\beta)=\delta$ too then, for each $n$, $h_{\alpha,\beta}$
maps $|M^{\alpha}_{\omega}|\cap[\gamma^{\delta}_{n},\gamma^{\delta}_{n+1})$
onto $|M^{\beta}_{\omega}|\cap[\gamma^{\delta}_{n},\gamma^{\delta}_{n+1}]$;
(iv):
if ${\dot{\zeta}}(\beta)=\delta={\dot{\zeta}}(\alpha)$ and
$\lambda=\lambda^{<\chi(\ast)}$, then
$|M^{\alpha}_{\omega}|\cap\gamma^{\delta}_{1}=|M^{\beta}_{\omega}|\cap\gamma^{\delta}_{1}$.
###### Remark 3.21.
(1) We use $\lambda\leq 2^{\mu}$ only to get
“$h_{\alpha,\beta}\restriction(|M^{\alpha}_{\omega}|\cap|M^{\beta}_{\omega}|)=\mathop{\mathrm{id}}$”
in condition (c1).
(2) Below we quote “guessing of clubs” that is clause (ii) in the proof,
without this we just get a somewhat weaker conclusion.
Proof: Let $S$ be the disjoint union of stationary
$S_{\alpha,\beta,\gamma}\quad(\alpha<\mu,\beta<\lambda,\gamma<\lambda).$
For each $\alpha$, $\beta$, $\gamma$ let $\langle C_{\delta}:\delta\in
S_{\alpha,\beta,\gamma}\rangle$ satisfy
$\boxtimes$ (i):
$C_{\delta}$ is an unbounded subset of $\delta$ of order type $\omega$, and
(ii):
for every club $C$ of $\lambda$, for stationarily many $\delta\in
S_{\alpha,\beta,\gamma}$ we have $C_{\delta}\subseteq C$
(iii):
$0\notin C_{\delta}$
(exists by [Sh 331, 2.2] or [Sh 365]). Let ${\mathbf{W}}^{\ast}$ be the family
of quadruples $(\delta,\bar{M},\eta,C)$ such that:
$(\alpha)$:
$(\bar{M},\eta)$ satisfies the requirement (a1) (so $\bar{M}=\langle
M_{n}:n<\omega\rangle)$;
$(\beta)$:
$0\notin C$, and letting $\\{\gamma_{n}:n<\omega\\}$ enumerate in increasing
order $C\cup\\{0\\}$ we have $\lambda\cap M_{n}$ is a subset of $\gamma_{n+1}$
but not of $\gamma_{n}$, and $\bigcup\limits_{n<\omega}\gamma_{n}=\delta$ and
$C\cap(\bigcup\limits_{n}M_{n})=\emptyset$;
$(\gamma)$:
$\bigcup\limits_{n}|M_{n}|\subseteq{\mathscr{H}}_{<\chi(\ast)}(\mu+\mu)$;
$(\delta)$:
in $\tau(M_{n})$ there are a two place relation $R$ and a one place function
${\mathord{\mathrm{cd}}}$ (not necessarily
${\mathord{\mathrm{cd}}}\restriction M_{n}={\mathord{\mathrm{cd}}}^{M_{n}}$,
similarly for $R$, see below recall that as usual,
$\tau(M_{n}))\in{\mathscr{H}}_{<\chi(\ast)}(\chi(\ast))$ for transparency.)
As $\mu^{<\chi(\ast)}=\mu$ clearly $|{\mathbf{W}}^{\ast}|=\mu$, so let
${\mathbf{W}}^{\ast}=\left\\{(\delta^{j},\langle
M_{j,n}:n<\omega\rangle,\eta_{j},C^{j}):j<\mu\right\\}.$
If $\lambda=\lambda^{<\chi(\ast)}$ let $\\{N_{\beta}:\beta<\lambda\\}$ list
the models $N\in{\mathscr{H}}_{<\chi(\ast)}(\lambda)$ with
$\tau(N)\in{\mathscr{H}}_{<\chi(\ast)}(\chi(\ast))$.
Also, let $\langle A_{\alpha}:\alpha<\lambda\rangle$ be a sequence of pairwise
distinct subsets of $\mu$, and define the two place relation $R$ on $\lambda$
by
$\left[\gamma_{1}\;R\;\gamma_{2}\ \iff\ \gamma_{1}<\mu\ \&\ \gamma_{1}\in
A_{\gamma_{2}}\right].$
Lastly, for $\delta\in S_{\alpha,\beta,\gamma}$ let
$\begin{array}[]{ll}{\mathbf{W}}^{0}_{\delta}=:&\left\\{(\bar{M},\eta):\bar{M}=\langle
M_{n}:n<\omega\rangle,\ \eta\in{}^{\omega}\lambda,\mbox{ satisfy {\bf(a1)},
so}\right.\\\ &\quad\eta\mbox{ is increasing with limit $\delta$, and}\\\
&\quad\mbox{there is an isomorphism $h$ from
}\bigcup\limits_{n<\omega}M_{n}\mbox{ onto
}\bigcup\limits_{n<\omega}M_{\alpha,n},\\\ &\quad\mbox{mapping $\eta(n)$ to
$\eta^{\alpha}(n)$ and $M_{n}$ onto $M_{\alpha,n}$, preserving $\in,R$,
${\mathord{\mathrm{cd}}}(x)=y$ and}\\\ &\quad\mbox{their negations; (for $R$
and ${\mathord{\mathrm{cd}}}$: in $\bigcup\limits_{n<\omega}M_{n}$ we mean}\\\
&\quad\mbox{the standard ${\mathord{\mathrm{cd}}}$ over
$\bigcup\limits_{n<\omega}M_{\alpha,n}$ as in $(\delta)$ above); and}\\\
&\quad(\forall\epsilon<\lambda)[\epsilon\in\bigcup\limits_{n}M_{n}\
\Rightarrow\
\mathop{\mathrm{otp}}(C_{\delta}\cap\epsilon)=\mathop{\mathrm{otp}}(C^{\alpha}\cap
h(\epsilon))].\\\ &\quad\mbox{Also, if $\lambda=\lambda^{<\chi(\ast)}$ then
}\\\ &\quad
N_{\beta}=(\bigcup\limits_{n}M_{n})\restriction\\{x\in\bigcup\limits_{n}M_{n}:{\mathord{\mathrm{cd}}}(x)<\mathop{\mathrm{min}}(C_{\delta})\\}\left.\right\\}.\end{array}$
We proceed as in the proof of 3.11 after ${{\mathbf{W}}}^{0}_{\delta}$ was
defined (only ${{\dot{\zeta}}}(\alpha)=\delta\in$
$S_{\alpha_{1},\beta_{1},\gamma_{1}}\ \Rightarrow\ h(\alpha)=\gamma_{1})$.
Suppose $G$ is a winning strategy for player I. So suppose that if player II
has chosen $\eta(0),\eta(1),\ldots,\eta(n-1)$, player I will choose
$M_{\eta}$. So $|M_{\eta}|$ is a subset of
${\mathscr{H}}_{<\chi(\ast)}(\lambda)$ of cardinality $<\chi(\ast)$ and
$\mathop{\mathrm{Rang}}(\eta)\subseteq M_{\eta}$. For
$\eta\in{}^{\omega}\lambda$ we define
$M_{\eta}=\bigcup\limits_{\ell<\omega}M_{\eta\restriction\ell}$. Let
${\mathscr{T}}_{n}$ be the set of $\eta\in{}^{n}\lambda$ such that $M_{\eta}$
is well defined; so $\cup\\{{\mathscr{T}}_{n}:n<\omega\\}$ is a subtree of
$({}^{\omega>}\lambda,\triangleleft)$ with each node having $\lambda$
immediate successors.
We can find a function ${\bf c}_{n}$ from ${\mathscr{T}}_{n}$ into $\mu$ such
that ${\bf c}_{n}(\eta)={\bf c}_{n}(\nu)$ iff there is an isomorphism $h$ from
$M_{\eta}$ onto $M_{\nu}$ mapping $M_{\eta{\restriction}k}$ onto
$M_{\nu{\restriction}k}$ for every $k<n$. By [RuSh 117] or see [Sh:f,
Ch.XI,3.5], [Sh:E62, §1] or the proof of 3.22 below, there is ${\mathscr{T}}$
such that
${\mathscr{T}}\subseteq{}^{\omega>}\lambda$, ${\mathscr{T}}$ is closed under
initial segments,
$\langle\rangle\in{\mathscr{T}}$, $\left[\eta\in{\mathscr{T}}\ \Rightarrow\
(\exists^{\lambda}\alpha)[\eta{}^{\frown}\\!\langle\alpha\rangle\in{\mathscr{T}}]\right]$,
${\bf c}_{n}{\restriction}({\mathscr{T}}\cap{\mathscr{T}}_{n})$ is constant.
It follows that fixing any $\nu_{*}\in\lim({\mathscr{T}})$ we can find
$\langle h_{\eta};\eta\in{\mathscr{T}}\rangle$ such that $h_{\eta}$ is an
isomorphism from $M_{\nu_{*}{\restriction}\mathop{\rm\ell g}(\eta)}$ onto
$M_{\eta}$ increasing with $\eta$.
Note that above all those isomorphisms are unique as the interpretation of
$\in$ satisfies comprehension. Also clause (c1) follows from the use of $R$.
The rest should be clear. $\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{6.11}}$
###### Lemma 3.22.
Let $S$, $\lambda$, $\mu$, $\theta$ be as in (*) of 3.20 and in addition:
$\aleph_{0}\leq\kappa=\mathop{\mathrm{cf}}(\kappa)<\chi(\ast),\qquad(\forall\chi<\chi(\ast))[\chi^{<\chi(\ast}<\chi(\ast)].$
Then we can find
${\mathbf{W}}=\\{(\bar{M}^{\alpha},\eta^{\alpha}):\alpha<\alpha(\ast)\\}$ and
functions ${\dot{\zeta}}:\alpha(\ast)\longrightarrow S$ and
$h:\alpha(\ast)\longrightarrow\lambda$ such that:
(a0), (b0), (b2):
as in 3.20 (i.e. as in 3.11),
(b1)∗, (c1), (c2):
as in 3.20,
(a1)∗:
as (a1) in 3.11 except that we omit “$\langle M_{j}:j\leq i\rangle\in
M_{i+1}$” and add: $\left[a\subseteq|M_{i}|\ \&\ |a|<\kappa\ \Rightarrow\ a\in
M_{i}\right]$ and for $i<j$, $M_{i}\cap\lambda$ is an initial segment of
$M_{j}\cap\lambda$,
(a2)∗:
for every expansion ${\mathscr{A}}$ of
$({\mathscr{H}}_{<\chi(\ast)}(\lambda),\in,<)$ by $\chi<\chi(\ast)$ relations,
for some $\alpha<\alpha(\ast)$, for every $n$,
$M^{\alpha}_{n}\prec{\mathscr{A}}$ in fact, for stationarily many $\zeta\in
S$, there is such $\alpha$ satisfying ${\dot{\zeta}}(\alpha)=\zeta$.
###### Remark 3.23.
We can retain (a1)∗ and add $a\subseteq M_{i}\wedge|a|<\kappa\Rightarrow a\in
M_{i}$.
Proof: Similar to 3.20, use the proof of [Sh 247], but for completeness we
give details.
We choose $\langle
S_{\alpha,\beta,\gamma}:\alpha<\mu,\beta<\lambda,\gamma<\lambda\rangle$ as
there. The main point is that defining ${{\mathbf{W}}}^{\ast}$ we have one
additional demand:
$(\epsilon)$:
if $n<\omega$ and $u\subseteq M_{n}$ has cardinality $<\kappa$ then $u\in
M_{n}$.
We then define ${{\mathbf{W}}}^{0}_{\delta}$ and $\langle
N_{\alpha}:\alpha<\lambda\rangle$ as there.
This give the changed demand in $(a1)^{*}$, but it give extra work in
verifying the demand $(a2)^{*}$.
So let a model ${\mathscr{A}}$ and cardinal $\chi=\chi^{<\kappa}<\chi(*)$ as
there be given ; as usual,
$\tau({\mathscr{A}})\in{\mathscr{H}}_{<\chi(\ast)}(\chi(\ast))$ and
${\mathscr{A}}$ expand $({\mathscr{H}}_{<\chi(\ast)}(\lambda),\in,<)$. For
every ${\bf x}=(\delta_{\bf x},\bar{M}_{\bf x},\eta_{\bf x},C_{\bf
x})\in{{\mathbf{W}}}^{\ast}$ we define a family ${\mathscr{F}}_{\bf x}$ , a
function $n:{\mathscr{F}}\rightarrow\omega$ and a function ${\rm rank}_{\bf
x}$ from ${\mathscr{F}}_{\bf x}$ into ${\rm Ord}\cup\\{\infty\\}$ as follows:
$(\alpha)$:
${\mathscr{F}}_{\bf x}=\Cup\\{{\mathscr{F}}_{{\bf x},n}:n<\omega\\}$
$(\beta)$:
${\mathscr{F}}_{{\bf x},n}=\\{f:f\mbox{ is an elementary embedding of }M_{{\bf
x},n}\mbox{ into }{\mathscr{A}}\\}$
$(\gamma)$:
$n(f)=k$ if and only if $f\in{\mathscr{F}}_{{\bf x},k}$
$(\delta)$:
${\rm rank}(f)=\cup\\{\epsilon+1:\mbox{ for every }\alpha<\lambda\mbox{ there
is }g\in{\mathcal{F}}_{{\bf x},n(f)}\mbox{ extending }f,\mbox{ such that
}\beta={\rm rank}_{\bf x}(g)\mbox{ and
}\mathop{\mathrm{Rang}}(g)\cap\alpha=\mathop{\mathrm{Rang}}(f)\cap\lambda\\}$
Now
Case 1: for no ${\bf x}\in{{\mathbf{W}}}^{\ast}$ and $f\in{\mathscr{F}}_{{\bf
x},0}$ do we have ${\rm rank}_{\bf x}(f)=\infty$
For every ${\bf x}\in{{\mathbf{W}}}^{\ast}$ and $f\in{\mathscr{F}}_{\bf x}$
let $\beta(f,{\bf x})$ be the first ordinal $\alpha<\lambda$ such that if
${\rm rank}_{\bf x}(f)=\epsilon$ then there is no $g\in{\mathscr{F}}_{{\bf
x},n(f)+1}$ extending $f$ with ${\rm rank}_{\bf x}(g)=\epsilon$ and
$\mathop{\mathrm{Rang}}(g)\cap\alpha=\mathop{\mathrm{Rang}}(f)\cap\lambda$.
Next let $\langle{\mathscr{A}}_{i}:i<\lambda\rangle$ be an increasing
continuous sequence of elementary submodels of ${\mathscr{A}}$, each of
cardinality $<\lambda$ such that $\langle{\mathscr{A}}_{j}:j\leq
i\rangle\in{\mathscr{A}}_{i+1}$.
Easily the set $E=\\{i<\lambda:{\mathscr{A}}_{i}\cap\lambda=i>\mu\\}$ is a
club of $\lambda$.
Choose by induction on $n<\omega$ an ordinal $i_{n}$ increasing with $n$ such
that $i_{n}\in E$ is of cofinality $\kappa$ , possible as
$2^{<\kappa}<\lambda$ as $\kappa<\chi(*)$ and
$\alpha<\lambda\rightarrow|\alpha|^{<\chi(*)}<\lambda$ hence
${\mathscr{A}}_{i_{n}}$ is an elementary submodel of ${\mathscr{A}}$ of
cardinality $<\lambda$. Choose $M\prec{\mathscr{A}}$ of cardinality $\chi$,
including $\\{i_{n}:n<\omega\\}$ such that every $u\subseteq M$ of cardinality
$<\kappa$ belongs to $M$.
Note that, if $u\subseteq{\mathscr{A}}_{i_{n}}$ has cardinality $<\kappa$ then
$u\in{\mathscr{A}}_{i_{n}}$ because $i_{n}\in E$ and
$\mathop{\mathrm{cf}}(i_{n})=\kappa$.
Let $M^{*}_{n}$ be ${\mathscr{A}}\upharpoonright({\mathscr{A}}_{i_{n}}\cap
M)$, easily $M^{*}_{n}\in{\mathscr{A}}_{i_{n}}$, so $[u\subseteq
M^{*}_{n}\wedge|u|<\kappa\Rightarrow u\in M^{*}_{n}]$. We can find ${\bf
x}\in{{\mathbf{W}}}$, and isomorphism $f_{n}$ from $M_{{\bf x},n}$ onto
$M^{*}_{n}$ increasing with $n$. Now clearly ${\bf
x}\in{\mathscr{A}}_{i_{n}}$, (why? as s
${{\mathbf{W}}}^{*}\in{\mathscr{A}}_{i_{n}}$ and $|{{\mathbf{W}}}^{*}|\leq\mu$
and $\mu+1\subseteq{\mathscr{A}}_{i_{n}}$). Also $f_{n}\in{\mathscr{F}}_{{\bf
x},n}$ and $f_{n}\in{\mathscr{A}}_{i_{n}}$, (as $M^{*}_{n},M_{{\bf
x},n}\in{\mathscr{A}}_{i_{n}}$) and the uniqueness of $f_{n}$ as those models
expand a submodel of $({\mathscr{H}}_{<\chi(\ast)}(\lambda),\in,<)$ and
necessarily are transitive over the ordinals). Similarly by the choice of
${\bf x}$, we have $f_{n}\subseteq f_{n+1}$. So $\langle{\rm rank}_{\bf
x}(f_{n}):n<\omega\rangle$ is constantly $\infty$ as otherwise we get an
infinite decreasing sequence of ordinals.
But this contradict our case assumption.
Case 2: not case 1,
So we choose ${\bf x}\in{{\mathbf{W}}}^{\ast}$ and $f\in{\mathscr{F}}_{{\bf
x},0}$ such that ${\rm rank}_{\bf x}(f)=\infty$
We easily get the desired contradiction and even a $\Delta$-system tree of
models. How? let $\langle\eta_{\alpha}:\alpha<\lambda\rangle$ list
${}^{\omega>}\lambda$ such that $\eta_{\alpha}\triangleleft\eta_{\beta}$
implies $\alpha<\beta$. Now we choose a pair
$(f_{\eta_{\alpha}},\gamma_{\alpha})$ by induction on $\alpha<\lambda$ such
that
(i):
$f_{\eta_{\alpha}}\in{\mathscr{F}}_{{\bf x},\mathop{\rm\ell
g}(\eta_{\alpha})}$
(ii:
$\gamma_{\alpha}=\mathop{\mathrm{sup}}\cup\\{\lambda\cap\mathop{\mathrm{Rang}}(f_{\eta_{\beta}}):\beta<\alpha\\}$
(iii):
if $\eta_{\beta}\triangleleft\eta_{\alpha}$ and $\mathop{\rm\ell
g}(\eta_{\alpha})=(\mathop{\rm\ell g}(\eta_{\beta})+1$ then
$\gamma_{\alpha}\cap\mathop{\mathrm{Rang}}(f_{\eta_{\alpha}})=\lambda\cap\mathop{\mathrm{Rang}}(f_{\eta_{\beta}})$
There is no problem to carry the induction. This finish the proof.
$\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{6.11A}}$
###### Lemma 3.24.
(1):
In 3.22 if in addition $\lambda=\mu^{+}$ then we can add
(c3):
if ${\dot{\zeta}}(\alpha)={\dot{\zeta}}(\beta)$, then
$|M^{\alpha}_{\omega}|\cap|M^{\beta}_{\omega}|\cap\lambda$ is an initial
segment of $|M^{\alpha}_{\omega}|\cap\lambda$ and of
$|M^{\beta}_{\omega}|\cap\lambda$, so when $\alpha\neq\beta$ it is a bounded
subset of ${\dot{\zeta}}(\alpha)$.
(2):
In 3.22 (and 3.24) , when $\kappa>{\aleph_{0}}$ then it follows that
(c4)∗:
if $\alpha\neq\beta$ and $\\{\eta^{\alpha}\restriction n:n<\omega\\}\subseteq
M^{\beta}_{\omega}$ then $\bar{M}^{\alpha}$, $\bar{\eta}^{\alpha}\in
M^{\beta}_{\omega}$.
(3):
Assume $\lambda=\mu^{+}$ and $\lambda=\mu^{\theta}$ and
$S\subseteq\\{\delta:\delta<\lambda,\mathop{\mathrm{cf}}(\delta)={\aleph_{0}}\\}$
is a stationary subset of $\lambda$ and $\langle C_{\delta}:\delta\in
S\rangle$ guess clubs (and $C_{\delta}$ is an unbounded subset of $\delta$ of
order type $\omega$, of course). Then we can find
$\langle\bar{N}_{\eta}:\eta\in\Gamma\rangle$ such that
(a):
$\Gamma=\cup\\{\Gamma_{\delta}:\delta\in S\\}$ where
$\Gamma_{\delta}\subseteq\\{\eta:\eta$ in an increasing $\omega$-sequence of
ordinals $<\delta$ with limit $\delta\\}$ and $\delta(\eta)=\delta$ when
$\eta\in\Gamma_{\delta},\delta\in S$.
(b):
$\bar{N_{\eta}}$ is $\langle N_{\eta,n}:n\leq\omega\rangle$ in
$\prec$-increasing, and we let $N_{\eta}=N_{\eta,\omega}$
(c):
each $N_{\eta}$ is a model of cardinality $\kappa$ with vocabulary
$\subseteq{\mathscr{H}}(\kappa^{+})$ for notational simplicity, and universe
$\subseteq\delta:=\delta(\eta)$ and
$N_{\eta,n}=N_{\eta}{\restriction}\gamma^{\delta}_{n}$ where
$\gamma^{\delta}_{n}$ is the $n$-the member of $C_{\delta}$
(d):
for every distinct $\eta,\nu\in\Gamma_{\delta}$ where $\delta\in S$, for some
$n<\omega$ we have $N_{\eta}\cap N_{\nu}=N_{\eta,n}=N_{\nu,n}$
(e):
for every $\eta,\nu\in\Gamma_{\delta}$ the models $N_{\eta},N_{\nu}$ are
isomorphic moreover there is such isomorphism $f$ which preserve the order of
the ordinals and maps $N_{\eta,n}$ onto $N_{\nu,n}$
(f):
if ${\mathscr{A}}$ is a model with universe $\lambda$ and vocabulary
$\subseteq{\mathscr{H}}(\kappa^{+})$ then for stationarily many $\delta\in S$
for some $\eta\in\Gamma_{\delta}\subseteq\Gamma$ we have
$N_{\eta}\prec{\mathscr{A}}$ . Moreover, if $\kappa^{\partial}=\kappa$ and $h$
is a one to one function from ${}^{\partial}\lambda$ into $\lambda$ then we
can add: if $\rho\in{}^{\partial}(N_{\eta,n})$ then $h(\rho)\in N_{\eta,n}$
Proof: (1) Let $g^{0},g^{1}$ be two place functions from
$\lambda\times\lambda$ to $\lambda$ such that for
$\alpha\in[\mu,\lambda]:\langle g^{0}(\alpha,i):i<\mu\rangle$ enumerate
$\\{j:j<\mu\\}$ without repetitions, and $g^{1}(\alpha,g^{0}(\alpha,i))=i$ for
$i<\lambda$.
Now we can restrict ourselves to $\bar{M}^{\alpha}$ such that each
$M^{\alpha}_{i}$ (for $i\leq\omega$) is closed under $g^{0},g^{1}$. Then (c3)
follows immediately from
$[{\dot{\zeta}}(\alpha)={\dot{\zeta}}(\beta)\quad\Rightarrow\quad|M^{\alpha}_{\omega}|\cap\mu=|M^{\beta}_{\omega}|\cap\mu]$
(required in (c1)).
(2) Should be clear.
(3) This just rephrase what we have proved above.
$\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{6.12}}$
###### Lemma 3.25.
Suppose that $\lambda=\mu^{+}$,
$\mu=\kappa^{\aleph_{0}}=2^{\kappa}>2^{\aleph_{0}}$,
$\mathop{\mathrm{cf}}(\kappa)=\aleph_{0}$ and
$S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\aleph_{0}\\}$ is
stationary, $\theta=\aleph_{0}$,
$\aleph_{0}<\chi(\ast)=\mathop{\mathrm{cf}}(\chi(\ast))<\kappa$. Then we can
find ${\mathbf{W}}=\\{(\bar{M}^{\alpha},\eta^{\alpha}):\alpha<\alpha(\ast)\\}$
and functions
${\dot{\zeta}}:\alpha(\ast)\longrightarrow S,\qquad
h:\alpha(\ast)\longrightarrow\lambda$
and $\langle C_{\delta}:\delta\in S\rangle$ with
$\langle\gamma^{\delta}_{n}:n<\omega\rangle$ listing $C_{\delta}$ in
increasing order such that:
(a0)-(a1):
as in 3.11,
(a2)∗ :
as in 3.22,
(b0)-(b2):
as in 3.11 and even
(b1)∗:
$\alpha\neq\beta$, $\\{\eta^{\alpha}\restriction n:n<\omega\\}\subseteq
M^{\beta}_{\omega}$ implies $\alpha<\beta$ and even
${\dot{\zeta}}(\alpha)<{\dot{\zeta}}(\beta)$,
(c1)-(c3):
as in 3.20 \+ 3.24(1),
(c4):
if ${\dot{\zeta}}(\alpha)={\dot{\zeta}}(\beta)=\delta$ but $\alpha\neq\beta$
then for some $n_{0}\geq 1$, there are no $n>n_{0}$ and
$\alpha_{1}\leq\beta_{2}\leq\alpha_{3}$ satisfying:
$\begin{array}[]{l}\alpha_{1}\in|M^{\alpha}_{\omega}|\cap[\gamma^{\delta}_{n},\gamma^{\delta}_{n+1}),\\\
\beta_{2}\in|M^{\beta}_{\omega}|\cap[\gamma^{\delta}_{n},\gamma^{\delta}_{n+1}),\\\
\alpha_{3}\in|M^{\alpha}_{\omega}|\cap[\gamma^{\delta}_{n},\gamma^{\delta}_{n+1}),\end{array}$
i.e., either
$\mathop{\mathrm{sup}}\left([\gamma^{\delta}_{n},\gamma^{\delta}_{n+1})\cap|M^{\alpha}_{\omega}|\right)<\mathop{\mathrm{min}}\left([\gamma^{\delta}_{n},\gamma^{\delta}_{n+1})\cap|M^{\beta}_{\omega}|\right)$
or
$\mathop{\mathrm{sup}}\left([\gamma^{\delta}_{n},\gamma^{\delta}_{n+1})\cap|M^{\beta}_{\omega}|\right)<\mathop{\mathrm{min}}\left([\gamma^{\delta}_{n},\gamma^{\delta}_{n+1})\cap|M^{\alpha}_{\omega}|\right)$;
(c5):
if $\Upsilon<\kappa$ and there is $B\subseteq{}^{\omega}\kappa$,
$|B|=\kappa^{\aleph_{0}}$ which contains no perfect set with density
$\Upsilon$ (holds trivially if $\kappa$ is strong limit), then also
$\\{\eta^{\alpha}:\alpha<\alpha(\ast)\\}$ does not contain such a set. (See
3.26).
Proof: We repeat the proof of 3.20 with some changes.
Let $\langle S_{\alpha,\beta,\gamma}:\alpha<\mu,\ \beta<\lambda,\
\gamma<\lambda\rangle$ be pairwise disjoint stationary subsets of $S$. Let
$g^{0},g^{1}$ be as in the proof of 3.24. By 3.7 there is a sequence $\langle
C_{\delta}:\delta\in S\rangle$ such that:
(i):
$C_{\delta}$ is a club of $\delta$ of order type $\kappa$, not $\omega$!,
$0\notin C_{\delta}$,
(ii):
for $\alpha<\mu$, $\beta<\lambda$, $\gamma<\lambda$, for every club $C$ of
$\lambda$, the set
$\\{\delta\in S_{\alpha,\beta,\gamma}:C_{\delta}\subseteq C\\}$
is stationary.
We then define ${\mathbf{W}}^{\ast}$, $(\delta^{j},\langle
M_{j,n}:n<\omega\rangle,\eta_{j},C^{j})$ for $j<\mu$,$A_{\alpha}$ for
$\alpha<\lambda$, and $R$ as in the proof of 3.20.
Now, for $\delta\in S_{\alpha,\beta,\gamma}$ let ${\mathbf{W}}^{1}_{\delta}$
be the collection of all systems $\langle
M_{\rho},\eta_{\rho}:\rho\in{}^{\omega>}\kappa\rangle$ such that
(i):
$\eta_{\rho}$ is an increasing sequence of ordinals of length $\mathop{\rm\ell
g}(\rho)$,
(ii):
$\mathop{\mathrm{otp}}\left(C_{\delta}\cap\eta_{\rho}(\ell)\right)=1+\rho(\ell)$
for $\ell<\mathop{\rm\ell g}(\rho)$,
(iii):
there are isomorphisms $\langle h_{\rho}:\rho\in{}^{\omega>}\kappa\rangle$
such that $h_{\rho}$ maps $M_{\rho}$ onto $M_{\alpha,\mathop{\rm\ell
g}(\rho)}$ preserving $\in,R$, ${\mathord{\mathrm{cd}}}(x)=y$,
$g^{0}(x_{1},x_{2})=y$, $g^{1}(x_{1},x_{2})=y$ (and their negations),
(iv):
if $\rho\triangleleft\nu$ then $h_{\rho}\subseteq h_{\nu}$, $M_{\rho}\prec
M_{\sigma}$, $M_{\rho}\in M_{\nu}$,
(v):
$M_{\rho}\cap C_{\delta}=\emptyset$, and
$M_{\rho}\cap\lambda\subseteq\bigcup\limits_{\ell}[\gamma_{\rho(\ell)},\gamma_{\rho(\ell)+1})$,
where $\gamma_{\zeta}$ is the $\zeta$-th member of $C_{\delta}$,
(vi):
if $\rho\in{}^{\omega>}\kappa$, $\ell<\mathop{\rm\ell g}(\rho)$, $\gamma$ is
the $(1+\rho(\ell))$-th member of $C_{\delta}$ then $M_{\ell}\cap\gamma$
depends only on $\rho\restriction\ell$, and $M_{\rho}{\restriction}\gamma\prec
M_{\rho}$,
(vii):
$N_{\beta}=M_{\langle\rangle}$.
Now clearly $|{\mathbf{W}}^{1}_{\delta}|\leq\mu$, so let
${\mathbf{W}}^{1}_{\delta}=\\{\langle(M^{j}_{\rho},\eta^{j}_{\rho}):,\rho\in{}^{\omega>}\kappa\rangle:j<\mu\\}$.
Let $\langle\rho_{j}:j<\mu\rangle$ be a list of distinct members of
${}^{\omega}\kappa$; for (c5) — choose as there.
Let
$M^{j}_{\ell}=\bigcup\limits_{\ell<\omega}M^{j}_{\rho_{j}\restriction\ell},\quad\eta^{j}=\langle\eta^{j}_{\rho_{j}\restriction(\ell+1)}(\ell+1):\ell\leq\omega\rangle.$
Now,
$\\{\langle{M}^{j}_{\ell}:\ell<\omega\rangle:j<\mu\\}$
is as required in (c4). Also (c5) is straightforward (as taking union for all
$\delta$’s change little), (of course, we are omitting $\delta$’s where we get
unreasonable pairs).
The rest is as before. $\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{6.13}}$
###### Remark 3.26.
The existence of $B$ as in (c5) is proved, for some $\Upsilon$ for all strong
limit $\kappa$ of cofinality $\aleph_{0}$ in [Sh:g, Ch II, 6.9, p 104], really
stronger conclusions hold. If $2^{\kappa}$ is regular and belongs to
$\\{\mathop{\mathrm{cf}}(\prod\kappa_{n}/D):D$ an ultrafilter on $\omega$,
$\kappa_{n}<\kappa\\}$ or $2^{\kappa}$ is singular and is the supremum of this
set, then it exists for $\Upsilon=(2^{\aleph_{0}})^{+}$. Now, if above we
replace $D$ by the filter of co-bounded subsets of $\omega$, then we get it
even for $\Upsilon=\aleph_{0}$; by [Sh:E12, Part D] the requirement holds,
e.g., for $\beth_{\delta}$ for a club of $\delta<\omega_{1}$
Moreover, under this assumption on $\kappa$ we can demand (essentially, this
is expanded in 3.30)
$(c4)^{*}$ :
we strengthen clause (c4) to:
if ${\dot{\zeta}}(\alpha)={\dot{\zeta}}(\beta)=\delta$ but $\alpha\neq\beta$
then for some $n_{0}\geq 1$, we have
either for every $n\in[n_{1},\omega)$ we have
$\mathop{\mathrm{sup}}\left([\gamma^{\delta}_{n},\gamma^{\delta}_{n+1})\cap|M^{\alpha}_{\omega}|\right)<\mathop{\mathrm{min}}\left([\gamma^{\delta}_{n},\gamma^{\delta}_{n+1})\cap|M^{\beta}_{\omega}|\right)$
or for every $n\in[n_{1},\omega)$ we have
$\mathop{\mathrm{sup}}\left([\gamma^{\delta}_{n},\gamma^{\delta}_{n+1})\cap|M^{\beta}_{\omega}|\right)<\mathop{\mathrm{min}}\left([\gamma^{\delta}_{n},\gamma^{\delta}_{n+1})\cap|M^{\alpha}_{\omega}|\right)$;
###### Lemma 3.27.
We can combine 3.25 with 3.22.
Proof: Left to the reader.
###### Lemma 3.28.
Suppose $\aleph_{0}=\theta<\chi(\ast)=\mathop{\mathrm{cf}}(\chi(\ast))$ and:
$\lambda^{\aleph_{0}}=\lambda^{<\chi(\ast)}$, $\chi(\ast)\leq\lambda$ and:
$\lambda=\lambda^{+}_{1}$, and $(\ast)_{\lambda_{1}}$ (see below) holds.
Then
$(\ast)_{\lambda}$:
we can find
${\mathbf{W}}=\\{(\bar{M}^{\alpha},\eta^{\alpha}):\alpha<\alpha(\ast)\\}$ and
functions ${\dot{\zeta}}:\alpha(\ast)\longrightarrow S$ and
$h:\alpha(\ast)\longrightarrow\lambda$ such that:
(a0)-(a2):
as in 3.11,
(b0)-(b2):
as in 3.11, and even
(c3):
if ${\dot{\zeta}}(\alpha)={\dot{\zeta}}(\beta)$ then
$|M_{\alpha}|\cap|M_{\beta}|$ is a bounded subset of ${\dot{\zeta}}(\alpha)$.
Proof: Left to the reader.
###### Lemma 3.29.
Suppose that $\lambda$ is a strongly inaccessible uncountable cardinal,
$\mathop{\mathrm{cf}}(\lambda)\geq\chi(\ast)=\mathop{\mathrm{cf}}(\chi(\ast))>\theta=\aleph_{0},$
and let $S\subseteq\lambda$ consist of strong limit singular cardinals of
cofinality $\aleph_{0}$ and be stationary. Then we can find
${\mathbf{W}}=\\{(\bar{M}^{\alpha},\eta^{\alpha}):\alpha<\alpha(\ast)\\}$ and
functions ${\dot{\zeta}}:\alpha(\ast)\longrightarrow S$ and
$h:\alpha(\ast)\longrightarrow\lambda$ such that:
(a0)-(a2):
of 3.11 (except that $h(\alpha)$ depends not only on ${\dot{\zeta}}(\alpha))$,
(b0),(b3):
of 3.11,
(b1)+:
of 3.18,
(c3)-:
if ${\dot{\zeta}}(\alpha)=\delta={\dot{\zeta}}(\beta)$ then
$|M^{\alpha}_{\omega}|\cap|M^{\beta}_{\omega}|\cap\delta$ is a bounded subset
of $\delta$.
Remark:
(1):
See [Sh 45] for essentially a use of a weaker version.
(2):
We can generalize 3.22.
Proof: See the proof of [Sh 331, 1.10(3)] but there
$\mathop{\mathrm{sup}}(N_{\langle\rangle}\cap\lambda)<\delta$.
###### Lemma 3.30.
(1):
Suppose that $\lambda=\mu^{+}$, $\mu=\kappa^{\theta}=2^{\kappa}$,
$\theta<\mathop{\mathrm{cf}}(\chi(\ast))=\chi(\ast)<\kappa$, $\kappa$ is
strong limit, $\kappa>\mathop{\mathrm{cf}}(\kappa)=\theta>\aleph_{0}$,
$S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\theta\\}$ is
stationary. Then we can find
${\mathbf{W}}=\\{(\bar{M}^{\alpha},\eta^{\alpha}):\alpha<\alpha(\ast)\\}$
(actually, a sequence), functions ${\dot{\zeta}}:\alpha(\ast)\longrightarrow
S$ and $h:\alpha(\ast)\longrightarrow\lambda$ and $\langle
C_{\delta}:\delta\in S\rangle$ such that:
(a1)-(a2):
as in 3.11,
(b0):
$\eta^{\alpha}\neq\eta^{\beta}$ for $\alpha\neq\beta$,
(b1):
if $\\{\eta^{\alpha}\restriction i:i<\theta\\}\subseteq M^{\beta}_{\theta}$
and $\alpha\neq\beta$ then $\alpha<\beta$ and even
${\dot{\zeta}}(\alpha)<{\dot{\zeta}}(\beta)$,
(b2):
if $\eta^{\alpha}\restriction(j+1)\in M^{\beta}_{\theta}$ then
$M^{\alpha}_{j}\in M^{\beta}_{\theta}$,
(c2):
$\bar{C}=\langle C_{\delta}:\delta\in S\rangle$, $C_{\delta}$ a club of
$\delta$ of order type $\theta$, and every club of $\lambda$ contains
$C_{\delta}$ for stationarily many $\delta\in S$,
(c3):
if $\delta\in S$, $C_{\delta}=\\{\gamma_{\delta,i}:i<\theta\\}$ is the
increasing enumeration, $\alpha<\alpha^{\ast}$ satisfies
${\dot{\zeta}}(\alpha)=\delta$, then there is
$\langle\langle\gamma^{-}_{\alpha,i},\gamma^{+}_{\alpha,i}\rangle:i<\theta\mbox{
odd }\rangle$ such that
$\gamma^{-}_{\alpha,i}\in M^{\alpha}_{i},\quad
M^{\alpha}_{i}\cap\lambda\subseteq\gamma^{+}_{\alpha,i},\quad\gamma_{\delta,i}<\gamma^{-}_{\alpha,i}<\gamma^{+}_{\alpha,i}<\gamma_{\delta,i+1}$
and
$(\ast)$:
if ${\dot{\zeta}}(\alpha)={\dot{\zeta}}(\beta)$, $\alpha<\beta$ then for every
large enough odd $i<\theta$, $\gamma^{+}_{\alpha,i}<\gamma^{-}_{\beta,i}$
(hence
$[\gamma^{-}_{\alpha,i},\gamma^{+}_{\alpha,i})\cap[\gamma^{-}_{\beta,i},\gamma^{+}_{\beta,i})=\emptyset$)
and
$[\gamma^{-}_{\beta,i},\gamma^{+}_{\beta,i})\cap
M^{\alpha}_{\theta}=\emptyset.$
(2):
In part (1), assume $\theta=\aleph_{0}$ and
$\mathop{\mathrm{pp}}(\kappa)=^{+}2^{\kappa}$. Then the conclusion holds;
moreover, (c3) (from 3.24).
Remark: The assumption $\mathop{\mathrm{pp}}(\kappa)=2^{\kappa}$ holds, for
example, for $\kappa=\beth_{\delta}$ for a club of $\delta<\omega$ (see
[Sh400, §5]).
Proof:
(1) By 3.6 we can find $\bar{C}=\langle C_{\delta}:\delta\in S\rangle$,
$C_{\delta}$ a club of $\delta$, of order type $\kappa$ such that for any club
$C$ of $\lambda$ for stationarily many $\delta\in S$, we have:
$C_{\delta}\subseteq C$.
First case: assume $\mu(=2^{\kappa})$ is regular.
By [Sh:g, II, 5.9], we can find an increasing sequence
$\langle\kappa_{i}:i<\theta\rangle$ of regular cardinals $>\chi(\ast)$ such
that $\kappa=\sum\limits_{i<\theta}\kappa_{i}$, and
$\prod\limits_{i<\theta}\kappa_{i}/J^{\mathop{\mathrm{bd}}}_{\theta}$ has true
cofinality $\mu$, and let $\langle f_{\epsilon}:\epsilon<\mu\rangle$ exemplify
this, which means:
$\epsilon<\zeta<\mu\quad\Rightarrow\quad f_{\epsilon}<f_{\zeta}\mod
J^{\mathop{\mathrm{bd}}}_{\theta},$
and for every $f\in\prod\limits_{i<\theta}\kappa_{i}$, for some $\epsilon<\mu$
we have $f<f_{\epsilon}\mod J^{\mathop{\mathrm{bd}}}_{\theta}$. We may assume
that if $\epsilon$ is limit and $\bar{f}\restriction\epsilon$ has
$<_{J^{\mathop{\mathrm{bd}}}_{\theta}}$–l.u.b., then $f_{\epsilon}$ is a
$<_{J^{\mathop{\mathrm{bd}}}_{\theta}}$–l.u.b., and we know that if
$\mathop{\mathrm{cf}}(\epsilon)>2^{\theta}$ then this holds, and that w.l.o.g
$\bigwedge\limits_{i<\theta}\mathop{\mathrm{cf}}(f_{\epsilon}(i))=\mathop{\mathrm{cf}}(\epsilon)$.
Without loss of generality
$\kappa_{i}>f_{\epsilon}(i)>\bigcup\limits_{j<i}\kappa_{j}$.
We shall define $W$ later. Let ${\rm St}$ be a strategy for player I. By the
choice of $\bar{C}$, for some $\delta\in S$, for every $\alpha\in C_{\delta}$
of cofinality $>\theta$, ${\mathscr{H}}_{<\chi(\ast)}(\alpha)$ is closed under
the strategy ${\rm St}$. Let $C_{\delta}=\\{\alpha_{i}:i<\kappa\\}$ be
increasing continuous. For each $\epsilon<\mu$ we choose a play of the game,
player I using ${\rm St}$, $\langle
M^{\epsilon}_{j},\eta^{\epsilon}_{j}:j<\theta\rangle$ such that:
$\begin{array}[]{l}\langle M^{\epsilon}_{j}:j\leq
j_{1}\rangle\in{\mathscr{H}}_{<\chi(\ast)}\left(\alpha_{f_{\epsilon}(j_{1})+1}\right),\\\
\eta^{\epsilon}_{\gamma}=\langle{\mathord{\mathrm{cd}}}\left(\alpha_{f_{\epsilon}(i)},\langle
M^{\epsilon}_{i}:i\leq j\rangle\right):j<\gamma\rangle,\quad\mbox{ and}\\\
\eta^{\epsilon}_{j+1}\in M^{\epsilon}_{j+1}.\end{array}$
Then let $g_{\epsilon}\in\prod\limits_{i<\theta}\kappa_{i}$ be:
$g_{\epsilon}(i)=\mathop{\mathrm{sup}}\bigl{(}\kappa_{i}\cap\bigcup\limits_{j<\theta}M^{\epsilon}_{j}\bigr{)},$
so for some $\beta_{\epsilon}\in(\epsilon,\mu)$, we have
$g_{\epsilon}<f_{\beta_{\epsilon}}\mod J^{\mathop{\mathrm{bd}}}_{\theta}$.
On the other hand, if $\mathop{\mathrm{cf}}(\epsilon)=(2^{\theta})^{+}$,
without loss of generality,
$\mathop{\mathrm{cf}}\big{(}f_{\epsilon}(i)\big{)}=\mathop{\mathrm{cf}}(\epsilon)$
for every $i<\theta$ (see [Sh:g, II, §1]), so there is
$\gamma_{\epsilon}<\epsilon$ such that
$h_{\epsilon}<f_{\gamma_{\epsilon}}\mod
J^{\mathop{\mathrm{bd}}}_{\theta}\quad\mbox{ where
}h_{\epsilon}(i)=\mathop{\mathrm{sup}}\bigl{(}f_{\epsilon}(i)\cap\bigcup\limits_{j<\theta}M^{\epsilon}_{j}\bigr{)}.$
So for some $\gamma(\ast)<\mu$ we have:
$S_{\delta}[{\rm
St}]=\\{\epsilon<\mu:\mathop{\mathrm{cf}}(\epsilon)=(2^{\theta})^{+},\mbox{
and }\gamma_{\epsilon}=\gamma(\ast)\\}\ \mbox{ is stationary.}$
Now, for each $\delta\in S$ we can consider the set ${\bf C}_{\delta}$ of all
possible such $\langle\bar{M}^{\epsilon},\eta^{\epsilon}:\epsilon<\mu\rangle$,
where $\bar{M}^{\epsilon}=\langle M^{\epsilon}_{j}:j<i\rangle$,
$\eta^{\epsilon}_{\theta}$ are as above (letting ${\rm St}$ vary on all
strategies of player I for which
$\alpha\in C_{\delta}\ \&\
\mathop{\mathrm{cf}}(\alpha)>\theta\quad\Rightarrow\quad{\mathscr{H}}_{<\chi(\ast)}(\alpha)\mbox{
is closed under }{\rm St}).$
A better way to write the members of ${\bf C}_{\delta}$ is
$\langle\langle\bar{M}^{\epsilon}_{j},\eta^{\epsilon}_{j}:j<\theta\rangle:\epsilon<\mu\rangle$,
but for $j<\theta$,
$f_{\epsilon(1)}\restriction j=f_{\epsilon(2)}\restriction
j\quad\Rightarrow\quad\bar{M}^{\epsilon(1)}_{j}=M^{\epsilon(2)}_{j}\ \&\
\eta^{\epsilon(1)}_{j}=\eta^{\epsilon(2)}_{j};$
actually it is a function from $\\{f_{\epsilon}\restriction
j:\epsilon<\mu,j<\theta\\}$ to ${\mathscr{H}}_{<\chi(\ast)}(\delta)$. But the
domain has cardinality $\kappa$, the range has cardinality $|\delta|\leq\mu$.
So $|{\bf C}_{\delta}|\leq\mu^{\kappa}=(2^{\kappa})^{\kappa}=2^{\kappa}=\mu$.
So we can well order ${\bf C}_{\delta}$ in a sequence of length $\mu$, and
choose by induction on $\epsilon<\mu$ a representative of each for
${{\mathbf{W}}}$ satisfying the requirements.
Second case: assume $\mu$ is singular.
So let $\mu=\sum\limits_{\xi<\mathop{\mathrm{cf}}(\mu)}\mu_{\xi}$, $\mu_{\xi}$
regular, without loss of generality
$\mu_{\xi}>(\sum\\{\mu_{\epsilon}:\epsilon<\xi\\})^{+}+(\mathop{\mathrm{cf}}(\mu))^{+}$.
We know that $\mathop{\mathrm{cf}}(\mu)>\kappa$, and again by [Sh:g, VIII §1]
there are $\langle\kappa_{\xi,i}:i<\theta\rangle$,
$\langle\kappa_{i}:i<\theta\rangle$ such that:
$\mathop{\mathrm{tcf}}\left(\prod_{i<\theta}\kappa_{\xi,i}/J^{\mathop{\mathrm{bd}}}_{\theta}\right)=\mu_{\xi},\qquad\mathop{\mathrm{tcf}}\left(\prod_{i<\theta}\kappa_{i}/J^{\mathop{\mathrm{bd}}}_{\theta}\right)=\mathop{\mathrm{cf}}(\mu),$
$\kappa^{a}_{i}<\kappa_{\xi
i}<\kappa^{b}_{i},\qquad\kappa^{a}_{i}<\kappa_{i}<\kappa^{b}_{i}\qquad\mbox{
and }\qquad i<j\quad\Rightarrow\quad\kappa^{b}_{i}<\kappa^{a}_{j}$
(we can even get $\kappa^{a}_{i}>\prod\limits_{j<i}\kappa^{b}_{j}$ as we can
uniformize on $\xi$).
Let $\langle f^{\xi}_{\epsilon}:\epsilon<\mu_{\xi}\rangle$, $\langle
f_{\epsilon}:\epsilon<\mathop{\mathrm{cf}}(\mu)\rangle$ witness the true
cofinalities. Now, for every $f\in\prod\limits_{i<\theta}\kappa_{i}$ (for
simplicity such that $f(i)>\sum\limits_{j<i}\kappa_{j}$,
$\bigwedge\limits_{i}\mathop{\mathrm{cf}}(f(i))=(2^{\theta})^{+}$) and $\xi$
we can repeat the previous argument for $\langle
f+f^{\xi}_{\epsilon}:\epsilon<\mu_{\epsilon}\rangle$. After “cleaning inside”,
replacing by a subset of cardinality $\mu_{\xi}$ we find a common bound below
$\prod\limits_{i<\theta}\kappa_{i}$ and below $\prod f$, and we can uniformize
on $\xi$.
Thus we apply on every $f_{\epsilon}$,
$\mathop{\mathrm{cf}}(\epsilon)=(2^{\theta})^{+}$ and use the same argument on
the bound we have just gotten.
(2) Should be clear.
Similarly to 3.20 with $\omega^{2}$ for $\theta$ , (not a cardinal!) we have
###### Claim 3.31.
Suppose that
(*):
$\lambda$ is a regular cardinal, $\theta=\aleph_{0}$,
$\mu=\mu^{<\chi(\ast)}<\lambda\leq 2^{\mu}$,
$S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\aleph_{0}\\}$ is
stationary and $\aleph_{0}<\chi(\ast)=\mathop{\mathrm{cf}}(\chi(\ast))$.
Then we can find
${\mathbf{W}}=\\{(\bar{M}^{\alpha},\eta^{\alpha}):\alpha<\alpha(\ast)\\}$
and functions
${\dot{\zeta}}:\alpha(\ast)\longrightarrow S\ \mbox{ and }\
h:\alpha(\ast)\longrightarrow\lambda$
such that:
(a0):
like 3.11,
(a1):
$\bar{M}^{\alpha}=\langle M^{\alpha}_{i}:i\leq\omega^{2}\rangle$ is an
increasing continuous elementary chain ($\tau(M^{\alpha}_{i})$, the
vocabulary, may be increasing too and belongs to
${\mathscr{H}}_{<\chi(\ast)}(\chi(\ast))$, each $M^{\alpha}_{i}$ is a model
belonging to ${\mathscr{H}}_{<\chi(\ast)}(\lambda)$ [so necessarily has
cardinality $<\chi(\ast)$], $M^{\alpha}_{i}\cap\chi(\ast)$ is an ordinal,
$[\chi(\ast)=\chi^{+}\ \Rightarrow\ \chi+1\subseteq M^{\alpha}_{i}]$,
$\eta^{\alpha}\in{}^{\omega^{2}}\lambda$ is increasing with limit
${\dot{\zeta}}(\alpha)\in S$, $\eta^{\alpha}\restriction i\in
M^{\alpha}_{i+1}$, $M^{\alpha}_{i}$ belongs to
${\mathscr{H}}_{<\chi(\ast)}(\eta^{\alpha}(i))$ and $\langle
M^{\alpha}_{i}:i\leq j\rangle$ belongs to $M^{\alpha}_{j+1}$,
(a2):
like 3.11 (with $\omega^{2}$ instead $\theta)$,
(b0), (b1), (b2):
as in 3.11,
(b1)∗:
as in 3.20,
(c1):
if ${\dot{\zeta}}(\alpha)={\dot{\zeta}}(\beta)$ then
$M^{\alpha}_{\omega^{2}}\cap\mu=M^{\beta}_{\omega^{2}}\cap\mu$ and there is an
isomorphism $h_{\alpha,\beta}$ from $M^{\alpha}_{\omega^{2}}$ onto
$M^{\beta}_{\omega^{2}}$ mapping $\eta^{\alpha}(i)$ to $\eta^{\beta}(i)$,
$M^{\alpha}_{i}$ to $M^{\beta}_{i}$ for $i<\omega^{2}$,
$h_{\alpha,\beta}\restriction\left(\left|M^{\alpha}_{\omega^{2}}\right|\cap\left|M^{\beta}_{\omega^{2}}\right|\right)$
is the identity,
(c2):
as in 3.20 using $\langle M^{\alpha}_{\omega n}:n<\omega\rangle$,
(c3):
as in 3.24 assuming $\lambda=\mu^{+}$,
(c4):
$\eta^{\alpha}(i)>\mathop{\mathrm{sup}}(|M^{\alpha}_{i}|\cap\lambda)$ (so
$\mathop{\mathrm{sup}}(|M^{\alpha}_{\omega(n+1)}|\cap\lambda)=\bigcup\limits_{\ell}\eta^{\alpha}(\omega
n+\ell)$)
Proof: We use $\langle\bar{M}^{\alpha,0}:\alpha<\alpha(\ast)\rangle$ which we
got in 3.20. Now for each $\alpha$ we look at
$\bigcup\limits_{n<\omega}M^{\alpha,0}_{n}$ as an elementary submodel of
$({\mathscr{H}}_{<\chi(\ast)}(\lambda),\in)$ with a function ${\rm St}$
(intended as strategy for player I, in the play for (a2) above).
Play in $\bigcup\limits_{n<\omega}M^{\alpha,0}_{n}$ and get
$\begin{array}[]{c}\langle M^{\alpha}_{i},\eta^{\alpha}(i):i<\omega
n\rangle\in M^{\alpha,0}_{n},\\\
\mathop{\mathrm{sup}}\\{\eta^{\alpha}(i):i<\omega n\\}\in
M^{\alpha,0}_{n+1},\\\ \eta^{\alpha}(\omega
n)>\mathop{\mathrm{sup}}(M^{\alpha,0}_{n}\cap\lambda).\end{array}$
$\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{6.14B}}$
###### Lemma 3.32.
Assume that $\lambda\geq\chi(\ast)>\theta$ are regular cardinals,
$S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\theta\\}$ is a
stationary set, $\lambda^{<\chi(\ast)}=\lambda$, and the conclusion of 3.30
holds for them. Then it holds also for $\lambda^{+}$ instead of $\lambda$.
Proof: By [Sh 331, 2.10](2) or see [Sh 365], we know
$(\ast)$:
there are $\langle C_{\delta}:\delta<\lambda^{+}\mbox{ and
}\mathop{\mathrm{cf}}(\delta)=\theta\rangle$, $\langle
e_{\alpha}:\alpha<\lambda^{+}\rangle$ such that:
(i):
$C_{\delta}$ is a club of $\delta$ of order type $\theta$,
$\alpha\in C_{\delta}\ \&\
\alpha>\mathop{\mathrm{sup}}(C_{\delta}\cap\alpha)\quad\Rightarrow\quad\mathop{\mathrm{cf}}(\alpha)=\lambda,$
(ii):
$e_{\alpha}$ is a club of $\alpha$ of order type
$\mathop{\mathrm{cf}}(\alpha)$,
$e_{\alpha}=\\{\beta^{\alpha}_{i}:i<\mathop{\mathrm{cf}}(\alpha)\\}$
(increasing continuous),
(iii):
if $E$ is a club of $\lambda^{+}$ then for stationarily many
$\delta<\lambda^{+}$, $\mathop{\mathrm{cf}}(\delta)=\theta$,
$C_{\delta}\subseteq E$ and the set
$\\{i<\lambda:\mbox{ for every }\alpha\in C_{\delta},\
\mathop{\mathrm{cf}}(\alpha)=\lambda\ \Rightarrow\ \beta^{\alpha}_{i+1}\in
E\\}$
is unbounded in $\lambda$.
Now copying the black box of $\lambda$ on each $\delta<\lambda^{+}$,
$\mathop{\mathrm{cf}}(\delta)=\theta$, we can finish easily.
###### Lemma 3.33.
If $\lambda$, $\mu$, $\kappa$, $\theta$, $\chi(\ast)$, $S$ are as in 3.30, and
$\alpha<\chi(\ast)\quad\Rightarrow\quad|\alpha|^{\theta}<\chi(\ast)$
then there is a stationary
$S^{\ast}\subseteq\\{A\subseteq\lambda:|A|<\chi(\ast)\\}$ and a one-to-one
function ${\mathord{\mathrm{cd}}}$ from $S^{\ast}$ to $\lambda$ such that:
$A\in S^{\ast}\ \&\ B\in S^{\ast}\ \&\ A\neq B\ \&\ A\subset
B\quad\Rightarrow\quad{\mathord{\mathrm{cd}}}(A)\in B.$
Remark: This gives another positive instance to a problem of Zwicker. (See [Sh
247].)
Proof: Similar to the proof of 3.30 only choose
${\mathord{\mathrm{cd}}}:\\{A:A\subseteq\lambda\mbox{ and
}|A|<\chi(\ast)\\}\longrightarrow\lambda$
one-to-one, and then define
$S^{\ast}\cap\\{A:A\subseteq\alpha,\ |A|<\chi(\ast)\\}$
by induction on $\alpha$. $\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{6.14D}}$
###### Problem 3.34.
(1):
Can we prove in ZFC that for some regular $\lambda>\theta$
$(\ast)_{\lambda,\theta,\chi(\ast)}$:
we can define for $\alpha\in
S^{\lambda}_{\theta}=\\{\delta<\lambda:\aleph_{0}\leq\mathop{\mathrm{cf}}(\delta)=\theta\\}$
a model $M_{\alpha}$ with a countable vocabulary and universe an unbounded
subset of $\alpha$ of cardinality $<\chi(\ast)$, $M_{\delta}\cap\chi(\ast)$ is
an ordinal such that: for every model $M$ with countable vocabulary and
universe $\lambda$, for some (equivalently: stationarily many) $\delta\in
S^{\lambda}_{\kappa}$, $M_{\delta}\subseteq M$.
(2):
The same dealing with relational vocabularies only (we call it $(\ast)^{{\rm
rel}}_{\lambda,\theta,\kappa}$).
###### Remark 3.35.
Note that by 3.8 if $(\ast)_{\lambda,\theta,\kappa}$,
$\mu=\mathop{\mathrm{cf}}(\mu)>\lambda$ then $(\ast)_{\mu^{+},\theta,\kappa}$.
* * *
Now (3.36—3.40) we return to black boxes for singular $\lambda$, i.e., we deal
with the case $\mathop{\mathrm{cf}}(\lambda)\leq\theta$.
###### Lemma 3.36.
Suppose that $\lambda^{\theta}=\lambda^{<\chi(\ast)}$, $\lambda$ is a singular
cardinal, $\theta$ is regular, and $\chi(\ast)$ is regular $>\theta$. Assume
further
$(\alpha)$:
$\mathop{\mathrm{cf}}(\lambda)\leq\theta$,
$(\beta)$:
$\lambda=\sum\limits_{i\in w}\mu_{i}$, $|w|\leq\theta$, $w\subseteq\theta^{+}$
(usually $w=\mathop{\mathrm{cf}}(\lambda)$) and $[i<j\ \Rightarrow\
\mu_{i}<\mu_{j}]$, and each $\mu_{i}$ is regular $<\lambda$ and
$\mathop{\mathrm{cf}}(\lambda)>\aleph_{0}\ \wedge\
\mathop{\mathrm{cf}}(\lambda)=\theta\quad\Rightarrow\quad
w=\mathop{\mathrm{cf}}(\lambda),$
$(\gamma)$:
$\mu>\lambda$, $\mu$ is a regular cardinal, $D$ is a uniform filter on $w$ (so
$\\{\alpha\in w:\alpha>\beta\\}\in D$ for each $\beta\in w$), $\mu$ is the
true cofinality of $\prod\limits_{i\in w}(\mu_{i},<)/D$ (see [Sh:E62, §3] or
[Sh:g]),
$(\delta)$:
$\bar{f}=\langle f_{i}/D:i<\mu\rangle$ exemplifies “the true cofinality of
$\prod\limits_{i}(\mu_{i},<)/D$ is $\mu$”, i.e.,
$\begin{array}[]{c}\alpha<\beta<\lambda\quad\Rightarrow\quad
f_{\alpha}/D<f_{\beta}/D,\\\
f\in\prod\limits_{i}\mu_{i}\quad\Rightarrow\quad\bigvee\limits_{\alpha}f/D<f_{\alpha}/D,\end{array}$
$(\epsilon)$:
$S\subseteq\\{\delta<\mu:\mathop{\mathrm{cf}}(\delta)=\theta\\}$ is good for
$(\mu,\theta,\chi(\ast))$, and
$(\zeta)$:
if $\theta>\mathop{\mathrm{cf}}(\lambda)$, $\delta\in S$, then for some
$A_{\delta}\in D$ and unbounded $B_{\delta}\subseteq\delta$ we have
$\alpha\in B_{\delta}\ \wedge\ \beta\in B_{\delta}\ \wedge\ \alpha<\beta\
\wedge\ i\in A_{\delta}\quad\Rightarrow\quad f_{\alpha}(i)<f_{\beta}(i),$
i.e., $\langle f_{\alpha}\restriction A_{\delta}:\alpha\in B_{\delta}\rangle$
is $<$–increasing.
Then we can find
${\mathbf{W}}=\\{(\bar{M}^{\alpha},\eta^{\alpha}):\alpha<\alpha(\ast)\\}$
(pedantically a sequence) and functions ${\dot{\zeta}}$ from $\alpha(\ast)$ to
$S$ and $h$ from $\alpha(\ast)$ to $\mu$ such that:
(a0),(a1),(a2):
as in 3.11 except that we replace $(\ast)$ of (a1) by
$(\ast)^{\prime}$ (i):
$\eta^{\alpha}\in{}^{\theta}\lambda$,
(ii):
if $i<\mathop{\mathrm{cf}}(\lambda)$ then
$\mathop{\mathrm{sup}}(\mu_{i}\cap\mathop{\mathrm{Rang}}(\eta^{\alpha}))=\mathop{\mathrm{sup}}(\mu_{i}\cap
M^{\alpha}_{\theta})$, and
(iii):
if $\xi<{\dot{\zeta}}(\alpha)$ then
$f_{\xi}/E<\langle\mathop{\mathrm{sup}}(\mu_{i}\cap
M^{\alpha}_{\theta}):i<\mathop{\mathrm{cf}}(\lambda)\rangle/E\leq
f_{{\dot{\zeta}}(\alpha)}/E$,
(b0)-(b3):
as in 3.11.
Proof: For $A\subseteq\theta$ of cardinality $\theta$ let
${\mathord{\mathrm{cd}}}^{A}_{\lambda,\chi(\ast)}:{\mathscr{H}}_{<\chi(\ast)}(\lambda)\longrightarrow{}^{A}\lambda$
be one-to-one, and $G:\lambda\longrightarrow\lambda$ be such that for $\gamma$
divisible by $|\gamma|$, $\alpha<\gamma\leq\lambda$ ($\mu\geq\aleph_{0}$), the
set $\\{\beta<\gamma:G(\beta)=\alpha\\}$ is unbounded in $\gamma$ and of order
type $\gamma$. Let $\bar{A}=\langle A_{i}:i<\theta\rangle$ be a sequence of
pairwise disjoint subsets of $\theta$ each of cardinality $\theta$. Let for
$\delta\in S$
$\begin{array}[]{ll}{\mathbf{W}}^{0}_{\delta}=\left\\{(\bar{M},\eta):\right.&\bar{M},\eta\mbox{
satisfy {\bf(a1)}, and for some }y\in{\mathscr{H}}_{<\chi(\ast)}(\lambda),\\\
&\mbox{for every }i<\theta\mbox{ we have}\\\ &\langle G(\eta(i)):i\in
A_{j}\rangle={\mathord{\mathrm{cd}}}^{A}_{\lambda,\chi(\ast)}(\langle\bar{M}\restriction
j,\eta\restriction j,y\rangle)\left.\right\\}.\end{array}$
The rest is as before.
###### Claim 3.37.
Suppose that $\lambda^{\theta}=\lambda^{<\chi(\ast)}$, $\lambda$ is singular,
$\theta,\chi(\ast)$ are regular, $\chi(\ast)>\theta$.
(1):
If $(\forall\alpha<\lambda)[|\alpha|^{<\chi(\ast)}<\lambda]$ then by
$\lambda^{\theta}=\lambda^{<\chi(\ast)}$ we know that either
$\mathop{\mathrm{cf}}(\lambda)\geq\chi(\ast)$ (and so lemma 3.17 applies) or
$\mathop{\mathrm{cf}}(\lambda)\leq\theta$.
(2):
We can find regular $\mu_{i}$ ($i<\mathop{\mathrm{cf}}(\lambda)$) increasing
with $i$, $\lambda=\sum\limits_{i<\mathop{\mathrm{cf}}(\lambda)}\mu_{i}$.
(3):
For $\langle\mu_{i}:i\in w\rangle$ as in 3.36($\beta$) we can find
$D,\mu,\bar{f}$ as in 3.36($\gamma$),($\delta$), $D$ the co-bounded filter
plus one unbounded subset of $\omega$.
(4):
For $\langle\mu_{i}:i\in w\rangle$, $D,\mu,\bar{f}$ as in ($\beta$),
($\gamma$), ($\delta$) of 3.36 we can find $\mu$ and pairwise disjoint
$S\subseteq\mu$ as required in ($\epsilon$), ($\delta$) of 3.36 provided that
$\theta>\mathop{\mathrm{cf}}(\lambda)\ \Rightarrow\ 2^{\theta}<\mu$
[equivalently $<\lambda$].
(5):
If $\mathop{\mathrm{cf}}(\lambda)>\aleph_{0}$,
$(\forall\alpha<\lambda)[|\alpha|^{\mathop{\mathrm{cf}}(\lambda)}<\lambda]$,
$\lambda<\mu=\mathop{\mathrm{cf}}(\mu)\leq\lambda^{\mathop{\mathrm{cf}}(\lambda)}$
then we can find $\langle\mu_{i}:i<\mathop{\mathrm{cf}}(\lambda)\rangle$, and
the co-bounded filter $D$ on $\mathop{\mathrm{cf}}(\lambda)$ as required in
$(\beta),(\gamma)$ of 3.29.
Proof: Now 1),2),3) are trivial, for (5) see [Sh 345, §9]. As for 4), we
should recall [Sh 345, §5] actually say:
###### Fact 3.38.
If $\langle\mu_{i}:i\in w\rangle,\bar{f},D$ are as in 3.36, then
$\begin{array}[]{ll}S=\left\\{\right.\delta<\mu:&\mathop{\mathrm{cf}}(\delta)=\theta\mbox{
and there are }A_{\delta}\in D,\ \beta\mbox{ and unbounded
}B_{\delta}\subseteq\delta\\\ &\mbox{such that }[\alpha\in
B_{\delta}\wedge\beta\in B_{\delta}\wedge\alpha<\beta\wedge i\in
A_{\delta}f_{\alpha}(i)<f_{\beta}(i)]\left.\right\\}\end{array}$
is good for $(\mu,\theta,\chi(\ast))$.
###### Lemma 3.39.
Let $\chi(1)=\chi(\ast)+(<\chi(\ast))^{\theta}$.
In 3.36, if $\lambda^{\theta}=\lambda^{\chi(1)}$, we can strengthen (b1) to
(b1)+ (of 3.18).
Proof: Combine proofs of 3.36, 3.18.
###### Lemma 3.40.
${\ref{6.9}\over\ref{6.5}}\times\ref{6.14}$ and
${\ref{6.10}\over\ref{6.5}}\times\ref{6.16}$ hold (we need also the parallel
to 3.30).
Proof: Left to the reader.
* * *
Now we draw some conclusions.
The first, 3.41, gives what we need in 2.7 (so 2.3 ).
###### Conclusion 3.41.
Suppose $\lambda^{\theta}=\lambda^{<\chi(\ast)}$,
$\mathop{\mathrm{cf}}(\lambda)\geq\chi(\ast)+\theta^{+}$,
$\theta=\mathop{\mathrm{cf}}(\theta)<\chi(\ast)=\mathop{\mathrm{cf}}(\chi(\ast))$.
Then we can find
${\mathbf{W}}=\\{(\bar{M}^{\alpha},\eta^{\alpha}):\alpha<\alpha(\ast)\\},\qquad
M^{\alpha}_{i}=(N^{\alpha}_{i},A^{\alpha}_{i},B^{\alpha}_{i}),$
where
$A^{\alpha}_{i}\subseteq\lambda\cap|N^{\alpha}_{i}|,\quad
B^{\alpha}_{i}\subseteq\lambda\cap|N^{\alpha}_{i}|,\quad A^{\alpha}_{i}\neq
B^{\alpha}_{i},$
and functions ${\dot{\zeta}},h$ such that:
(a0),(a1):
as in 3.11;
(a2):
as in 3.11 except that in the game, player I can choose $M_{i}$, only as
above;
(b0),(b1),(b2):
as in 3.11;
(b1)′′:
if $\\{\eta^{\alpha}\restriction i:i<\theta\\}\subseteq M^{\beta}$ but
$\alpha<\beta$ (so $\beta<\alpha+(<\chi(\ast))^{\theta}$ then
$\begin{array}[]{c}A^{\alpha}_{\theta}\cap(|M^{\alpha}_{\theta}|\cap|M^{\beta}_{\theta}|)\neq
B^{\beta}_{\theta}\cap(|M^{\alpha}_{\theta}|\cap|M^{\beta}_{\theta}|),\\\
B^{\alpha}_{\theta}\cap(|M^{\alpha}_{\theta}|\cap|M^{\beta}_{\theta}|)\neq
A^{\beta}_{\theta}\cap(|M^{\alpha}_{\theta}|\cap|M^{\beta}_{\theta}|).\end{array}$
Proof: First assume $\lambda$ is regular, and
${\mathbf{W}}=\\{(\bar{M}^{\alpha},\eta^{\alpha}):\alpha<\alpha(\ast)\\}$,
${\dot{\zeta}}$, $h$ be as in the conclusion of 3.11. Let
$w=\\{{\mathord{\mathrm{cd}}}(\alpha,\beta):\alpha,\beta<\lambda\\}$, and
$G_{1},G_{2}:w\longrightarrow\lambda$ be such that for $\alpha\in E$,
$\alpha={\mathord{\mathrm{cd}}}(G_{1}(\alpha),G_{2}(\alpha))$. Let
$\begin{array}[]{ll}Y=\left\\{\alpha<\alpha(\ast):\right.&\bar{M}^{\alpha}_{i}\mbox{
has the form }(N^{\alpha}_{i},A^{\alpha}_{i},B^{\alpha}_{i}),\\\
&A^{\alpha}_{i},B^{\alpha}_{i}\mbox{ distinct subsets of
}\lambda\cap|N^{\alpha}_{i}|\mbox{ (equivalently,}\\\ &\mbox{monadic
relations), }\\\ &h(\alpha)\in E,\mbox{ and }\\\
&G_{2}(h(\alpha))=\mathop{\mathrm{min}}\\{\gamma:\gamma\in
A^{\alpha}_{i}\setminus B^{\alpha}_{i}\mbox{ or }\gamma\in
B^{\alpha}_{i}\setminus A^{\alpha}_{i}\\}\left.\right\\}.\end{array}$
Now we let
${\mathbf{W}}^{\ast}=\\{(\bar{M}^{\alpha},\eta^{\alpha}):\alpha\in
Y\\},\quad{\dot{\zeta}}^{\ast}={\dot{\zeta}}\restriction Y,\quad
h^{\ast}=G_{1}\circ h.$
They exemplify that 3.41 holds.
What if $\lambda$ is singular? Still
$\mathop{\mathrm{cf}}(\lambda)\geq\chi(\ast)+\theta^{\ast}$ and we can just
use 3.17 instead 3.11. $\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{6.20}}$
###### Claim 3.42.
(1):
In 3.11, if $\lambda=\lambda^{<\chi(\ast)}$, we can let
$h:S\longrightarrow{\mathscr{H}}_{<\chi(\ast)}(\lambda)$ be onto; generally we
can still make $\mathop{\mathrm{Rang}}(h)$ be $\subseteq A$, whenever
$|A|=\lambda$.
(2):
In 3.11, by its proof, whenever $S^{\prime}\subseteq S$ is stationary, and
$\bigwedge\limits_{\zeta}(h^{-1}(\zeta)\cap S^{\prime}$ stationary) then
$\\{(\bar{M}^{\alpha},\eta^{\alpha}):\alpha<\alpha(\ast),\
{\dot{\zeta}}(\alpha)\in S^{\prime}\\}$
satisfies the same conclusion.
(3):
For any unbounded $a\subseteq\theta$ we can let player I choose also $\eta(i)$
for $i\in\theta\setminus a$, without changing our conclusions.
(4):
Similar statements hold for the parallel claims.
(5):
It is natural to have $\chi(\ast)=\chi^{+}$.
Proof: Straightforward.
###### Fact 3.43.
We can make the following changes in (a1), (a2) of 3.11 (and in all similar
lemmas here) getting equivalent statements.
$(\ast)$:
$M^{\alpha}_{i}\in{\mathscr{H}}_{<\chi(\ast)}(\lambda+\lambda)$; in the game,
for some arbitrary $\lambda^{\ast}\geq\lambda$ (but fix during the game)
player I chooses the $M^{\alpha}_{i}\in{\mathscr{H}}(\lambda^{\ast})$ (of
cardinality $<\chi(\ast))$, and in the end instead
“$\bigwedge\limits_{i}M_{i}=M^{\alpha}_{i}$” we have “there is an isomorphism
from $M_{\theta}$ onto $M^{\alpha}_{\theta}$ taking $M_{i}$ onto
$M^{\alpha}_{i}$ and is the identity on
$M_{\theta}\cap{\mathscr{H}}_{<\chi(\ast)}(\lambda)$ and maps
$|M_{\theta}|\setminus{\mathscr{H}}(\lambda)$ into
${\mathscr{H}}_{<\chi(\ast)}(\lambda+\lambda)\setminus{\mathscr{H}}_{<\chi(\ast)}(\lambda)$
and preserves $\in$ and $\notin$ and “being an ordinal” and “not being an
ordinal”.
###### Exercise 3.44.
If $D$ is a normal fine filter on ${\mathscr{P}}(\mu)$, $\lambda$ is regular,
$\lambda\leq\mu$,
$S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\theta\\}$ is
stationary, moreover:
$(\ast)_{D,S}$:
$\\{a\in{\mathscr{P}}(\mu):\mathop{\mathrm{sup}}(a\cap\lambda)\in
S\\}\neq\emptyset\;\mod\;D$
then we can partition $S$ to $\lambda$ stationary disjoint subsets each
satisfying $(\ast)$. [ Hint: like the proof of 3.3.]
###### Notation 3.45.
(1):
Let $\kappa$ be an uncountable regular cardinal. We let ${\rm
seq}^{\alpha}_{<\kappa}({\mathscr{A}})$, where ${\mathscr{A}}$ is an expansion
of a submodel of some ${\mathscr{H}}_{\leq\mu}(\lambda)$ with
$|\tau({\mathscr{A}})|\leq\chi$, be the set of sequences $\langle
M_{i}:i<\alpha\rangle$, which are increasing continuous,
$M_{i}\prec{\mathscr{A}}$, $\|M_{i}\|<\kappa$, $M_{i}\cap\kappa\in\kappa$,
$\kappa=\kappa_{1}^{+}\ \Rightarrow\ \kappa_{1}+1\subseteq M_{i}$, $\langle
M_{j}:j\leq i\rangle\in M_{i+1}$. (If $\alpha=\delta$ is limit,
$M_{\delta}=:\bigcup\limits_{i<\delta}M_{i})$.
(2):
If $\kappa=\kappa^{+}_{1}$, we may write $\leq\kappa_{1}$ instead $<\kappa$.
We repeat the definition of filters introduced in [Sh 52, Definition 3.2].
###### Definition 3.46.
(1):
${\mathscr{E}}^{\theta}_{<\kappa}(A)$ is the following filter on
$[A]^{<\kappa}$:
$Y\in{\mathscr{E}}^{\theta}_{<\kappa}(A)$ if and only if for (every) $\chi$
large enough, for some $x\in{\mathscr{H}}(\chi)$ the set
$\\{(\bigcup\limits_{i<\theta}M_{i})\cap A:\langle
M_{i}:i<\theta\rangle\in{\rm
seq}^{\theta}_{<\kappa}({\mathscr{H}}(\chi),\in,x)\\}$
is included in $Y$.
###### Exercise 3.47.
Let $\lambda$, $\kappa$, $\theta$, and $Y\subseteq[\lambda]^{<\kappa}$ be
given. Then
${\bf(a)}\quad\Rightarrow\quad{\bf(b)}\quad\Rightarrow\ {\bf(c)},$
where
(a):
For some
${\mathbf{W}}=\\{(\bar{M}^{\alpha},\eta^{\alpha}):\alpha<\alpha(\ast)\\}$,
${\dot{\zeta}},h$ satisfy 3.11,
$Y=\\{M^{\alpha}_{\theta}\cap\lambda:\alpha<\alpha(\ast)\\},$
and
$(\ast)$:
$\alpha\neq\beta\wedge\bigwedge\limits_{i<\theta}\eta^{\alpha}_{i}\in
M^{\beta}_{\theta}\quad\Rightarrow\quad\alpha<\beta$.
(b):
$\diamondsuit_{E^{\theta}_{<\kappa}(\lambda)}$ holds.
(c):
Like (a) without $(\ast)$.
###### Exercise 3.48.
If $\lambda^{2^{\kappa}}=\lambda$, $\theta\leq\kappa$ then
$\diamondsuit_{E^{\theta}_{<\kappa}}$ (main case: $\kappa=\theta)$.
###### Exercise 3.49.
If $\lambda=\mu^{+}$, $\lambda^{\kappa}=\lambda$, $\theta=\aleph_{0}$,
$\kappa=\kappa^{\theta}$, then there is a coding set with diamond (see [Sh
247]).
###### Exercise 3.50.
Suppose that $\mathop{\mathrm{cf}}(\lambda)>\aleph_{0}$,
$2^{\lambda}=\lambda^{\mathop{\mathrm{cf}}(\lambda)}$,
$\chi(\ast)\geq\theta>\mathop{\mathrm{cf}}(\lambda)$,
$(\forall\alpha<\lambda)[|\alpha|^{\chi(\ast)}<\lambda]$, $\mathfrak{C}$ is a
model expanding $({\mathscr{H}}_{<\chi(\ast)}(\lambda),\in)$,
$|\tau({\mathfrak{C}})|\leq\aleph_{0}$. Then we can find
$\\{\bar{M}^{\alpha}:\alpha<\alpha(\ast)\\}$ such that:
(i):
$\bar{M}^{\alpha}=\langle M^{\alpha}_{i}:i<\sigma\rangle$,
$M^{\alpha}_{i}\in{\mathscr{H}}_{<\chi(\ast)}(\lambda)$,
$M^{\alpha}_{i}\cap\chi(\ast)$ is an ordinal,
$M^{\alpha}_{i}\restriction\tau({\mathfrak{C}})\prec{\mathfrak{C}}$, $[i<j\
\Rightarrow\ M^{\alpha}_{i}\prec M^{\alpha}_{j}]$, $\langle
M^{\alpha}_{j}:j\leq i\rangle\in M^{\alpha}_{i+1}$,
(ii):
if $f_{n}$ is a $k_{n}$–place function from $\lambda$ to
${\mathscr{H}}_{<\chi(\ast)}(\lambda)$ then for some $\alpha$,
$M^{\alpha}_{\sigma}\prec({\mathfrak{C}},f_{n})_{n<\omega}$.
###### Exercise 3.51.
Suppose $\theta=\mathop{\mathrm{cf}}(\mu)<\mu$,
$(\forall\alpha<\mu)[|\alpha|^{\theta}<\mu]$, $2^{\mu}=\mu^{\theta}$ and
$\lambda=(2^{\mu})^{+}$,
$S\subseteq\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\theta\\}$. Let
$\mu=\sum\limits_{i<\theta}\mu_{i}$, $\mu_{i}$ regular strictly increasing,
and $\mathop{\mathrm{cf}}(\prod\mu_{i}/E)=2^{\mu}$. Then we can find
${\mathbf{W}}=\\{(\bar{M}^{\alpha},\eta^{\alpha}):\alpha<\alpha(\ast)\\},\quad{\dot{\zeta}}:\alpha(\ast)\longrightarrow
S,\quad h:\alpha(\ast)\longrightarrow\lambda$
such that:
$(\ast)$:
for $\delta\in S$ there is a club $C_{\delta}$ of $\delta$ of order type
$\theta$ such that
$\alpha\in C_{\delta}\wedge\mathop{\mathrm{otp}}(\alpha\cap
C_{\delta})=\gamma+1\quad\Rightarrow\quad\mathop{\mathrm{cf}}(\alpha)=\mu_{\gamma}.$
###### Remark 3.52.
We do not know if the existence of a Black Box for $\lambda^{+}$ with $h$ one-
to-one follows from ZFC (of course it is a consequence of $\diamondsuit$). On
the other hand, it is difficult to get rid of such a Black Box (i.e., prove
the consistency of non-existence).
If $\lambda=\lambda^{<\lambda}$ then we have $h:S\longrightarrow\lambda$,
$S\subseteq\\{\delta<\lambda^{+}:\mathop{\mathrm{cf}}(\delta)<\lambda\\}$ such
that $C_{\delta}$ is a club of $\delta$,
$\mathop{\mathrm{otp}}(C_{\delta})=\mathop{\mathrm{cf}}(\delta)$ and
$\forall\mbox{ club }\ C\subseteq\alpha\in C_{\delta})$
$[\mathop{\mathrm{cf}}(\alpha)>\aleph_{0}\ \wedge\
\mathop{\mathrm{min}}\limits_{C^{\prime}{\rm\;club\;of\;}C^{\prime}_{\alpha}}\mathop{\mathrm{sup}}(h\restriction
C^{\prime})=\mathop{\mathrm{otp}}(C\cap\alpha)].$
This is hard to get rid of, (i.e., hard to find a forcing notion making it no
longer a black box, without collapsing too many cardinals); compare with
Mekler- Shelah [MkSh 274].
## 4\. On Partitions to stationary sets
We present some results on the club filter on $[\kappa]^{\aleph_{0}}$ and
$[\kappa]^{\theta}$ and some relatives, and $\diamondsuit$ (see Def. [Sh:E62,
§3] or 4.4(2) below). There are overlaps of the claims hence redundant parts
which still have some interest.
###### Claim 4.1.
Assume $\kappa$ is a cardinal $>\aleph_{1}$, then $[\kappa]^{\aleph_{0}}$ can
be partitioned to $\kappa^{\aleph_{0}}$ (pairwise disjoint) stationary sets.
Proof: Follows by 4.2 below, [in details, let $\tau$ be the vocabulary
$\\{c_{n}:n<\omega\\}$ where each $c_{n}$ is an individual constant. By 4.2
below there is a sequence $\bar{M}=\langle
M_{u}:u\in[\kappa]^{\aleph_{0}}\rangle$ of $\tau$-models, with $M_{u}$ having
universe $u$ such that $\bar{M}$ is a diamond sequence.
For each $\eta\in{}^{\omega}\lambda$ let ${\mathscr{S}}_{\eta}$ be the set
$u\in[\kappa]^{\aleph_{0}}$ such that for every $n<\omega$ we have
$c^{M_{u}}_{n}=\eta(n)\\}$.
By the choice of $\bar{M}$ necessarily each set ${\mathscr{S}}_{\eta}$ is a
stationary subset of $[\kappa]^{\aleph_{0}}$, and trivially those set s are
pairwise disjoint.]
###### Claim 4.2.
Let $\kappa>\aleph_{1}$. Then we have diamond on $[\kappa]^{\aleph_{0}}$
(modulo the filter of clubs on it, see 4.4(2) or [Sh:E62]), and we can find
$A_{\alpha}\subseteq[\kappa]^{\aleph_{0}}$ for $\alpha<\lambda\buildrel\rm
def\over{=}2^{\kappa^{\aleph_{0}}}$ such that each is stationary but the
intersection of any two is not.
Proof: The existence of the $A_{\alpha}$-s for $\alpha<\lambda$ follows from
the other result. Let $\tau$ be a countable vocabulary,
$\tau_{1}=\tau\cup\\{<\\}$. First we prove it when
$\kappa=\aleph_{2}\in[\aleph_{2},2^{\aleph_{0}})$. Let
$\omega\setminus\\{0\\}$ be the disjoint union of $s_{n}$ for $n<\omega$, each
$s_{n}$ is infinite with the first element $>n+3$ when $n>0$. Let $\langle
C_{\delta}:\delta\in S^{2}_{0}\rangle$ be club guessing, where
$S^{2}_{0}=\\{\delta<\omega_{2}:\mathop{\mathrm{cf}}(\delta)=\aleph_{0}\\}$,
such that $C_{\delta}\subseteq\delta=\mathop{\mathrm{sup}}(C_{\delta})$ has
order type $\omega$.
Let
$\langle({\mathfrak{A}}^{\zeta},\bar{\alpha}^{\zeta}):\zeta<2^{\aleph_{0}}\rangle$
list without repetitions the pairs $({\mathfrak{A}},\bar{\alpha})$,
${\mathfrak{A}}$ a model with vocabulary $\tau_{1}$ and universe a limit
countable ordinal and $\bar{\alpha}=\langle\alpha_{n}:n<\omega\rangle$ an
increasing sequence of ordinals with limit ${\rm sup}({\mathfrak{A}})$ and
${\mathfrak{A}}{\restriction}\alpha_{n}\prec{\mathfrak{A}}$. Let $E_{n}$ be
the following equivalence relation relation on $2^{\aleph_{0}}$ : $\epsilon
E_{n}\zeta$ iff
$({\mathfrak{A}}^{\epsilon}{\restriction}\alpha^{\epsilon}_{n},\bar{\alpha}^{\epsilon}{\restriction}n)$
is isomorphic to
$({\mathfrak{A}}^{\zeta}{\restriction}\alpha^{\zeta}_{n},\bar{\alpha}^{\zeta}\upharpoonright
n)$ which means: there is an isomorphism $f$ from
${\mathfrak{A}}\upharpoonright\alpha^{\epsilon}_{n}$ onto
${\mathfrak{A}}^{\zeta}\upharpoonright\alpha^{\zeta}_{n}$ which maps
${\mathfrak{A}}^{\epsilon}\upharpoonright\alpha^{\epsilon}_{k}$ onto
${\mathfrak{A}}^{\zeta}\upharpoonright\alpha^{\zeta}_{k}$ for $k<n$ and is an
order preserving function (for the ordinals, alternatively we restrict
ourselves to the case $<$ is interpreted as a well ordering.
We can find subsets $t^{\zeta}$ of $\omega$ such that
$(*)$:
for $\zeta,\epsilon<2^{\aleph_{0}}$ we have $t^{\zeta}\cap
s_{n}=t^{\epsilon}\cap s_{n}$ iff
${\mathfrak{A}}^{\zeta}\restriction\alpha^{\zeta}_{n}={\mathfrak{A}}^{\epsilon}\restriction\alpha^{\epsilon}_{n}$
and $\alpha^{\zeta}_{k}=\alpha^{\epsilon}_{k}$ for $k\leq n$. Also
$t^{\zeta}\cap s_{n}$ is infinite and
$\epsilon\neq\zeta\Rightarrow{\aleph_{0}}>|t^{\epsilon}\cap t^{\zeta}|$ for
simplicity ( so $t^{\zeta}\cap s_{n}$ depend just on $\zeta/E_{n}$, in fact
code it).
For $\zeta<2^{\aleph_{0}}$ let
${\mathscr{S}}_{\zeta}\buildrel\rm
def\over{=}\big{\\{}a\in[\kappa]^{\aleph_{0}}:t_{\zeta}=\\{|C_{{\rm
sup}(a)}\cap\beta|:\beta\in a\\}\big{\\}},$
and let
${\mathscr{S}}^{\prime}_{\zeta}=\\{a\in{\mathscr{S}}_{t}:\mathop{\mathrm{otp}}(a)=\mathop{\mathrm{otp}}({\mathfrak{A}}^{\zeta})\\},$
and for $a\in{\mathscr{S}}^{\prime}_{\zeta}$ let $N_{a}$ be the model
isomorphic to ${\mathfrak{A}}^{\zeta}$ by the function $f_{a}$, where
$\mathop{\mathrm{Dom}}(f_{a})=a$,
$f_{a}(\gamma)=\mathop{\mathrm{otp}}(\gamma\cap a)$.
Let ${\mathscr{S}}$ be the union of ${\mathscr{S}}^{\prime}_{\zeta}$ for
$\zeta<2^{\aleph_{0}}$. Clearly $\zeta\neq\xi\ \Rightarrow\
{\mathscr{S}}_{\zeta}\cap{\mathscr{S}}_{\xi}=\emptyset$, and so
${\mathscr{S}}^{\prime}_{\zeta}\cap{\mathscr{S}}^{\prime}_{\xi}=\emptyset$.
Hence $N_{a}$ is well defined for $a\in{\mathscr{S}}$
Let $K_{n}$ be the set of pairs $({\mathfrak{A}},\bar{\alpha})$ such that
${\mathfrak{A}}$ is a $\tau_{1}$-model with universe a countable subset of
$\kappa$ with no last member, and $\bar{\alpha}$ is an increasing sequence of
ordinals $<\kappa$ of length $n$ such that $\alpha_{k}<{\rm
sup}({\mathfrak{A}})$ and
$[\alpha_{k},\alpha_{k+1})\cap{\mathfrak{A}}\neq\emptyset$ and
${\mathfrak{A}}\restriction\alpha_{k}\prec{\mathfrak{A}}$. So clearly there is
a function ${\rm cd}_{n}:K_{n}\rightarrow{\mathscr{P}}(s_{n})$ such that: if
$\zeta<2^{\aleph_{0}}$ then ${\rm
cd}_{n}({\mathfrak{A}},\bar{\alpha})=t^{\zeta}\cap s_{n}$ iff the pairs
$({\mathfrak{A}},\bar{\alpha}),(({\mathfrak{A}}^{\zeta},\bar{\alpha}^{\zeta}{\restriction}n))$
are isomorphic.
Let $M$ be a $\tau_{1}$–model with universe $\kappa$. Now (see [Sh:E62], or
history in the introduction of §3, and the proof of 3.22) we can find a full
subtree ${\mathscr{T}}$ of ${}^{\omega>}(\aleph_{2})$ (i.e., it is non-empty,
closed under initial segments and each member has $\aleph_{2}$ immediate
successors) and elementary submodels $N_{\eta}$ of $M$ for
$\eta\in{\mathscr{T}}$ such that:
(a):
Rang$(\eta)\subseteq N_{\eta}$,
(b):
if $\eta$ is an initial segment of $\rho$ then $N_{\eta}$ is a submodel
$N_{\rho}$, moreover $N_{\eta}\cap\aleph_{2}$ is an initial segment of
$N_{\rho}\cap\aleph_{2}$.
Now let $E$ be the set of $\delta<\aleph_{2}$ satisfying: if
$\rho\in{\mathscr{T}}$ and $\rho\in{}^{\omega>}\delta$ then
$N_{\rho}\cap\aleph_{2}$ is a bounded subset of $\delta$, and $\delta$ is a
limit ordinal. Let $E_{1}$ be the set of $\delta\in E$ such that if
$\rho\in{\mathscr{T}}\cap{}^{\omega>}\delta$ then for every $\beta<\delta$
there is $\gamma$ such that $\beta<\gamma<\delta$ and
$\eta{}^{\frown}\\!\langle\gamma\rangle\in{\mathscr{T}}$. So by the choice of
$\langle C_{\delta}:\delta\in S\rangle$ for some $\delta\in S$ we have
$C_{\delta}\subset E_{1}$
Let $\langle\alpha_{\delta,k}:k<\omega\rangle$ list $C_{\delta}$ in increasing
order.
Now we choose by induction on $n$ a triple
$(\eta_{n},s^{*}_{n},\alpha_{n},k_{n})$ such that
1. (*)
1. (a)
$\eta_{n}\in{\mathscr{T}}$ has length $n$ (so $\eta_{0}$ is necessarily
$\langle\rangle$).
2. (b)
if $n=m+1$ then $\eta_{n}$ is a successor of $\eta_{m}$
3. (c)
$s^{*}_{n}$ is ${\bf
cd}_{n}((N_{\eta_{n}},\langle\alpha_{\ell}:\ell<n\rangle))$ if the pair
$(N_{\eta_{n}},\langle\alpha_{\ell}:\ell<n\rangle)$ belongs to $K_{n}$ and is
$s_{n}$ otherwise; actually it is so,
4. (d)
$\alpha_{n}={\rm sup}(N_{\eta_{n}})+1$
5. (e)
$k_{n}={\rm min}\\{k:N_{\eta_{n}}\subseteq\alpha_{\delta,k}\\}$ and $k_{0}=0$
and $\bar{n}[0,k_{n}]\subseteq\bigcup\limits_{\ell<n}s_{\ell}\cup\\{0\\}$
6. (f)
if $n=m+1$ and $k_{m}<k_{n}$ then
1. $(\alpha)$
${\rm min}(N_{\eta_{n}}\setminus N_{\eta_{m}})>\alpha_{\delta,k_{n}-1}$
2. $(\beta)$
$(k_{m},k_{n})$ is disjoint to $\bigcup\limits_{\ell<n}s^{*}_{\ell}$
3. $(\delta)$
$k_{n}\in\cup\\{s^{*}_{\ell}:\ell<n\\}$
4. ($\epsilon$)
$k_{n}$ is minimal under those restrictions.
7. (g)
if $n=m+1$ and $k_{n}=k_{m}$ then we cannot find $k\in(k_{m},\omega)$
satisfying $(\beta),(\gamma)$ of clause (f).
There is no problem to carry the induction. In the end let
$\eta=\bigcup\limits_{n}\eta_{n}\in{\rm lim}(i{\mathscr{T}})$, so we get a
$\tau_{1}$-model $N_{\eta}=:\cup\\{N_{\eta_{n}}:n<\omega\\}$, and an
increasing sequence $\langle\alpha_{n}:n<\omega\rangle$ of ordinals with limit
${\rm sup}({\mathfrak{A}})$. Now by the choice of
$\langle({\mathfrak{A}}^{\zeta},\bar{\alpha}^{\zeta}):\zeta<2^{\aleph_{0}}\rangle$
clearly for some $\zeta$ we have
$(N_{\eta},\bar{\alpha}),({\mathfrak{A}}^{\zeta},\bar{\alpha}^{\zeta})$ are
isomorphic, so necessarily
$(N_{\eta}{\restriction}\alpha_{n},\bar{\alpha}{\restriction}n)$ belongs to
$K_{n}$ and necessarily ${\rm
cd}_{n}(N_{\eta},\langle\alpha_{\ell}:\ell<n\rangle)=s^{*}_{n}$. Also clearly
${\rm sup}(N_{\eta})=\delta$ and
$\\{k_{n}:n<\omega\\}=\\{|C_{\delta}\cap\beta|:\beta\in
N_{\eta}\\}=\\{\alpha_{\delta,k_{n}}:n<\omega\\}\\}$
Letting $a$ be the universe of $N_{\eta}$ it follows that
$a\in{\mathscr{S}}_{\zeta}$ so $N_{a}$ is well defined and isomorphism to
${\mathfrak{A}}^{\zeta}$ hence to $N_{\eta}$ using $<^{M}$ we get
$N_{a}=N_{\eta}$. But $N_{\eta}\prec M$. So $\langle
N_{a}:a\in{\mathscr{S}}\rangle$ is really a diamond seq, well for
$\tau_{1}$-models rather then $\tau$-models, but this does no harm and will
help for $\kappa>{\aleph_{2}}$.
Second, we consider the case $\kappa>\aleph_{2}$. For each
$c\in[\kappa]^{\aleph_{0}}$, if
$\mathop{\mathrm{otp}}(c)=\mathop{\mathrm{otp}}(c\cap\omega_{2},<^{N_{c\cap\omega_{2}}})$,
let $g_{c}$ be the unique isomorphism from
$(c\cap\omega_{2},<^{N_{c\cap\omega_{2}}})$ onto $(c,<),<$ the usual order,
and let $M_{c}$ be the $\tau$–model with universe $c$ such that $g$ is an
isomorphism from $N_{c\cap\omega_{2}}{\restriction}\tau$ onto $M_{c}$. Clearly
it is an isomorphism and the $M_{c}$’s form a diamond sequence.
[Why? For notational simplicity $\tau$ has predicates only. Let $M_{0}=M$ be a
$\tau$-model with universe $\kappa$, let $M_{1}$ be an elementary submodel of
$M$ of cardinality $\aleph_{2}$ such that $\omega_{2}\subseteq M_{1}$, let $h$
be a one-to-one function from $M_{1}$ onto $\omega_{2}$ let $M_{2}$ be a
$\tau$-model with universe $\omega_{2}$ such that $h$ is an isomorphism from
$M_{1}$ onto $M_{2}$, and let $M_{3}$ be the $\tau_{1}$-model expanding
$M_{2}$ such that $<^{M_{3}}=\\{(h(\alpha),h(\beta)):\alpha<\beta$ are from
$M_{1}\\}$. So for some $a\in{\mathscr{S}}\subseteq[\kappa]^{\aleph_{0}}$ we
have $N_{a}\prec M_{3}$ and $h(\alpha)=\beta\in
N_{a}\wedge\alpha<\omega_{2}\Rightarrow\alpha\in a$ (the set of $a$-s
satisfying this contains a club of $[{\aleph_{2}}]^{\aleph_{0}}$). Let
$c=\\{\alpha:h(\alpha)\in a\\}$, so clearly $c\cap\omega_{2}=a$ and
$M_{c}\prec M_{1}$ hence $M_{c}\prec M$, so we are done.] $\hskip
5.0pt\hbox{\hskip 5.0pt\vrule width=4.0pt,height=6.0pt,depth=1.5pt\hskip
1.0pt}_{\ref{4.2new}}$
###### Discussion 4.3.
Some concluding remark are:
1.:
We can use other cardinals, but it is natural if we deal with
$D_{\kappa,<\theta,\aleph_{0}}$ (see below).
2.:
The context is very near to §3, but the stress is different.
###### Definition 4.4.
Let $\kappa\geq\theta\geq\sigma$, $\theta$ uncountable regular. If
$\theta=\mu^{+}$ we may write $\mu$ instead of $<\theta$.
(1):
Let $D=D_{1}=D^{1}_{\kappa,<\theta,\aleph_{0}}$ be the filter
$[\kappa]^{<\theta}$ generated by $\\{A^{1}_{x}:x\in{\mathscr{H}}(\chi)\\}$
where
$\begin{array}[]{ll}A^{1}_{x}=\\{N\cap\kappa:&N\mbox{ is an elementary
submodel of }({\mathscr{H}}(\chi),\in)\mbox{ and}\\\ &N\mbox{ is
}\bigcup\limits_{n<\omega}N_{n},\ N_{n}\mbox{ increasing and }N_{n}\in
N_{n+1}\\\ &\mbox{and }\|N_{n}\|<\theta\mbox{ and
}N_{n}\cap\theta\in\theta\\}.\end{array}$
(2):
Let $D=D_{2}=D^{2}_{\kappa,<\theta,\sigma}$ be the filter on
$[\kappa]^{<\theta}$ generated by $\\{A^{2}_{x}:x\in{\mathscr{H}}(\chi)\\}$
where
$\begin{array}[]{ll}A^{2}_{x}=\\{N\cap\kappa:&N\mbox{ is an elementary
submodel of }({\mathscr{H}}(\chi),\in)\mbox{ and}\\\ &N\mbox{ is
}\bigcup\limits_{\zeta<\sigma}N_{\zeta},\ N_{\zeta}\mbox{ increasing and}\\\
&\langle N_{\varepsilon}:\varepsilon\leq\zeta\rangle\in N_{\zeta+1}\mbox{ and
}N_{\varepsilon}\cap\theta\in\theta\\}.\end{array}$
(3):
For a filter $D$ on $[\kappa]^{<\theta}$ let $\diamondsuit_{D}$ mean:
fixing any countable vocabulary $\tau$ there are $S\in D$ and $N=\langle
N_{a}:a\in S\rangle$, each $N_{a}$ a $\tau$–model with universe $a$, such that
for every $\tau$–model $M$ with universe $\lambda$ we have
$\\{a\in S:N_{a}\subseteq M\\}\neq\emptyset\mod D.$
(4):
Instead $<\theta$ we may write $\theta$.
###### Claim 4.5.
Assume $\theta\leq\sigma$ and $\kappa>\sigma^{+}$ and let
$D=D_{\kappa,\theta,\aleph_{0}}$.
(1):
$[\kappa]^{\theta}$ can be partitioned to $\sigma^{\aleph_{0}}$ (pairwise
disjoint) $D$–positive sets.
(2):
Assume in addition that $\sigma^{\aleph_{0}}\geq 2^{\theta}$. Then
$(\alpha)$:
we can find $A_{\alpha}\subseteq[\kappa]^{\theta}$ for
$\alpha<\lambda\buildrel\rm def\over{=}2^{\kappa^{\theta}}$ such that each is
$D$–positive but they are pairwise disjoint $\mod D$,
$(\beta)$:
if $\lambda=\kappa^{\theta}$ and $\tau$ is a countable vocabulary then
$\diamondsuit_{\lambda,\theta,\aleph_{0}}$; moreover there are
$S^{*}\subseteq[\lambda]^{\theta}$ and function $N^{*}$ with domain $S^{*}$
such that
(a):
for distinct $a,b$ from $S^{*}$ we have $a\cap\kappa\neq b\cap\kappa$,
(b):
for $a\in S^{*}$ we have $N^{*}(a)=N^{*}_{a}$ is a $\tau$–model with universe
$a$,
(c):
for a $\tau$–model $M$ with universe $\lambda$, the set
$\\{a:N^{*}_{a}=M\restriction a\\}$ is stationary.
Proof: Similar to earlier ones : part (1) like Claim 4.1 case (a), part (2)
like the proof of Claim 4.2.
$\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{4.5new}}$
###### Claim 4.6.
(1):
If $\theta\leq\kappa_{0}\leq\kappa_{1}$ and $\diamondsuit_{S_{0}}$ i.e.
$\diamondsuit_{D_{\kappa_{0},\theta,\sigma}S_{0}}$, where $S_{0}$ is a subset
of $[\kappa_{0}]^{\theta}$ which is $D_{\kappa_{0},\theta,\sigma}$–positive
and $S_{1}\buildrel\rm
def\over{=}\\{a\in[\kappa_{1}]^{\theta}:a\cap\kappa_{0}\in S_{0}\\}$, then
$\diamondsuit_{S_{1}}$, i.e.
$\diamondsuit_{D_{\kappa_{1},\theta,\sigma}S_{1}}$.
(2):
In part (1), if in addition $\kappa_{0}=(\kappa_{0})^{\theta}$ and
$\kappa_{2}=(\kappa_{1})^{\theta}$ then we can find
$S_{2}\subseteq[\kappa_{2}]^{\theta}$ such that:
(a):
$a\in S_{2}$ implies $a\cap\kappa_{0}\in S_{0}$,
(b):
if $b$, $c$ are distinct members of $S_{2}$ then $b\cap\kappa_{1}$,
$c\cap\kappa_{1}$ are distinct, and
(c):
$\diamondsuit_{S_{2}}$.
(3):
If $\kappa=\kappa^{\theta}$ then $\diamondsuit_{D_{\kappa,\theta,\sigma}}$.
###### Remark 4.7.
This works for other uniform definition of normal filters.
Above, $\kappa^{\theta^{\sigma}}=\kappa$ can be replaced by: every tree with
$\leq\theta$ nodes has at most $\theta^{*}$–branches and
$\kappa^{\theta^{*}}=\kappa$.
Proof: 1) Easy.
2) Implicit in earlier proof, 4.2.
3) See [Sh 212], [Sh 247]
$\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}_{\ref{4.6new}}$
## References
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|
arxiv-papers
| 2008-12-03T05:42:22 |
2024-09-04T02:48:59.129684
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Saharon Shelah",
"submitter": "Saharon Shelah",
"url": "https://arxiv.org/abs/0812.0656"
}
|
0812.0727
|
# Lepton Flavor Violating $\tau^{-}\to\mu^{-}V^{0}$ Decays in the Two Higgs
Doublet Model III
Wenjun Li liwj24@163.com Department of Physics, Henan Normal University,
XinXiang, Henan, 453007, P.R.China
Kavli Institute for Theoretical Physics China, CAS, Beijing 100190, China
Yanqin Ma, Gongwei Liu, Wei Guo Department of Physics, Henan Normal
University, XinXiang, Henan, 453007, P.R.China
###### Abstract
In this paper, the lepton flavor violating
$\tau^{-}\to\mu^{-}V^{0}(V^{0}=\rho^{0},\phi,\omega)$ decays are studied in
the framework of the two Higgs doublet model(2HDM) III. We present a
computation of the the $\gamma-$, $Z$ penguin and box diagrams contributions,
and make an analysis of their impacts. Our results show that, among the
$\gamma-$ penguins, the penguins with neutral Higgs in the loop are very
larger than those with charged Higgs in the loop. We find that the model
parameter $\lambda_{\tau\mu}$ is tightly constrained at the order of
$O(10^{-3})$ and the branching ratios of these decays are available at the
experiment measure. With the high luminosity, the B factories have
considerable capability to find these LFV processes. On the other hand, these
processes can also provide some valuable information to future research and
furthermore present the reliable evidence to test the 2HDM III model.
###### pacs:
13.35.Dx, 12.15.Mm, 12.60.-i
## I Introduction
The flavor physics is always the hot subject in particle physics. Recently,
with rapid development of neutrinos experimentsuperK , the lepton flavor
violation(LFV) processes of charged-lepton sector have attracted many people’s
attention. In the standard model(SM), the LFV processes are forbidden. Hence,
the LFV decays are expected to be a powerful probe to many extensions of the
SM with new LFV source and/or new particles.
The LFV $\tau$ decays have become a seeking goal in experiment. Due to the
comparability of $e^{+}e^{-}\to b\bar{b}$ and $e^{+}e^{-}\to\tau^{-}\tau^{-}$
cross section ($\sigma\sim 0.99nb$) around the $\Upsilon(4s)$ energy region,
large events of $\tau$ leptons are available at BaBar and Belle (${\cal
L}_{BaBar}~{}470fb^{-1},{\cal L}_{Belle}~{}710fb^{-1}$). And now the tau pairs
production has attained the reach of $10^{-9}$. The tau factory has performed
the experimental search for the tau radiative decays and $\tau\to 3l$ decays,
as well as $\tau\to lV^{0}$ decaysbebar . The current experimental upper
limits of the $\tau^{-}\to\mu^{-}\rho^{0}(\phi,\omega)$ decays with
543$fb^{-1}$ of data at Belle laboratory are belle :
$\displaystyle{\cal B}(\tau^{-}\to\mu^{-}\rho^{0})<6.8\times
10^{-8},\,\,\,\,90\%CL$ $\displaystyle{\cal
B}(\tau^{-}\to\mu^{-}\phi)<1.3\times 10^{-7},\,\,\,\,90\%CL$
$\displaystyle{\cal B}(\tau^{-}\to\mu^{-}\omega)<8.9\times
10^{-8},\,\,\,\,90\%CL$ (1)
There are also lots of theoretical researches on $\tau\to lV^{0}$ decays in
many possible extensions of the SM. For example, Saha et al. have deliberated
constraints on the parameters from $\tau\to
l\rho^{0}(\phi,K^{*0},\bar{K}^{*0})$ decays in RPV SUSY modelSaha . Ilakovac
et al. found only the ratios of $\tau^{-}\to e^{-}\rho^{0}(\phi,\pi^{0})$
decays reach the order of $10^{-6}$ in models with heavy Dirac or Majorana
neutrinosIlakovac . The case of $\tau\to lP(V^{0})$ decays in topcolor model
have been considered by Yue Chongxing $\textit{et al}.$Yue . Such
investigations also have been presented in MSSM and minimal susysemmetry
$SO(10)$ modelsFukuyama , a general unconstrained MSSM modelRossi and two
constrained MSSM seesaw modelsArganda as well.
In our previous workli , we have studied the $\tau\to\mu
P(P=\pi^{0},\eta,\eta^{\prime})$ decays in 2HDM model III. In this model,
there exist flavor-changing neutral currents(FCNCs) at tree level. In order to
satisfy the current experiment constrains, the tree-level FCNCs are suppressed
in low-energy experiments for the first two generation fermions. While
processes concerning with the third generation fermions would be larger. These
FCNCs with neutral Higgs bosons mediated may produce sizable effects to the
$\tau-\mu$ transition. The $\tau\to\mu P$ decays could yield one pseudoscalar
meson from the vacuum state through the scalar and pseudoscalar currents.
Hence, this type decay could occur at the tree level through the neutral Higgs
bosons exchange. In this paper, we extend our discussion to the case of one
vector meson in the hadronic final state. Different from pseudoscalar meson,
the vector meson is only generated through vector currents and therefore
receive no contributions of the neutral Higgs at tree level. So we consider
the effects with Higgs bosons in the loop. There are the $\gamma-,$ $Z$
penguin and the box diagrams for the $\tau\to\rho^{0}(\phi,\omega)$ decays.
For the instance of vector meson $K^{*0}(\bar{K}^{*0})$, the LFV processes
could occur at loop level likewise but the additional loop at the hadronic
vertex would generate one suppressed factor. So these two decays are not
discussed in this paper. Our results suggest that, in the $\gamma-$ penguins,
the contributions of penguin with neutral Higgs bosons in the loop is greater
than those of penguin with charged Higgs bosons in the loop. The model
parameter $\lambda_{\tau\mu}$ is restrained at $O(10^{-3})$ and the decay
branching ratios could as large as the current upper limits of $O(10^{-7})$.
For $\tau^{-}\to\mu^{-}PP$ processes, we will make further study in our later
work.
The paper is organized as follows: In section II, we make a brief introduction
of the theoretical framework for the two-Higgs-doublet model III. In section
III, we present the decay amplitudes and the numerical predictions for the
branching ratios. Our conclusions are listed in the last section.
## II The Two-Higgs -Doublet Model III
As the simplest extension of the SM, the Two-Higgs-Doublet Model has an
additional Higgs doublet. In order to ensure the forbidden FCNCs at tree
level, it requires either the same doublet couple to the u-type and d-type
quarks(2HDM I) or one scalar doublet couple to the u-type quarks and the other
to d-type quarks(2HDM II). While in the 2HDM IIIcheng ; 2hdm3 , two Higgs
doublets could couple to the u-type and d-type quarks simultaneously.
Particularly, without an ad hoc discrete symmetry exerted, this model permits
flavor changing neutral currents occur at the tree level.
The Yukawa Lagrangian is generally expressed as the following form:
$\displaystyle{\cal
L}_{Y}=\eta^{U}_{ij}\bar{Q}_{i,L}\tilde{H}_{1}U_{j,R}+\eta^{D}_{ij}\bar{Q}_{i,L}H_{1}D_{j,R}+\xi^{U}_{ij}\bar{Q}_{i,L}\tilde{H}_{2}U_{j,R}+\xi^{D}_{ij}\bar{Q}_{i,L}H_{2}D_{j,R}\,+\,h.c.,$
(2)
where $H_{i}(i=1,2)$ are the two Higgs doublets. $Q_{i,L}$ is the left-handed
fermion doublet, $U_{j,R}$ and $D_{j,R}$ are the right-handed singlets,
respectively. These $Q_{i,L},U_{j,R}$ and $D_{j,R}$ are weak eigenstates,
which can be rotated into mass eigenstates. While $\eta^{U,D}$ and $\xi^{U,D}$
are the non-diagonal matrices of the Yukawa couplings.
We can conveniently choose a suitable basis to denote $H_{1}$ and $H_{2}$ as:
$\displaystyle H_{1}=\frac{1}{\sqrt{2}}\left[\left(\begin{array}[]{c}0\\\
v+\phi^{0}_{1}\end{array}\right)+\left(\begin{array}[]{c}\sqrt{2}\,G^{+}\\\
iG^{0}\end{array}\right)\right],\,\,\,\,\,\,\,\,H_{2}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\sqrt{2}\,H^{+}\\\
\phi^{0}_{2}+iA^{0}\end{array}\right),$ (9)
where $G^{0,\pm}$ are the Goldstone bosons, $H^{\pm}$ and $A^{0}$ are the
physical charged-Higgs boson and CP-odd neutral Higgs boson, respectively. Its
virtue is the first doublet $H_{1}$ corresponds to the scalar doublet of the
SM while the new Higgs fields arise from the second doublet $H_{2}$.
The CP-even neutral Higgs boson mass eigenstates $H^{0}$ and $h^{0}$ are
linear combinations of $\phi_{1}^{0}$ and $\phi^{0}_{2}$ in Eq.(9),
$\displaystyle H^{0}$ $\displaystyle=$
$\displaystyle\phi_{1}^{0}\cos\alpha+\phi^{0}_{2}\sin\alpha,\,\,\,h^{0}=-\phi^{0}_{1}\sin\alpha+\phi^{0}_{2}\cos\alpha,$
(10)
where $\alpha$ is the mixing angle.
After diagonalizing the mass matrix of the fermion fields, the Yukawa
Lagrangian becomesDavid
$\displaystyle L_{Y}$ $\displaystyle=$
$\displaystyle-\overline{U}M_{U}U-\overline{D}M_{D}D+\frac{i}{\upsilon}\chi^{0}\left(\overline{U}M_{U}\gamma_{5}U-\overline{D}M_{D}\gamma_{5}D\right)$
(11) $\displaystyle+$
$\displaystyle\frac{\sqrt{2}}{\upsilon}\chi^{-}\overline{D}V^{\dagger}_{CKM}\left[M_{U}R-M_{D}L\right]U-\frac{\sqrt{2}}{\upsilon}\chi^{+}\overline{U}V_{CKM}\left[M_{D}R-M_{U}L\right]D$
$\displaystyle+$
$\displaystyle\frac{iA^{0}}{\sqrt{2}}\left\\{\overline{U}\left[\widehat{\xi}^{U}R-\widehat{\xi}^{U{\dagger}}L\right]U+\overline{D}\left[\widehat{\xi}^{D{\dagger}}L-\widehat{\xi}^{D}R\right]D\right\\}$
$\displaystyle-$
$\displaystyle\frac{H^{0}}{\sqrt{2}}\overline{U}\left\\{\frac{\sqrt{2}}{\upsilon}M_{U}\cos\alpha+\left[\widehat{\xi}^{U}R+\widehat{\xi}^{U{\dagger}}L\right]\sin\alpha\right\\}U-\frac{H^{0}}{\sqrt{2}}\overline{D}\left\\{\frac{\sqrt{2}}{\upsilon}M_{D}\cos\alpha+\left[\widehat{\xi}^{D}R+\widehat{\xi}^{D{\dagger}}L\right]\sin\alpha\right\\}D$
$\displaystyle-$
$\displaystyle\frac{h^{0}}{\sqrt{2}}\overline{U}\left\\{-\frac{\sqrt{2}}{\upsilon}M_{U}\sin\alpha+\left[\widehat{\xi}^{U}R+\widehat{\xi}^{U{\dagger}}L\right]\cos\alpha\right\\}U-\frac{h^{0}}{\sqrt{2}}\overline{D}\left\\{\frac{\sqrt{2}}{\upsilon}M_{D}\sin\alpha+\left[\widehat{\xi}^{D}R+\widehat{\xi}^{D{\dagger}}L\right]\cos\alpha\right\\}D$
$\displaystyle-$ $\displaystyle
H^{+}\overline{U}\left[V_{CKM}\widehat{\xi}^{D}R-\widehat{\xi}^{U{\dagger}}V_{CKM}L\right]D-H^{-}\overline{D}\left[\widehat{\xi}^{D{\dagger}}V^{\dagger}_{CKM}L-V^{\dagger}_{CKM}\widehat{\xi}^{U}R\right]U$
where U and D now are the fermion mass eigenstates and
$\displaystyle\hat{\eta}^{U,D}$ $\displaystyle=$
$\displaystyle(V_{L}^{U,D})^{-1}\cdot\eta^{U,D}\cdot
V_{R}^{U,D}=\frac{\sqrt{2}}{v}M^{U,D}(M^{U,D}_{ij}=\delta_{ij}m_{j}^{U,D}),$
(12) $\displaystyle\hat{\xi}^{U,D}$ $\displaystyle=$
$\displaystyle(V_{L}^{U,D})^{-1}\cdot\xi^{U,D}\cdot V_{R}^{U,D},$ (13)
where $V_{L,R}^{U,D}$ are the rotation matrices acting on up and down-type
quarks, with left and right chiralities respectively. Thus
$V_{CKM}=(V_{L}^{U})^{{\dagger}}V_{L}^{D}$ is the usual Cabibbo-Kobayashi-
Maskawa (CKM) matrix. In general, the matrices $\hat{\eta}^{U,D}$ of Eq.(12)
are diagonal, while the matrices $\hat{\xi}^{U,D}$ are non-diagonal which
could induce scalar-mediated FCNC. Seen from Eq.(11), the coupling of neutral
Higgs bosons to the fermions could generate FCNC parts. For the arbitrariness
of definition for $\xi^{U,D}_{ij}$ couplings, we can adopt the rotated
couplings expressed $\xi^{U,D}$ in stead of $\hat{\xi}^{U,D}$ hereafter.
In this work, we use the Cheng-Sher ansatzcheng
$\xi^{U,D}_{ij}=\lambda_{ij}\,\frac{\sqrt{m_{i}m_{j}}}{v}$ (14)
which ensures that the FCNCs within the first two generations are naturally
suppressed by small fermions masses. This ansatz suggests that LFV couplings
involving the electron are suppressed, while LFV transitions involving muon
and tau are much less suppressed and may lead to some loop effects which are
promising to be tested by the future B factory experiments. In Eq.(14), the
parameter $\lambda_{ij}$ is complex and $i,j$ are the generation indexes. In
this study, we shall discuss the phenomenological applications of the type III
2HDM.
## III The discussion for $\tau^{-}\to\mu^{-}V^{0}$ decays
As we have mentioned above, one vector meson could not be generated from the
vacuum state through the scalar and/or pseudoscalar currents. In 2HDM model
III, the neutral Higgs bosons mediated tree and penguin diagrams have no
contributions to $\tau^{-}\to\mu^{-}V^{0}$ processes. Accordingly, their decay
amplitudes acquire contributions from the $\gamma-,Z-$ penguin and box
diagrams. Comparing to the $\tau\to\mu P$decays, in addition to neutral Higgs
bosons, the penguin with charged Higgs bosons in the loop also contribute to
these decays. We will make a detail analysis of their effects in the later
paragraphs. The penguin diagrams at the quark level pertinent to these decays
are list in Fig.1.
Figure 1: The $\gamma$ and $Z^{0}$ penguin diagrams for
$\tau^{-}\to\mu^{-}q\bar{q}$ decay, where the neutral and charged Higgs bosons
are in the loop.
The amplitudes could be factorized into leptonic vertex corrections and
hadronic parts described with hadronic matrix elements. In dealing with
hadronic matrix elements, we take the generalized factorization approach and
write the hadronic matrix elements as
$<V|\bar{q}\gamma_{\mu}q|0>=-m_{V}f_{V}\varepsilon_{\mu}^{*}$ with the decay
constant $f_{V}$. The quark contents of $\rho^{0}$ meson are chosen as
$\rho^{0}=\frac{1}{\sqrt{2}}(-u\bar{u}+d\bar{d})$. For the vector
$\phi-\omega$ meson system, we employ the ideal mixing scheme between
$\phi(1020)$ and $\omega(782)$ which is supported by existing data:
$\phi=-s\bar{s},\omega=\frac{1}{\sqrt{2}}(u\bar{u}+d\bar{d})$xiaoz . Then, the
total amplitudes could be expressed as:
$\displaystyle{\cal M}(\tau^{-}\to\mu^{-}V^{0})$ $\displaystyle=$
$\displaystyle{\cal M}_{\gamma}+{\cal M}_{Z}+{\cal M}_{box}$
$\displaystyle{\cal M}_{\gamma}(\tau^{-}\to\mu^{-}V^{0})$ $\displaystyle=$
$\displaystyle\frac{i\alpha^{2}_{w}S_{w}^{2}}{2m^{2}_{w}}\cdot\bar{\mu}\cdot[F1_{\gamma}\cdot
L+F2_{\gamma}\cdot
R+F3_{\gamma}\cdot\gamma_{\rho}L+F4_{\gamma}\cdot\gamma_{\rho}R]\cdot\tau$
$\displaystyle\otimes\langle
V^{0}|\frac{2}{3}\bar{u}\gamma^{\rho}u-\frac{1}{3}\bar{d}\gamma^{\rho}d-\frac{1}{3}\bar{s}\gamma^{\rho}s|0\rangle$
$\displaystyle{\cal M}_{Z}$ $\displaystyle=$
$\displaystyle\frac{i\alpha_{w}^{2}}{8m^{4}_{W}}\cdot\bar{\mu}\cdot\biggl{[}F_{1}^{z}\cdot
L+F_{2}^{z}\cdot
R+F_{3}^{z}\cdot\gamma_{\rho}L+F_{4}^{z}\cdot\gamma_{\rho}R]\cdot\tau$
$\displaystyle\otimes\langle
V^{0}|g^{u}_{V}(\bar{u}\gamma^{\rho}u)-g^{d}_{V}(\bar{d}\gamma^{\rho}d)-g^{s}_{V}(\bar{s}\gamma^{\rho}s)|0\rangle$
$\displaystyle{\cal M}_{box}$ $\displaystyle=$
$\displaystyle\frac{i\alpha_{w}^{2}}{m^{4}_{W}}\cdot\bar{\mu}\cdot\biggl{[}F_{1}^{box}\cdot
L+F_{2}^{box}\cdot
R+F_{3}^{box}\cdot\gamma_{\rho}L+F_{4}^{box}\cdot\gamma_{\rho}R+F_{5}^{box}\cdot
i\sigma_{\rho\lambda}L+F_{6}^{box}\cdot
i\sigma_{\rho\lambda}R\biggl{]}\cdot\tau$ (15) $\displaystyle\otimes\langle
V^{0}|\bar{u}\gamma^{\rho}u-\bar{d}\gamma^{\rho}d-\bar{s}\gamma^{\rho}s|\rangle$
where ${\cal M}_{\gamma},{\cal M}_{Z}$ and ${\cal M}_{box}$ are the amplitudes
of the $\gamma-$ penguin, Z penguin and box diagrams. The relevant auxiliary
functions are listed in Appendix.
In our calculation, the input parameters are the Higgs masses, mixing angle
$\alpha$, $|\lambda_{ij}|$ and their phase angles $\theta_{ij}$. Given the
constraints from the current experiment permits and theoretical
considerationsli ; dyb ; Atwood ; csh ; Rodolfo ; zhuang ; Rozo , we assume
$\displaystyle m_{H^{\pm}}$ $\displaystyle=$ $\displaystyle
200GeV,\,\,\,m_{H^{0}}=160GeV,\,\,\,m_{h^{0}}=115GeV,\,\,\,m_{A^{0}}=120GeV,\,\,\,\alpha=\pi/4,\,\,$
$\displaystyle|\lambda_{uu}|$ $\displaystyle=$ $\displaystyle
150,\,\,\,|\lambda_{dd}|=120,\,\,\,|\lambda_{\tau\tau}|=10,\,\,\,\,|\lambda_{tt}|=|\lambda_{tc}|=|\lambda_{ut}|=0.03,\,\,$
$\displaystyle|\lambda_{ss}|$ $\displaystyle=$
$\displaystyle|\lambda_{bb}|=|\lambda_{db}|=|\lambda_{bs}|=100,\,\,\theta=\pi/4,\,\,$
(16)
where the Higgs masses satisfy the relation $115GeV\leq
m_{h^{0}}<m_{A^{0}}<m_{H^{0}}\leq 200GeV$dyb ; csh ; Rodolfo , and the
absolute value of $\lambda_{tt}\cdot\lambda_{bb}$ is approximate to threecsh ;
zhuang .
Using the above parameters, we could get the contributions of three diagrams
to these decays. As we expected, the contributions of box diagrams are
$O(10^{-25})$ order or so which are very smaller than those of $\gamma-$ and
$Z-$ penguins. Hence, we neglect the box diagrams contributions. We have
studied the relation of branching ratio and $\lambda_{\tau\mu}$. The
computation indicate that the variation of $\theta_{\tau\mu}$, the phase angle
of parameters $\lambda_{\tau\mu}$, does almost not affect the values of
branching ratios. So we take $\theta_{\tau\mu}=\pi/4$ as literatures do.
The Fig.2 gives the total penguin contributions denoted by the solid line. We
denote the $\gamma-$ penguin and the $Z-$ penguin contributions by the dash
line and the dot line, respectively. Due to the suppressed factor
$O(1/m^{2}_{Z})$ from the Z propagator, the $Z$ penguin contributions are
supposed to be lower than those of the $\gamma-$ penguin. These decay
amplitudes have common leptonic parts, so the differences of decay amplitudes
mainly come from the hadronic parts. For the similar contents of $\rho^{0}$
and $\omega$, the curves of $\tau^{-}\to\mu^{-}\rho^{0}$ and
$\tau^{-}\to\mu^{-}\omega$ decays display similar trend, namely, their $Z$
penguin contributions are lower one order than those of their $\gamma-$
penguin. However, for $\tau^{-}\to\mu^{-}\phi$ decay, the magnitudes of $Z$
penguin are close to the $\gamma-$ penguin contributions.
The relations of ratios versus $|\lambda_{\tau\mu}|$ are also presented in
Fig.2, where the horizon lines denote the experimental upper limits.
Evidently, one can see from Fig.2 that these branching ratios rise with the
increase of $|\lambda_{\tau\mu}|$. We have got the constraints on
$|\lambda_{\tau\mu}|$ from the experimental data, which are list in Table.I.
It is obviously that the parameter $|\lambda_{\tau\mu}|$ is restrained at the
order of $O(10^{-3})$. The $|\lambda_{\tau\mu}|$ constraints for
$\tau^{-}\to\mu^{-}\rho^{0}(\omega)$ are little severe than that of
$\tau^{-}\to\mu^{-}\phi$ decay. The bounds of $|\lambda_{\tau\mu}|$ from
different phenomenological considerations li ; sher ; Martin ; Rodolfo ; Zhou
; Cotti are demonstrated in Tab.I, too. Comparing the values of
$\lambda_{\tau\mu}$ in Tab.I, one can see that our constraint is stringenter
than the limits in literatures.
Table 1: Constraints on the $\lambda_{\tau\mu}$ from $\tau^{-}\to\mu^{-}\rho^{0}(\phi,\omega)$ decays in the 2HDM III. Decay modes | Bounds on $\lambda_{\tau\mu}$ | Previous Bounds
---|---|---
$\tau\to\mu\rho^{0}$ | $\leq 1.26\times 10^{-3}$ | $\lambda_{\tau\mu}\sim O(1)$ sher
$\tau\to\mu\phi$ | $\leq 2.45\times 10^{-3}$ | $\lambda_{\tau\mu}\sim O(10)$ li ; Martin ; Zhou , $\lambda_{\tau\mu}\sim O(10)-O(10^{2})$ Rodolfo
$\tau\to\mu\omega$ | $\leq 1.48\times 10^{-3}$ | $\lambda_{\tau\mu}\sim O(10^{2})-O(10^{3})$ Cotti
Figure 2: The branching ratios versus model parameter $|\lambda_{\tau\mu}|$
with $\theta_{\tau\mu}=\pi/4$, (a) for $\tau^{-}\to\mu^{-}\rho^{0}$ decay, (b)
for $\tau^{-}\to\mu^{-}\phi$ decay, and (c) for $\tau^{-}\to\mu^{-}\omega$
decay. The solid line denotes the total contributions; the dash line and the
dot line denote the $\gamma-$ and Z penguin contributions, respectively. The
horizontal lines are the experimental upper limits. Figure 3: The branching
ratios versus model parameter $|\lambda_{\tau\mu}|$ with
$\theta_{\tau\mu}=\pi/4$, (a) for $\tau^{-}\to\mu^{-}\rho^{0}$ decay (b) for
$\tau^{-}\to\mu^{-}\phi$ decay, and (c) for $\tau^{-}\to\mu^{-}\omega$ decay.
The solid line denotes the $\gamma$ penguin contributions; the dash line and
the dot line denote the contributions of $\gamma-$ penguin with neutral Higgs
bosons in the loop and those of $\gamma-$ penguin with charged Higgs bosons in
the loop, respectively.
Now we illustrate the contributions of $\gamma-$ penguin with neutral and
charged Higgs bosons in the loop. In Fig.3, the solid line denotes the
$\gamma-$ penguin contributions, the dash line and the dot line denote the
contributions of $\gamma-$ penguin with neutral Higgs bosons in the loop and
those of $\gamma-$ penguin with charged Higgs bosons in the loop,
respectively. Apparently, the contributions of $\gamma-$ penguin with neutral
Higgs in the loop are quite higher by nearly four order magnitudes than those
of $\gamma-$ penguin with charged Higgs in the loop. And the dot line and the
solid line coincide with each other for three decays. As a result, the
contributions of $\gamma-$ penguin with neutral Higgs in the loop are
dominated one.
Figure 4: The branching ratios versus model parameter $|\lambda_{\tau\mu}|$
with $\theta_{\tau\mu}=\pi/4$, (a) for $\tau^{-}\to\mu^{-}\rho^{0}$ decay (b)
for $\tau^{-}\to\mu^{-}\phi$ decay, and (c) for $\tau^{-}\to\mu^{-}\omega$
decay. The solid line denotes the Z penguin contributions; the dash line and
the dot line denote the contributions of $Z$ penguin with charged Higgs bosons
in the loop and those of $Z$ penguin with neutral Higgs bosons in the loop,
respectively
The contributions of $Z$ penguin with charged and neutral Higgs bosons in the
loop are demonstrated in figure 4. The solid line denotes the $Z$ penguins
contributions, the dash line and the dot line denote the contributions of $Z$
penguin with charged Higgs bosons in the loop and those of $Z$ penguin with
neutral Higgs bosons in the loop, respectively. Unlike the case of $\gamma$
penguin, the contributions of $Z$ penguin with neutral Higgs in the loop are
rather smaller by nearly eight order magnitudes than those of $Z$ penguin with
charged Higgs in the loop. So the contributions of $Z$ penguin with neutral
Higgs in the loop are subordinate one. In a word, the $\gamma-$ penguin with
neutral Higgs in the loop plays a main role in these decays.
## IV Conclusion
In summary, we have calculated the branching ratios of
$\tau^{-}\to\mu^{-}\rho^{0}(\phi,\omega)$ decays in the model III 2HDM.
Comparing to the $\tau^{-}\to\mu^{-}P$ decays, besides the neutral higgs
bosons in the loop, an additional charged Higgs boson in the loop offer
contributions to $\tau^{-}\to\mu^{-}V^{0}$ decays. The impacts of the
$\gamma-$ penguin, $Z-$ penguin and those of two types Higgs in loop are
formulated. It is concluded that the $\gamma-$ penguin with neutral Higgs
bosons in loop are dominated in the $\gamma-$ penguin, while the $Z-$ penguin
with charged Higgs bosons in loop mainly contributes to the $Z-$ penguins. Our
work suggests that the parameter $|\lambda_{\tau\mu}|$ is constrained at the
order of $O(10^{-3})$. And in the rational parameters space, the
$Br(\tau^{-}\to\mu^{-}V^{0})$ can reach the experimental values. With the
experiment luminosity increasing, these LFV decays are available to the
collider’s measure capability. Our study is hoped to supply good information
for the future experiment and explore the structure of the 2HDM III model.
## Appendix
For simplicity, we only list the amplitude of $\gamma-$ penguin.
The amplitude of $\gamma-$ penguin diagrams is
$\displaystyle{\cal M}_{\gamma}(\tau^{-}\to\mu^{-}V^{0})$ $\displaystyle=$
$\displaystyle\frac{i\alpha^{2}_{w}S_{w}^{2}}{2m^{2}_{w}}\cdot\bar{\mu}\cdot[F1_{\gamma}\cdot
L+F2_{\gamma}\cdot
R+F3_{\gamma}\cdot\gamma_{\rho}L+F4_{\gamma}\cdot\gamma_{\rho}R]\cdot\tau$
(17) $\displaystyle\otimes\langle
V^{0}|\frac{2}{3}\bar{u}\gamma^{\rho}u-\frac{1}{3}\bar{d}\gamma^{\rho}d-\frac{1}{3}\bar{s}\gamma^{\rho}s|0\rangle.$
Where the auxiliary functions $F_{\gamma}$ are written as:
$\displaystyle F1_{\gamma}$ $\displaystyle=$
$\displaystyle\frac{m_{\tau}\sqrt{m_{\tau}m_{\mu}}\cdot}{k^{2}}\cdot\int_{0}^{1}dx\int_{0}^{1-x}dy\left\\{\biggl{(}-\frac{m_{\tau}\lambda^{*}_{\tau\mu}\lambda_{\tau\tau}\cdot
x}{S_{c}(x,y,m^{2}_{H^{-}},x_{tc})}+\frac{m_{\tau}\lambda^{*}_{\tau\mu}}{2}\sum_{i}J_{i}\times
x\biggl{)}\cdot p_{1}^{\rho}\right.$ (18)
$\displaystyle\left.+\biggl{(}-\frac{m_{\tau}\cdot\lambda^{*}_{\tau\mu}\lambda_{\tau\tau}\cdot
x}{S_{c}(x,y,m^{2}_{H^{-}},x_{tc})}+\frac{1}{2}\sum_{i}K_{i}\times
y\biggl{)}\cdot p_{2}^{\rho}\right\\},$ $\displaystyle F2_{\gamma}$
$\displaystyle=$
$\displaystyle\frac{m_{\tau}\sqrt{m_{\tau}m_{\mu}}}{k^{2}}\cdot\int_{0}^{1}dx\int_{0}^{1-x}dy\left\\{\biggl{(}-\frac{m_{\mu}\cdot\lambda^{*}_{\tau\mu}\lambda_{\tau\tau}\cdot
y}{S_{c}(x,y,m^{2}_{H^{-}},x_{tc})}+\frac{m_{\tau}\cdot\lambda_{\tau\mu}}{2}\cdot\sum_{i}J^{*}_{i}\times
x\biggl{)}\cdot p_{1}^{\rho}\right.$ (19)
$\displaystyle\left.+\biggl{(}-\frac{m_{\mu}\cdot\lambda^{*}_{\tau\mu}\lambda_{\tau\tau}\cdot
y}{S_{c}(x,y,m^{2}_{H^{-}},x_{tc})}+\frac{1}{2}\cdot\sum_{i}K^{*}_{i}\times
y\biggl{)}\cdot p_{2}^{\rho}\right\\},$ $\displaystyle F3_{\gamma}$
$\displaystyle=$
$\displaystyle\frac{m_{\tau}\sqrt{m_{\tau}m_{\mu}}}{k^{2}}\cdot\left\\{\frac{1}{(m^{2}_{\tau}-m^{2}_{\mu})}\cdot\int_{0}^{1}dx\biggl{[}m_{\mu}\cdot\lambda^{*}_{\tau\mu}\lambda_{\tau\tau}(x-1)\ln\frac{S_{a}(x,x_{tc})}{S_{b}(x)}+\frac{1}{2}M_{i}\biggl{]}+N+\int_{0}^{1}dx\int_{0}^{1-x}dyQ_{i}\right\\},$
$\displaystyle F4_{\gamma}$ $\displaystyle=$
$\displaystyle\frac{m_{\tau}\sqrt{m_{\tau}m_{\mu}}}{k^{2}}\cdot\left\\{\frac{1}{(m^{2}_{\tau}-m^{2}_{\mu})}\cdot\int_{0}^{1}dx\biggl{[}\lambda^{*}_{\tau\mu}\lambda_{\tau\tau}(x-1)\cdot\biggl{(}m_{\tau}^{2}\ln
S_{a}(x,x_{tc})-m_{\mu}^{2}\ln
S_{b}(x)\biggl{)}+\frac{1}{2}M^{*}_{i}\biggl{]}+N^{*}\right.$ (21)
$\displaystyle\left.+\int_{0}^{1}dx\int_{0}^{1-x}dy[\ln\frac{S_{c}(x,y,m^{2}_{H^{-}},x_{tc})}{\mu^{2}}+Q^{*}_{i}]\right\\}.$
The followings are expressions of $J_{i},K_{i},M_{i},N$ and $Q_{i}$.
$\displaystyle J_{H^{0}}$ $\displaystyle=$
$\displaystyle\frac{\omega_{s}}{S^{H^{0}}_{c}(x,y,m^{2}_{H^{0}},x^{H^{0}}_{tn})},\,\,\,\,\,J_{h^{0}}=\frac{\upsilon_{s}}{S^{h^{0}}_{c}(x,y,m^{2}_{h^{0}},x^{h^{0}}_{tn})},\,\,\,\,\,J_{A^{0}}=\frac{2iIm\lambda_{\tau\tau}}{S^{A^{0}}_{c}(x,y,m^{2}_{A^{0}},x^{A^{0}}_{tn})},$
$\displaystyle K_{H^{0}}$ $\displaystyle=$
$\displaystyle(m_{\tau}\lambda^{*}_{\tau\mu}+m_{\mu}\lambda_{\tau\mu})\times
J^{*}_{H^{0}},\,\,\,\,\,K_{h^{0}}=(m_{\tau}\lambda^{*}_{\tau\mu}+m_{\mu}\lambda_{\tau\mu})\times
J^{*}_{h^{0}},\,\,\,\,\,\,K_{A^{0}}=\frac{\lambda^{*}_{\tau\tau}(m_{\mu}\lambda_{\tau\mu}-m_{\tau}\lambda^{*}_{\tau\mu})}{S^{A^{0}}_{c}(x,y,m^{2}_{A^{0}},x^{A^{0}}_{tn})}$
$\displaystyle M_{H^{0}}$ $\displaystyle=$
$\displaystyle\biggl{[}[x(m_{\tau}^{2}\omega^{*}\lambda_{\tau\mu}+m_{\tau}m_{\mu}\omega\lambda^{*}_{\tau\mu})-\omega_{s}(m_{\tau}^{2}\lambda_{\tau\mu}+m_{\tau}m_{\mu}\lambda^{*}_{\tau\mu})]\ln
S^{H^{0}}_{a}(x,x^{H^{0}}_{tn})$
$\displaystyle-[x(m_{\mu}^{2}\omega^{*}\lambda_{\tau\mu}+m_{\tau}m_{\mu}\omega\lambda^{*}_{\tau\mu})-(m_{\mu}^{2}\omega^{*}+m_{\tau}^{2}\omega)\lambda_{\tau\mu}-m_{\tau}m_{\mu}\omega_{s}\lambda^{*}_{\tau\mu}]\ln
S^{H^{0}}_{b}(x)\biggl{]}$ $\displaystyle Q_{H^{0}}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\int_{0}^{1-x}dy\biggl{(}\omega^{*}\lambda_{\tau\mu}[\ln\frac{S_{c}(x,y,m^{2}_{H^{-}},x_{tc})}{\mu^{2}}+\frac{m^{2}_{\tau}(x^{2}-x-1)-m^{2}_{\mu}y}{S_{c}^{H^{0}}(x,y,m^{2}_{H^{0}},x^{H^{0}}_{tn})}]$
$\displaystyle+\frac{m_{\tau}m_{\mu}\lambda^{*}_{\tau\mu}[(x+y)\omega-\omega_{s}]-m^{2}_{\tau}\omega\lambda_{\tau\mu}}{S_{c}^{H^{0}}(x,y,m^{2}_{H^{0}},x^{H^{0}}_{tn})}\biggl{)}$
$\displaystyle M_{h^{0}}$ $\displaystyle=$
$\displaystyle[x(m_{\tau}^{2}\upsilon^{*}\lambda_{\tau\mu}+m_{\tau}m_{\mu}\upsilon\lambda^{*}_{\tau\mu})-\upsilon_{s}(m_{\tau}^{2}\lambda_{\tau\mu}+m_{\tau}m_{\mu}\lambda^{*}_{\tau\mu})]\ln
S^{h^{0}}_{a}(x,x^{h^{0}}_{tn})$
$\displaystyle-[x(m_{\mu}^{2}\upsilon^{*}\lambda_{\tau\mu}+m_{\tau}m_{\mu}\upsilon\lambda^{*}_{\tau\mu})-(m_{\mu}^{2}\upsilon^{*}+m_{\tau}^{2}\upsilon)\lambda_{\tau\mu}-m_{\tau}m_{\mu}\upsilon_{s}\lambda^{*}_{\tau\mu}]\ln
S^{h^{0}}_{b}(x)\biggl{]}$ $\displaystyle Q_{h^{0}}$ $\displaystyle=$
$\displaystyle\biggl{(}[\ln\frac{S_{c}(x,y,m^{2}_{H^{-}},x_{tc})}{\mu^{2}}+\frac{m^{2}_{\tau}(x^{2}-x-1)-m^{2}_{\mu}y}{S_{c}^{h^{0}}(x,y,m^{2}_{h^{0}},x^{h^{0}}_{tn})}]\upsilon^{*}\lambda_{\tau\mu}$
$\displaystyle+\frac{m_{\tau}m_{\mu}\lambda^{*}_{\tau\mu}[(x+y)\upsilon-\upsilon_{s}]-m^{2}_{\tau}\upsilon\lambda_{\tau\mu}}{S_{c}^{h^{0}}(x,y,m^{2}_{h^{0}},x^{h^{0}}_{tn})}\biggl{)}$
$\displaystyle M_{A^{0}}$ $\displaystyle=$
$\displaystyle\biggl{[}[x(m_{\tau}^{2}\lambda_{\tau\tau}^{*}\lambda_{\tau\mu}+m_{\tau}m_{\mu}\lambda_{\tau\tau}\lambda^{*}_{\tau\mu})-2iIm\lambda_{\tau\tau}(m_{\tau}m_{\mu}\lambda^{*}_{\tau\mu}-m_{\tau}^{2}\lambda_{\tau\mu})]\ln
S^{A^{0}}_{a}(x,x^{A^{0}}_{tn})$
$\displaystyle-[x(m_{\mu}^{2}\lambda_{\tau\tau}^{*}\lambda_{\tau\mu}+m_{\tau}m_{\mu}\lambda_{\tau\tau}\lambda^{*}_{\tau\mu})-(m_{\mu}^{2}\lambda_{\tau\tau}^{*}-m_{\tau}^{2}\lambda_{\tau\tau})\lambda_{\tau\mu}-2im_{\tau}m_{\mu}\lambda^{*}_{\tau\mu}Im\lambda_{\tau\tau}]\ln
S^{A^{0}}_{b}(x)\biggl{]}$ $\displaystyle Q_{A^{0}}$ $\displaystyle=$
$\displaystyle\biggl{(}[\ln\frac{S^{A^{0}}_{c}(x,y,m^{2}_{A^{0}},x^{A^{0}}_{tn})}{\mu^{2}}+\frac{m^{2}_{\tau}(x^{2}-x-1)-m^{2}_{\mu}y}{S^{A^{0}}_{c}(x,y,m^{2}_{A^{0}},x^{A^{0}}_{tn})}]\lambda^{*}_{\tau\tau}\lambda_{\tau\mu}$
$\displaystyle+\frac{m^{2}_{\tau}\lambda_{\tau\tau}\lambda_{\tau\mu}+m_{\tau}m_{\mu}\lambda^{*}_{\tau\mu}[(x+y)\lambda_{\tau\tau}-2iIm\lambda_{\tau\tau}]}{S^{A^{0}}_{c}(x,y,m^{2}_{A^{0}},x^{A^{0}}_{tn})}\biggl{)}$
$\displaystyle N$ $\displaystyle=$
$\displaystyle\frac{1}{2}\lambda_{\tau\mu}(\omega^{*}+\upsilon^{*}+\lambda^{*}_{\tau\tau})$
$\displaystyle\omega$ $\displaystyle=$
$\displaystyle(\lambda_{\tau\tau}\sin^{2}\alpha+\sin\alpha\cos\alpha),\,\,\,\,\upsilon=(\lambda_{\tau\tau}\cos^{2}\alpha-\sin\alpha\cos\alpha)$
$\displaystyle\omega_{s}$ $\displaystyle=$ $\displaystyle 2\sin^{2}\alpha
Re\lambda_{\tau\tau}+\cos 2\alpha,\,\,\,\,\upsilon_{s}=2\cos^{2}\alpha
Re\lambda_{\tau\tau}-\cos 2\alpha$
The integrate function expressions are :
$\displaystyle S_{a}(x,x_{tc})$ $\displaystyle=$
$\displaystyle(x-1)(x_{tc}x-1),\,\,\,\,\,S_{b}(x)=1-x,\,\,\,S_{c}(x,y,m^{2}_{H^{-}},x_{tc})=m^{2}_{H^{-}}[x+(x^{2}-x+y)x_{tc}],\,\,\,x_{tc}=\frac{m^{2}_{\tau}}{m^{2}_{H^{-}}},$
$\displaystyle S^{i}_{a}(x,x^{i}_{tn})$ $\displaystyle=$
$\displaystyle(x-1)(x^{i}_{tn}x-1)\,\,\,\,\,S^{i}_{b}(x)=S_{b}(x),\,\,\,S^{i}_{c}(x,y,m^{2}_{i},x^{i}_{tn})=m^{2}_{i}[y+x+x^{i}_{tn}x(x-1)],\,\,x^{i}_{tn}=\frac{m^{2}_{\tau}}{m^{2}_{i}}$
(22)
###### Acknowledgements.
I thank Prof.Chaoshang Huang for discussion. The work is supported by National
Science Foundation under contract No.10547110, He nan Educational Committee
Foundation under contract No.2007140007, the Project of Knowledge Innovation
Program (PKIP) of Chinese Academy of Sciences under Grant No.KJCX2.YW.W10.
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|
arxiv-papers
| 2008-12-03T14:27:54 |
2024-09-04T02:48:59.156742
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wenjun Li, Yanqin ma, Gongwei Liu, Wei Guo",
"submitter": "Li WenJun",
"url": "https://arxiv.org/abs/0812.0727"
}
|
0812.0754
|
# Strong Spatial Mixing and Approximating Partition Functions of Two-State
Spin Systems without Hard Constrains
Jinshan Zhang Department of
Mathematical Sciences
Tsinghua University
Beijing, China 100084
Email: zjs02@mails.tsinghua.edu.cn
Abstract: We prove Gibbs distribution of two-state spin systems(also known as
binary Markov random fields) without hard constrains on a tree exhibits strong
spatial mixing(also known as strong correlation decay), under the assumption
that, for arbitrary ‘external field’, the absolute value of ‘inverse
temperature’ is small, or the ‘external field’ is uniformly large or small.
The first condition on ‘inverse temperature’ is tight if the distribution is
restricted to ferromagnetic or antiferromagnetic Ising models.
Thanks to Weitz’s self-avoiding tree, we extends the result for sparse on
average graphs, which generalizes part of the recent work of Mossel and
Sly[15], who proved the strong spatial mixing property for ferromagnetic Ising
model. Our proof yields a different approach, carefully exploiting the
monotonicity of local recursion. To our best knowledge, the second condition
of ‘external field’ for strong spatial mixing in this paper is first
considered and stated in term of ‘maximum average degree’ and ‘interaction
energy’. As an application, we present an FPTAS for partition functions of
two-state spin models without hard constrains under the above assumptions in a
general family of graphs including interesting bounded degree graphs.
Keywords: Strong Spatial Mixing; Self-Avoiding Trees; Two-State Spin Systems;
Ising Models; FPTAS; Partition Function
## I Introduction
Counting problem has played an important role in theoretic computer science
since Valiant[19] introduced $\\#$P-Complete conception and proved many
enumeration problems are computationally intractable. The most successful and
powerful existing method for counting problem is due to Markov Chain method,
which has been successfully used to provide a fully polynomial randomized
approximation schemes (FPRAS)(which approximates the real value within a
factor of $\epsilon$ in polynomial time of the input and $\epsilon^{-1}$ with
the probability $\geq$ 3/4) for convex bodies[3] and the number of perfect
matchings on bipartite graphs[9]. Since many counting problems such as the
number of matchings, independent sets, circuits[14] etc. can be viewed as
special cases of computing partition functions associated with Gibbs measures
in statistical physics. Hence studying the computation of partition function
is a natural extension of counting problems.
Self-reducing [10] or conditional probability method is a well known method to
compute partition functions if the marginal probability of a vertex can be
efficiently approximated. Gibbs sampling also known as Glauber dynamics is a
popular used method to approximate marginal probability. This is a Markov
Chain approach locally updating the chain according to conditional Gibbs
measure. Hence studying the convergence rate(also known as mixing time) of
Glauber dynamics becomes a major research direction. Recently the problem
whether the Glauber dynamics converges ‘fast’(in a polynomial time of the
input and logarithm of reciprocal of sampling error) deeply related to whether
a phase transition takes place in statistical model has been extensively
studied, see [16] for hard core model(also known as independent set model) and
[15][6] for ferromagnetic Ising model. Another approach to approximate
marginal probability comes from the property of the structure of Gibbs
measures on various graphs. This method utilizes local recursion and leads to
deterministic approximation schemes rather than random approximation schemes
of Markov Chain method. Our paper focuses on this recursive approach.
The recursive approach for counting problems is introduced by Weitz[21] and
Bandyopadhyay, Gamarnik [1] for counting the number of independent sets and
colorings. The key of this method is to establish the $strong$ $spatial$
$mixing$ property also known as $strong$ $correlation$ $decay$ on certain
defined rooted trees, which means the marginal probability of the root is
asymptotically independent of the configuration on the leaves far below.
Usually the exponential decay with the distance implies a deterministic
polynomial time approximating algorithm for marginal probability of the root.
In [21], Weitz establishes the equivalence between the marginal probability of
a vertex in a general graph $G$ and that of the root of a tree named
$self$-$avoiding$ $tree$ associated with $G$ for two-state spin systems and
shows the correlations on any graph decay at least as fast as its
corresponding self-avoiding tree. He also proves the strong correlation decay
for hard-core model on bounded degree trees. Later Gamarnik et.al.[5] and
Bayati et.al.[2] bypass the construction of a self-avoiding tree, by instead
creating a certain $computation$ $tree$ and establishing the strong
correlation decay on the corresponding computation tree for list coloring and
matchings problems. An interesting relation between self-avoiding tree and
computation tree is that they share the same recursive formula for hard-core
model. Considering the motivation of construction of the self-avoiding tree,
Jung and Shah[8] and Nair and Tetali [17] generalize Weitz’s work for certain
Markov random field models, and Lu et.al.[12] for TP decoding problem. Mossel
and Sly[15] show ferromagnetic Ising model exhibits strong correlation decay
on ‘sparse on average’ graph under the tight assumption that the ‘inverse
temperature’ in term of ‘maximum average degree’ is small.
In this paper, based on self-avoiding tree, we establish the strong spatial
mixing for general two-state spin systems also know binary Markov random field
without hard constrains on a graph that are sparse on average under certain
assumptions. Our first condition is on the ‘inverse temperature’. We show that
there exits a value $J_{d}$ in term of ‘maximum average degree’ $d$, if the
absolute value of the ‘inverse temperature’ is smaller than $J_{d}$, for
arbitrary ‘external field’, the Gibbs measure exhibits strong spatial mixing
on a sparse on average graph. Since for (anti)ferromagnetic Ising model,
strong spatial mixing on a finite regular tree implies uniqueness of Gibbs
measures of infinite regular tree[4][20]. $J_{d}$ in our setting is the
critical point for uniqueness of Gibbs measures of infinite regular tree with
degree of each vertex $d$, implying our condition is also necessary on trees.
The condition is the same as that of Mossel and Sly[15] when ferromagnetic
Ising model is the only focus. This makes part of their work in our framework.
Our proof yields a different approach, and also avoids the argument between
weak spatial mixing and strong spatial mixing employed in [21]. In fact our
proof is based an inequality similar to the one in [13] and carefully exploits
monotonicity of the recursive formula. The recursive formula on trees is well
known. Recently Pemantle and Peres [18] use it to present the exact capacity
criteria that govern behavior at critical point of ferromagnetic Ising model
on trees under various boundary conditions. Our second condition is for
‘external field’. We prove for any ‘inverse temperature’ on a graph which are
sparse on average Gibbs distribution exhibits strong spatial mixing when the
‘external field’ is uniformly larger than $B(d,\alpha_{\max},\gamma)$ or
smaller than $-B(d,-\alpha_{\min},\gamma)$, where $d$ is ‘maximum average
degree’ and $\alpha_{\min}$, $\alpha_{\max}$, $\gamma$ are parameters of the
system. To our best knowledge, this condition on ‘external field’ is first
considered for strong spatial mixing. The technique employed in the proof is
Lipchitz approach which has been used in [1][2][5]. The novelty here is that
we propose a ‘path’ characterization of this method, allowing us to give the
‘external field’ condition in term of ‘maximum average degree’ rather than
maximum degree. Some notations of the ‘sparse on average’ graphs have appeared
in [15]. These are graphs where the sum degrees along each self-avoiding
path(a path with distinct vertices) with length $O($log$n)$ is $O($log$n)$.
As an application of our results, we present a fully polynomial time
approximation schemes(FPTAS)(which approximates the real value within a factor
of $\epsilon$ in polynomial time of the input and $\epsilon^{-1}$) for
partition functions of two-state spin systems without hard constrains under
our assumptions on the graph $G=(V,E)$, where, for each vertex $v\in V$ of
$G$, the number of total vertices of its associated self-avoiding tree
$T_{saw(v)}$ with hight $O($log$n)$ is $O(n^{O(1)})$. This includes bounded
degree graph and especially $Z^{d}$ lattice more concerned in statistical
physics. Jerrum and Sinclair [7] provided an FPRAS for partition function of
ferromangetic Ising model for any graph with any positive ‘inverse
temperature’ and identical external field for all the vertices. Their results
do not include the case where different vertices have different external
field, and are not applied to antiferromagnetic Ising model where the ‘inverse
temperature’ is negative either.
The remainder of the paper has the following structure. In Section II, we
present some preliminary definitions and main results. We go on to prove the
theorems in Section III. Section IV is devoted to propose an FPTAS for the
partition functions under our conditions. Further work and conclusion are
given in Section IV.
## II Preliminaries and Main Results
### II-A Two-State Spin Systems
In the two-state spin systems on a finite graph $G=(V,E)$ with vertices
$V=\\{1,2,\cdots,n\\}$ and edge set $E$, a configuration consists of an
assignment $\sigma=(\sigma_{i})$ of $\Omega=\\{\pm 1\\}$ values, or “spins”,
to each vertex(or“sites”) of $V$. Each vertex $i\in V$ is associated with a
random variable $X_{i}$ with range ${\pm 1}$. We often refer to the spin
values $\pm 1$ as $(+)$ and $(-)$. The probability of finding the system in
configuration $\sigma\in\Omega^{n}$ is given by the joint distribution of $n$
dimensional random vector $X=\\{X_{1},X_{2},\cdots,X_{n}\\}$(also known as the
Gibbs distribution with the nearest neighbor interaction)
$P_{G}(X=\sigma)=\frac{1}{Z(G)}\exp(\sum\limits_{(i,j)\in
E}\beta_{ij}(\sigma_{i},\sigma_{j})+\sum\limits_{i\in V}h_{i}(\sigma_{i})).$
Here $Z(G)$ is called partition function of the system and a normalized factor
such that $\sum_{\sigma\in\Omega^{n}}P_{G}(X=\sigma)=1$, and $h_{i}$ and
$\beta_{ij}$ are defined as a function from $\Omega$ and $\Omega^{2}$ to
$R\cup\\{\pm\infty\\}$ respectively. We use notation
$\beta_{ij}(a,b)=\beta_{ji}(b,a)$. We say the system has hard constraints if
there exit an edge $(i,j)\in E$ or a vertex $k$, and an assignment
$\sigma_{i}$, $\sigma_{j}$ or $\sigma_{k}$ such that
$\beta_{ij}(\sigma_{i};\sigma_{j})=\infty$ or $h_{k}(\sigma_{k})=\infty$ (e.g.
hard-core model is one of the systems with hard constrains where
$\beta_{ij}(+,+)=-\infty$, $\beta_{ij}(+,-)=\beta_{ij}(-,+)=\beta_{ij}(-,-)=0$
and $h_{i}(-)=0$). In this paper we focus to the systems without hard
constrains. We call the function $\beta_{ij}$ ‘interaction energy’ and $h_{i}$
‘applied field’ . If
$\beta_{ij}(\sigma_{i},\sigma_{j})=J_{ij}\sigma_{i}\sigma_{j}$ and
$h_{i}=B_{i}\sigma_{i}$ for all the edge $(i,j)\in E$ and vertex $i\in V$,
where $J_{ij}$ and $B_{i}$ are constant numbers varying with edges or
vertices, the system is called Ising model. Further, if $J_{ij}$ is uniformly
(negative)positive for all $(i,j)\in E$, the system is called
(anti)ferromagnetic Ising model. $J_{ij}$ and $B_{i}$ are called $inverse$
$temperature$ and $external$ $field$ respectively. To match the notation of
Ising model, set
$J_{ij}=\frac{\beta_{ij}(+,+)+\beta_{ij}(-,-)-\beta_{ij}(-,+)-\beta_{ij}(+,-)}{4}$
and $B_{i}=\frac{h_{i}(+)-h_{i}(-)}{2}$ for all edges and vertices , in this
paper we call $J_{ij}$ and $B_{i}$ are ‘inverse temperature’ and ‘external
field’ of general two-state spin systems without hard constrains (denoted by
TSSHC for abbreviation)respectively. For any $\Lambda\subseteq V$,
$\sigma_{\Lambda}$ denotes the set $\\{\sigma_{i},i\in\Lambda\\}$. With a
little abuse of notations, $\sigma_{\Lambda}$ also denotes the configuration
that $i$ is fixed $\sigma_{i}$, $\forall i\in\Lambda$. Let $Z(G,\Phi)$ denote
the partition function under the condition $\Phi$, e.g. $Z(G,X_{1}=+)$
represent the partition function under the condition the vertex $1$ is fixed
$+$.
### II-B Definitions and Notations
Definition 2.1 (Self-Avoiding Tree) Consider a graph $G=(V,E)$ and a vertex
$v\in V$ in $G$. Given any order of all the vertices in $G$. There is
associated partial order on $E$ of the order on $V$ defined as $(i,j)>(k,l)$
iff $(i,j)$, $(k,l)$ share a common vertex and $i+j>k+l$. The self-avoiding
tree $T_{saw(v)}(G)$(for simplicity denoted by $T_{saw(v)}$) corresponding to
the vertex $v$ is the tree of self-avoiding walks originating at $v$ except
that the vertices closing a cycle are also included in the tree and are fixed
to be either $+$ or $-$. Specifically, the vertex of the $T_{saw(v)}$ closing
a cycle is fixed $+$ if the edge ending the cycle is larger than the edge
starting the cycle and $-$ otherwise. Given any configuration
$\sigma_{\Lambda}$ of $G$, $\Lambda\subset V$, the self-avoiding tree is
constructed the same as the above procedure except that the vertex which is a
copy of the vertex $i$ in $\Lambda$ is fixed to the same spin $\sigma_{i}$ as
$i$ and the subtree below it is not constructed(See Figure 1). Hence, for any
configuration $\sigma_{\Lambda}$ of $G$, $\Lambda\subset V$, we also use
$\sigma_{\Lambda}$ to denote the configuration of $T_{saw(v)}$ obtained by
imposing the condition corresponding to $\sigma_{\Lambda}$ as above. For any
$(i,j)\in E$ and $i\in V$ of $G$, the ‘interaction energy’ function and
‘applied field’ function on all their copies of the induced system on
$T_{saw(v)}$ by $G$ are the same as $\beta_{ij}$ and $h_{i}$ respectively.
We now provide the remarkable property of the self-avoiding tree, one of two
main results of [21], which is one of the essential techniques of our proofs.
Figure 1: The graph with one vertex assigned + (Right)and its corresponding
self-avoiding tree $T_{saw(1)}$(Left)
Proposition 2.1 _For two-state spin systems on $G=(V,E)$, for any
configuration $\sigma_{\Lambda}$, $\Lambda\subset V$ and any vertex $v\in V$,
then_
$P_{G}(X_{v}=+|\sigma_{\Lambda})=P_{T_{saw(v)}}(X_{v}=+|\sigma_{\Lambda}).$
In order to generalize our result to more general families of graphs , which
are sparse on average, we need some definitions and notation of these graphs.
Definition 2.2 Let $|A|$ denote the cardinality of the set $A$. The length of
a path is the number of edges it contains. The distance of two vertices in a
graph is the length of shortest path connecting these two vertices. A path
$v_{1},v_{2},\cdots$ is called a self-avoiding path if for all $i\neq j$ ,
$v_{i}\neq v_{j}$. In a graph $G=(V,E)$, let $d(u,v)$ denote the distance
between $u$ and $v$, $u$,$v\in V$. The distance between a vertex $v\in V$ and
a subset $\Lambda\subset V$ is defined by
$d(v,\Lambda)=$min$\\{d(v,u):u\in\Lambda\\}$. The set of vertices within
distance $l$ of $v$ is denoted by $V(G,v,l)=\\{u:d(v,u)\leq l\\}$. Similarly,
the set of vertices with distance $l$ of $v$ is denoted by
$S(G,v,l)=\\{u:d(v,u)=l\\}$. We call a vertex at the height $t$ of a rooted
tree if the distance between it and the root is $t$. Let $\delta_{v}$ denote
the degree of $v$ in $G$. The $maximal$ $path$ $density$ $m$ is defined by
$m(G,v,l)=\max\limits_{\Gamma}\sum\limits_{u\in\Gamma}\delta_{u}$, where the
maximum is taken over all self-avoiding paths $\Gamma$ starting at $v$ with
length at most $l$. The $maximum$ $average$ $path$ $degree$ $\delta(G,v,l)$ is
defined by $\delta(G,v,l)=(m(G,v,l)-\delta_{v})/l,l\geq 1$. The $maximum$
$average$ $degree$ of $G$ is defined by $\Delta(G,l)=\max_{v\in
V}\delta(G,v,l)$. Roughly speaking, in this paper, a family of graphs
$\mathcal{G}$ is sparse on average if there exits a constant number $a$ and
$d$ such that $\Delta(G,a\log n)\leq d$ for any $G\in\mathcal{G}$.
Some properties of the above definitions are useful in our proof, we present
them. Most of proofs are simply obtained by induction and can be found in
[15].
Proposition 2.2 _Let $j$, $l$ denote positive natural numbers, then_
$m(G,v,jl)\leq j\max\limits_{u\in G}\\{m(G,u,l)-\delta_{u}\\}+\delta_{v}.$
Proposition 2.3 _Let $l$ be natural numbers, then_
$|S(T_{saw(v)},v,l+1)|\leq\delta_{v}(\delta(G,v,l)-1)^{l}.$
Proposition 2.4 _Let $j$, $l$ be natural numbers, then_
$|V(T_{saw(v)},v,jl)|\leq(\max\limits_{u\in V}|V(T_{saw(u)},u,l)|)^{j}.$
Definition 2.3 ((Exponential) Strong Spatial Mixing) Let $G=(V,E)$ be a graph
with $n$ vertices. The Gibbs distribution of two-state spin systems on $G$
exhibits strong spatial mixing iff for any vertex $v\in V$, subset
$\Lambda\subset V$, any two configurations $\sigma_{\Lambda}$ and
$\eta_{\Lambda}$ on $\Lambda$, denote
$\Theta=\\{v\in\Lambda:\sigma_{v}\neq\eta_{v}\\}$ and $t=d(v,\Theta)$,
$|P_{G}(X_{v}=+|\sigma_{\Lambda})-P_{G}(X_{v}=+|\eta_{\Lambda})|\leq f(t),$
where $f(t)$ goes to zero if $t$ goes to infinity and is called decay
function.
For the purpose of our settings, we present a weak form of exponential strong
spatial mixing. We say the distribution exhibits exponential strong spatial
mixing if there exits positive numbers $a$, $b$, $c$ independent of $n$ such
that $f(t)=b\exp(-ct)$ when $t=ka$log$n$, $k=1,2,\cdots$.
Remark: In the above definition of (exponential) strong spatial mixing,
$P_{G}(X_{v}=+|\sigma_{\Lambda})$ and $P_{G}(X_{v}=+|\eta_{\Lambda})$ can be
replaced by $\log(P_{G}(X_{v}=+|\sigma_{\Lambda}))$ and
$\log(P_{G}(X_{v}=+|\eta_{\Lambda}))$ respectively if $d(v,\Lambda)$ is large
than a constant number, due to the inequality $2x\leq$log$(1+x)\leq x$ when
$|x|\leq.5$, and we call it the logarithmic form exponential strong spatial
mixing. )
Definition 2.4 (FPTAS) An approximation algorithm is called a fully polynomial
time approximation scheme(FPTAS) iff for any $\epsilon>0$, it takes a
polynomial time of input and $\epsilon^{-1}$ to output a value $\bar{M}$
satisfying
$1-\epsilon\leq\frac{\bar{M}}{M}\leq 1+\epsilon,$
where $M$ is the real value.
Remark: In the above definition $1-\epsilon$ and $1+\epsilon$ can be replaced
by $e^{-\epsilon}$ and $e^{\epsilon}$.
### II-C Main Results
For simplicity , We use the following notations. Consider a two-state spin
systems with hard constrains(TSSHC) on a graph $G=(V,E)$ with $n$ vertices
$V=\\{1,2,\cdots,n\\}$ and edge set $E$. Let $J=\max_{(i,j)\in E}|J_{ij}|$,
$B_{\min}=\min_{i\in V}B_{i}$, $B_{\max}=\max_{i\in V}B_{i}$,
$\alpha_{\max}=\max\limits_{(i,j)\in
E}\\{\beta_{ij}(-,-)-\beta_{ij}(+,-),\beta_{ij}(-,+)-\beta_{ij}(+,+)\\}$,
$\alpha_{\min}=\min\limits_{(i,j)\in
E}\\{\beta_{ij}(-,-)-\beta_{ij}(+,-),\beta_{ij}(-,+)-\beta_{ij}(+,+)\\}$,
$\gamma_{ij}=\max_{(i,j)\in
E}\\{\frac{|b_{ij}c_{ij}-a_{ij}d_{ij}|}{a_{ij}c_{ij}},\frac{|b_{ij}c_{ij}-a_{ij}d_{ij}|}{b_{ij}d_{ij}}\\}$,
$\gamma=\max_{(i,j)\in E}\\{\gamma_{ij}\\}$, where
$J_{ij}=\frac{\beta_{ij}(+,+)+\beta_{ij}(-,-)-\beta_{ij}(-,+)-\beta_{ij}(+,-)}{4}$,
$B_{i}=\frac{h_{i}(+)-h_{i}(-)}{2}$ are ‘inverse temperature’ and ‘external
field’ respectively, and $a_{ij}=\exp(\beta_{ij}(+,+))$,
$b_{ij}=\exp(\beta_{ij}(+,-))$, $c_{ij}=\exp(\beta_{ij}(-,+))$,
$d_{ij}=\exp(\beta_{ij}(-,-))$.
Theorem 2.1 _Let $G=(V,E)$ be a graph with vertices $V=\\{1,2,\cdots,n\\}$,
edges set $E$ and TSSHC on it. If there exit two positive numbers $a>0$ and
$d>0$ such that $\Delta(G,a\log n)\leq d$, and when_
$(d-1)\tanh{J}<1$
_or equivalently $J<J_{d}=\frac{1}{2}\log(\frac{d}{d-2})$, then the Gibbs
distribution of TSSHC exhibits logarithmic form exponential strong spatial
mixing for arbitrary ‘external field’, specifically, for any $i\in V$, any two
configurations $\sigma_{\Lambda}$ and $\eta_{\Lambda}$ on $\Lambda$, denote
$\Theta=\\{j\in\Lambda:\sigma_{j}\neq\eta_{j}\\}$ and $t=d(i,\Theta)=ka\log
n+1$, $k=1,2,\cdots,$_
$|\log(P_{G}(X_{i}=+|\sigma_{\Lambda}))-\log(P_{G}(X_{i}=+|\eta_{\Lambda}))|\leq
f(t),$
_where $f(t)=4J\delta_{i}((d-1)\tanh J)^{t-1}$._
Remark: If the graph is bounded with the maximum degree $D$, then $d$ can be
replaced by $D$ while for any $a>0$, and $J_{ij}$ is the ‘inverse temperature’
in (anti)ferromagnetic Ising model, then theorem 2.1 still holds and $J_{D}$
is the critical point for uniqueness of Gibbs measures on a infinite tree with
maximum degree $D$[13]. Note the decay function is slight different from the
definition since $\delta_{i}$ may be $O(\log n)$, however, in this case we can
choose $k$ large enough independent of $n$ such that $f(t)=e^{-bt}$ when $n$
is large, where $b$ is a positive number independent of $n$, then replace $a$
by $ka$ as required. In fact in the application of the algorithm, this is not
important.
Theorem 2.2 _Let $G=(V,E)$ be a graph with vertices $V=\\{1,2,\cdots,n\\}$,
edges set $E$ and TSSHC on it. If there exit two positive numbers $a>0$ and
$d>0$ such that $\Delta(G,a\log n)\leq d$, and $(d-1)\tanh{J}\geq 1$, and
when_
$B_{\min}>B(d,\alpha_{\max},\gamma)\ \ \ \ \ \ or\ \ \ \ \
B_{\max}<-B(d,-\alpha_{\min},\gamma)$
_where
$B(d,\alpha,\gamma)=\frac{(d-1)\alpha}{2}+\log(\frac{\sqrt{\gamma(d-1)}+\sqrt{\gamma(d-1)-4}}{2})$,
the Gibbs distribution of TSSHC exhibits exponential strong spatial mixing,
specifically, for any $i\in V$, any two configurations $\sigma_{\Lambda}$ and
$\eta_{\Lambda}$ on $\Lambda$, denote
$\Theta=\\{j\in\Lambda:\sigma_{j}\neq\eta_{j}\\}$ and $t=d(i,\Theta)=ka\log
n+1$, $k=1,2,\cdots,$_
$|P_{G}(X_{i}=+|\sigma_{\Lambda})-P_{G}(X_{i}=+|\eta_{\Lambda})|\leq f(t),$
_where
$f(t)=\frac{\delta_{i}\gamma}{4}(\frac{(d-1)\gamma\exp(2B_{\min}-(d-1)\alpha_{\max})}{(1+\exp(2B_{\min}-(d-1)\alpha_{\max}))^{2}})^{t-1}$
or
$f(t)=\frac{\delta_{i}\gamma}{4}(\frac{(d-1)\gamma\exp(2B_{\max}-(d-1)\alpha_{\min})}{(1+\exp(2B_{\max}-(d-1)\alpha_{\min}))^{2}})^{t-1}$
respectively._
Remark: It’s easy to check $\gamma\geq 4\tanh J$, hence in theorem 2.2, if
$(d-1)\tanh{J}\geq 1$, then $\gamma(d-1)-4\geq 0$. As a corollary of Theorem
2.2, from its proof in section III, we know if the graph is bounded degree
with maximum degree is $d$, the condition for ‘external field’ can be relaxed
to $B_{i}>B(d,\alpha_{\max},\gamma)$ or $B_{i}<-B(d,-\alpha_{\min},\gamma)$
for any $i\in V$, which does not require that ‘external field’ is uniformly
large or uniformly small.
Theorem 2.3 _Let $G=(V,E)$ be a graph with $n$ vertices
$V=\\{1,2,\cdots,n\\}$, edges set $E$ and TSSHC on it. If there exit two
positive numbers $a>0$ and $d>0$ such that for any $i\in V$_
$V(T_{saw(i)},i,a\log{n})\leq(d-1)^{a\log{n}},$
_where $|V(T_{saw(i)},i,l)|=\\{j\in T_{saw(i)}:d(i,j)\leq l\\}$, then when
$J<J_{d}$ or $J\geq J_{d}$, $B_{\min}>B(d,\alpha_{\max},\gamma)$ or
$B_{\max}<-B(d,-\alpha_{\min},\gamma)$, there exits an FPTAS for partition
function of TSSHC on $G$._
## III Proofs
We now proceed to prove the theorems. One of the technical lemmas for the
theorem 2.1 is an inequality similar to [13]. We present it now.
Lemma 3.1 _Let $a$, $b$, $c$, $d$, $x$, $y$ be positive numbers and
$g(x)=\frac{ax+b}{cx+d}$ and
$t=|\frac{\sqrt{ad}-\sqrt{bc}}{\sqrt{ad}+\sqrt{bc}}|$, then_
$\max(\frac{g(x)}{g(y)},\frac{g(y)}{g(x)})\leq(\max(\frac{x}{y},\frac{y}{x}))^{t}.$
Proof: Case 1. $ad\geq bc$. Since
$g(x)=\frac{ax+b}{cx+d}=\frac{a}{c}-\frac{ad-bc}{c(cx+d)}$ is an increasing
function, w.l.o.g. suppose $x\geq y$ and let $x=zy$, where $z\geq 1$, then
$\begin{split}\log(\frac{g(x)}{g(y)})&=\int^{z}_{1}\frac{d(\log(\frac{g(\alpha
y)}{g(y)}))}{d\alpha}d\alpha\\\ &=\int^{z}_{1}(\frac{ay}{a\alpha
y+b}-\frac{cy}{c\alpha y+d})d\alpha\\\ &=\int^{z}_{1}\frac{(ad-bc)y}{(a\alpha
y+b)(c\alpha y+d)}d\alpha\\\ &=\int^{z}_{1}\frac{(ad-bc)y}{(\sqrt{ac}\alpha
y-\sqrt{bd})^{2}+(\sqrt{bc}+\sqrt{ad})^{2}\alpha y}d\alpha\\\
&\leq\int^{z}_{1}\frac{(ad-bc)y}{(\sqrt{bc}+\sqrt{ad})^{2}\alpha
y}d\alpha=\frac{\sqrt{ad}-\sqrt{bc}}{\sqrt{ad}+\sqrt{bc}}\log z.\end{split}$
Hence,
$\max(\frac{g(x)}{g(y)},\frac{g(y)}{g(x)})=\frac{g(x)}{g(y)}\leq(\frac{x}{y})^{t}=(\max(\frac{x}{y},\frac{y}{x}))^{t},$
where $t=\frac{\sqrt{ad}-\sqrt{bc}}{\sqrt{ad}+\sqrt{bc}}$.
Case 2. $ad\leq bc$. Similar to the first case, $g(x)$ is a decreasing
function, let $h(x)=1/g(x)$, then $h(x)$ is an increasing function, w.l.o.g.
suppose $x\geq y$, repeat the process of Case 1 for $\frac{h(x)}{h(y)}$, then
$\frac{h(x)}{h(y)}\leq(\frac{x}{y})^{\frac{\sqrt{bc}-\sqrt{ad}}{\sqrt{ad}+\sqrt{bc}}}.$
Hence,
$\max(\frac{g(x)}{g(y)},\frac{g(y)}{g(x)})=\frac{g(y)}{g(x)}=\frac{h(x)}{h(y)}\leq(\frac{x}{y})^{t}=(\max(\frac{x}{y},\frac{y}{x}))^{t},$
where $t=\frac{\sqrt{bc}-\sqrt{ad}}{\sqrt{ad}+\sqrt{bc}}$. $\Box$
Lemma 3.2 _Let $T=(V,E)$ be a tree rooted at $0$ with vertices
$V=\\{0,1,2,\cdots,n\\}$, edge set $E$ and TSSHC on it. Suppose some vertices
are fixed. Let $T_{k}$ and $T_{l}$ be two subtrees of $T$ including vertex $k$
and $l$ respectively by removing an edge $(k,l)$ where $d(k,0)<d(l,0)$. The
fixed vertices remain fixed on $T_{k}$ and $T_{l}$. Then the probability of
$X_{0}=+$ on $T$ equals the probability of $X_{0}=+$ on the subtree $T_{k}$
except changing the ‘external field’ $h_{k}$ to certain value
$h^{{}^{\prime}}_{k}$._
Proof: Let $\Omega_{T_{l}}$ denote the configuration spaces, $E_{l}$ and
$V_{l}$ the edge set and vertices on $T_{l}$. Setting
$\begin{split}h^{{}^{\prime}}_{k}(\sigma_{k})&=h_{k}(\sigma_{k})+\\\
&\log(\sum\limits_{\sigma\in\Omega_{T_{l}}}e^{\beta_{kl}(\sigma_{k},\sigma_{l})+\sum\limits_{(i,j)\in
E_{l}}\beta_{ij}(\sigma_{i},\sigma_{j})+\sum\limits_{i\in
V_{l}}h_{i}(\sigma_{i})})\end{split}$
completes the proof. $\Box$
With Lemma 3.1 and Lemma 3.2, we now proceed to prove (exponential) strong
spatial mixing property on trees.
Theorem 3.1 _Let $T$ be a tree rooted at $0$ with vertices
$V=\\{0,1,2,\cdots,n\\}$, edge set $E$ and TSSHC on it. Let $\Lambda\subset V$
, $\zeta_{\Lambda}$ and $\eta_{\Lambda}$ be any two configurations on
$\Lambda$. Let $\Theta=\\{i:\zeta_{i}\neq\eta_{i},i\in\Lambda\\}$,
$t=d(0,\Theta)$ and $s=|S(T,0,t)|=|\\{i:d(0,i)=t,i\in T\\}|$. Then_
$\max(\frac{P_{T}(X_{0}=+|\zeta_{\Lambda})}{P_{T}(X_{0}=+|\eta_{\Lambda})},\frac{P_{T}(X_{0}=+|\eta_{\Lambda})}{P_{T}(X_{0}=+|\zeta_{\Lambda})})\leq\exp(4Js(\tanh{J})^{t-1})$
Proof: For any $i\in V$, let $T_{i}$ denote the subtree with $i$ as its root
and $Z(i)$ be the TSSHC induced on $T_{i}$ by $T$. Noting $T_{0}$ is $T$. To
prove the theorem, it’s convenient to deal with the ratio
$\frac{P_{T}(X_{0}=+|\zeta_{\Lambda})}{P_{T}(X_{0}=-|\zeta_{\Lambda})}$ rather
than $P_{T}(X_{0}=+|\zeta_{\Lambda})$ itself. Denote
$R^{\zeta_{\Lambda}}_{i}\equiv\frac{P_{T_{i}}(X_{i}=+|\zeta_{\Lambda_{i}})}{P_{T_{i}}(X_{i}=-|\zeta_{\Lambda_{i}})}$
where $\zeta_{\Lambda_{i}}$ is the condition by imposing the configuration
$\zeta_{\Lambda}$ on $T_{i}$, and note a simple relation if
$x_{1},x_{2}\in(0,1)$, then $\frac{x_{1}}{x_{2}}\geq 1$ iff
$\frac{x_{1}}{1-x_{1}}\geq\frac{x_{2}}{1-x_{2}}$, further
$\max\\{\frac{x_{1}}{x_{2}},\frac{x_{2}}{x_{1}}\\}\leq\max\\{\frac{x_{1}/(1-x_{1})}{x_{2}/(1-x_{2})},\frac{x_{2}/(1-x_{2})}{x_{1}/(1-x_{1})}\\}$.
Hence replace $x_{1}$ and $x_{2}$ by $P_{T}(X_{0}=+|\zeta_{\Lambda})$ and
$P_{T}(X_{0}=+|\eta_{\Lambda})$, we need only to show
$\max(\frac{R^{\zeta_{\Lambda}}_{0}}{R^{\eta_{\Lambda}}_{0}},\frac{R^{\eta_{\Lambda}}_{0}}{R^{\zeta_{\Lambda}}_{0}})\leq\exp(4Js(\tanh{J})^{t-1}).$
(1)
Theorem 3.1 follows by
$\max(\frac{P_{T}(X_{0}=+|\zeta_{\Lambda})}{P_{T}(X_{0}=+|\eta_{\Lambda})},\frac{P_{T}(X_{0}=+|\eta_{\Lambda})}{P_{T}(X_{0}=+|\zeta_{\Lambda})})\leq\max(\frac{R^{\zeta_{\Lambda}}_{0}}{R^{\eta_{\Lambda}}_{0}},\frac{R^{\eta_{\Lambda}}_{0}}{R^{\zeta_{\Lambda}}_{0}})$.
We go on to prove (1) by induction on $t$. Before we doing this, some trivial
cases need to be clarified. We are interested in the case $t\geq 1$ and $0$ is
unfixed. Let $\Gamma_{kl}$ denote the unique self-avoiding path from $k$ to
$l$ on $T$. If $i$ is a leave on $T$ and $d(0,i)<t$, where $t=d(0,\Theta)$.
Define $U=\\{j\in V:j\in\Gamma_{0i},\exists k\in
S(T,0,t),s.t.j\in\Gamma_{0k}\\}$. Note $U\neq\emptyset$ since $0\in U$. Let
$j_{i}\in U$ such that $d(i,j_{i})=d(i,U)$. By lemma 3.2, we can remove the
subtree bellow $j_{i}$ and change external field $h_{j_{i}}$ at $j_{i}$ to
$h^{{}^{\prime}}_{j_{i}}$ without changing the probability of $X_{0}=+$. More
importantly, this process removes at least one leave at the hight $<t$, and
does not remove any vertex at the hight $\geq t$. Thus, w.l.o.g. suppose $T$
is a tree rooted at $0$ where any leave on it at the height $\geq t$. Let
$0_{1},0_{2},\cdots,0_{q}$ be the neighbors connected to $0$. A trivial
calculation then gives that
$\begin{split}&R^{\zeta_{\Lambda}}_{0}=\frac{Z(T_{0},X_{0}=+,\zeta_{\Lambda})}{Z(T_{0},X_{0}=-,\zeta_{\Lambda})}\\\
&=\frac{e^{h_{0}(+)}\sum\limits_{\sigma\in\Omega_{0}}e^{\sum\limits_{i=1}^{q}(\beta_{00_{i}}(+,\sigma_{0_{i}})+\sum\limits_{(k,l)\in
T_{i}}\beta_{kl}(\sigma_{k},\sigma_{l})+\sum\limits_{k\in
T_{i}}h_{k}(\sigma_{k}))}}{e^{h_{0}(-)}\sum\limits_{\sigma\in\Omega_{0}}e^{\sum\limits_{i=1}^{q}(\beta_{00_{i}}(-,\sigma_{0_{i}})+\sum\limits_{(k,l)\in
T_{i}}\beta_{kl}(\sigma_{k},\sigma_{l})+\sum\limits_{k\in
T_{i}}h_{k}(\sigma_{k}))}}\\\
&=e^{2B_{0}}\prod\limits^{q}_{i=1}\frac{\sum\limits_{\sigma\in\Omega_{T_{i}}}e^{\beta_{00_{i}}(+,\sigma_{0_{i}})+\sum\limits_{(k,l)\in
T_{i}}\beta_{kl}(\sigma_{k},\sigma_{l})+\sum\limits_{k\in
T_{i}}h_{k}(\sigma_{k})}}{\sum\limits_{\sigma\in\Omega_{T_{i}}}e^{\beta_{00_{i}}(-,\sigma_{0_{i}})+\sum\limits_{(k,l)\in
T_{i}}\beta_{kl}(\sigma_{k},\sigma_{l})+\sum\limits_{k\in
T_{i}}h_{k}(\sigma_{k})}}\\\ \end{split}$
$\begin{split}&=e^{2B_{0}}\prod\limits^{q}_{i=1}\frac{a_{i}Z(T_{0_{i}},X_{i}=+,\zeta_{\Lambda_{i}})+b_{i}Z(T_{0_{i}},X_{i}=-,\zeta_{\Lambda_{i}})}{c_{i}Z(T_{0_{i}},X_{i}=+,\zeta_{\Lambda_{i}})+d_{i}Z(T_{0_{i}},X_{i}=-,\zeta_{\Lambda_{i}})}\\\
&=e^{2B_{0}}\prod\limits^{q}_{i=1}\frac{a_{i}R^{\zeta_{\Lambda}}_{0_{i}}+b_{i}}{c_{i}R^{\zeta_{\Lambda}}_{0_{i}}+d_{i}}.\end{split}$
(2)
where $B_{0}=\frac{h_{0}(+)-h_{0}(-)}{2}$, $a_{i}=e^{\beta_{00_{i}}(+,+)}$,
$b_{i}=e^{\beta_{00_{i}}(+,-)}$, $c_{i}=e^{\beta_{00_{i}}(-,+)}$,
$d_{i}=e^{\beta_{00_{i}}(-,-)}$ . Now we check the base case $t=1$ where
$R^{\zeta_{\Lambda}}_{0_{i}},R^{\eta_{\Lambda}}_{0_{i}}\in[0,+\infty]$, by the
monotonicity of
$\frac{a_{i}R^{\zeta_{\Lambda}}_{0_{i}}+b_{i}}{c_{i}R^{\zeta_{\Lambda}}_{0_{i}}+d_{i}}$
and
$\frac{a_{i}R^{\eta_{\Lambda}}_{0_{i}}+b_{i}}{c_{i}R^{\eta_{\Lambda}}_{0_{i}}+d_{i}}$,
$\begin{split}\max(\frac{R^{\zeta_{\Lambda}}_{0}}{R^{\eta_{\Lambda}}_{0}},\frac{R^{\eta_{\Lambda}}_{0}}{R^{\zeta_{\Lambda}}_{0}})&\leq\prod\limits^{q}_{i=1}\max(\frac{a_{i}d_{i}}{b_{i}c_{i}},\frac{b_{i}c_{i}}{a_{i}d_{i}})\leq
e^{4qJ}\end{split}$
Hence $t=1$, (1) holds. Assume by induction that (1) holds for $t-1$, and we
will show it holds for $t$. Let $s_{i}=|S(T_{0_{i}},0_{i},t-1)|$,
$i=1,2,\cdots,q$, still using above recursive procedure, then
$\begin{split}\max(\frac{R^{\zeta_{\Lambda}}_{0}}{R^{\eta_{\Lambda}}_{0}},\frac{R^{\eta_{\Lambda}}_{0}}{R^{\zeta_{\Lambda}}_{0}})&\leq\prod\limits^{q}_{i=1}\max(\frac{\frac{a_{i}R^{\zeta_{\Lambda}}_{0_{i}}+b_{i}}{c_{i}R^{\zeta_{\Lambda}}_{0_{i}}+d_{i}}}{\frac{a_{i}R^{\eta_{\Lambda}}_{0_{i}}+b_{i}}{c_{i}R^{\eta_{\Lambda}}_{0_{i}}+d_{i}}},\frac{\frac{a_{i}R^{\eta_{\Lambda}}_{0_{i}}+b_{i}}{c_{i}R^{\eta_{\Lambda}}_{0_{i}}+d_{i}}}{\frac{a_{i}R^{\zeta_{\Lambda}}_{0_{i}}+b_{i}}{c_{i}R^{\zeta_{\Lambda}}_{0_{i}}+d_{i}}})\\\
&\leq\prod\limits^{q}_{i=1}\max(\frac{R^{\zeta_{\Lambda}}_{0_{i}}}{R^{\eta_{\Lambda}}_{0_{i}}},\frac{R^{\eta_{\Lambda}}_{0_{i}}}{R^{\zeta_{\Lambda}}_{0_{i}}})^{|\frac{\sqrt{a_{i}d_{i}}-\sqrt{b_{i}c_{i}}}{\sqrt{a_{i}d_{i}}+\sqrt{b_{i}c_{i}}}|}\\\
&\leq\prod\limits^{q}_{i=1}\max(\frac{R^{\zeta_{\Lambda}}_{0_{i}}}{R^{\eta_{\Lambda}}_{0_{i}}},\frac{R^{\eta_{\Lambda}}_{0_{i}}}{R^{\zeta_{\Lambda}}_{0_{i}}})^{\tanh
J}\end{split}$
where the second inequality comes from the Lemma 3.1. According to the
hypothesis of induction
$\max(\frac{R^{\zeta_{\Lambda}}_{0_{i}}}{R^{\eta_{\Lambda}}_{0_{i}}},\frac{R^{\eta_{\Lambda}}_{0_{i}}}{R^{\zeta_{\Lambda}}_{0_{i}}})\leq\exp(4Js_{i}(\tanh{J})^{t-2})$,
it’s sufficient to show
$\begin{split}\max(\frac{R^{\zeta_{\Lambda}}_{0}}{R^{\eta_{\Lambda}}_{0}},\frac{R^{\eta_{\Lambda}}_{0}}{R^{\zeta_{\Lambda}}_{0}})&\leq\prod\limits^{q}_{i=1}\exp(4Js_{i}(\tanh{J})^{t-1})\\\
&=\exp(4Js(\tanh{J})^{t-1})\end{split}$
where the last equation follows by $\sum_{i=1}^{q}s_{i}=s$. This completes the
proof of Theorem 3.1. $\Box$
With Theorem 3.1 and self-avoiding tree, it’s enough to prove Theorem 2.1.
Proof of Theorem 2.1: Due to Proposition 2.1, the only thing left is to verify
$|S(T_{saw(i)},i,t)|=\delta_{i}(d-1)^{t-1}$ when $t=d(i,\Theta)=ka\log n+1$,
$k=1,2,\cdots$, under the condition that there exit two positive numbers $a>0$
and $d>0$ such that $\Delta(G,a\log n)\leq d$. By Proposition 2.2, we know
$m(G,i,ka\log n)\leq ka\log n\Delta(G,a\log n)+\delta_{i}\leq ka\log
nd+\delta_{i}$, hence, $\delta(G,i,ka\log n)\leq d$ follows from the
definition $\delta(G,i,l)=(m(G,i,l)-\delta_{i})/l$. By Proposition 2.3, it’s
sufficient to show $|S(T_{saw(i)},i,ka\log
n+1)|\leq\delta_{i}(\delta(G,i,ka\log n)-1)^{ka\log
n}\leq\delta_{i}(d-1)^{ka\log n}=\delta_{i}(d-1)^{t-1}$. This is exactly what
we need. $\Box$
Next we will proceed to prove Theorem 2.2, we still use the recursive formula
but with another form. The technique used is a well known method, Lipchitz
approach. A ‘path’ version of it will be presented, which allow us to bound
the ‘external field’ with maximum average degree. Before presenting it, we
need some notation for simplicity. Let $T=(V,E)$ be a tree rooted at $0$ with
vertices ${0,1,2,\cdots,n}$, edge set $E$ and $TSSHC$ on it. For each edge
$(i,j)\in E$, recall the notation in main results,
$a_{i,j}=e^{\beta_{ij}(+,+)}$, $b_{i,j}=e^{\beta_{ij}(+,-)}$,
$c_{i,j}=e^{\beta_{ij}(-,+)}$, and $d_{i,j}=e^{\beta_{ij}(-,-)}$. Let
$M_{ij}=c_{ij}-d_{ij}$, $N_{ij}=a_{i,j}-b_{i,j}$ Define
$f_{ij}(x)=\frac{M_{ij}x+d_{ij}}{N_{ij}x+b_{ij}},\ \ \ \
h_{ij}(x)=\frac{a_{ij}d_{ij}-b_{ij}c_{ij}}{(M_{ij}x+d_{ij})(N_{ij}x+b_{ij})}.$
Recall
$\alpha_{\max}=\max\limits_{(i,j)\in
E}\\{\beta_{ij}(-,-)-\beta_{ij}(+,-),\beta_{ij}(-,+)-\beta_{ij}(+,+)\\}$,
$\alpha_{\min}=\min\limits_{(i,j)\in
E}\\{\beta_{ij}(-,-)-\beta_{ij}(+,-),\beta_{ij}(-,+)-\beta_{ij}(+,+)\\}$,
$\gamma_{ij}=\max\\{\frac{|b_{ij}c_{ij}-a_{ij}d_{ij}|}{a_{ij}c_{ij}},\frac{|b_{ij}c_{ij}-a_{ij}d_{ij}|}{b_{ij}d_{ij}}\\}$,
$\gamma=\max_{(i,j)\in E}\\{\gamma_{ij}\\}$. For any $i\in V$, let $T_{i}$
denote the subtree with $i$ as its root and $Z(i)$ be the TSSHC induced on
$T_{i}$ by $T$. Recall $B_{i}=\frac{h_{i}(+)-h_{i}(-)}{2}$ is the ‘external
field’, denote $\lambda_{i}=e^{-2B_{i}}$, and let $\Gamma_{ij}$ be the unique
self-avoiding path from $i$ to $j$ on $T$.
Lemma 3.3 _For any $(i,j)\in E$,
$\max\limits_{x\in[0,1]}|h_{ij}(x)|\leq\gamma_{ij}$ _
Proof: The proof is technique and left to the appendix.
With the above notations, we present a ‘path’ version Lipchitz approach.
Lemma 3.4 _Let $\Lambda\subset V$ , $\zeta_{\Lambda}$ and $\eta_{\Lambda}$ be
any two configurations on $\Lambda$. Let
$\Theta=\\{i:\zeta_{i}\neq\eta_{i},i\in\Lambda\\}$, $t=d(0,\Theta)$ and
$S(T,0,t)=\\{i:d(0,i)=t,i\in T\\}$. Then_
$\begin{split}&|P_{T}(X_{0}=+|\zeta_{\Lambda})-P_{T}(X_{0}=+|\eta_{\Lambda})|\\\
&\leq\gamma^{t}\sum\limits_{k\in S(T,0,t)}\prod\limits_{i\in\Gamma_{0k}i\neq
k}g_{i}(z_{i})(1-g_{i}(z_{i}))\end{split}$
_where $g_{i}(x_{i})=(1+\lambda_{i}\prod\limits_{(i,i_{j})\in
T_{i}}f_{ii_{j}}(x_{ii_{j}}))^{-1}$ and $x_{i}$ is a vector with elements
$x_{ii_{j}}\in[0,1],i\in V$, and $z_{i}$ are constant vectors with elements in
$[0,1]$. _
Proof: For any $i$ in $T$, let $p^{\zeta_{\Lambda}}_{i}\equiv
P_{T_{i}}(X_{i}=+|\zeta_{\Lambda_{i}})$ and
$R^{\zeta_{\Lambda}}_{i}\equiv\frac{P_{T_{i}}(X_{i}=+|\zeta_{\Lambda_{i}})}{P_{T_{i}}(X_{i}=-|\zeta_{\Lambda_{i}})}$,
where $\zeta_{\Lambda_{i}}$ is configuration by restriction of
$\zeta_{\Lambda}$ on $T_{i}$. Then we have the following equality
$\begin{split}&p^{\zeta_{\Lambda}}_{0}=P_{T}(X_{0}=+|\zeta_{\Lambda})=\frac{1}{1+\frac{P_{T}(X_{0}=-|\zeta_{\Lambda})}{P_{T}(X_{0}=+|\zeta_{\Lambda})}}=\frac{1}{1+1/R^{\zeta_{\Lambda}}_{0}}\\\
&=\frac{1}{1+\lambda_{0}\prod\limits_{(0,0_{j})\in
T}\frac{c_{00_{j}}R^{\zeta_{\Lambda}}_{0_{j}}+d_{00_{j}}}{a_{00_{j}}R^{\zeta_{\Lambda}}_{0_{j}}+b_{00_{j}}}}=\frac{1}{1+\lambda_{0}\prod\limits_{(0,0_{j})\in
T}\frac{M_{00_{j}}p^{\zeta_{\Lambda}}_{0_{j}}+d_{00_{j}}}{N_{00_{j}}p^{\zeta_{\Lambda}}_{0_{j}}+b_{00_{j}}}}\\\
&=g_{0}(x_{0})\end{split}$
where
$x_{0}=(p^{\zeta_{\Lambda}}_{0_{1}},p^{\zeta_{\Lambda}}_{0_{2}},\cdots,p^{\zeta_{\Lambda}}_{0_{\delta_{0}}})$.
First, note for any $x=(x_{1},x_{2},\cdots,x_{q})$ and
$y=(y_{1},y_{2},\cdots,y_{q})$, $q=\delta_{0}$, first order Taylor expansion
at $y$ gives that there exists a $\theta\in[0,1]$ such that
$g_{0}(x)-g_{0}(y)=\nabla g_{0}(y+\theta(x-y))(x-y)^{T},$
where $(x-y)^{T}$ denotes the transportation of the vector $(x-y)$. Calculate
the $\frac{\partial g_{0}(x)}{\partial x_{i}}$, we have
$\begin{split}\frac{\partial g_{0}(x)}{\partial
x_{i}}&=-\frac{\lambda_{0}\prod\limits^{q}_{j=1}f_{00_{j}}(x_{j})(\frac{d\log(f_{00_{i}}(x_{i})}{dx_{i}})}{(1+\lambda_{0}\prod\limits^{q}_{j=1}f_{00_{j}}(x_{j}))^{2}}\\\
&=-g_{0}(x)(1-g_{0}(x))(\frac{M_{00_{i}}}{M_{00_{i}}x_{i}+d_{00_{i}}}-\frac{N_{00_{i}}}{N_{00_{i}}x_{i}+b_{00_{i}}})\\\
&=g_{0}(x)(1-g_{0}(x))\frac{a_{00_{i}}d_{00_{i}}-b_{00_{i}}c_{00_{i}}}{(M_{00_{i}}x_{i}+d_{00_{i}})(N_{00_{i}}x_{i}+b_{00_{i}})}\\\
&=g_{0}(x)(1-g_{0}(x))h_{00_{i}}(x_{i}).\end{split}$
Hence, there’s exits $\theta_{0}\in[0,1]$ such that
$\begin{split}|p^{\zeta_{\Lambda}}_{0}-p^{\eta_{\Lambda}}_{0}|&\leq\sum\limits_{j=1}^{q}|g_{0}(z_{0})(1-g_{0}(z_{0}))h_{00_{j}}(x_{j})||p^{\zeta_{\Lambda}}_{0_{j}}-p^{\eta_{\Lambda}}_{0_{j}}|\\\
&\leq\sum\limits_{j=1}^{q}g_{0}(z_{0})(1-g_{0}(z_{0}))\gamma_{00_{j}}|p^{\zeta_{\Lambda}}_{0_{j}}-p^{\eta_{\Lambda}}_{0_{j}}|\\\
&\leq\gamma\sum\limits_{j=1}^{q}g_{0}(z_{0})(1-g_{0}(z_{0}))|p^{\zeta_{\Lambda}}_{0_{j}}-p^{\eta_{\Lambda}}_{0_{j}}|\end{split}$
where
$z_{0}=p^{\eta_{\Lambda}}_{0}+\theta_{0}(p^{\zeta_{\Lambda}}_{0}-p^{\eta_{\Lambda}}_{0})$
and the second inequality follows by Lemma 3.3. Now repeat the procedure for
$|p^{\zeta_{\Lambda}}_{0_{j}}-p^{\eta_{\Lambda}}_{0_{j}}|$, $j=1,2,\cdots,q$,
it is easy to see that the summation is over all the self-avoiding paths
emitting from the root $0$. For each path $\Gamma$, if the end point of
$\Gamma$ is a leave $j$ with $d(0,j)\leq t-1$ or there is a vertex $i$ on
$\Gamma$ with $d(0,i)\leq t-1$ being fixed, the contribution of the path to
the summation is zero since
$p^{\zeta_{\Lambda}}_{i}-p^{\eta_{\Lambda}}_{i}=p^{\zeta_{\Lambda}}_{j}-p^{\eta_{\Lambda}}_{j}=0$.
Hence the remaining path with length $t$ is in the set $\\{\Gamma_{0k}:k\in
S(T,0,t)\\}$. This completes the proof of lemma 3.4. $\Box$
In order to prove the Theorem 2.2, we need the following lemma.
Lemma 3.5 _Let $\lambda_{i}\geq 0$, $i=1,2,\cdots,n$. Then_
$\prod\limits_{i=1}^{n}(1+\lambda_{i})\geq(1+\sqrt[n]{\prod\limits^{n}_{i=1}\lambda_{i}})^{n}.$
Proof: The proof is technique and left to the appendix.
With Lemma 3.4 and 3.5, it is sufficient to prove Theorem 2.2.
Proof of Theorem 2.2: Following the notation of Lemma 3.4, let $s=|S(T,0,t)|$,
we have
$\begin{split}|p^{\zeta_{\Lambda}}_{0}-p^{\eta_{\Lambda}}_{0}|&\leq\gamma^{t}\sum\limits_{k\in
S(T,0,t)}\prod\limits_{i\in\Gamma_{0k}i\neq k}g_{i}(z_{i})(1-g_{i}(z_{i}))\\\
&\leq s\gamma^{t}\max\limits_{\begin{subarray}{c}k\in
S(T,0,t)\end{subarray}}\prod\limits_{i\in\Gamma_{0k}i\neq
k}g_{i}(z_{i})(1-g_{i}(z_{i}))\\\ &\leq
s\frac{\gamma^{t}}{4}\max\limits_{\begin{subarray}{c}(0,0_{j})\in T\\\ k\in
S(T,0,t)\end{subarray}}\prod\limits_{i\in\Gamma_{0_{j}k}i\neq
k}g_{i}(z_{i})(1-g_{i}(z_{i})).\end{split}$
By Lemma 3.5, for each $\Gamma_{0_{j}k}$, $(0,0_{j})\in T$, $k\in S(T,0,t)$,
$\begin{split}&\prod\limits_{i\in\Gamma_{0_{j}k}i\neq
k}g_{i}(z_{i})(1-g_{i}(z_{i}))\\\ &=\prod\limits_{i\in\Gamma_{0_{j}k}i\neq
k}\frac{\lambda_{i}\prod\limits_{(i,i_{l})\in
T_{i}}f_{ii_{l}}(z_{ii_{l}})}{(1+\lambda_{i}\prod\limits_{(i,i_{l})\in
T_{i}}f_{ii_{l}}(z_{ii_{l}}))^{2}}\\\
&\leq(\frac{r_{jk}}{(1+r_{jk})^{2}})^{t-1}\end{split}$
where $r_{jk}=(\prod\limits_{i\in\Gamma_{0_{j}k}i\neq
k}\lambda_{i}\prod\limits_{(i,i_{l})\in
T_{i}}f_{ii_{l}}(z_{ii_{l}}))^{1/(t-1)}$. A simple calculation gives that
$e^{\alpha_{\min}}\leq f_{ij}(x)\leq e^{\alpha_{\max}}$, for any $(i,j)\in T$.
Hence,
$\begin{split}e^{\alpha_{\min}(\delta(T,0,t-1)-1)}&\leq(\prod\limits_{i\in\Gamma_{0_{j}k}i\neq
k}\prod\limits_{(i,i_{l})\in T_{i}}f_{ii_{l}}(z_{ii_{l}}))^{1/(t-1)}\\\ &\leq
e^{\alpha_{\max}(\delta(T,0,t-1)-1)}.\end{split}$
Now we prove the exponential strong spatial mixing under assumption of Theorem
2.2. Suppose $T$ is a self-avoiding tree of $G$.
$\delta(T,0,t-1)\leq\Delta(G,t-1)\leq d$ when $t=ka\log n+1$, $k=1,2,\cdots$.
If $B_{\min}>B(d,\alpha_{\max},\gamma)$, then
$\frac{\gamma(d-1)\exp(2B_{\min}-\alpha_{\max}(d-1))}{(1+\exp(2B_{\min}-\alpha_{\max}(d-1)))^{2}}<1.$
Noting $s\leq\delta_{0}(d-1)^{t-1}$ and
$(\prod\limits_{i\in\Gamma_{0_{j}k}i\neq k}\lambda_{i})^{1/(t-1)}\leq
e^{-2B_{\min}}$ , now we can see
$\begin{split}&|p^{\zeta_{\Lambda}}_{0}-p^{\eta_{\Lambda}}_{0}|\leq
s\frac{\gamma^{t}}{4}(\frac{r_{jk}}{(1+r_{jk})^{2}})^{t-1}\\\
&\leq\frac{\delta_{0}\gamma}{4}(\frac{\gamma(d-1)\exp(2B_{\min}-\alpha_{\max}(d-1))}{(1+\exp(2B_{\min}-\alpha_{\max}(d-1)))^{2}})^{t-1}.\end{split}$
(3)
The similar case holds for $B_{\max}<-B(d,-\alpha_{\min},\gamma)$. This
completes the proof. $\Box$
Remark: As we point out in section II that if the graph $G$ is a bounded
degree graph with the maximum degree $d$, the condition in Theorem 2.2 can be
relaxed to $B_{i}>B(d,\alpha_{\max},\gamma)$ or
$B_{i}<-B(d,-\alpha_{\min},\gamma)$ for any $i\in V$. The reason for this
comes from the upper bound for $g_{i}(z_{i})(1-g_{i}(z_{i}))$ in the Lemma 3.4
since $\gamma(d-1)g_{i}(z_{i})(1-g_{i}(z_{i}))<1$ for any
$i\in\Gamma_{0_{j}k},i\neq k$ where ${(0,0_{j})\in T,k\in S(T,0,t)}$. We
emphasize that one way to improve the condition by this method is to carefully
analyze the bound of $f_{ij}(x)$ for each iterative step according to the
range of $x$ since this will give better bound for $g_{i}(x)$. We do not
optimize the parameter here and do not know whether dealing with the bound of
$f_{ij}(x)$ carefully makes the $B(d,\alpha_{\max},\gamma)$ or
$-B(d,-\alpha_{\min},\gamma)$ optimally approximate the critical point of
‘external field’ for uniqueness of Gibbs measures either if there does exit
one(note that the critical points of ‘external field’ for ferromagnetic and
antiferromagnetic Ising model are different on Cayley tree, an infinite
regular tree with the same degree for each vertex [4]).
The proof of Theorem 2.3 will be shown in Section IV.
## IV Approximating Partition Function
In the proof of Theorem 2.1 and 2.2, the calculation of the marginal
probability of the root yields a local recursive procedure. If we truncate the
tree at height $t$, and then use the recursive method to compute the marginal
probability at root, it is easy to see the complexity of this procedure is the
number of vertices of truncated tree. We now present the algorithm based on
the above procedure and self-avoiding tree.
Let $G=(V,E)$ be a graph with vertices $V=\\{1,2,\cdots,n\\}$, edge set $E$
and $TSSHC$ on it. Let $\Phi_{1}$ denote the whole state space ( which means
$P_{G}(X_{1}=+|\Phi_{1})=P_{G}(X_{1}=+)$), and $\Phi_{j}=\\{X_{i}=+,1\leq
i\leq j-1\\}$, $2\leq j\leq n+1$.
Algorithm for Partition Function $Z(G$)
Input: $G$ with the TSSHC, $\epsilon>0$ precision.
Output: $\widehat{Z(G)}$, the estimator of partition function $Z(G)$.
For j=1:n
compute $\widehat{p_{j}}$, an estimator of conditional marginal probability
$p_{j}=P_{G}(X_{j}=+|\Phi_{j})$, through self-avoiding tree $T_{saw(j)}$
truncated at a certain height $t_{j}$ under the condition $\Phi_{j}$ such that
$(1-\frac{\epsilon}{2n})\leq\frac{p_{j}}{\widehat{p_{j}}}\leq(1-\frac{\epsilon}{2n})$.
(The initial values of iteration at height $t_{j}$ are arbitrary nonnegative
numbers, if we adopt the recursive formula in the proof of Lemma 3.4 where
$P_{T}(X_{0}=+|\zeta_{\Lambda})=g_{0}(x_{0})$)
Output:$\widehat{Z(G)}=Z(G,\Phi_{n+1})\prod\limits^{n}_{i=1}\widehat{p_{i}}^{-1}$.
With the above algorithm, it is enough to prove Theorem 2.3.
Proof of Theorem 2.3: First we show under the assumption of the theorem, the
Gibbs distribution exhibits exponential strong spatial mixing. Since(by
Proposition 2.4) $|V(T_{saw(i)},i,ka\log n)|\leq|\max\limits_{j\in
V}(V(T_{saw(j)},j,ka\log n))|^{k}\leq(d-1)^{ka\log n}$ for any $i\in V$ and
$k=1,2,\cdots$, we can obtain the trivial bound of the number of vertices at
height $ka\log n$, that is , $|S(T_{saw(i)},i,ka\log
n)\leq|V(T_{saw(i)},i,ka\log n)|\leq(d-1)^{ka\log n}$. Let $t=ka\log n$ from
the proof of Theorem 2.1 and 2.2(see Formula (1) and (2)), substituting
$(d-1)^{t}$ to $s$ in (1) (2), we get the exponential strong spatial mixing of
Theorem 2.3. Specifically, if $J<J_{d}$, the decay function
$f(t)=4J(d-1)((d-1)\tanh J)^{t-1}$ which corresponds to the logarithmic form
exponential strong spatial mixing, and if $J\geq J_{d}$,
$B_{\min}>B(d,\alpha_{\max},\gamma)$ or
$B_{\max}<-B(d,-\alpha_{\min},\gamma)$, the decay function has the same form
as in Theorem 2.2 except replacing $\delta_{i}$ by $d-1$. In both cases, we
suppose decay function $f(t)=be^{-ct}$ where $b$, $c$ are constant positive
numbers independent of $n$, $t=ka\log n$, $k=1,2,\cdots$. Through exponential
decay property, it’s sufficient to show the above algorithm provides an FPTAS
for $Z(G)$. Now we check the output $\widehat{Z(G)}$ satisfying
$(1-\epsilon)\leq\frac{\widehat{Z(G)}}{Z(G)}\leq(1+\epsilon)$. Since
$p_{j}=\frac{Z(G,\Phi_{j+1})}{Z(G,\Phi_{j})}$, multiplying them gives
$Z(G)=Z(G,\Phi_{n+1})\prod\limits^{n}_{i=1}p_{i}^{-1}$. Hence,
$1-\epsilon\leq(1-\frac{\epsilon}{2n})^{n}\leq\prod\limits^{n}_{i=1}\frac{\widehat{p_{i}}^{-1}}{p_{i}^{-1}}=\frac{\widehat{Z(G)}}{Z(G)}\leq(1+\frac{\epsilon}{2n})^{n}\leq
1+\epsilon$. As we point out previously that the complexity of the algorithm
at each step is $O(|V(T_{saw(j)},j,t_{j})|)=O((d-1)^{t_{j}})$ when
$t_{j}=ka\log n$, $k=1,2,\cdots$, we only need to set $f(t_{j})\leq
O(\frac{\epsilon}{2n})$ to promise
$(1-\frac{\epsilon}{2n})\leq\frac{p_{j}}{\widehat{p_{j}}}\leq(1-\frac{\epsilon}{2n})$
which requires $t_{j}=O(\log n+\log(\epsilon^{-1}))$. Thus, the complexity of
the algorithm is $nO((d-1)^{O(\log
n+\log(\epsilon^{-1}))})=O(n^{O(1)}+n(\epsilon^{-1})^{O(1)})$, which completes
the proof. $\Box$
## V Conclusion and Further Work
We have shown that the Gibbs distribution of TSSHC on a ‘sparse on average’
graph $G=(V,E)$ with ‘maximum average degree’ $d$ exhibits the (exponential)
strong spatial mixing when the absolute value of ‘inverse temperature’
$|J_{ij}|<J_{d}$ or the ‘external field’ $B_{i}$ is uniformly larger than
$B(d,\alpha_{\max},\gamma)$ or smaller than $-B(d,-\alpha_{\min},\gamma)$, for
any $(i,j)\in E$, $i\in V$. Here $J_{d}$ is the critical point for uniqueness
of Gibbs measure on a infinite $d$ regular tree of Ising model, implying the
condition for inverse temperature is tight when restricting it on Ising model,
$B(d,\alpha,\gamma)$ is constant with parameter $d$, $\alpha$, $\gamma$. It is
not difficult to apply our results to Erd$\ddot{o}$-R$\dot{e}$nyi random graph
$G(n,d/n)$, where each edge is chosen independently with probability $d/n$,
since the average degree in $G(n,d/n)$ is $d(1-o(1))$ while it contains many
vertices with degree $\log n/\log\log n$[15]. As an application of strong
spatial mixing property, we present an FPTAS for partition functions on a
little modified sparse graphs, which includes interesting bounded degree
graph.
For future work, we expect to improve the condition on ‘external field’. We
have presented a way to improve it in the remark, however, we believe the
essential improvement needs other method. Maybe the approach of analysis of
the fixed point in[11] works here.
## References
* [1] A. Bandyopadhyay and D. Gamarnik. _Counting without sampling: New algorithms for enumeration problems using statistical physics_ , Proceedings of 17th ACM-SIAM Symposium on Discrete Algorithms (SODA) (2006).
* [2] M. Bayati, D. Gamarnik, D. Katz, C. Nair, and P. Tetali. _Simple deterministic approximation algorithms for counting matchings_ , Proceedings of the 39th annual ACM symposium on Theory of computing(STOC) (2007).
* [3] M. Dyer, A. Frieze and R. Kannan. _A random polynomial-time algorithm for approximating the volume of convex bodies_ , J. Assoc. Comput. Mach. 38, (1991), 1-17.
* [4] H. O. Georgii. _Gibbs measures and phase transitions_ , volume 9 of de Gruyter Studies in Mathematics. Walter de Gruyter $\&$ Co., Berlin, (1988).
* [5] D. Gamarnik and D. Katz. _Correlation decay and deterministic FPTAS for counting list-colorings of a graph_ , Proceedings of 18th ACM-SIAM Symposium on Discrete Algorithms (SODA) (2007).
* [6] A. Gershchenfeld and A. Montanari. _Reconstruction for models on random graphs_ , At arXiv:0704.3293. To Appear at FOCS 2007.
* [7] M. Jerrum and A. Sinclair. _Polynomial-time approximation algorithms for Ising model_ SIAM J. Comput. Vol22, No. 5, (1993), 1087-1116.
* [8] K. Jung and D. Shah. _Inference in Binary Pair-wise Markov Random Field through Self-Avoiding Walk_ , Preprint on http://arxiv.org/abs/cs.AI/0610111v2.
* [9] M. Jerrum, A. Sinclair, and E. Vigoda. _A polynomial-time approximation algorithms for permanent of a matrix with non-negative entries_ , Journal of the Association for Computing Machinery 51 (2004), no. 4, 671-697.
* [10] M. Jerrum, L. Valiant, and V. Vazirani. _Random generation of combinatorial structures from a uniform distribution_ , Theoret. Comput. Sci. 43. (1986), 169-188.
* [11] F. P. Kelly. _Stochastic models of computer communication systems_ , Journal of the Royal Statistical Society B47, (1985), 379-395.
* [12] Y. Lu, C. M$\acute{e}$asson and A. Montanari. _TP decoding_ , http://arxiv.org/PS_cache/arxiv/pdf/0710/0710.0564v1.
* [13] R. Lyons. _The Ising model and percolation on trees and treelike graphs_ , Comm. Math. Phys., 125(2), (1989), 337-353.
* [14] E. Marinariand and G. Semerjian. _On the number of circuits in random graphs_ , http://arxiv.org/PS_cache/cond-mat/pdf/0603/0603657v1.
* [15] E. Mossel and A. Sly. _Rapid mixing of gibbs sampling on graphs that are sparse on average_ , To Appear in SODA 2008.
* [16] E. Mossel, D. Weitz, and N. Wormald. _On the hardness of sampling independent sets beyond the tree threshold_ , To Appear in Prob. Theory Related. Fields, (2007).
* [17] C. Nair and P. Tetali. _The correlation decay (CD) tree and strong spatial mixing in multi-spin systems_ , Preprint on http://front.math.ucdavis.edu/math.PR/0701494.
* [18] R. Pemantle and Y. Peres. _The critical Ising model on trees, concave recursions and nonlinear capacity_ , http://arxiv.org/PS_cache/math/pdf/0503/0503137v2.
* [19] L. G. Valiant. _The complexity of computing the permanent_ , Theoretical Computer Science 8 (1979), 189-201.
* [20] D. Weitz. _combinatorial cirteria for uniqueness of Gibbs measures_ , Random Structures and Algorithms 27, (2005), 445-475.
* [21] D. Weitz. _Counting indpendent sets up to the tree threshold_ , Proceedings of the 38th annual ACM symposium on Theory of computing(STOC), (2006), 140-149.
## Appendix
Proof of Lemma 3.3:
Since $M_{ij}x+d_{ij}\geq 0$ and $N_{ij}x+b_{ij}\geq 0$, $\forall x\in[0,1]$,
we need only to show
$\min\limits_{x\in[0,1]}w(x)=\min(a_{ij}c_{ij},b_{ij}d_{ij})$, where
$w(x)=(M_{ij}x+d_{ij})(N_{ij}x+b_{ij})$. The case $M_{ij}N_{ij}=0$ is trivial,
so w.l.o.g. suppose $M_{ij}N_{ij}\neq 0$. Noting
$x_{l}=-\frac{d_{ij}N_{ij}+b_{ij}M_{ij}}{2M_{ij}N_{ij}}$ is an extremum of
$w(x)$ on $R$. There are three cases needed to be discussed.
Case 1. $M_{ij}N_{ij}<0$, then $w(x)$ reaches its minimum at boundary. Then
$\min\limits_{x\in[0,1]}w(x)\leq\min(w(0),w(1))=\min(a_{ij}c_{ij},b_{ij}d_{ij})$.
Case 2. $M_{ij}>0,N_{ij}>0$, then $x_{l}\leq 0$, $w(x)$ is increasing on
$[0,1]$, then $\min\limits_{x\in[0,1]}w(x)=w(0)=b_{ij}d_{ij}$.
Case 3. $M_{ij}<0,N_{ij}<0$, then $x_{l}\geq 1$, $w(x)$ is decreasing on
$[0,1]$, hence $\min\limits_{x\in[0,1]}w(x)=w(1)=a_{ij}c_{ij}$. $\Box$
Proof of Lemma 3.5:
$\begin{split}\prod\limits_{i=1}^{n}(1+\lambda_{i})&=1+\sum\limits^{n}_{k=1}(\sum\limits_{i_{1}<i_{2}<\cdots<i_{k}}\prod\limits^{k}_{j=1}\lambda_{i_{j}})\\\
&\geq
1+\sum\limits^{n}_{k=1}(C^{k}_{n}(\prod\limits^{n}_{i=1}\lambda_{i})^{\frac{C_{n-1}^{k-1}}{C^{k}_{n}}})\\\
&=1+\sum\limits^{n}_{k=1}(C^{k}_{n}(\prod\limits^{n}_{i=1}\lambda_{i})^{\frac{k}{n}})\\\
&=(1+\sqrt[n]{\prod\limits^{n}_{i=1}\lambda_{i}})^{n},\end{split}$
where $C^{k}_{n}=\frac{n!}{k!(n-k)!}$. The first inequality uses the
arithmetic-geometric average inequality.
|
arxiv-papers
| 2008-12-03T16:56:53 |
2024-09-04T02:48:59.164027
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jinshan Zhang",
"submitter": "Jinshan Zhang",
"url": "https://arxiv.org/abs/0812.0754"
}
|
0812.0834
|
# Stochastic Volterra Equations in Banach Spaces and Stochastic Partial
Differential Equations*
Xicheng ZHANG School of Mathematics and Statistics
The University of New South Wales, Sydney, 2052, Australia,
Department of Mathematics, Huazhong University of Science and Technology
Wuhan, Hubei 430074, P.R.China
email: XichengZhang@gmail.com.
###### Abstract.
In this paper, we first study the existence-uniqueness and large deviation
estimate of solutions for stochastic Volterra integral equations with singular
kernels in $2$-smooth Banach spaces. Then, we apply them to a large class of
semilinear stochastic partial differential equations (SPDE) driven by Brownian
motions as well as by fractional Brownian motions, and obtain the existence of
unique maximal strong solutions (in the sense of SDE and PDE) under local
Lipschitz conditions. Lastly, high order SPDEs in a bounded domain of
Euclidean space, second order SPDEs on complete Riemannian manifolds, as well
as stochastic Navier-Stokes equations are investigated.
###### Key words and phrases:
Stochastic Volterra Equation, Large Deviation, Fractional Brownian Motion,
Stochastic Reaction-Diffusion Equation, Stochastic Navier-Stokes Equation
###### Contents
1. 1 Introduction
2. 2 Preliminaries
1. 2.1 Gronwall’s inequality of Volterra type
2. 2.2 Ito’s integral in $2$-smooth Banach spaces
3. 2.3 A local non-linear interpolation lemma
4. 2.4 A criterion for Laplace principles
3. 3 Abstract stochastic Volterra integral equations
1. 3.1 Global existence and uniqueness
2. 3.2 Path continuity of solutions
3. 3.3 Local existence and uniqueness
4. 3.4 Continuous dependence of solutions with respect to data
4. 4 Large deviation for stochastic Volterra equations
5. 5 Semilinear stochastic evolutionary integral equations
6. 6 Semilinear stochastic partial differential equations
1. 6.1 Mild solutions of SPDEs driven by Brownian motions
2. 6.2 Strong solutions of SPDEs driven by Brownian motions
3. 6.3 SPDEs driven by fractional Brownian motions
7. 7 Application to SPDEs in bounded domains of ${\mathbb{R}}^{d}$
8. 8 Application to SPDEs on complete Riemannian manifolds
9. 9 Application to stochastic Navier-Stokes equations
1. 9.1 Unique maximal strong solution for SNSEs
2. 9.2 Non-explosion and large deviation for 2D SNSEs
## 1\. Introduction
The aims of this paper are three folds: First of all, we prove the existence
and uniqueness of solutions with continuous paths for stochastic Volterra
integral equations with singular kernels in a $2$-smooth Banach space.
Secondly, the large deviation principles (abbrev. LDP) of Freidlin-Wentzell
type for stochastic Volterra equations are established under small
perturbations of multiplicative noises. Thirdly, we apply them to several
classes of semilinear stochastic partial differential equations (abbrev.
SPDE). In particular, we give a unified treatment in certain sense for the
LDPs of a large class of SPDEs.
In finite dimensional space, stochastic Volterra integral equations with
regular kernels and driven by Brownian motions were first studied by Berger
and Mizel in [6]. Later, Protter [62] studied the stochastic Volterra
equations driven by general semimartingales. Using the Skorohod integral,
Pardoux and Protter [57] also investigated the stochastic Volterra equations
with anticipating coefficients. The study of stochastic Volterra equations
with singular kernels can be found in [18, 20, 78, 45, 53, etc.]. Recently,
the present author [82] studied the approximation of Euler type and the LDP of
Freidlin-Wentzell type for stochastic Volterra equations with singular
kernels. In particular, the kernels in [82] may deal with the fractional
Brownian motion kernels as well as the fractional order integral kernels. The
study of LDP for stochastic Volterra equations is also referred to [53, 45].
Since the work of Freidlin and Wentzell [26], the theory of small perturbation
large deviations for stochastic differential equations (abbrev. SDE) has been
studied extensively (cf. [3, 74, etc.]). In the classical method, to establish
such an LDP for SDE, one usually needs to discretize the time variable and
then prove various necessary exponential continuity and tightness for
approximation equations in different spaces by using comparison principle.
However, such verifications would become rather complicated and even
impossible in some cases, e.g., stochastic evolution equations with
multiplicative noises.
Recently, Dupuis and Ellis [24] systematically developed a weak convergence
approach to the theory of large deviation. The central idea is to prove some
variational representation formula for the Laplace transform of bounded
continuous functionals, which will lead to proving a Laplace principle which
is equivalent to the LDP. In particular, for Brownian functionals, an elegant
variational representation formula has been established by Boué-Dupuis [9] and
Budhiraja-Dupuis [14]. A simplified proof was given by the present author
[81]. This variational representation has already been proved to be very
effective for various finite and infinite dimensional stochastic dynamical
systems even with irregular coefficients (cf. [64, 65, 15, 82, 66, etc.]). One
of the main advantages of this argument is that one only needs to make some
simple moment estimates (see Section 4 below).
On the other hand, it is well known that in the deterministic case, many PDE
problems of parabolic and hyperbolic types can be written as Volterra type
integral equations in Banach spaces by using the corresponding semigroup and
the variation-of-constants formula (cf. [27, 37, 58]). An obvious merit of
this procedure is that the unbounded operators in PDEs no longer appear and
the analysis is entirely analogous to the ODE case. Thus, one naturally
expects to take the same advantages for SPDEs in Banach spaces. However, it is
not all Banach spaces in which stochastic integrals are well defined. One can
only work in a class of $2$-smooth Banach spaces. The definition of stochastic
integrals in $2$-smooth Banach spaces and related properties such as
Burkholder-Davis-Gundy’s (abbrev. BDG) inequality, Girsanov’s theorem,
stochastic Fubini’s theorem and the distribution of stochastic integrals can
be found in [52, 10, 11, 54, etc. ]. Thus, similar to the deterministic case,
we can develop a parallel theory in $2$-smooth Banach spaces for SPDEs. It
should be emphasized that besides the usual SPDEs driven by multiplicative
Brownian noises, a class of stochastic evolutionary integral equations
appearing in viscoelasticity and heat conduction with memory (cf. [63]) as
well as a class of SPDEs driven by additive fractional Brownian noise, can
also be written as abstract stochastic Volterra equations in Banach spaces.
In the past three decades, the theory of general SPDEs has been developed
extensively by numerous authors mainly based on two different approaches:
semigroup method based on the variation-of-constants formula (as said above)
(cf. [77, 19, 10, 11, 12, 80, etc.]) and variation method based on Galerkin’s
finite dimensional approximation (cf. [56, 44, 68, 43, 50, 61, 83, 31, etc.]).
A new regularization method is given in [86]. An overview for the
classification and applications of SPDEs are referred to the recent book of
Kotelenez [42]. In the author’s knowledge, most of the well known results are
primarily concentrated on the mild or weak solutions, even measure-valued
solutions. Such notions of solutions naturally appear in the study of SPDEs
driven by the space-time white noises, and in this case one cannot obtain any
differentiability of the solutions in the spatial variable.
Nevertheless, when one considers an SPDE driven by the spatial regular and
time white noises, it is reasonable to require the existence of spatial
regular solutions or classical solutions in the sense of PDE. For linear
SPDEs, such regular solutions are easy and well known (cf. [44, 68, 25,
etc.]). However, for nonlinear SPDEs, there seems to be few results (cf. [43,
48, 81, 86]). A major difficulty to prove the spatial regularity of solutions
is that one cannot use the usual bootstrap method in the theory of PDE since
there is no any differentiability of solutions with respect to the time
variable. The present author [81] (see also [32, 86]) solves this problem by
using a non-linear interpolation result due to Tartar [75]. Obviously, for the
regularity theory of SPDEs, by using Sobolev’s embedding theorem (cf. [1]), it
is natural to consider the $L^{p}$-solution of SPDEs. This is also why we need
to work in $2$-smooth Banach spaces. It should be remarked that the
$L^{p}$-theory for SPDEs has been established in [10, 11, 12, 43, 22, 23, 80,
etc.]. But, there are few results to deal with the $L^{p}$-strong solution in
the sense of PDE. In the present paper, we shall prove a general result about
the existence of strong solutions in the sense of both SDE and PDE (see
Theorem 6.9).
We now describe our structure of this paper: In Section 2, we prepare some
preliminaries for later use, and divide it into four subsections. In
Subsection 2.1, we prove a Gronwall’s lemma of Volterra type under rather weak
assumptions on kernel functions. Moreover, two simple examples are provided to
show this lemma. In Subsection 2.2, we recall the Itô integral in $2$-smooth
Banach spaces and Burkholder-Davies-Gundy’s inequality as well as Kolmogorov’s
continuity criterion of random fields in random intervals. In Subsection 2.3,
we recall the properties of analytic semigroups and prove a local non-linear
interpolation lemma, see also [75] for other related non-linear interpolation
results. This lemma will play an important role in proving the existence of
strong solutions (in the PDE’s sense) in Theorems 7.2 and 8.2 below. In
Subsection 2.4, we recall the criterion of Laplace principle established by
Budihiraja and Dupuis [9, 14] (see also [84]).
In Section 3, using the Gronwall inequality of Volterra type in Subsection
2.1, we first prove the existence and uniqueness of solutions for stochastic
Volterra equations in $2$-smooth Banach spaces under global Lipschitz
conditions and singular kernels. Next, in Subsection 3.2, we study the
regularity of solutions under slightly stronger assumptions on kernels.
Moreover, a BDG type of inequality for stochastic Volterra type integral is
also proved. In Subsection 3.3, employing the usual localizing method, we
prove the existence of a unique maximal solution for stochastic Volterra
equation under local Lipschitz conditions. Lastly, in Subsection 3.4, we
discuss the continuous dependence of solutions with respect to the
coefficients.
In Section 4, using the weak convergence method, we prove the Freidlin-
Wentzell large deviation principle for the small perturbations of stochastic
Volterra equations under a compactness assumption and some uniform non-
explosion conditions for the controlled equations. We also refer to [47, 66]
for the application of weak convergence approach in the LDPs of stochastic
evolution equations (the case of evolution triple). In the proof of Section 4,
we need to use the Yamada-Watanabe Theorem in infinite dimensional space,
which has been established by Ondreját [54] (see also [67] for the case of
evolution triple). We want to say that although Ondreját only considered the
case of convolution semigroup, their proofs are also adapted to more general
stochastic Volterra equations. Moreover, since we are considering the path
continuous solution, the proof in [54] can be simplified .
In Section 5, a simple application in a class of semilinear stochastic
evolutionary integral equations is presented, which has been studied in [17,
8, 40, etc.] for additive noises. Such type of stochastic evolution equations
appears in viscoelasticity, heat conduction in materials with memory, and
electrodynamics with memory [63].
In Section 6, we apply our general results to a large class of semilinear
stochastic evolution equations driven by multiplicative Brownian noise and
additive fractional Brownian noise. A basic result in semigroup theory states
that if $f$ is a Hölder continuous function in the Banach space
${\mathbb{X}}$, then
$t\mapsto\int^{t}_{0}{\mathfrak{T}}_{t-s}f(s){\mathord{{\rm d}}}s\mbox{ is
continuous in ${\mathscr{D}}({\mathfrak{L}})$},$
where ${\mathfrak{T}}_{t}$ is an analytic semigroup and ${\mathfrak{L}}$ is
the generator of ${\mathfrak{T}}_{t}$. We will use this result to prove the
existence of strong solutions (in the sense of PDE) for semilinear SPDEs.
Moreover, we also give a simple result about the SPDE driven by additive
fractional Brownian noises. The corresponding LDPs are also obtained (see also
[71, 59, 15, 66, 47, etc.] for the study of LDPs of stochastic evolution
equations). We remark that the skeleton equation for the LDP of SPDEs driven
by fractional Brownian motion is a non-convolution type of Volterra integral
equation.
In Section 7, high order SPDEs in a bounded domain of Euclidean space are
studied. Our stochastic version may be regarded as a parallel result in the
deterministic case (cf. [58, p.246, Theorem 4.5]). Moreover, the LDP is also
obtained.
In Section 8, we in particular study the second order stochastic parabolic
equations on complete Riemannian manifolds. Under one-side Lipschitz and
polynomial growth conditions, we obtain the global existence-uniqueness of
strong solutions. When the manifold is compact, the LDP also holds in this
case. In particular, stochastic reaction diffusion equations with polynomial
growth coefficients are included.
In Section 9, we first prove the existence and uniqueness of local
$L^{p}$-strong solutions for stochastic Navier-Stokes equations (SNSE) in any
dimensional case. In the two dimensional case, we also obtain the non-
explosion of solutions. Moveover, the LDPs for $2$-dimensional SNSEs are also
established in the case of both Dirichlet boundary and periodic boundary. We
remark that the $L^{p}$-solutions for SNSEs have been studied by Brzezniak and
Peszat [13] (bounded domain) and Mikulevicius and Rozovskii [49] (the whole
space). The large deviation result for two dimensional SNSEs with additive
noise was proved by Chang [16] using Girsanov’s transformation. In [70], the
authors also used the weak convergence method to prove the large deviation
estimate for two dimensional SNSEs with multiplicative noises. But, it seems
that there is a gap in their proofs [70, p.1655 line 6 and p.1658 line 2].
Therein, the $\mathord{{\bf v}}_{n}$ only weakly converges to $\mathord{{\bf
v}}$ in $S_{M}$. This seems not enough to derive their limits.
We conclude this introduction by making the following Convention: Throughout
this paper, the letter $C$ with or without subscripts will denote a positive
constant, whose value may change from one place to another. Moreover, we also
use the notation $E_{1}\preceq E_{2}$ to denote $E_{1}\leqslant C\cdot E_{2}$,
where $C>0$ is an unimportant constant.
## 2\. Preliminaries
### 2.1. Gronwall’s inequality of Volterra type
Let $\triangle:=\\{(t,s)\in{\mathbb{R}}^{2}_{+}:s\leqslant t\\}$. We first
recall the following result due to Gripenberg [33, Theorem 1 and p.88].
###### Lemma 2.1.
Let $\kappa:\triangle\to{\mathbb{R}}_{+}$ be a measurable function. Assume
that for any $T>0$
$t\mapsto\int^{t}_{0}\kappa(t,s){\mathord{{\rm d}}}s\in L^{\infty}(0,T)$
and
$\limsup_{\epsilon\downarrow
0}\left\|\int^{\cdot+\epsilon}_{\cdot}\kappa(\cdot+\epsilon,s){\mathord{{\rm
d}}}s\right\|_{L^{\infty}(0,T)}<1.$
Define
$\displaystyle r_{1}(t,s):=\kappa(t,s),\ \
r_{n+1}(t,s):=\int^{t}_{s}\kappa(t,u)r_{n}(u,s){\mathord{{\rm d}}}u,\ \
n\in{\mathbb{N}}.$ (2.1)
Then for any $T>0$, there exist constants $C_{T}>0$ and $\gamma\in(0,1)$ such
that
$\displaystyle\left\|\int^{\cdot}_{0}r_{n}(\cdot,s){\mathord{{\rm
d}}}s\right\|_{L^{\infty}(0,T)}\leqslant C_{T}n\gamma^{n},\ \ \forall
n\in{\mathbb{N}}.$ (2.2)
In particular, the series
$\displaystyle r(t,s):=\sum_{n=1}^{\infty}r_{n}(t,s)$ (2.3)
converges for almost all $(t,s)\in\triangle$, and
$\displaystyle r(t,s)-k(t,s)=\int^{t}_{s}k(t,u)r(u,s){\mathord{{\rm
d}}}u=\int^{t}_{s}r(t,u)k(u,s){\mathord{{\rm d}}}u$ (2.4)
and for any $T>0$
$\displaystyle t\mapsto\int^{t}_{0}r(t,s){\mathord{{\rm d}}}s\in
L^{\infty}(0,T).$ (2.5)
The function $r$ defined by (2.3) is called the resolvent of $\kappa$. All the
functions $\kappa$ in Lemma 2.1 will be denoted by ${\mathscr{K}}$. In what
follows, we shall denote by ${\mathscr{K}}_{0}$ the subclass of
${\mathscr{K}}$ with the property that
$\limsup_{\epsilon\downarrow
0}\left\|\int^{\cdot+\epsilon}_{\cdot}\kappa(\cdot+\epsilon,s){\mathord{{\rm
d}}}s\right\|_{L^{\infty}(0,T)}=0.$
We also denote by ${\mathscr{K}}_{>1}$ the set of all positive measurable
functions $\kappa$ on $\triangle$ with the property that for any $T>0$ and
some $\beta=\beta(T)>1$
$\displaystyle t\mapsto\int^{t}_{0}\kappa^{\beta}(t,s){\mathord{{\rm d}}}s\in
L^{\infty}(0,T).$ (2.6)
It is clear that
${\mathscr{K}}_{>1}\subset{\mathscr{K}}_{0}\subset{\mathscr{K}}$ and for any
$\kappa_{1},\kappa_{2}\in{\mathscr{K}}_{0}\ (\mathrm{resp.}\
{\mathscr{K}}_{>1})$ and $C_{1},C_{2}\geqslant 0$,
$C_{1}\kappa_{1}+C_{2}\kappa_{2}\in{\mathscr{K}}_{0}\ (\mathrm{resp.}\
{\mathscr{K}}_{>1}).$
Let $0\leqslant h\in L^{1}_{loc}({\mathbb{R}}_{+})$. If $\kappa(t,s)=h(s)$,
then $\kappa\in{\mathscr{K}}_{0}$ and
$r(t,s)=h(s)\exp\left\\{\int^{t}_{s}h(u){\mathord{{\rm d}}}u\right\\};$
if $\kappa(t,s)=h(t-s)$, then $\kappa\in{\mathscr{K}}_{0}$ and
$\displaystyle r(t,s)=a(t-s):=\sum_{n=1}^{\infty}a_{n}(t-s),$ (2.7)
where
$a_{1}(t)=h(t),\ \ a_{n+1}(t):=\int^{t}_{0}h(t-s)a_{n}(s){\mathord{{\rm
d}}}s.$
When $0\leqslant h\in L^{1}({\mathbb{R}}_{+})$, a classical result due to
Paley and Wiener (cf. [51, p.207, Theorem 5.2]) says that
$\displaystyle a\in L^{1}({\mathbb{R}}_{+})\mbox{ if and only if
}\int^{\infty}_{0}h(t){\mathord{{\rm d}}}t<1.$ (2.8)
In this case, $\hat{a}(s)=\hat{h}(s)/(1-\hat{h}(s))$, where the hat denotes
the Laplace transform, i.e.:
$\hat{h}(s):=\int^{\infty}_{0}e^{-st}h(t){\mathord{{\rm d}}}t,\ \ s\geqslant
0.$
We want to say that (2.8) is useful in the study of large time asymptotic
behavior of solutions for Volterra equations. An important extension to
nonintegrable convolution kernel can be found in [69, 38] (see also [33]). A
simple example is provided in Example 3.2 below.
We now prove the following Gronwall lemma of Volterra type (see also [37,
Lemma 7.1.1] for a case of special convolution kernel).
###### Lemma 2.2.
Let $\kappa\in{\mathscr{K}}$ and $r_{n}$ and $r$ be defined respectively by
(2.1) and (2.3). Let $f,g:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ be two
measurable functions satisfying that for any $T>0$ and some $n\in{\mathbb{N}}$
$\displaystyle t\mapsto\int^{t}_{0}r_{n}(t,s)f(s){\mathord{{\rm d}}}s\in
L^{\infty}(0,T)$ (2.9)
and for almost all $t\in(0,\infty)$
$\displaystyle\int^{t}_{0}r(t,s)g(s){\mathord{{\rm d}}}s<+\infty.$ (2.10)
If for almost all $t\in(0,\infty)$
$\displaystyle f(t)\leqslant g(t)+\int^{t}_{0}\kappa(t,s)f(s){\mathord{{\rm
d}}}s,$ (2.11)
then for almost all $t\in(0,\infty)$
$\displaystyle f(t)\leqslant g(t)+\int^{t}_{0}r(t,s)g(s){\mathord{{\rm d}}}s.$
(2.12)
###### Proof.
First of all, if we define
$h(t):=g(t)+\int^{t}_{0}r(t,s)g(s){\mathord{{\rm d}}}s,$
then by (2.4) and (2.10)
$h(t)=g(t)+\int^{t}_{0}\kappa(t,s)h(s){\mathord{{\rm d}}}s\ \ \mbox{ for a.a.
$t\in(0,\infty)$}.$
Thus, by (2.11) we have
$\displaystyle
f(t)-h(t)\leqslant\int^{t}_{0}\kappa(t,s)(f(s)-h(s)){\mathord{{\rm d}}}s\ \
\mbox{ for a.a. $t\in(0,\infty)$}.$ (2.13)
Set $\tilde{f}(t):=f(t)-h(t)$ and define
$\tilde{f}^{*}(t):=\mathrm{ess}\\!\\!\sup_{s\in[0,t]}\tilde{f}(s),\ \ t>0$
and
$\tau_{0}:=\inf\\{t>0:\tilde{f}^{*}(t)>0\\}.$
Clearly, $t\mapsto\tilde{f}^{*}(t)$ is increasing and
$\displaystyle\tilde{f}(t)\leqslant 0\ \ \mbox{ for a.a. $t\in[0,\tau_{0})$}.$
(2.14)
We want to prove that
$\tau_{0}=+\infty.$
Suppose $\tau_{0}<+\infty$. Iterating inequality (2.13), we obtain
$\tilde{f}(t)\leqslant\int^{t}_{0}r_{n}(t,s)\tilde{f}(s){\mathord{{\rm
d}}}s\leqslant\int^{t}_{0}r_{n}(t,s)f(s){\mathord{{\rm d}}}s,\ \ \forall
n\in{\mathbb{N}}.$
By (2.9), one knows that $0<\tilde{f}^{*}(t)<+\infty$ for any $t>\tau_{0}$.
Moreover, we have
$\tilde{f}(t)\stackrel{{\scriptstyle(\ref{E33})}}{{\leqslant}}\int^{t}_{\tau_{0}}r_{n}(t,s)\tilde{f}(s){\mathord{{\rm
d}}}s\leqslant\tilde{f}^{*}(t)\int^{t}_{\tau_{0}}r_{n}(t,s){\mathord{{\rm
d}}}s,\ \ \forall n\in{\mathbb{N}}.$
So, for any $T>\tau_{0}$
$0<\tilde{f}^{*}(T)\leqslant\tilde{f}^{*}(T)\cdot\left\|\int^{\cdot}_{\tau_{0}}r_{n}(\cdot,s){\mathord{{\rm
d}}}s\right\|_{L^{\infty}(\tau_{0},T)}\stackrel{{\scriptstyle(\ref{QW2})}}{{\longrightarrow}}0$
as $n\to\infty$, which is impossible. So, $\tau_{0}=+\infty$. ∎
The following two examples show that (2.12) is sensitive to
$\kappa\in{\mathscr{K}}$.
###### Example 2.3.
For $C_{0}>0$, set
$\kappa_{C_{0}}(t,s):=\frac{C_{0}}{\sqrt{t^{2}-s^{2}}},\ \ s<t.$
It is clear that
$\int^{t}_{s}\kappa_{C_{0}}(t,u){\mathord{{\rm
d}}}u=C_{0}((\pi/2)-\arcsin(s/t)).$
From this, one sees that
$\displaystyle\left\\{\begin{aligned} &\kappa_{C_{0}}\notin{\mathscr{K}},\ \
\mbox{ if $C_{0}\geqslant 2/\pi$};\\\
&\kappa_{C_{0}}\in{\mathscr{K}}\cap{\mathscr{K}}^{c}_{0},\ \ \mbox{ if
$0<C_{0}<2/\pi$}.\end{aligned}\right.$
Consider the following Volterra equation
$x(t)=\int^{t}_{0}\kappa_{C_{0}}(t,s)x(s){\mathord{{\rm d}}}s,\ \ t\geqslant
0.$
If $C_{0}=1$, there are at least two solutions $x(t)\equiv 0$ and $x(t)=t$; if
$C_{0}=\frac{2}{\pi}$, there are infinitely many solutions
$x(t)\equiv\mathrm{constant}$; if $0<C_{0}<2/\pi$, by Lemma 2.2 there is only
one solution $x(t)\equiv 0$ in $L^{\infty}_{loc}({\mathbb{R}}_{+})$.
###### Example 2.4.
For $C_{0}>0$ and $\alpha,\beta\in[0,1)$, set
$\kappa_{C_{0}}^{\alpha,\beta}(t,s):=\frac{C_{0}}{(t-s)^{\alpha}s^{\beta}},\ \
s<t.$
It is clear that
$\displaystyle\int^{t}_{u}\kappa_{C_{0}}^{\alpha,\beta}(t,s){\mathord{{\rm
d}}}s=C_{0}t^{1-\alpha-\beta}\int^{1}_{u/t}\frac{1}{(1-s)^{\alpha}s^{\beta}}{\mathord{{\rm
d}}}s.$ (2.15)
From this, one sees that
$\displaystyle\left\\{\begin{aligned}
&\kappa_{C_{0}}^{\alpha,\beta}\notin{\mathscr{K}},\ \ \mbox{ if
$\alpha+\beta>1$ and $C_{0}>0$};\\\
&\kappa_{C_{0}}^{\alpha,\beta}\notin{\mathscr{K}},\ \ \mbox{ if
$\alpha+\beta=1$ and
$C_{0}\geqslant\int^{1}_{0}\frac{1}{(1-s)^{\alpha}s^{\beta}}{\mathord{{\rm
d}}}s$};\\\
&\kappa_{C_{0}}^{\alpha,\beta}\in{\mathscr{K}}\cap{\mathscr{K}}^{c}_{0},\ \
\mbox{ if $\alpha+\beta=1$ and
$C_{0}<\int^{1}_{0}\frac{1}{(1-s)^{\alpha}s^{\beta}}{\mathord{{\rm d}}}s$};\\\
&\kappa_{C_{0}}^{\alpha,\beta}\in{\mathscr{K}}_{>1},\ \ \mbox{ if
$\alpha+\beta<1$ and $C_{0}>0$}.\end{aligned}\right.$
Consider the following Volterra equation
$\displaystyle
x(t)=\int^{t}_{0}k^{\alpha,\beta}_{C_{0}}(t,s)x(s){\mathord{{\rm d}}}s,\ \
t\geqslant 0.$
If $\alpha+\beta<1$, by Lemma 2.2 there is only one solution $x(t)\equiv 0$ in
$L^{\infty}_{loc}({\mathbb{R}}_{+})$; if $\alpha=\beta=C_{0}=1/2$, there are
at least two solutions $x(t)\equiv 0$ and $x(t)=\sqrt{t}$.
### 2.2. Ito’s integral in $2$-smooth Banach spaces
Throughout this paper, we shall fix a stochastic basis
$(\Omega,{\mathcal{F}},P;({\mathcal{F}}_{t})_{t\geqslant 0})$, i.e., a
complete probability space with a family of right-continuous filterations. In
what follows, without special declarations, all expectations ${\mathbb{E}}$
are taken with respect to the probability measure $P$.
Let $\\{W^{k}(t):t\geqslant 0,k\in{\mathbb{N}}\\}$ be a sequence of
independent one dimensional standard Brownian motions on
$(\Omega,{\mathcal{F}},P;({\mathcal{F}}_{t})_{t\geqslant 0})$. Let $l^{2}$ be
the usual Hilbert space of all square summable real number sequences,
$\\{e_{k},k\in{\mathbb{N}}\\}$ the usual orthonormal basis of $l^{2}$. Let
${\mathbb{X}}$ be a separable Banach space, and $L(l^{2};{\mathbb{X}})$ the
set of all bounded linear operators from $l^{2}$ to ${\mathbb{X}}$. For an
operator $B\in L(l^{2};{\mathbb{X}})$, we also write
$B=(B_{1},B_{2},\cdots)\in{\mathbb{X}}^{\mathbb{N}},\ \ B_{k}=Be_{k}.$
###### Definition 2.5.
An operator $B\in L(l^{2};{\mathbb{X}})$ is called radonifying if
$\mbox{ the series }\sum_{k}Be_{k}\cdot W^{k}(1)\ \mbox{ converges in $\
L^{2}(\Omega;{\mathbb{X}})$}.$
We shall denote by $L_{2}(l^{2};{\mathbb{X}})$ the space of all radonifying
operators, and write for $B\in L_{2}(l^{2};{\mathbb{X}})$
$\displaystyle\|B\|_{L_{2}(l^{2};{\mathbb{X}})}:=\left({\mathbb{E}}\big{\|}Be_{k}\cdot
W^{k}(1)\big{\|}^{2}_{\mathbb{X}}\right)^{1/2}.$ (2.16)
Here and below, we use the convention that the repeated indices will be
summed.
The following proposition is well known, and a detailed proof was given in
[54, Proposition 2.5].
###### Proposition 2.6.
The space $L_{2}(l^{2};{\mathbb{X}})$ with norm (2.16) is a separable Banach
space.
In order to introduce the stochastic integral of an ${\mathbb{X}}$-valued
measurable $({\mathcal{F}}_{t})$-adapted process with respect to $W$, in the
sequel, we assume that ${\mathbb{X}}$ is $2$-smooth (cf. [60]), i.e., there
exists a constant $C_{\mathbb{X}}\geqslant 2$ such that for all
$x,y\in{\mathbb{X}}$
$\|x+y\|^{2}_{\mathbb{X}}+\|x-y\|^{2}_{\mathbb{X}}\leqslant
2\|x\|^{2}_{\mathbb{X}}+C_{\mathbb{X}}\|y\|^{2}_{\mathbb{X}}.$
Let now $s\mapsto B(s)$ be an $L_{2}(l^{2};{\mathbb{X}})$-valued measurable
and $({\mathcal{F}}_{t})$-adapted process with
$\int^{T}_{0}\|B(s)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}{\mathord{{\rm
d}}}s<+\infty,\ \ a.s.,\ \ \forall T>0.$
One can define the Itô stochastic integral (cf. [54, Section 3])
$t\mapsto{\mathcal{I}}_{t}(B):=\int^{t}_{0}B(s){\mathord{{\rm
d}}}W(s)=\int^{t}_{0}B_{k}(s)\cdot{\mathord{{\rm d}}}W^{k}(s)\in{\mathbb{X}}$
such that $t\mapsto{\mathcal{I}}_{t}(B)$ is an ${\mathbb{X}}$-valued
continuous local $({\mathcal{F}}_{t})$-martingale. Moreover, let $\tau$ be any
(${\mathcal{F}}_{t}$)-stopping time, then
$\int^{t\wedge\tau}_{0}B(s){\mathord{{\rm
d}}}W(s)=\int^{t}_{0}1_{\\{s<\tau\\}}\cdot B(s){\mathord{{\rm d}}}W(s).$
The following BDG inequality for ${\mathcal{I}}_{t}(B)$ holds (cf. [54,
Section 5]).
###### Theorem 2.7.
For any $p>0$, there exists a constant $C_{p}>0$ depending only on $p$ such
that
$\displaystyle{\mathbb{E}}\left(\sup_{t\in[0,T]}\Big{\|}\int^{t}_{0}B(s){\mathord{{\rm
d}}}W(s)\Big{\|}^{p}_{\mathbb{X}}\right)\leqslant
C_{p}{\mathbb{E}}\left(\int^{T}_{0}\|B(s)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}{\mathord{{\rm
d}}}s\right)^{p/2}.$ (2.17)
The following two typical examples of $2$-smooth Banach spaces are usually met
in applications.
###### Example 2.8.
Let ${\mathbb{X}}$ be a separable Hilbert space. Clearly, ${\mathbb{X}}$ is
$2$-smooth. In this case, $L_{2}(l^{2};{\mathbb{X}})$ consists of all Hilbert-
Schmidt operators of mapping $l^{2}$ into ${\mathbb{X}}$, and
$\displaystyle\|B\|_{L_{2}(l^{2};{\mathbb{X}})}=\left(\sum_{k=1}^{\infty}\|Be_{k}\|_{\mathbb{X}}^{2}\right)^{1/2}.$
###### Example 2.9.
Let $(E,{\mathcal{E}},\mu)$ be a measure space, ${\mathbb{H}}$ a separable
Hilbert space. For $p\geqslant 2$, let $L^{p}(E,\mu;{\mathbb{H}})$ be the
usual ${\mathbb{H}}$-valued $L^{p}$-space over $(E,{\mathcal{E}},\mu)$. Then
${\mathbb{X}}=L^{p}(E,\mu;{\mathbb{H}})$ is $2$-smooth (cf. [60, 10]). In this
case, by BDG’s inequality for Hilbert space valued martingale we have
$\displaystyle\|B\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}$ $\displaystyle=$
$\displaystyle{\mathbb{E}}\left(\int_{E}\big{\|}B_{k}(x)\cdot
W^{k}(1)\big{\|}^{p}_{\mathbb{H}}\mu({\mathord{{\rm d}}}x)\right)^{2/p}$
(2.18) $\displaystyle\leqslant$
$\displaystyle\left(\int_{E}{\mathbb{E}}\big{\|}B_{k}(x)\cdot
W^{k}(1)\big{\|}^{p}_{\mathbb{H}}\mu({\mathord{{\rm d}}}x)\right)^{2/p}$
$\displaystyle\leqslant$ $\displaystyle
C_{p}\left(\int_{E}\Big{(}\sum_{k=1}^{\infty}\|B_{k}(x)\|^{2}_{\mathbb{H}}\Big{)}^{p/2}\mu({\mathord{{\rm
d}}}x)\right)^{2/p}$ $\displaystyle=$ $\displaystyle
C_{p}\|B\|^{2}_{L^{p}(E,\mu;l^{2}\otimes{\mathbb{H}})}.$
Hence
$L^{p}(E,\mu;l^{2}\otimes{\mathbb{H}})\hookrightarrow
L_{2}(l^{2};{\mathbb{X}})=L_{2}(l^{2};L^{p}(E,\mu;{\mathbb{H}})).$
We also recall the following Kolmogorov continuity criterion, which can be
derived directly by Garsia’s inequality (cf. [77]).
###### Theorem 2.10.
Let $\\{X(t),t\geqslant 0\\}$ be an ${\mathbb{X}}$-valued stochastic process,
and $\tau$ a bounded random time. Suppose that for some $C_{0},p>0$ and
$\delta>1$
${\mathbb{E}}\|(X(t)-X(s))\cdot
1_{\\{s,t\in[0,\tau]\\}}\|^{p}_{\mathbb{X}}\leqslant C_{0}|t-s|^{\delta}.$
Then there exist constants $C_{1}>0$ and $a\in(0,(\delta-1)/p)$ independent of
$C_{0}$ and a continuous version $\tilde{X}$ of $X$ such that
${\mathbb{E}}\left(\sup_{s\not=t\in[0,\tau]}\frac{\|\tilde{X}(t)-\tilde{X}(s)\|^{p}_{\mathbb{X}}}{|t-s|^{ap}}\right)\leqslant
C_{1}\cdot C_{0}.$
### 2.3. A local non-linear interpolation lemma
In what follows, we fix a densely defined closed linear operator
${\mathfrak{L}}$ on ${\mathbb{X}}$ for which
$\displaystyle
S_{\phi}:=\\{\lambda\in{\mathbb{C}}:0<\phi\leqslant|\arg\lambda|\leqslant\pi\\}\subset\rho({\mathfrak{L}}),$
(2.19)
and for some $C\geqslant 1$
$\|(\lambda-{\mathfrak{L}})^{-1}\|_{L({\mathbb{X}})}\leqslant\frac{C}{1+|\lambda|},\
\ \lambda\in S_{\phi},$
where $\rho({\mathfrak{L}})$ denotes the resolvent set of ${\mathfrak{L}}$.
The above operator ${\mathfrak{L}}$ is also called sectorial (cf. [37, p.18]).
It is well known that ${\mathfrak{L}}$ generates an analytic semigroup
${\mathfrak{T}}_{t}=e^{-{\mathfrak{L}}t},\ \ t\geqslant 0.$
Moreover, we also assume that ${\mathfrak{L}}^{-1}$ is a bounded linear
operator on ${\mathbb{X}}$, i.e.,
$0\in\rho({\mathfrak{L}}).$
Thus, for any $\alpha\in{\mathbb{R}}$, the fractional power
${\mathfrak{L}}^{\alpha}$ is well defined (cf. [37, 58]). For $\alpha>0$, we
define the fractional Sobolev space ${\mathbb{X}}_{\alpha}$ by
${\mathbb{X}}_{\alpha}:={\mathscr{D}}({\mathfrak{L}}^{\alpha})$
with the norm
$\|x\|_{{\mathbb{X}}_{\alpha}}:=\|{\mathfrak{L}}^{\alpha}x\|_{\mathbb{X}}.$
For $\alpha<0$, ${\mathbb{X}}_{\alpha}$ is defined as the completion of
${\mathbb{X}}$ with respect to the above norm. It is clear that
${\mathbb{X}}_{\alpha}$ is still $2$-smooth, and $B\in
L_{2}(l^{2};{\mathbb{X}}_{\alpha})$ if and only if
${\mathfrak{L}}^{\alpha}B\in L_{2}(l^{2};{\mathbb{X}})$, i.e.,
$\displaystyle\|B\|_{L_{2}(l^{2};{\mathbb{X}}_{\alpha})}=\|{\mathfrak{L}}^{\alpha}B\|_{L_{2}(l^{2};{\mathbb{X}})}.$
(2.20)
The following properties are well known (cf. [37, p.24-27] or [58, p.74]).
###### Proposition 2.11.
1. (i)
${\mathfrak{T}}_{t}:{\mathbb{X}}\to{\mathbb{X}}_{\alpha}$ for each $t>0$ and
$\alpha>0$.
2. (ii)
For each $t>0$, $\alpha\in{\mathbb{R}}$ and every $x\in{\mathbb{X}}_{\alpha}$,
${\mathfrak{T}}_{t}{\mathfrak{L}}^{\alpha}x={\mathfrak{L}}^{\alpha}{\mathfrak{T}}_{t}x$.
3. (iii)
For some $\delta>0$ and each $t,\alpha>0$, the operator
${\mathfrak{L}}^{\alpha}{\mathfrak{T}}_{t}$ is bounded in ${\mathbb{X}}$ and
$\|{\mathfrak{L}}^{\alpha}{\mathfrak{T}}_{t}x\|_{\mathbb{X}}\leqslant
C_{\alpha}t^{-\alpha}e^{-\delta t}\|x\|_{\mathbb{X}},\ \ \forall
x\in{\mathbb{X}}.$
4. (iv)
Let $\alpha\in(0,1]$ and $x\in{\mathbb{X}}_{\alpha}$, then
$\|{\mathfrak{T}}_{t}x-x\|_{\mathbb{X}}\leqslant
C_{\alpha}t^{\alpha}\|x\|_{{\mathbb{X}}_{\alpha}}.$
5. (v)
For any $0\leqslant\beta<\alpha$
$\|x\|_{{\mathbb{X}}_{\beta}}\leqslant
C_{\alpha,\beta}\|x\|^{1-\frac{\beta}{\alpha}}\|x\|^{\frac{\beta}{\alpha}}_{{\mathbb{X}}_{\alpha}},\
\ \forall x\in{\mathbb{X}}_{\alpha}.$
We need the following embedding result.
###### Proposition 2.12.
For any $0<\theta<1$ and $\alpha>0$
$\displaystyle(L_{2}(l^{2};{\mathbb{X}}),L_{2}(l^{2};{\mathbb{X}}_{\alpha}))_{\theta,1}\subset
L_{2}(l^{2};({\mathbb{X}},{\mathbb{X}}_{\alpha})_{\theta,1})\subset
L_{2}(l^{2};{\mathbb{X}}_{\theta\alpha}),$ (2.21)
where $(\cdot,\cdot)_{\theta,1}$ stands for the real interpolation space
between two Banach spaces.
###### Proof.
We only prove the first embedding. The second embedding follows from [76,
p.101, (d) and (f)], i.e.,
$({\mathbb{X}},{\mathbb{X}}_{\alpha})_{\theta,1}\subset{\mathbb{X}}_{\theta\alpha}.$
Let
$B\in(L_{2}(l^{2};{\mathbb{X}}),L_{2}(l^{2};{\mathbb{X}}_{\alpha}))_{\theta,1}=:{\mathbb{B}}_{\theta,1}.$
By the $K$-method of real interpolation space, we have (cf. [76, p.24])
$\|B\|_{{\mathbb{B}}_{\theta,1}}=\int^{\infty}_{0}\frac{t^{-\theta}K(t,B)}{t}{\mathord{{\rm
d}}}t,$
where the $K$-function of $B$ is defined by
$K(t,B):=\inf_{B=B_{1}+B_{2}}\Big{\\{}\|B_{1}\|_{L_{2}(l^{2};{\mathbb{X}})}+t\|B_{2}\|_{L_{2}(l^{2};{\mathbb{X}}_{\alpha})}\Big{\\}},\
\ t\geqslant 0.$
By Definition 2.5 we have
$\displaystyle K(t,B)$ $\displaystyle=$
$\displaystyle\inf_{B=B_{1}+B_{2}}\Big{\\{}\Big{(}{\mathbb{E}}\|B_{1}e_{k}\cdot
W^{k}(1)\|^{2}_{{\mathbb{X}}}\Big{)}^{\frac{1}{2}}+t\Big{(}{\mathbb{E}}\|B_{2}e_{k}\cdot
W^{k}(1)\|^{2}_{{\mathbb{X}}_{\alpha}}\Big{)}^{\frac{1}{2}}\Big{\\}}$
$\displaystyle\geqslant$
$\displaystyle\inf_{B=B_{1}+B_{2}}\Big{\\{}\Big{(}{\mathbb{E}}\big{[}\|B_{1}e_{k}\cdot
W^{k}(1)\|_{{\mathbb{X}}}+t\|B_{2}e_{k}\cdot
W^{k}(1)\|_{{\mathbb{X}}_{\alpha}}\big{]}^{2}\Big{)}^{\frac{1}{2}}\Big{\\}}$
$\displaystyle\geqslant$
$\displaystyle\Big{(}{\mathbb{E}}\Big{[}\inf_{B=B_{1}+B_{2}}\Big{\\{}\|B_{1}e_{k}\cdot
W^{k}(1)\|_{{\mathbb{X}}}+t\|B_{2}e_{k}\cdot
W^{k}(1)\|_{{\mathbb{X}}_{\alpha}}\Big{\\}}\Big{]}^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\geqslant$
$\displaystyle\Big{(}{\mathbb{E}}\Big{[}K\big{(}t,Be_{k}\cdot
W^{k}(1)\big{)}\Big{]}^{2}\Big{)}^{\frac{1}{2}},$
where
$K\big{(}t,Be_{k}\cdot W^{k}(1)\big{)}:=\inf_{Be_{k}\cdot
W^{k}(1)=x_{1}+x_{2}}\Big{\\{}\|x_{1}\|_{{\mathbb{X}}}+t\|x_{2}\|_{{\mathbb{X}}_{\alpha}}\Big{\\}}.$
Therefore, by Minkowski’s inequality we obtain
$\displaystyle\|B\|_{{\mathbb{B}}_{\theta,1}}$ $\displaystyle\geqslant$
$\displaystyle\left({\mathbb{E}}\left[\int^{\infty}_{0}\frac{t^{-\theta}K\big{(}t,Be_{k}\cdot
W^{k}(1)\big{)}}{t}{\mathord{{\rm d}}}t\right]^{2}\right)^{\frac{1}{2}}$
$\displaystyle=$ $\displaystyle\left({\mathbb{E}}\|Be_{k}\cdot
W^{k}(1)\|^{2}_{({\mathbb{X}},{\mathbb{X}}_{\alpha})_{\theta,1}}\right)^{\frac{1}{2}}=\|B\|_{L_{2}(l^{2};({\mathbb{X}},{\mathbb{X}}_{\alpha})_{\theta,1})}.$
The result follows. ∎
The following local non-linear interpolation lemma will play a crucial role in
the proofs of Theorems 7.2 and 8.2 below. We refer to [75] for some other
nonlinear interpolation results.
###### Lemma 2.13.
Let $0\leqslant\alpha_{0}<\alpha_{1}\leqslant 1$ and
$0\leqslant\alpha_{2}<\alpha_{3}\leqslant 1$. Let
$\Psi:{\mathbb{X}}_{\alpha_{0}}\to L_{2}(l^{2};{\mathbb{X}}_{\alpha_{2}})$ be
a locally Lipschitz continuous map, and satisfy that for all $R>0$ and
$x\in{\mathbb{X}}_{\alpha_{1}}$ with
$\|x\|_{{\mathbb{X}}_{\alpha_{0}}}\leqslant R$
$\|\Psi(x)\|_{L_{2}(l^{2};{\mathbb{X}}_{\alpha_{3}})}\leqslant
C_{R}(1+\|x\|_{{\mathbb{X}}_{\alpha_{1}}}).$
Then for any $0<\theta^{\prime}<\theta<1$ and $R>0$
$\sup_{\|x\|_{{\mathbb{X}}_{\alpha_{0}+\theta(\alpha_{1}-\alpha_{0})}}\leqslant
R}\|\Psi(x)\|_{L_{2}(l^{2};{\mathbb{X}}_{\alpha_{2}+\theta^{\prime}(\alpha_{3}-\alpha_{2})})}\leqslant
C_{R}.$
###### Proof.
By (2.20), we may assume that $\alpha_{2}=0$. Fix $R>0$ and
$x\in{\mathbb{X}}_{\alpha_{0}+\theta(\alpha_{1}-\alpha_{0})}$ with
$\|x\|_{{\mathbb{X}}_{\alpha_{0}+\theta(\alpha_{1}-\alpha_{0})}}\leqslant R.$
Set for $t\geqslant 0$
$K(t,\Psi(x)):=\inf_{\Psi(x)=\Psi_{1}+\Psi_{2}}\Big{\\{}\|\Psi_{1}\|_{L_{2}(l^{2};{\mathbb{X}})}+t\|\Psi_{2}\|_{L_{2}(l^{2};{\mathbb{X}}_{\alpha_{3}})}\Big{\\}}.$
For $\delta>0$ and $t\in[0,1]$, noting that
$\|{\mathfrak{T}}_{t^{\delta}}x\|_{{\mathbb{X}}_{\alpha_{0}}}\preceq\|x\|_{{\mathbb{X}}_{\alpha_{0}}}\preceq\|x\|_{{\mathbb{X}}_{\alpha_{0}+\theta(\alpha_{1}-\alpha_{0})}}\preceq
R,$
by the assumptions and (iii) and (iv) of Proposition 2.11 we have
$\displaystyle K(t,\Psi(x))$ $\displaystyle\leqslant$
$\displaystyle\|\Psi(x)-\Psi({\mathfrak{T}}_{t^{\delta}}x)\|_{L_{2}(l^{2};{\mathbb{X}})}+t\|\Psi({\mathfrak{T}}_{t^{\delta}}x)\|_{L_{2}(l^{2};{\mathbb{X}}_{\alpha_{3}})}$
$\displaystyle\leqslant$ $\displaystyle
C_{R}\|{\mathfrak{T}}_{t^{\delta}}x-x\|_{{\mathbb{X}}_{\alpha_{0}}}+C_{R}t\cdot(1+\|{\mathfrak{T}}_{t^{\delta}}x\|_{{\mathbb{X}}_{\alpha_{1}}})$
$\displaystyle\leqslant$ $\displaystyle
C_{R}t^{\delta\theta(\alpha_{1}-\alpha_{0})}\cdot\|x\|_{{\mathbb{X}}_{\alpha_{0}+\theta(\alpha_{1}-\alpha_{0})}}$
$\displaystyle+C_{R}t\cdot\big{(}1+t^{-\delta(1-\theta)(\alpha_{1}-\alpha_{0})}\|x\|_{{\mathbb{X}}_{\alpha_{0}+\theta(\alpha_{1}-\alpha_{0})}}\big{)}$
$\displaystyle\leqslant$ $\displaystyle
C_{R}\big{(}t^{\delta\theta(\alpha_{1}-\alpha_{0})}+t+t^{1-\delta(1-\theta)(\alpha_{1}-\alpha_{0})}\big{)}.$
Letting $\delta=\frac{1}{\alpha_{1}-\alpha_{0}}$, we obtain that for
$t\in[0,1]$
$K(t,\Psi(x))\leqslant C_{R}(t^{\theta}+t)\leqslant C_{R}t^{\theta}.$
Moveover, it is clear that for $t\geqslant 1$
$K(t,\Psi(x))\leqslant\|\Psi(x)\|_{L_{2}(l^{2};{\mathbb{X}})}\leqslant
C_{R}\|x\|_{\alpha_{0}}+\|\Psi(0)\|_{L_{2}(l^{2};{\mathbb{X}})}\leqslant
C_{R}.$
Hence, for any $0<\theta^{\prime}<\theta<1$
$\displaystyle\|\Psi(x)\|_{(L_{2}(l^{2};{\mathbb{X}}),L_{2}(l^{2};{\mathbb{X}}_{\alpha_{3}}))_{\theta^{\prime},1}}$
$\displaystyle=$
$\displaystyle\int^{\infty}_{0}\frac{t^{-\theta^{\prime}}K(t;\Psi(x))}{t}{\mathord{{\rm
d}}}t$ $\displaystyle\leqslant$ $\displaystyle
C_{R}\left[\int^{1}_{0}\frac{t^{\theta-\theta^{\prime}}}{t}{\mathord{{\rm
d}}}t+\int^{\infty}_{1}\frac{t^{-\theta^{\prime}}}{t}{\mathord{{\rm
d}}}t\right]\leqslant C_{R}.$
The result follows by (2.21). ∎
### 2.4. A criterion for Laplace principles
It is well known that there exists a Hilbert space so that
$l^{2}\subset{\mathbb{U}}$ is Hilbert-Schmidt with embedding operator $J$ and
$\\{W^{k}(t),k\in{\mathbb{N}}\\}$ is a Brownian motion with values in
${\mathbb{U}}$, whose covariance operator is given by $Q=J\circ J^{*}$. For
example, one can take ${\mathbb{U}}$ as the completion of $l^{2}$ with respect
to the norm generated by scalar product
${\langle}h,h^{\prime}{\rangle}_{\mathbb{U}}:=\left(\sum_{k=1}^{\infty}\frac{h_{k}h^{\prime}_{k}}{k^{2}}\right)^{\frac{1}{2}},\
\ h,h^{\prime}\in l^{2}.$
For $T>0$ and a Banach space ${\mathbb{B}}$, we denote by
${\mathcal{B}}({\mathbb{B}})$ the Borel $\sigma$-field, and by
${\mathbb{C}}_{T}({\mathbb{B}})$ the continuous function space from $[0,T]$ to
${\mathbb{B}}$, which is endowed with the uniform norm. Define
$\displaystyle\ell^{2}_{T}:=\left\\{h=\int^{\cdot}_{0}\dot{h}(s){\mathord{{\rm
d}}}s:~{}~{}\dot{h}\in L^{2}(0,T;l^{2})\right\\}$ (2.22)
with the norm
$\|h\|_{\ell^{2}_{T}}:=\left(\int^{T}_{0}\|\dot{h}(s)\|_{l^{2}}^{2}{\mathord{{\rm
d}}}s\right)^{1/2},$
where the dot denotes the generalized derivative. Let $\mu$ be the law of the
Brownian motion $W$ in ${\mathbb{C}}_{T}({\mathbb{U}})$. Then
$({\mathbb{C}}_{T}({\mathbb{U}}),\ell^{2}_{T},\mu)$
forms an abstract Wiener space.
For $T,N>0$, set
${\mathbb{D}}_{N}:=\\{h\in\ell^{2}_{T}:\|h\|_{\ell^{2}_{T}}\leqslant N\\}$
and
$\displaystyle{\mathcal{A}}^{T}_{N}:=\left\\{\begin{aligned} &\mbox{
$h:[0,T]\to l^{2}$ is a continuous and $({\mathcal{F}}_{t})$-adapted }\\\
&\mbox{ process, and for almost all $\omega$},\ \
h(\cdot,\omega)\in{\mathbb{D}}_{N}\end{aligned}\right\\}.$ (2.23)
It is well known that with respect to the weak convergence topology in
$\ell^{2}_{T}$ (cf. [41]),
$\displaystyle\mbox{${\mathbb{D}}_{N}$ is metrizable as a compact Polish
space}.$ (2.24)
Let ${\mathbb{S}}$ be a Polish space. A function $I:{\mathbb{S}}\to[0,\infty]$
is given.
###### Definition 2.14.
The function $I$ is called a rate function if for every $a<\infty$, the set
$\\{f\in{\mathbb{S}}:I(f)\leqslant a\\}$ is compact in ${\mathbb{S}}$.
Let
$\\{Z_{\epsilon}:{\mathbb{C}}_{T}({\mathbb{U}})\to{\mathbb{S}},\epsilon\in(0,1)\\}$
be a family of measurable mappings. Assume that there is a measurable map
$Z_{0}:\ell^{2}_{T}\mapsto{\mathbb{S}}$ such that
1. (LD)1
For any $N>0$, if a family
$\\{h^{\epsilon},\epsilon\in(0,1)\\}\subset{\mathcal{A}}^{T}_{N}$ (as random
variables in ${\mathbb{D}}_{N}$) converges in distribution to
$h\in{\mathcal{A}}^{T}_{N}$, then for some subsequence $\epsilon_{k}$,
$Z_{\epsilon_{k}}\Big{(}\cdot+\frac{h^{\epsilon_{k}}(\cdot)}{\sqrt{\epsilon_{k}}}\Big{)}$
converges in distribution to $Z_{0}(h)$ in ${\mathbb{S}}$.
1. (LD)2
For any $N>0$, if $\\{h_{n},n\in{\mathbb{N}}\\}\subset{\mathbb{D}}_{N}$ weakly
converges to $h\in\ell^{2}_{T}$, then for some subsequence $h_{n_{k}}$,
$Z_{0}(h_{n_{k}})$ converges to $Z_{0}(h)$ in ${\mathbb{S}}$.
For each $f\in{\mathbb{S}}$, define
$\displaystyle
I(f):=\frac{1}{2}\inf_{\\{h\in\ell^{2}_{T}:~{}f=Z_{0}(h)\\}}\|h\|^{2}_{\ell^{2}_{T}},$
(2.25)
where $\inf\emptyset=\infty$ by convention. Then under (LD)2, $I(f)$ is a rate
function. In fact, assume that $I(f_{n})\leqslant a$. By the definition of
$I(f_{n})$, there exists a sequence $h_{n}\in\ell_{2}$ such that
$Z_{0}(h_{n})=f_{n}$ and
$\frac{1}{2}\|h_{n}\|_{\ell^{2}_{T}}^{2}\leqslant a+\frac{1}{n}.$
By the weak compactness of ${\mathbb{D}}_{2a+2}$, there exist a subsequence
$n_{k}$ (still denoted by $n$) and $h\in\ell^{2}_{T}$ such that $h_{n}$ weakly
converges to $h$ and
$\|h\|_{\ell^{2}_{T}}^{2}\leqslant\varliminf_{n\rightarrow\infty}\|h_{n}\|_{\ell^{2}_{T}}^{2}\leqslant
2a.$
Hence, by (LD)2 we have
$\lim_{k\to\infty}\|Z_{0}(h_{n_{k}})-Z_{0}(h)\|_{\mathbb{S}}=0$
and
$I(Z_{0}(h))\leqslant a.$
We recall the following result due to [9, 14] (see also [81, Theorem 4.4]).
###### Theorem 2.15.
Under (LD)1 and (LD)2, $\\{Z_{\epsilon},\epsilon\in(0,1)\\}$ satisfies the
Laplace principle with the rate function $I(f)$ given by (2.25). More
precisely, for each real bounded continuous function $g$ on ${\mathbb{S}}$:
$\displaystyle\lim_{\epsilon\rightarrow
0}\epsilon\log{\mathbb{E}}^{\mu}\left(\exp\left[-\frac{g(Z_{\epsilon})}{\epsilon}\right]\right)=-\inf_{f\in{\mathbb{S}}}\\{g(f)+I(f)\\}.$
(2.26)
In particular, the family of $\\{Z_{\epsilon},\epsilon\in(0,1)\\}$ satisfies
the large deviation principle in $({\mathbb{S}},{\mathcal{B}}({\mathbb{S}}))$
with the rate function $I(f)$. More precisely, let $\nu_{\epsilon}$ be the law
of $Z_{\epsilon}$ in $({\mathbb{S}},{\mathcal{B}}({\mathbb{S}}))$, then for
any $A\in{\mathcal{B}}({\mathbb{S}})$
$-\inf_{f\in A^{o}}I(f)\leqslant\liminf_{\epsilon\rightarrow
0}\epsilon\log\nu_{\epsilon}(A)\leqslant\limsup_{\epsilon\rightarrow
0}\epsilon\log\nu_{\epsilon}(A)\leqslant-\inf_{f\in\bar{A}}I(f),$
where the closure and the interior are taken in ${\mathbb{S}}$, and $I(f)$ is
defined by (2.25).
## 3\. Abstract stochastic Volterra integral equations
In this section, we consider the following stochastic Volterra integral
equation in a $2$-smooth Banach space ${\mathbb{X}}$:
$\displaystyle X(t)=g(t)+\int^{t}_{0}A(t,s,X(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}B(t,s,X(s)){\mathord{{\rm d}}}W(s),$ (3.1)
where $g(t)$ is an ${\mathbb{X}}$-valued measurable
$({\mathcal{F}}_{t})$-adapted process, and
$A:\triangle\times\Omega\times{\mathbb{X}}\to{\mathbb{X}}\in{\mathcal{M}}_{\triangle}\times{\mathcal{B}}({\mathbb{X}})/{\mathcal{B}}({\mathbb{X}})$
and
$B:\triangle\times\Omega\times{\mathbb{X}}\to
L_{2}(l^{2};{\mathbb{X}})\in{\mathcal{M}}_{\triangle}\times{\mathcal{B}}({\mathbb{X}})/{\mathcal{B}}(L_{2}(l^{2};{\mathbb{X}})).$
Here and below, $\triangle:=\\{(t,s)\in{\mathbb{R}}^{2}_{+}:s\leqslant t\\}$,
and ${\mathcal{M}}_{\triangle}$ denotes the progressively measurable
$\sigma$-field on $\triangle\times\Omega$ generated by the sets
$E\in{\mathcal{B}}(\triangle)\times{\mathcal{F}}$ with properties:
$1_{E}(t,s,\cdot)\in{\mathcal{F}}_{s}$ for all $(t,s)\in\triangle$, and
$s\mapsto 1_{E}(t,s,\omega)$ is right continuous for any
$t\in{\mathbb{R}}_{+}$ and $\omega\in\Omega$.
We start with the global existence and uniqueness of solutions of Eq.(3.1)
under global Lipschitz conditions and singular kernels.
### 3.1. Global existence and uniqueness
In this subsection, we make the following global Lipschitz and linear growth
conditions on the coefficients:
1. (H1)
For some $p\geqslant 2$ and any $T>0$
$\mathrm{ess}\\!\\!\sup_{t\in[0,T]}\int^{t}_{0}[\kappa_{1}(t,s)+\kappa_{2}(t,s)]\cdot{\mathbb{E}}\|g(s)\|_{\mathbb{X}}^{p}{\mathord{{\rm
d}}}s<+\infty,$
where $\kappa_{1}$ and $\kappa_{2}$ are from (H2) and (H3) below.
2. (H2)
There exists $\kappa_{1}\in{\mathscr{K}}_{0}$ such that for all
$(t,s)\in\triangle$, $\omega\in\Omega$ and $x\in{\mathbb{X}}$
$\|A(t,s,\omega,x)\|_{\mathbb{X}}\leqslant\kappa_{1}(t,s)\cdot(\|x\|_{\mathbb{X}}+1)$
and
$\|B(t,s,\omega,x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}\leqslant\kappa_{1}(t,s)\cdot(\|x\|_{\mathbb{X}}^{2}+1).$
3. (H3)
There exists $\kappa_{2}\in{\mathscr{K}}_{0}$ such that for all
$(t,s)\in\triangle$, $\omega\in\Omega$ and $x,y\in{\mathbb{X}}$
$\|A(t,s,\omega,x)-A(t,s,\omega,y)\|_{\mathbb{X}}\leqslant\kappa_{2}(t,s)\cdot\|x-y\|_{\mathbb{X}}$
and
$\|B(t,s,\omega,x)-B(t,s,\omega,y)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}\leqslant\kappa_{2}(t,s)\cdot\|x-y\|_{\mathbb{X}}^{2}.$
We now prove the following basic existence and uniqueness result.
###### Theorem 3.1.
Assume that (H1)-(H3) hold. Then there exists a unique measurable
$({\mathcal{F}}_{t})$-adapted process $X(t)$ such that for almost all
$t\geqslant 0$,
$\displaystyle X(t)=g(t)+\int^{t}_{0}A(t,s,X(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}B(t,s,X(s)){\mathord{{\rm d}}}W(s),\ \ \mbox{ $P$-a.s.}$
(3.2)
and for any $T>0$ and some $C_{T,p,\kappa_{1}}>0$,
$\displaystyle{\mathbb{E}}\|X(t)\|^{p}_{\mathbb{X}}\leqslant
C_{T,p,\kappa_{1}}\left[{\mathbb{E}}\|g(t)\|_{\mathbb{X}}^{p}+\mathrm{ess\\!\\!}\sup_{t\in[0,T]}\int^{t}_{0}\kappa_{1}(t,s)\cdot{\mathbb{E}}\|g(s)\|_{\mathbb{X}}^{p}{\mathord{{\rm
d}}}s\right]$ (3.3)
for almost all $t\in[0,T]$, where $p$ is from (H1). Moreover, if
$\displaystyle t\mapsto\int^{t}_{0}\kappa_{1}(t,s){\mathord{{\rm d}}}s\in
L^{\infty}({\mathbb{R}}_{+}),$ (3.4)
then for almost all $t\geqslant 0$
$\displaystyle{\mathbb{E}}\|X(t)\|^{p}_{\mathbb{X}}$ $\displaystyle\leqslant$
$\displaystyle
C_{p,\kappa_{1}}\bigg{(}{\mathbb{E}}\|g(t)\|_{\mathbb{X}}^{p}+\int^{t}_{0}\tilde{\kappa}_{1}(t,s)\cdot{\mathbb{E}}\|g(s)\|_{\mathbb{X}}^{p}{\mathord{{\rm
d}}}s$ (3.5)
$\displaystyle+\int^{t}_{0}r_{\tilde{\kappa}_{1}}(t,u)\cdot\left[\int^{u}_{0}\tilde{\kappa}_{1}(u,s)\cdot{\mathbb{E}}\|g(s)\|_{\mathbb{X}}^{p}{\mathord{{\rm
d}}}s\right]{\mathord{{\rm d}}}u\bigg{)},$
where $\tilde{\kappa}_{1}=\tilde{C}_{p,\kappa_{1}}\cdot\kappa_{1}$,
$r_{\tilde{\kappa}_{1}}$ is defined by (2.3) in terms of $\tilde{\kappa}_{1}$,
and $C_{p,\kappa_{1}},\tilde{C}_{p,\kappa_{1}}$ are constants only depending
on $p,\kappa_{1}$.
###### Proof.
We use Picard’s iteration to prove the existence. Let $X_{1}(t):=g(t)$, and
define recursively for $n\in{\mathbb{N}}$
$\displaystyle X_{n+1}(t)=g(t)+\int^{t}_{0}A(t,s,X_{n}(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}B(t,s,X_{n}(s)){\mathord{{\rm d}}}W(s).$ (3.6)
Fix $T>0$ below. By (H2), the BDG inequality (2.17) and Hölder’s inequality we
have
$\displaystyle{\mathbb{E}}\|X_{n+1}(t)\|^{p}_{\mathbb{X}}$
$\displaystyle\preceq$
$\displaystyle{\mathbb{E}}\|g(t)\|_{\mathbb{X}}^{p}+{\mathbb{E}}\left(\int^{t}_{0}\|A(t,s,X_{n}(s))\|_{\mathbb{X}}{\mathord{{\rm
d}}}s\right)^{p}$ (3.7)
$\displaystyle+{\mathbb{E}}\left\|\int^{t}_{0}B(t,s,X_{n}(s)){\mathord{{\rm
d}}}W(s)\right\|^{p}_{\mathbb{X}}$ $\displaystyle\preceq$
$\displaystyle{\mathbb{E}}\|g(t)\|_{\mathbb{X}}^{p}+{\mathbb{E}}\left(\int^{t}_{0}\kappa_{1}(t,s)\cdot(\|X_{n}(s)\|_{\mathbb{X}}+1){\mathord{{\rm
d}}}s\right)^{p}$
$\displaystyle+{\mathbb{E}}\left(\int^{t}_{0}\|B(t,s,X_{n}(s))\|_{L_{2}(l^{2};{\mathbb{X}})}^{2}{\mathord{{\rm
d}}}s\right)^{\frac{p}{2}}$ $\displaystyle\preceq$
$\displaystyle{\mathbb{E}}\|g(t)\|_{\mathbb{X}}^{p}+\int^{t}_{0}\kappa_{1}(t,s)\cdot{\mathbb{E}}(\|X_{n}(s)\|^{p}_{\mathbb{X}}+1){\mathord{{\rm
d}}}s\cdot\left(\int^{t}_{0}\kappa_{1}(t,s){\mathord{{\rm d}}}s\right)^{p-1}$
$\displaystyle+\int^{t}_{0}\kappa_{1}(t,s)\cdot{\mathbb{E}}(\|X_{n}(s)\|^{p}_{{\mathbb{X}}}+1){\mathord{{\rm
d}}}s\cdot\left(\int^{t}_{0}\kappa_{1}(t,s){\mathord{{\rm
d}}}s\right)^{\frac{p}{2}-1}$ $\displaystyle\preceq$
$\displaystyle{\mathbb{E}}\|g(t)\|_{\mathbb{X}}^{p}+C_{T,p}\cdot
C_{T}+C_{T,p}\int^{t}_{0}\kappa_{1}(t,s)\cdot{\mathbb{E}}\|X_{n}(s)\|^{p}_{\mathbb{X}}{\mathord{{\rm
d}}}s,$
where
$C_{T}:=\mathrm{ess}\sup_{t\in[0,T]}|\int^{t}_{0}\kappa_{1}(t,s){\mathord{{\rm
d}}}s|$ and $C_{T,p}:=C_{T}^{p-1}+C^{(p-2)/2}_{T}$.
Set
$f_{m}(t):=\sup_{n=1,\cdots,m}{\mathbb{E}}\|X_{n}(t)\|^{p}_{\mathbb{X}}.$
Then
$f_{m}(t)\leqslant
C_{T,p,\kappa_{1}}\Big{(}{\mathbb{E}}\|g(t)\|_{\mathbb{X}}^{p}+1\Big{)}+\int^{t}_{0}\tilde{\kappa}_{1}(t,s)\cdot
f_{m}(s){\mathord{{\rm d}}}s,$
where $\tilde{\kappa}_{1}=C_{T,p,\kappa_{1}}\cdot\kappa_{1}$ and the constant
$C_{T,p,\kappa_{1}}$ is independent of $m$.
Let $r_{\tilde{\kappa}_{1}}$ be defined by (2.3) in terms of
$\tilde{\kappa}_{1}$. Note that by (2.4)
$\displaystyle\int^{t}_{0}r_{\tilde{\kappa}_{1}}(t,s)\cdot{\mathbb{E}}\|g(s)\|_{\mathbb{X}}^{p}{\mathord{{\rm
d}}}s-\int^{t}_{0}\tilde{\kappa}_{1}(t,s)\cdot{\mathbb{E}}\|g(s)\|_{\mathbb{X}}^{p}{\mathord{{\rm
d}}}s$
$\displaystyle\quad=\int^{t}_{0}\left(\int^{t}_{s}r_{\tilde{\kappa}_{1}}(t,u)\tilde{\kappa}_{1}(u,s){\mathord{{\rm
d}}}u\right)\cdot{\mathbb{E}}\|g(s)\|_{\mathbb{X}}^{p}{\mathord{{\rm d}}}s$
$\displaystyle\quad=\int^{t}_{0}r_{\tilde{\kappa}_{1}}(t,u)\left(\int^{u}_{0}\tilde{\kappa}_{1}(u,s)\cdot{\mathbb{E}}\|g(s)\|_{\mathbb{X}}^{p}{\mathord{{\rm
d}}}s\right){\mathord{{\rm d}}}u.$
Hence, by Lemma 2.2 and (H1), we obtain that for almost all $t\in[0,T]$
$\displaystyle\sup_{n\in{\mathbb{N}}}{\mathbb{E}}\|X_{n}(t)\|^{p}_{\mathbb{X}}$
$\displaystyle=$ $\displaystyle\lim_{m\to\infty}f_{m}(t)\leqslant
C_{T,p,\kappa_{1}}\left({\mathbb{E}}\|g(t)\|_{\mathbb{X}}^{p}+\int^{t}_{0}r_{\tilde{\kappa}_{1}}(t,s)\cdot{\mathbb{E}}\|g(s)\|_{\mathbb{X}}^{p}{\mathord{{\rm
d}}}s\right)$ (3.8) $\displaystyle\leqslant$ $\displaystyle
C_{T,p,\kappa_{1}}\bigg{(}{\mathbb{E}}\|g(t)\|_{\mathbb{X}}^{p}+\int^{t}_{0}\tilde{\kappa}_{1}(t,s)\cdot{\mathbb{E}}\|g(s)\|_{\mathbb{X}}^{p}{\mathord{{\rm
d}}}s$
$\displaystyle\qquad+\int^{t}_{0}r_{\tilde{\kappa}_{1}}(t,u)\left(\int^{u}_{0}\tilde{\kappa}_{1}(u,s)\cdot{\mathbb{E}}\|g(s)\|_{\mathbb{X}}^{p}{\mathord{{\rm
d}}}s\right){\mathord{{\rm d}}}u\bigg{)}$
$\displaystyle\stackrel{{\scriptstyle(\ref{R3})}}{{\leqslant}}$ $\displaystyle
C_{T,p,\kappa_{1}}\left[{\mathbb{E}}\|g(t)\|_{\mathbb{X}}^{p}+\mathrm{ess\\!\\!}\sup_{t\in[0,T]}\int^{t}_{0}\kappa_{1}(t,s)\cdot{\mathbb{E}}\|g(s)\|_{\mathbb{X}}^{p}{\mathord{{\rm
d}}}s\right].$ (3.9)
On the other hand, set
$Z_{n,m}(t):=X_{n}(t)-X_{m}(t).$
As the above calculations, by (H3) we have
$\displaystyle{\mathbb{E}}\|Z_{n+1,m+1}(t)\|^{2}_{\mathbb{X}}$
$\displaystyle\preceq$
$\displaystyle{\mathbb{E}}\left\|\int^{t}_{0}(A(t,s,X_{n}(s))-A(t,s,X_{m}(s))){\mathord{{\rm
d}}}s\right\|^{2}_{\mathbb{X}}$
$\displaystyle+{\mathbb{E}}\left\|\int^{t}_{0}(B(t,s,X_{n}(s))-B(t,s,X_{m}(s))){\mathord{{\rm
d}}}W(s)\right\|^{2}_{\mathbb{X}}$ $\displaystyle\preceq$
$\displaystyle\int^{t}_{0}\kappa_{2}(t,s)\cdot{\mathbb{E}}\|Z_{n,m}(s)\|^{2}_{\mathbb{X}}{\mathord{{\rm
d}}}s.$
Set
$f(t):=\limsup_{n,m\rightarrow\infty}{\mathbb{E}}\|Z_{n,m}(t)\|^{2}_{\mathbb{X}}.$
By (3.9), (H1) and using Fatou’s lemma, we get
$f(t)\preceq\int^{t}_{0}\kappa_{2}(t,s)\cdot f(s){\mathord{{\rm d}}}s.$
By Lemma 2.2 again, we have for almost all $t\in[0,T]$
$f(t)=\limsup_{n,m\rightarrow\infty}{\mathbb{E}}\|Z_{n,m}(t)\|^{2}_{\mathbb{X}}=0.$
Hence, there exists an ${\mathbb{X}}$-valued $({\mathcal{F}}_{t})$-adapted
process $X(t)$ such that for almost all $t\in[0,T]$
$\lim_{n\rightarrow\infty}{\mathbb{E}}\|X_{n}(t)-X(t)\|^{2}_{\mathbb{X}}=0.$
Taking limits for (3.6), one finds that (3.2) holds.
Moreover, the estimate (3.3) follows from (3.9). Note that when (3.4) is
satisfied, the constant $C_{T,p}$ in (3.7) is independent of $T$. Hence, the
estimate (3.5) is direct from (3.8). The uniqueness follows by similar
calculations as above. ∎
###### Example 3.2.
Let for $\delta>0$
$h(s):=\frac{e^{-\delta s}}{s\log^{2}s},\ \ t>s\geqslant 0.$
It is easy to see that $h\in L^{1}({\mathbb{R}}_{+})$. Consider the following
stochastic Volterra equation:
$X(t)=x_{0}\sqrt{|\log(t\wedge 1)|}+\int^{t}_{0}h(t-s)A(X(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}\sqrt{h(t-s)}B(X(s)){\mathord{{\rm d}}}W(s),$
where $A:{\mathbb{X}}\to{\mathbb{X}}$ and $B:{\mathbb{X}}\to
L_{2}(l^{2};{\mathbb{X}})$ are global Lipschitz continuous functions. By
elementary calculations, one finds that
$\sup_{t\geqslant 0}\int^{t}_{0}\frac{e^{-\delta(t-s)}|\log(s\wedge
1)|}{(t-s)\log^{2}(t-s)}{\mathord{{\rm d}}}s<+\infty.$
So, (H1)-(H3) are satisfied with $p=2$. Moreover, by (2.8) and (3.5), one
finds that if $\delta$ is large enough, then for any $T>0$
$\sup_{t\geqslant T}{\mathbb{E}}\|X(t)\|^{2}_{\mathbb{X}}<+\infty.$
We remark that in this example, $X(0)=\infty$.
### 3.2. Path continuity of solutions
In this subsection, in addition to (H2) and (H3), we also assume that
1. (H1)′
The process $t\mapsto g(t)$ is continuous and $({\mathcal{F}}_{t})$-adapted,
and for any $p\geqslant 2$ and $T>0$
${\mathbb{E}}\left(\sup_{t\in[0,T]}\|g(t)\|^{p}_{\mathbb{X}}\right)<+\infty.$
1. (H4)
For all $s<t<t^{\prime}$, $\omega\in\Omega$ and $x\in{\mathbb{X}}$
$\|A(t^{\prime},s,\omega,x)-A(t,s,\omega,x)\|_{\mathbb{X}}\leqslant\lambda(t^{\prime},t,s)\cdot(\|x\|_{\mathbb{X}}+1)$
and
$\|B(t^{\prime},s,\omega,x)-B(t,s,\omega,x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}\leqslant\lambda(t^{\prime},t,s)\cdot(\|x\|_{\mathbb{X}}^{2}+1),$
where $\lambda$ is a positive measurable function satisfying that for any
$T>0$ and some $\gamma=\gamma(T),C=C(T)>0$
$\displaystyle\int^{t}_{0}\lambda(t^{\prime},t,s){\mathord{{\rm d}}}s\leqslant
C|t^{\prime}-t|^{\gamma},\ \ 0\leqslant t<t^{\prime}\leqslant T.$ (3.10)
###### Theorem 3.3.
Assume that (H1)′ and (H2)-(H4) hold, and the kernel function $\kappa_{1}$ in
(H2) belongs to ${\mathscr{K}}_{>1}$. Then there exists a unique
${\mathbb{X}}$-valued continuous $({\mathcal{F}}_{t})$-adapted process $X(t)$
such that $P$-a.s., for all $t\geqslant 0$
$\displaystyle X(t)=g(t)+\int^{t}_{0}A(t,s,X(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}B(t,s,X(s)){\mathord{{\rm d}}}W(s)$ (3.11)
and for any $p\geqslant 2$ and $T>0$,
$\displaystyle{\mathbb{E}}\left(\sup_{t\in[0,T]}\|X(t)\|^{p}_{\mathbb{X}}\right)<+\infty.$
(3.12)
Moreover, if for some $\delta>0$ and any $p\geqslant 2,T>0$, it holds that
${\mathbb{E}}\|g(t^{\prime})-g(t)\|^{p}_{\mathbb{X}}\leqslant
C_{T,p}|t^{\prime}-t|^{\delta p},$
then, $t\mapsto X(t)$ admits a Hölder continuous modification and for any
$p\geqslant 2,T>0$ and some $a>0$
${\mathbb{E}}\left(\sup_{t\not=t^{\prime}\in[0,T]}\frac{\|X(t^{\prime})-X(t)\|^{p}_{{\mathbb{X}}}}{|t^{\prime}-t|^{ap}}\right)\leqslant
C_{T,p,a}.$
###### Proof.
First of all, for any $p\geqslant 2$ and $T>0$, by (H1)′ and (3.3) we have
$\displaystyle\mathrm{ess\\!\\!}\sup_{t\in[0,T]}{\mathbb{E}}\|X(t)\|^{p}_{\mathbb{X}}<+\infty.$
(3.13)
Set
$J(t):=\int^{t}_{0}B(t,s,X(s)){\mathord{{\rm d}}}W(s)$
and write for $0\leqslant t<t^{\prime}\leqslant T$
$\displaystyle J(t^{\prime})-J(t)$ $\displaystyle=$
$\displaystyle\int^{t}_{0}\big{[}B(t^{\prime},s,X(s))-B(t,s,X(s))\big{]}{\mathord{{\rm
d}}}W(s)$
$\displaystyle+\int^{t^{\prime}}_{t}B(t^{\prime},s,X(s)){\mathord{{\rm
d}}}W(s)=:J_{1}(t^{\prime},t)+J_{2}(t^{\prime},t).$
In view of $\kappa_{1}\in{\mathscr{K}}_{>1}$, (2.6) holds for some $\beta>1$.
Fix $p\geqslant 2(\beta^{*}:=\beta/(\beta-1))$. By the BDG inequality (2.17),
(H2) and Hölder’s inequality we have
$\displaystyle{\mathbb{E}}\|J_{2}(t^{\prime},t)\|^{p}_{\mathbb{X}}$
$\displaystyle\preceq$
$\displaystyle{\mathbb{E}}\left(\int^{t^{\prime}}_{t}\kappa_{1}(t^{\prime},s)\cdot(\|X(s)\|^{2}_{\mathbb{X}}+1){\mathord{{\rm
d}}}s\right)^{\frac{p}{2}}$ $\displaystyle\preceq$
$\displaystyle\left(\int^{t^{\prime}}_{t}k^{\beta}_{1}(t^{\prime},s){\mathord{{\rm
d}}}s\right)^{\frac{p}{2\beta}}{\mathbb{E}}\left(\int^{t^{\prime}}_{t}(\|X(s)\|^{2\beta^{*}}_{\mathbb{X}}+1){\mathord{{\rm
d}}}s\right)^{\frac{p}{2\beta^{*}}}$
$\displaystyle\stackrel{{\scriptstyle(\ref{C2})}}{{\preceq}}$
$\displaystyle|t^{\prime}-t|^{\frac{p}{2\beta^{*}}-1}\int^{t^{\prime}}_{t}({\mathbb{E}}\|X(s)\|^{p}_{\mathbb{X}}+1){\mathord{{\rm
d}}}s$ $\displaystyle\stackrel{{\scriptstyle(\ref{PP0})}}{{\preceq}}$
$\displaystyle|t^{\prime}-t|^{\frac{p}{2\beta^{*}}},$
and by (H4) and Minkowski’s inequality
$\displaystyle{\mathbb{E}}\|J_{1}(t^{\prime},t)\|^{p}_{\mathbb{X}}$
$\displaystyle\preceq$
$\displaystyle{\mathbb{E}}\left(\int^{t}_{0}\lambda(t^{\prime},t,s)\cdot(\|X(s)\|^{2}_{\mathbb{X}}+1){\mathord{{\rm
d}}}s\right)^{\frac{p}{2}}$ $\displaystyle\preceq$
$\displaystyle\left(\int^{t}_{0}\lambda(t^{\prime},t,s)\cdot(({\mathbb{E}}\|X(s)\|^{p}_{\mathbb{X}})^{\frac{2}{p}}+1){\mathord{{\rm
d}}}s\right)^{\frac{p}{2}}$
$\displaystyle\stackrel{{\scriptstyle(\ref{PP0})}}{{\preceq}}$
$\displaystyle\left(\int^{t}_{0}\lambda(t^{\prime},t,s){\mathord{{\rm
d}}}s\right)^{\frac{p}{2}}$
$\displaystyle\stackrel{{\scriptstyle(\ref{C1})}}{{\preceq}}$
$\displaystyle|t^{\prime}-t|^{\frac{\gamma p}{2}}.$
Hence, for all $0\leqslant t<t^{\prime}\leqslant T$
${\mathbb{E}}\|J(t^{\prime})-J(t)\|^{p}_{\mathbb{X}}\preceq|t-t^{\prime}|^{\frac{\gamma
p}{2}}+|t-t^{\prime}|^{\frac{p}{2\beta^{*}}}.$
Similarly, we may prove that for all $0\leqslant t<t^{\prime}\leqslant T$ and
$p\geqslant\beta^{*}$
${\mathbb{E}}\left\|\int^{t^{\prime}}_{0}A(t^{\prime},s,X(s)){\mathord{{\rm
d}}}s-\int^{t}_{0}A(t,s,X(s)){\mathord{{\rm
d}}}s\right\|^{p}_{\mathbb{X}}\preceq|t-t^{\prime}|^{\gamma
p}+|t-t^{\prime}|^{\frac{p}{\beta^{*}}}.$
The desired conclusions follow from Theorem 2.10. ∎
We conclude this subsection by proving a lemma, which will be used frequently
later. We put it here since the proof is similar to Theorem 3.3.
###### Lemma 3.4.
Let $\tau$ be an (${\mathcal{F}}_{t}$)-stopping time and
$G:\triangle\times\Omega\to
L_{2}(l^{2};{\mathbb{X}})\in{\mathcal{M}}_{\triangle}/{\mathcal{B}}(L_{2}(l^{2};{\mathbb{X}})).$
Assume that for all $0\leqslant s<t<t^{\prime}$ and $\omega\in\Omega$
$\displaystyle\|G(t,s,\omega)\|_{L_{2}(l^{2};{\mathbb{X}})}^{2}$
$\displaystyle\leqslant$ $\displaystyle\kappa(t,s)\cdot f^{2}(s,\omega),$
(3.14)
$\displaystyle\|G(t^{\prime},s,\omega)-G(t,s,\omega)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}$
$\displaystyle\leqslant$ $\displaystyle\lambda(t^{\prime},t,s)\cdot
f^{2}(s,\omega),$ (3.15)
where $\kappa\in{\mathscr{K}}_{>1}$ and for any $T>0$ and some $\alpha>1$ and
$\gamma>0$
$\int^{t}_{0}\lambda^{\alpha}(t^{\prime},t,s){\mathord{{\rm d}}}s\leqslant
C_{T}|t^{\prime}-t|^{\gamma},\ \ \forall 0\leqslant t<t^{\prime}\leqslant T,$
and $(s,\omega)\mapsto f(s,\omega)$ is a positive measurable process with
${\mathbb{E}}\left(\int^{T\wedge\tau}_{0}f^{p}(s){\mathord{{\rm
d}}}s\right)<+\infty,\ \ \forall p\geqslant 2.$
Then $t\mapsto J(t):=\int^{t}_{0}G(t,s){\mathord{{\rm d}}}W(s)\in{\mathbb{X}}$
admits a continuous modification on $[0,\tau)$, and for any $T>0$ and $p$
large enough
$\displaystyle{\mathbb{E}}\left(\sup_{t\in[0,T\wedge\tau]}\left\|\int^{t}_{0}G(t,s){\mathord{{\rm
d}}}W(s)\right\|^{p}_{\mathbb{X}}\right)\leqslant
C_{T}{\mathbb{E}}\left(\int^{T\wedge\tau}_{0}f^{p}(s){\mathord{{\rm
d}}}s\right),$
where the constant $C_{T}$ is independent of $f$ and $\tau$.
###### Proof.
Fix $T>0$ and write for $0\leqslant t<t^{\prime}\leqslant T$
$\displaystyle J(t^{\prime})-J(t)$ $\displaystyle=$
$\displaystyle\int^{t^{\prime}}_{t}G(t^{\prime},s){\mathord{{\rm
d}}}W(s)+\int^{t}_{0}[G(t^{\prime},s)-G(t,s)]{\mathord{{\rm d}}}W(s)$
$\displaystyle=:$ $\displaystyle J_{1}(t^{\prime},t)+J_{2}(t^{\prime},t).$
In view of $\kappa\in{\mathscr{K}}_{>1}$ and (2.6), by the BDG inequality
(2.17) and Hölder’s inequality we have, for some $\beta>1$ and $p\geqslant
2(\beta^{*}=\beta/(\beta-1))$,
$\displaystyle{\mathbb{E}}\|J_{1}(t^{\prime},t)\cdot
1_{\\{t^{\prime},t\in[0,\tau)\\}}\|^{p}_{\mathbb{X}}$ $\displaystyle\leqslant$
$\displaystyle{\mathbb{E}}\left\|\int^{t^{\prime}\wedge\tau}_{t\wedge\tau}G(t^{\prime},s){\mathord{{\rm
d}}}W(s)\right\|_{\mathbb{X}}^{p}$ $\displaystyle\preceq$
$\displaystyle{\mathbb{E}}\left(\int^{t^{\prime}\wedge\tau}_{t\wedge\tau}\|G(t^{\prime},s)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}{\mathord{{\rm
d}}}s\right)^{p/2}$
$\displaystyle\stackrel{{\scriptstyle(\ref{PL3})}}{{\preceq}}$
$\displaystyle{\mathbb{E}}\left(\int^{t^{\prime}\wedge\tau}_{t\wedge\tau}\kappa(t^{\prime},s)\cdot
f^{2}(s){\mathord{{\rm d}}}s\right)^{p/2}$ $\displaystyle\preceq$
$\displaystyle\left(\int^{t^{\prime}}_{t}\kappa^{\beta}(t^{\prime},s){\mathord{{\rm
d}}}s\right)^{\frac{p}{2\beta}}\cdot{\mathbb{E}}\left(\int^{t^{\prime}\wedge\tau}_{t\wedge\tau}f^{2\beta^{*}}(s){\mathord{{\rm
d}}}s\right)^{\frac{p}{2\beta^{*}}}$ $\displaystyle\preceq$
$\displaystyle|t^{\prime}-t|^{\frac{p}{2\beta^{*}}-1}\cdot{\mathbb{E}}\left(\int^{T\wedge\tau}_{0}f^{p}(s){\mathord{{\rm
d}}}s\right)$
and for $p\geqslant 2(\alpha^{*}=\alpha/(\alpha-1))$,
$\displaystyle{\mathbb{E}}\|J_{1}(t^{\prime},t)\cdot
1_{\\{t^{\prime},t\in[0,\tau)\\}}\|^{p}_{\mathbb{X}}$ $\displaystyle\preceq$
$\displaystyle{\mathbb{E}}\left(\int^{t\wedge\tau}_{0}\|G(t^{\prime},s)-G(t,s)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}{\mathord{{\rm
d}}}s\right)^{p/2}$
$\displaystyle\stackrel{{\scriptstyle(\ref{PL4})}}{{\preceq}}$
$\displaystyle{\mathbb{E}}\left(\int^{t\wedge\tau}_{0}\lambda(t^{\prime},t,s)\cdot
f^{2}(s){\mathord{{\rm d}}}s\right)^{p/2}$ $\displaystyle\preceq$
$\displaystyle\left(\int^{t}_{0}\lambda^{\alpha}(t^{\prime},t,s){\mathord{{\rm
d}}}s\right)^{\frac{p}{2\alpha}}\cdot{\mathbb{E}}\left(\int^{t\wedge\tau}_{0}f^{2\alpha^{*}}(s){\mathord{{\rm
d}}}s\right)^{\frac{p}{2\alpha^{*}}}$ $\displaystyle\preceq$
$\displaystyle|t^{\prime}-t|^{\frac{\gamma
p}{2\alpha}}\cdot{\mathbb{E}}\left(\int^{T\wedge\tau}_{0}f^{p}(s){\mathord{{\rm
d}}}s\right).$
Hence, for any $p\geqslant 2(\alpha^{*}\vee\beta^{*})$ and $0\leqslant
t<t^{\prime}\leqslant T$,
${\mathbb{E}}\|(J(t^{\prime})-J(t))\cdot
1_{\\{t^{\prime},t\in[0,\tau)\\}}\|^{p}_{\mathbb{X}}\preceq|t^{\prime}-t|^{\left(\frac{p}{2\beta^{*}}-1\right)\wedge\frac{\gamma
p}{2\alpha}}\cdot{\mathbb{E}}\left(\int^{T\wedge\tau}_{0}f^{p}(s){\mathord{{\rm
d}}}s\right).$
The desired result now follows by Theorem 2.10. ∎
### 3.3. Local existence and uniqueness
In this subsection, we assume that
1. (H2)′
For any $R>0$, there exists $\kappa_{1,R}\in{\mathscr{K}}_{>1}$ such that for
all $(t,s)\in\triangle$, $\omega\in\Omega$ and $x\in{\mathbb{X}}$ with
$\|x\|_{\mathbb{X}}\leqslant R$
$\|A(t,s,\omega,x)\|_{\mathbb{X}}+\|B(t,s,\omega,x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}\leqslant\kappa_{1,R}(t,s).$
1. (H3)′
For any $R>0$, there exists $\kappa_{2,R}\in{\mathscr{K}}_{0}$ such that for
all $(t,s)\in\triangle$, $\omega\in\Omega$ and $x,y\in{\mathbb{X}}$ with
$\|x\|_{\mathbb{X}},\|y\|_{\mathbb{X}}\leqslant R$
$\|A(t,s,\omega,x)-A(t,s,\omega,y)\|_{\mathbb{X}}\leqslant\kappa_{2,R}(t,s)\cdot\|x-y\|_{\mathbb{X}}$
and
$\|B(t,s,\omega,x)-B(t,s,\omega,y)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}\leqslant\kappa_{2,R}(t,s)\cdot\|x-y\|_{\mathbb{X}}^{2}.$
1. (H4)′
For any $R>0$, there exists a measurable function $\lambda_{R}$ satisfying
that for any $T>0$ and some $\gamma,C>0$
$\int^{t}_{0}\lambda_{R}(t^{\prime},t,s){\mathord{{\rm d}}}s\leqslant
C|t^{\prime}-t|^{\gamma},\ \ 0\leqslant t<t^{\prime}\leqslant T,$
such that for all $s<t<t^{\prime}$, $\omega\in\Omega$ and $x\in{\mathbb{X}}$
with $\|x\|_{\mathbb{X}}\leqslant R$,
$\displaystyle\|A(t^{\prime},s,\omega,x)-A(t,s,\omega,x)\|_{\mathbb{X}}+\|B(t^{\prime},s,\omega,x)-B(t,s,\omega,x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}$
$\displaystyle\qquad\qquad\leqslant\lambda_{R}(t^{\prime},t,s).$
We first introduce the following notion of local solutions.
###### Definition 3.5.
Let $\tau$ be an $({\mathcal{F}}_{t})$-stopping time, and
$\\{X(t);t\in[0,\tau)\\}$ an ${\mathbb{X}}$-valued continuous
$({\mathcal{F}}_{t})$-adapted process. The pair of $(X,\tau)$ is called a
local solution of Eq.(3.1) if $P$-a.s., for all $t\in[0,\tau)$
$\displaystyle X(t)=g(t)+\int^{t}_{0}A(t,s,X(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}B(t,s,X(s)){\mathord{{\rm d}}}W(s);$
$(X,\tau)$ is called a maximal solution of Eq.(3.1) if
$\lim_{t\uparrow\tau(\omega)}\|X(t,\omega)\|_{\mathbb{X}}=+\infty\ \mbox{ on
$~{}~{}\\{\omega:\tau(\omega)<+\infty\\}$},\ P-a.s..$
We call $(X,\tau)$ a non-explosion solution of Eq.(3.1) if
$P\\{\omega:\tau(\omega)<+\infty\\}=0.$
###### Remark 3.6.
The stochastic integral in the above definition is defined on $[0,\tau)$ by
$\int^{t}_{0}B(t,s,X(s)){\mathord{{\rm
d}}}W(s)=\lim_{n\rightarrow\infty}\int^{t\wedge\tau_{n}}_{0}B(t,s,X(s)){\mathord{{\rm
d}}}W(s),\ \ t<\tau,$
where $\tau_{n}:=\inf\\{t>0:\|X(t)\|_{\mathbb{X}}>n\\}\nearrow\tau$.
We now prove the following main result in this section.
###### Theorem 3.7.
Under (H1)′-(H4)′, there exists a unique maximal solution $(X,\tau)$ for
Eq.(3.1) in the sense of Definition 3.5.
###### Proof.
For $n\in{\mathbb{N}}$, let $\chi_{n}$ be a positive smooth function on
${\mathbb{R}}_{+}$ with $\chi_{n}(s)=1,s\leqslant n$ and
$\chi_{n}(s)=0,s\geqslant n+1$. Define
$\displaystyle A_{n}(t,s,\omega,x)$ $\displaystyle:=$ $\displaystyle
A(t,s,\omega,x)\cdot\chi_{n}(\|x\|_{\mathbb{X}})$ $\displaystyle
B_{n}(t,s,\omega,x)$ $\displaystyle:=$ $\displaystyle
B(t,s,\omega,x)\cdot\chi_{n}(\|x\|_{\mathbb{X}}).$
It is easy to see that for $A_{n}$ and $B_{n}$, (H2) holds with
$\kappa_{1,n+1}$, (H4) holds with $\lambda_{n+1}$, and (H3) holds with some
$\kappa_{3,n}\in{\mathscr{K}}_{0}$. Thus, by Theorem 3.3 there exists a unique
continuous (${\mathcal{F}}_{t}$)-adapted process $X_{n}(t)$ such that for any
$p\geqslant 2$ and $T>0$
${\mathbb{E}}\left(\sup_{t\in[0,T]}\|X_{n}(t)\|^{p}_{\mathbb{X}}\right)\leqslant
C_{T,p,n}$
and
$\displaystyle X_{n}(t)=g(t)+\int^{t}_{0}A_{n}(t,s,X_{n}(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}B_{n}(t,s,X_{n}(s)){\mathord{{\rm d}}}W(s).$ (3.16)
We have the following claim:
Let $\tau$ be any stopping time. The uniqueness holds for (3.16) on
$[0,\tau)$.
We remark that when $\tau=T$ is non-random, it follows from Theorem 3.1. Let
$X_{i}(t),i=1,2$ be two ${\mathbb{X}}$-valued continuous
$({\mathcal{F}}_{t})$-adapted processes, and satisfy on $[0,\tau)$
$X_{i}(t)=g(t)+\int^{t}_{0}A_{n}(t,s,X_{i}(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}B_{n}(t,s,X_{i}(s)){\mathord{{\rm d}}}W(s),\ i=1,2.$
Set
$Z(t):=X_{1}(t)-X_{2}(t).$
Since $\kappa_{3,n}\in{\mathscr{K}}_{0}$, as the calculations in (3.7), by the
BDG inequality (2.17) and (H3) for $A_{n}$ and $B_{n}$, we have
$\displaystyle{\mathbb{E}}\|Z(t)\cdot 1_{\\{t<\tau\\}}\|^{p}_{\mathbb{X}}$
$\displaystyle\preceq$
$\displaystyle{\mathbb{E}}\left(\int^{t\wedge\tau}_{0}\kappa_{3,n}(t,s)\cdot\|Z(s)\|_{\mathbb{X}}{\mathord{{\rm
d}}}s\right)^{p}$ (3.17)
$\displaystyle+{\mathbb{E}}\left(\int^{t\wedge\tau}_{0}\kappa_{3,n}(t,s)\cdot\|Z(s)\|^{2}_{\mathbb{X}}{\mathord{{\rm
d}}}s\right)^{\frac{p}{2}}$ $\displaystyle=$
$\displaystyle{\mathbb{E}}\left(\int^{t}_{0}\kappa_{3,n}(t,s)\cdot
1_{\\{s<\tau\\}}\cdot\|Z(s)\|_{\mathbb{X}}{\mathord{{\rm d}}}s\right)^{p}$
$\displaystyle+{\mathbb{E}}\left(\int^{t}_{0}\kappa_{3,n}(t,s)\cdot
1_{\\{s<\tau\\}}\cdot\|Z(s)\|^{2}_{\mathbb{X}}{\mathord{{\rm
d}}}s\right)^{\frac{p}{2}}$ $\displaystyle\preceq$
$\displaystyle\int^{t}_{0}\kappa_{3,n}(t,s)\cdot{\mathbb{E}}\|Z(s)\cdot
1_{\\{s<\tau\\}}\|^{p}_{\mathbb{X}}{\mathord{{\rm d}}}s.$
By Lemma 2.2, we get
${\mathbb{E}}\|Z(t)\cdot 1_{\\{t<\tau\\}}\|^{p}_{\mathbb{X}}=0\ \ \mbox{ for
almost all $t\in[0,T]$},$
which implies by the arbitrariness of $T$ and the continuities of
$X_{i}(t),i=1,2$,
$X_{1}(\cdot)|_{[0,\tau)}=X_{2}(\cdot)|_{[0,\tau)}.$
The claim is proved.
Now, for $n\in{\mathbb{N}}$, define the stopping times
$\tau_{n}:=\inf\\{t>0:\|X_{n}(t)\|_{\mathbb{X}}>n\\}$
and
$\sigma_{n}:=\inf\\{t>0:\|X_{n+1}(t)\|_{\mathbb{X}}>n\\}.$
By the above claim, we have
$X_{n}(\cdot)|_{[0,\tau_{n}\wedge\sigma_{n})}=X_{n+1}(\cdot)|_{[0,\tau_{n}\wedge\sigma_{n})},$
which implies
$\tau_{n}\leqslant\sigma_{n}\leqslant\tau_{n+1},\ \ a.e..$
Hence, we may define
$\tau(\omega):=\lim_{n\to\infty}\tau_{n}(\omega)$
and for all $t<\tau(\omega)$
$X(t,\omega):=X_{n}(t,\omega),\ \ \mbox{ if $t<\tau_{n}(\omega)$}.$
Clearly, $(X,\tau)$ is a maximal solution of Eq.(3.1) in the sense of
Definition 3.5.
We next prove the uniqueness. Let $(\tilde{X},\tilde{\tau})$ be another
maximal solution of Eq.(3.1) in the sense of Definition 3.5. Define the
stopping times
$\tilde{\tau}_{n}:=\inf\\{t>0:\|\tilde{X}(t)\|_{\mathbb{X}}>n\\}$
and
$\hat{\tau}_{n}:=\tau_{n}\wedge\tilde{\tau}_{n},\ \
\hat{\tau}:=\tau\wedge\tilde{\tau}.$
It is clear that
$\hat{\tau}_{n}\nearrow\hat{\tau}\ \ a.s.\ \mbox{ as $n\to\infty$}$
and
$\displaystyle 1_{[0,\hat{\tau}_{n})}(t)\cdot\tilde{X}(t)$ $\displaystyle=$
$\displaystyle 1_{[0,\hat{\tau}_{n})}(t)\cdot
g(t)+1_{[0,\hat{\tau}_{n})}(t)\cdot\int^{t}_{0}A(t,s,\tilde{X}(s)){\mathord{{\rm
d}}}s$
$\displaystyle+1_{[0,\hat{\tau}_{n})}(t)\cdot\int^{t}_{0}B(t,s,\tilde{X}(s)){\mathord{{\rm
d}}}W(s)$ $\displaystyle=$ $\displaystyle 1_{[0,\hat{\tau}_{n})}(t)\cdot
g(t)+1_{[0,\hat{\tau}_{n})}(t)\cdot\int^{t}_{0}A_{n}(t,s,\tilde{X}(s)){\mathord{{\rm
d}}}s$
$\displaystyle+1_{[0,\hat{\tau}_{n})}(t)\cdot\int^{t}_{0}B_{n}(t,s,\tilde{X}(s)){\mathord{{\rm
d}}}W(s).$
By the above claim again, we have
$X(\cdot)|_{[0,\hat{\tau}_{n})}=\tilde{X}(\cdot)|_{[0,\hat{\tau}_{n})}.$
So
$X(\cdot)|_{[0,\hat{\tau})}=\tilde{X}(\cdot)|_{[0,\hat{\tau})}.$
By the definition of maximal solution we must have
$\hat{\tau}=\tau=\tilde{\tau}$. ∎
We have the following simple criterion of non explosion.
###### Theorem 3.8.
Assume that (H1)′, (H2) and (H4) hold, and $\kappa_{1}$ in (H2) belongs to
${\mathscr{K}}_{>1}$. Then there is no explosion for Eq.(3.1).
###### Proof.
Let $(X,\tau)$ be a maximal solution of Eq.(3.1). Define
$\tau_{n}:=\inf\\{t>0:\|X(t)\|_{\mathbb{X}}\geqslant n\\}.$
By the BDG inequality (2.17) and Hölder’s inequality, and using the same
method as estimating (3.17), we have, for any $T>0$, some $\beta>1$ and
$p\geqslant 2(\beta^{*}=\beta/(\beta-1))$
$\displaystyle{\mathbb{E}}\|X(t)\cdot
1_{\\{t\leqslant\tau_{n}\\}}\|^{p}_{\mathbb{X}}$ $\displaystyle\preceq$
$\displaystyle{\mathbb{E}}\|g(t)\|_{\mathbb{X}}^{p}+{\mathbb{E}}\left(\int^{t\wedge\tau_{n}}_{0}\|A(t,s,X(s))\|_{\mathbb{X}}{\mathord{{\rm
d}}}s\right)^{p}$
$\displaystyle+{\mathbb{E}}\left\|\int^{t\wedge\tau_{n}}_{0}B(t,s,X(s)){\mathord{{\rm
d}}}W(s)\right\|^{p}_{\mathbb{X}}$ $\displaystyle\preceq$
$\displaystyle{\mathbb{E}}\|g(t)\|_{\mathbb{X}}^{p}+{\mathbb{E}}\left(\int^{t\wedge\tau_{n}}_{0}\kappa_{1}(t,s)\cdot(\|X(s)\|_{\mathbb{X}}+1){\mathord{{\rm
d}}}s\right)^{p}$
$\displaystyle+{\mathbb{E}}\left(\int^{t\wedge\tau_{n}}_{0}\|B(t,s,X(s))\|_{L_{2}(l^{2};{\mathbb{X}})}^{2}{\mathord{{\rm
d}}}s\right)^{\frac{p}{2}}$ $\displaystyle\preceq$
$\displaystyle{\mathbb{E}}\|g(t)\|_{\mathbb{X}}^{p}+{\mathbb{E}}\left(\int^{t\wedge\tau_{n}}_{0}(\|X(s)\|^{\beta^{*}}_{\mathbb{X}}+1){\mathord{{\rm
d}}}s\right)^{\frac{p}{\beta^{*}}}$
$\displaystyle+{\mathbb{E}}\left(\int^{t\wedge\tau_{n}}_{0}(\|X(s)\|^{2\beta^{*}}_{\mathbb{X}}+1){\mathord{{\rm
d}}}s\right)^{\frac{p}{2\beta^{*}}}$ $\displaystyle\leqslant$ $\displaystyle
C_{T,p}\left[{\mathbb{E}}\|g(s)\|_{\mathbb{X}}^{p}+1+\int^{t}_{0}{\mathbb{E}}\|X(s)\cdot
1_{\\{s\leqslant\tau_{n}\\}}\|^{p}_{\mathbb{X}}{\mathord{{\rm d}}}s\right],$
where the constant $C_{T,p}$ is independent of $n$.
By Gronwall’s inequality, we obtain
$\sup_{t\in[0,T]}{\mathbb{E}}\|X(t)\cdot
1_{\\{t\leqslant\tau_{n}\\}}\|^{p}_{\mathbb{X}}\leqslant C_{T,p}.$
Using this estimate, as in the proofs of Theorem 3.3 and Lemma 3.4, we can
prove that for any $T>0$ and $p\geqslant 2$
$\sup_{n\in{\mathbb{N}}}{\mathbb{E}}\left(\sup_{t\in[0,T\wedge\tau_{n}]}\|X(t)\|^{p}_{\mathbb{X}}\right)\leqslant
C_{T,p}.$
Hence,
$\displaystyle\lim_{n\to\infty}P\\{\tau_{n}\leqslant T\\}$ $\displaystyle=$
$\displaystyle\lim_{n\to\infty}P\left\\{\sup_{t\in[0,T\wedge\tau_{n}]}\|X(t)\|_{\mathbb{X}}\geqslant
n\right\\}$ $\displaystyle\leqslant$
$\displaystyle\lim_{n\to\infty}{\mathbb{E}}\left(\sup_{t\in[0,T\wedge\tau_{n}]}\|X(t)\|^{p}_{\mathbb{X}}\right)/n^{p}$
$\displaystyle\leqslant$ $\displaystyle\lim_{n\to\infty}C_{T,p}/n^{p}=0,$
which produces the non-explosion, i.e., $P\\{\tau<\infty\\}=0$. ∎
###### Remark 3.9.
One cannot directly prove
$\sup_{n\in{\mathbb{N}}}{\mathbb{E}}\|X(t\wedge\tau_{n})\|^{p}_{\mathbb{X}}<+\infty,\
\ \forall t\geqslant 0$
to obtain the non-explosion, because it does not in general make sense to
write
$\int^{t\wedge\tau_{n}}_{0}B(t\wedge\tau_{n},s,X(s)){\mathord{{\rm d}}}W(s).$
### 3.4. Continuous dependence of solutions with respect to data
In this subsection, we study the continuous dependence of solutions for
Eq.(3.1) with respect to the coefficients.
Let $\\{(g_{m},A_{m},B_{m}),m\in{\mathbb{N}}\\}$ be a sequence of coefficients
associated to Eq.(3.1). Assume that for each $m\in{\mathbb{N}}$,
$(g_{m},A_{m},B_{m})$ satisfies (H1)′-(H4)′ with the same
$\kappa_{1,R},\kappa_{2,R}$ and $\lambda_{R}$ as $(g,A,B)$, and for each
$p\geqslant 2$
$\displaystyle\lim_{m\to\infty}\sup_{t\in[0,T]}{\mathbb{E}}\|g_{m}(t)-g(t)\|^{p}_{\mathbb{X}}=0$
(3.18)
and for each $T,R>0$,
$\displaystyle\lim_{m\to\infty}\sup_{t\in[0,T],\|x\|_{{\mathbb{X}}}\leqslant
R}\int^{t}_{0}\|A_{m}(t,s,x)-A(t,s,x)\|_{\mathbb{X}}{\mathord{{\rm d}}}s=0,$
(3.19)
$\displaystyle\lim_{m\to\infty}\sup_{t\in[0,T],\|x\|_{{\mathbb{X}}}\leqslant
R}\int^{t}_{0}\|B_{m}(t,s,x)-B(t,s,x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}{\mathord{{\rm
d}}}s=0.$ (3.20)
Let $(X_{m},\tau_{m})$ (resp. $(X,\tau)$) be the unique maximal solution
associated with $(g_{m},A_{m},B_{m})$ (resp. $(g,A,B)$). For each $R>0$ and
$m\in{\mathbb{N}}$, define
$\tau^{R}_{m}:=\inf\\{t>0:\|X(t)\|_{\mathbb{X}},\|X_{m}(t)\|_{\mathbb{X}}>R\\}.$
Suppose that for each $t>0$
$\displaystyle\lim_{R\to\infty}\sup_{m}P\\{\tau^{R}_{m}<t\\}=0.$ (3.21)
Then we have:
###### Theorem 3.10.
For each $t>0$ and $\epsilon>0$
$\lim_{m\to\infty}P\big{\\{}\|X_{m}(t)-X(t)\|_{\mathbb{X}}\geqslant\epsilon\big{\\}}=0.$
###### Proof.
For $R>0$ and $m\in{\mathbb{N}}$, set
$Z^{R}_{m}(t):=(X_{m}(t)-X(t))\cdot 1_{\\{t\leqslant\tau^{R}_{m}\\}}.$
Then
$Z^{R}_{m}(t)=J^{R}_{1,m}(t)+J^{R}_{2,m}(t)+J^{R}_{3,m}(t)+J^{R}_{4,m}(t)+J^{R}_{5,m}(t),$
where
$\displaystyle J^{R}_{1,m}(t)$ $\displaystyle:=$ $\displaystyle
1_{\\{t\leqslant\tau^{R}_{m}]}\cdot[g_{m}(t)-g(t)],$ $\displaystyle
J^{R}_{2,m}(t)$ $\displaystyle:=$ $\displaystyle
1_{\\{t\leqslant\tau^{R}_{m}]}\cdot\int^{t\wedge\tau^{R}_{m}}_{0}\big{[}A_{m}(t,s,X_{n}(s))-A_{m}(t,s,X(s))\big{]}{\mathord{{\rm
d}}}s,$ $\displaystyle J^{R}_{3,m}(t)$ $\displaystyle:=$ $\displaystyle
1_{\\{t\leqslant\tau^{R}_{m}]}\cdot\int^{t\wedge\tau^{R}_{m}}_{0}\big{[}A_{m}(t,s,X(s))-A(t,X(s))\big{]}{\mathord{{\rm
d}}}s,$ $\displaystyle J^{R}_{4,m}(t)$ $\displaystyle:=$ $\displaystyle
1_{\\{t\leqslant\tau^{R}_{m}]}\cdot\int^{t\wedge\tau^{R}_{m}}_{0}\big{[}B_{m}(t,s,X_{m}(s))-B_{m}(t,s,X(s))\big{]}{\mathord{{\rm
d}}}W(s),$ $\displaystyle J^{R}_{5,m}(t)$ $\displaystyle:=$ $\displaystyle
1_{\\{t\leqslant\tau^{R}_{m}]}\cdot\int^{t\wedge\tau^{R}_{m}}_{0}\big{[}B_{m}(t,s,X(s))-B(t,s,X(s))\big{]}{\mathord{{\rm
d}}}W(s).$
Fix $T>0$. Clearly, for any $p\geqslant 2$ and $t\in[0,T]$
${\mathbb{E}}\|J^{R}_{1,m}(t)\|^{p}_{\mathbb{X}}\leqslant\sup_{t\in[0,T]}{\mathbb{E}}\|g_{m}(t)-g(t)\|^{p}_{\mathbb{X}}=:{\mathcal{J}}_{1,m}.$
For $J^{R}_{2,m}(t)$, by (H3)′ and Hölder’s inequality we have, for $p$ large
enough ($\kappa_{2,R}\in{\mathscr{K}}_{>1}$)
$\displaystyle{\mathbb{E}}\|J^{R}_{2,m}(t)\|^{p}_{\mathbb{X}}$
$\displaystyle\leqslant$
$\displaystyle{\mathbb{E}}\left(\int^{t\wedge\tau^{R}_{m}}_{0}\kappa_{2,R}(t,s)\cdot\|X_{m}(s)-X(s)\|_{\mathbb{X}}{\mathord{{\rm
d}}}s\right)^{p}$ $\displaystyle\leqslant$
$\displaystyle\left[\int^{t}_{0}\kappa^{\beta}_{2,R}(t,s){\mathord{{\rm
d}}}s\right]^{\frac{p}{\beta}}\cdot{\mathbb{E}}\left[\int^{t}_{0}\|Z^{R}_{m}(s)\|^{\beta^{*}}_{\mathbb{X}}{\mathord{{\rm
d}}}s\right]^{\frac{p}{\beta^{*}}}$ $\displaystyle\leqslant$ $\displaystyle
C\int^{t}_{0}{\mathbb{E}}\|Z^{R}_{m}(s)\|^{p}_{\mathbb{X}}{\mathord{{\rm
d}}}s.$
For $J^{R}_{3,m}(t)$, we have
$\displaystyle{\mathbb{E}}\|J^{R}_{3,m}(t)\|^{p}_{\mathbb{X}}$
$\displaystyle\leqslant$
$\displaystyle{\mathbb{E}}\left(\sup_{\|x\|_{\mathbb{X}}\leqslant
R}\int^{t\wedge\tau^{R}_{m}}_{0}\|A_{m}(t,s,x)-A(t,s,x)\|_{\mathbb{X}}{\mathord{{\rm
d}}}s\right)^{p}$ $\displaystyle\leqslant$
$\displaystyle\left(\sup_{t\in[0,T]}\sup_{\|x\|_{\mathbb{X}}\leqslant
R}\int^{t}_{0}\|A_{m}(t,s,x)-A(t,s,x)\|_{\mathbb{X}}{\mathord{{\rm
d}}}s\right)^{p}=:{\mathcal{J}}^{R}_{3,m}.$
Similarly, by the BDG inequality (2.17) we have, for $p$ large enough
${\mathbb{E}}\|J^{R}_{4,m}(t)\|^{p}_{\mathbb{X}}\leqslant
C\int^{t}_{0}{\mathbb{E}}\|Z^{R}_{m}(s)\|^{p}_{\mathbb{X}}{\mathord{{\rm
d}}}s$
and
${\mathbb{E}}\|J^{R}_{5,m}(t)\|^{p}_{\mathbb{X}}\leqslant
C_{p}\left(\sup_{t\in[0,T]}\sup_{\|x\|_{\mathbb{X}}\leqslant
R}\int^{t}_{0}\|B_{m}(t,s,x)-B(t,s,x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}{\mathord{{\rm
d}}}s\right)^{\frac{p}{2}}=:{\mathcal{J}}^{R}_{5,m}.$
Combining the above calculations, we get
${\mathbb{E}}\|Z^{R}_{m}(t)\|^{p}_{\mathbb{X}}\leqslant{\mathcal{J}}_{1,m}+{\mathcal{J}}^{R}_{3,m}+{\mathcal{J}}^{R}_{5,m}+C\int^{t}_{0}{\mathbb{E}}\|Z^{R}_{m}(s)\|^{p}_{\mathbb{X}}{\mathord{{\rm
d}}}s.$
By Gronwall’s inequality and (3.18)-(3.20) we get, for any $R>0$ and $p$ large
enough
$\displaystyle\lim_{m\to\infty}{\mathbb{E}}\|Z^{R}_{m}(t)\|^{p}_{\mathbb{X}}=0.$
Hence
$\displaystyle
P\big{\\{}\|X_{m}(t)-X(t)\|_{\mathbb{X}}\geqslant\epsilon\big{\\}}$
$\displaystyle\leqslant$ $\displaystyle
P\big{\\{}\|X_{m}(t)-X(t)\|_{\mathbb{X}}\cdot
1_{\\{t\leqslant\tau^{R}_{m}\\}}\geqslant\epsilon\big{\\}}+P\big{\\{}\tau^{R}_{m}<t\big{\\}}$
$\displaystyle\leqslant$
$\displaystyle{\mathbb{E}}\|Z^{R}_{m}(t)\|^{p}_{\mathbb{X}}/\epsilon^{p}+P\big{\\{}\tau^{R}_{m}<t\big{\\}}.$
First letting $m\to\infty$, then $R\to\infty$, we then get the desired limit
by (3.21). ∎
## 4\. Large deviation for stochastic Volterra equations
In this section, we study the large deviation of small perturbations for
stochastic Volterra equations. In addition to (H2)′, (H3)′ and (H4)′, we
assume that $g$ and $A,B$ are non-random, and
1. (H1)′′
For any $T>0$ and some $\delta>0$,
$\|g(t)-g(t^{\prime})\|_{\mathbb{X}}\leqslant C|t-t^{\prime}|^{\delta},\ \
t,t^{\prime}\in[0,T]$
and for some $\alpha>0$,
$\sup_{t\in[0,T]}\|g(t)\|_{{\mathbb{X}}_{\alpha}}<+\infty.$
1. (H2)′′
For the same $\alpha$ as in (H1)′′ and any $R>0$, there exists a kernel
function $\kappa_{\alpha,R}\in{\mathscr{K}}_{0}$ such that for all
$(t,s)\in\triangle$ and $x\in{\mathbb{X}}$ with $\|x\|_{\mathbb{X}}\leqslant
R$
$\|A(t,s,x)\|_{{\mathbb{X}}_{\alpha}}+\|B(t,s,x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}}_{\frac{\alpha}{2}})}\leqslant\kappa_{\alpha,R}(t,s).$
###### Remark 4.1.
If the $\kappa_{\alpha,R}$ in (H2)′′ belongs to ${\mathscr{K}}_{>1}$, then
(H2)′′ implies (H2)′ in view of
${\mathbb{X}}_{\alpha}\hookrightarrow{\mathbb{X}}$.
Consider the following small perturbation of stochastic Volterra equation
(3.1)
$\displaystyle
X_{\epsilon}(t)=g(t)+\int^{t}_{0}A(t,s,X_{\epsilon}(s)){\mathord{{\rm
d}}}s+\sqrt{\epsilon}\int^{t}_{0}B(t,s,X_{\epsilon}(s)){\mathord{{\rm
d}}}W(s),$ (4.1)
where $\epsilon\in(0,1)$. By Theorem 3.7, there exists a unique maximal
solution $(X_{\epsilon},\tau_{\epsilon})$ for Eq.(4.1). Below, we fix $T>0$
and work in the finite time interval $[0,T]$, and assume that for each
$\epsilon\in(0,1)$
$\tau_{\epsilon}>T,\ \ a.s..$
By Yamada-Watanabe’s theorem (cf. [54, 67]), there exists a measurable mapping
$\Phi_{\epsilon}:{\mathbb{C}}_{T}({\mathbb{U}})\to{\mathbb{C}}_{T}({\mathbb{X}})$
such that
$X_{\epsilon}(t,\omega)=\Phi_{\epsilon}(W(\cdot,\omega))(t).$
It should be noticed that although the equation considered in [54] is a little
different from Eq.(3.1), the proof is obviously adapted to our more general
equation.
We now fix a family of processes $\\{h^{\epsilon},\epsilon\in(0,1)\\}$ in
${\mathcal{A}}^{T}_{N}$ (see (2.23) for the definition of
${\mathcal{A}}^{T}_{N}$), and put
$X^{\epsilon}(t,\omega):=\Phi_{\epsilon}\Big{(}W(\cdot,\omega)+\frac{h^{\epsilon}(\cdot,\omega)}{\sqrt{\epsilon}}\Big{)}(t).$
Here, we have used a little confused notations $X_{\epsilon}$ and
$X^{\epsilon}$, but they are clearly different. By Girsanov’s theorem (cf.
[54, Section 7]), $X^{\epsilon}(t)$ solves the following stochastic Volterra
equation (also called control equation):
$\displaystyle X^{\epsilon}(t)$ $\displaystyle=$ $\displaystyle
g(t)+\int^{t}_{0}A(t,s,X^{\epsilon}(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}B(t,s,X^{\epsilon}(s))\dot{h}^{\epsilon}(s){\mathord{{\rm
d}}}s$ (4.2)
$\displaystyle+\sqrt{\epsilon}\int^{t}_{0}B(t,s,X^{\epsilon}(s)){\mathord{{\rm
d}}}W(s).$
Although $h$ is defined only on $[0,T]$, we can extend it to
${\mathbb{R}}_{+}$ by setting $\dot{h}(t)=0$ for $t>T$ so that Eq.(4.2) can be
considered on ${\mathbb{R}}_{+}$. We shall always use this extension below.
Let $\tau^{\epsilon}$ be the explosion time of Eq.(4.2). For
$n\in{\mathbb{N}}$, define
$\displaystyle\tau^{\epsilon}_{n}:=\inf\\{t\geqslant
0:\|X^{\epsilon}(t)\|_{\mathbb{X}}>n\\}.$ (4.3)
Then $\tau^{\epsilon}_{n}\nearrow\tau^{\epsilon}$, and we have:
###### Lemma 4.2.
For any $\alpha_{0}\in(0,\alpha)$, there is an $a>0$ such that for $p$
sufficiently large
$\sup_{\epsilon\in(0,1)}{\mathbb{E}}\left(\sup_{t\not=t^{\prime}\in[0,T\wedge\tau^{\epsilon}_{n}]}\frac{\|X^{\epsilon}(t^{\prime})-X^{\epsilon}(t)\|^{p}_{{\mathbb{X}}_{\alpha_{0}}}}{|t^{\prime}-t|^{ap}}\right)\leqslant
C_{N,n,T,p,\kappa_{\alpha,n},\alpha_{0}}.$
###### Proof.
Note that
$\displaystyle\|X^{\epsilon}(t)\cdot
1_{\\{t\leqslant\tau^{\epsilon}_{n}\\}}\|_{{\mathbb{X}}_{\alpha}}$
$\displaystyle\leqslant$
$\displaystyle\|g(t)\|_{{\mathbb{X}}_{\alpha}}+\int^{t\wedge\tau^{\epsilon}_{n}}_{0}\|A(t,s,X^{\epsilon}(s))\|_{{\mathbb{X}}_{\alpha}}{\mathord{{\rm
d}}}s$
$\displaystyle+\int^{t\wedge\tau^{\epsilon}_{n}}_{0}\|B(t,s,X^{\epsilon}(s))\dot{h}^{\epsilon}(s)\|_{{\mathbb{X}}_{\alpha}}{\mathord{{\rm
d}}}s$
$\displaystyle+\sqrt{\epsilon}\left\|\int^{t\wedge\tau^{\epsilon}_{n}}_{0}B(t,s,X^{\epsilon}(s)){\mathord{{\rm
d}}}W(s)\right\|_{{\mathbb{X}}_{\alpha}}$ $\displaystyle=:$ $\displaystyle
J_{1}(t)+J_{2}(t)+J_{3}(t)+J_{4}(t).$
By (H2)′′ and (4.3) we have
${\mathbb{E}}|J_{2}(t)|^{p}\leqslant
C_{n}{\mathbb{E}}\left(\int^{t\wedge\tau^{\epsilon}_{n}}_{0}\kappa_{\alpha,n}(t,s){\mathord{{\rm
d}}}s\right)^{p}\leqslant C_{n,T,p,\kappa_{\alpha,n}}$
and by Hölder’s inequality
$\displaystyle{\mathbb{E}}|J_{3}(t)|^{p}$ $\displaystyle\leqslant$
$\displaystyle{\mathbb{E}}\left(\int^{t\wedge\tau^{\epsilon}_{n}}_{0}\|B(t,s,X^{\epsilon}(s))\dot{h}^{\epsilon}(s)\|_{{\mathbb{X}}_{\alpha}}{\mathord{{\rm
d}}}s\right)^{p}$ $\displaystyle\leqslant$
$\displaystyle{\mathbb{E}}\left(\int^{t\wedge\tau^{\epsilon}_{n}}_{0}\|B(t,s,X^{\epsilon}(s))\|_{L_{2}(l^{2};{\mathbb{X}}_{\alpha})}\cdot\|\dot{h}^{\epsilon}(s)\|_{l^{2}}{\mathord{{\rm
d}}}s\right)^{p}$ $\displaystyle\leqslant$ $\displaystyle
N^{\frac{p}{2}}{\mathbb{E}}\left(\int^{t\wedge\tau^{\epsilon}_{n}}_{0}\|B(t,s,X^{\epsilon}(s))\|^{2}_{L_{2}(l^{2};{\mathbb{X}}_{\alpha})}{\mathord{{\rm
d}}}s\right)^{\frac{p}{2}}$ $\displaystyle\leqslant$ $\displaystyle
C_{N,n,T,p,\kappa_{\alpha,n}},$
where we have used that $h^{\epsilon}\in{\mathcal{A}}^{T}_{N}$.
Similarly, by the BDG inequality (2.17) and (H2)′′ we have
${\mathbb{E}}|J_{4}(t)|^{p}\leqslant
C_{p}{\mathbb{E}}\left(\int^{t\wedge\tau^{\epsilon}_{n}}_{0}\|B(t,s,X^{\epsilon}(s))\|^{2}_{L_{2}(l^{2};{\mathbb{X}}_{\alpha})}{\mathord{{\rm
d}}}s\right)^{\frac{p}{2}}\leqslant C_{n,T,p,\kappa_{\alpha,n}}.$
Combining the above calculations, we get
$\displaystyle\sup_{\epsilon\in(0,1)}\sup_{t\in[0,T]}{\mathbb{E}}\|X^{\epsilon}(t)\cdot
1_{\\{t\leqslant\tau^{\epsilon}_{n}\\}}\|_{{\mathbb{X}}_{\alpha}}^{p}\leqslant
C_{N,n,T,p,\kappa_{\alpha,n}},\ \ p\geqslant 2.$ (4.4)
Moreover, as in the proofs of Theorem 3.3 and Lemma 3.4, by (H1)′′, (H2)′ and
(H4)′, for some $\beta_{3}>1$ and $p\geqslant
2(\beta_{3}^{*}:=\beta_{3}/(\beta_{3}-1)$), we have that for any $0\leqslant
t<t^{\prime}\leqslant T$
$\sup_{\epsilon\in(0,1)}{\mathbb{E}}\|(X^{\epsilon}(t^{\prime})-X^{\epsilon}(t))\cdot
1_{\\{t^{\prime},t\leqslant\tau^{\epsilon}_{n}\\}}\|^{p}_{{\mathbb{X}}}\leqslant
C_{T,p,n}\Big{(}|t-t^{\prime}|^{\delta p}+|t-t^{\prime}|^{\frac{\gamma
p}{2}}+|t-t^{\prime}|^{\frac{p}{2\beta^{*}_{3}}}\Big{)}.$
Thus, by (v) of Proposition 2.11 and (4.4), for any $\alpha_{0}\in(0,\alpha)$
and $p$ large enough we have
$\displaystyle\sup_{\epsilon\in(0,1)}{\mathbb{E}}\|(X^{\epsilon}(t^{\prime})-X^{\epsilon}(t))\cdot
1_{\\{t^{\prime},t\leqslant
T\wedge\tau^{\epsilon}_{n}\\}}\|^{p}_{{\mathbb{X}}_{\alpha_{0}}}$
$\displaystyle\qquad\leqslant
C_{N,n,T,p,\kappa_{\alpha,n},\alpha_{0}}\Big{(}|t-t^{\prime}|^{\delta
p}+|t-t^{\prime}|^{\frac{\gamma
p}{2}}+|t-t^{\prime}|^{\frac{p}{2\beta^{*}}}\Big{)}^{1-\frac{\alpha_{0}}{\alpha}}.$
The desired estimate now follows by Theorem 2.10. ∎
In order to obtain the tightness of the laws of
$\\{X^{\epsilon},\epsilon\in(0,1)\\}$ in ${\mathbb{C}}_{T}({\mathbb{X}})$, we
assume that
1. (C1)
${\mathfrak{L}}^{-1}$ is a compact operator on ${\mathbb{X}}$.
2. (C2)
$\lim_{n\rightarrow\infty}\sup_{\epsilon\in(0,1)}P\\{\omega:\tau^{\epsilon}_{n}(\omega)<T\\}=0.$
Note that (C2) implies
$P\\{\omega:\tau^{\epsilon}(\omega)>T\\}=1.$
We now prove the following key lemma for the large deviation principle of
Eq.(4.1).
###### Lemma 4.3.
Under (C1) and (C2), there exist subsequence $\epsilon_{k}\downarrow 0$, a
probability space $(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{P})$ and a
sequence
$\\{(\tilde{h}^{k},\tilde{X}^{k},\tilde{W}^{k})\\}_{k\in{\mathbb{N}}}$ as well
as $(h,X^{h},\tilde{W})$ defined on this probability space and taking values
in
${\mathbb{D}}_{N}\times{\mathbb{C}}_{T}({\mathbb{X}})\times{\mathbb{C}}_{T}({\mathbb{U}})$
such that
1. (i)
$(\tilde{h}^{k},\tilde{X}^{k},\tilde{W}^{k})$ has the same law as
$(h^{\epsilon_{k}},X^{\epsilon_{k}},W)$ for each $k\in{\mathbb{N}}$;
2. (ii)
$(\tilde{h}^{k},\tilde{X}^{k},\tilde{W}^{k})\rightarrow(h,X^{h},\tilde{W})$ in
${\mathbb{D}}_{N}\times{\mathbb{C}}_{T}({\mathbb{X}})\times{\mathbb{C}}_{T}({\mathbb{U}})$,
$\tilde{P}$-a.s. as $k\to\infty$;
3. (iii)
$(h,X^{h})$ uniquely solves the following Volterra equation:
$\displaystyle X^{h}(t)=g(t)+\int^{t}_{0}A(t,s,X^{h}(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}B(t,s,X^{h}(s))\dot{h}(s){\mathord{{\rm d}}}s.$ (4.5)
In particular, (LD)1 in Subsection 2.4 holds.
###### Proof.
Let $\alpha_{0}\in(0,\alpha)$ and $a>0$ be as in Lemma 4.2. For $R>0$, set
$K_{R}:=\left\\{x\in{\mathbb{C}}_{T}({\mathbb{X}}):\sup_{t\in[0,T]}\|x(t)\|_{\mathbb{X}}+\sup_{s\not=t\in[0,T]}\frac{\|x(t)-x(s)\|_{{\mathbb{X}}_{\alpha_{0}}}}{|t-s|^{a}}\leqslant
R\right\\}.$
By (C1), ${\mathbb{X}}_{\alpha_{0}}\hookrightarrow{\mathbb{X}}$ is compact
(cf. [37, p.29, Theorem 1.4.8]). Thus, by Ascoli-Arzelà’s theorem (cf. [39]),
the set $K_{R}$ is compact in ${\mathbb{C}}_{T}({\mathbb{X}})$. For any
$\delta>0$, by (C2) we can choose $n$ sufficiently large such that
$\displaystyle\sup_{\epsilon\in(0,1)}P\Big{\\{}\omega:\tau^{\epsilon}_{n}(\omega)<T\Big{\\}}\leqslant\delta.$
By Lemma 4.2 and Chebyschev’s inequality, for any $R>n$ we have
$\displaystyle P\\{X^{\epsilon}(\cdot)\notin K_{R}\\}$ $\displaystyle=$
$\displaystyle P\\{X^{\epsilon}(\cdot)\notin
K_{R},\tau^{\epsilon}_{n}\geqslant T\\}+P\\{X^{\epsilon}(\cdot)\notin
K_{R},\tau^{\epsilon}_{n}<T\\}$ $\displaystyle\leqslant$ $\displaystyle
P\left\\{\sup_{s\not=t\in[0,T\wedge\tau^{\epsilon}_{n}]}\frac{\|X^{\epsilon}(t)-X^{\epsilon}(s)\|_{{\mathbb{X}}_{\alpha_{0}}}}{|t-s|^{a}}\geqslant
R-n\right\\}+P\\{\tau^{\epsilon}_{n}<T\\}$ $\displaystyle\leqslant$
$\displaystyle{\mathbb{E}}\left[\sup_{s\not=t\in[0,T\wedge\tau^{\epsilon}_{n}]}\frac{\|X^{\epsilon}(t)-X^{\epsilon}(s)\|^{p}_{{\mathbb{X}}_{\alpha_{0}}}}{|t-s|^{ap}}\right]/(R-n)^{p}+\delta$
$\displaystyle\leqslant$ $\displaystyle
C_{N,n,T,p,\kappa_{\alpha,n},\alpha_{0}}/(R-n)^{p}+\epsilon^{\prime}.$
Therefore, for $R$ large enough we have
$\sup_{\epsilon\in(0,1)}P\\{X^{\epsilon}(\cdot)\notin K_{R}\\}\leqslant
2\delta.$
Thus, by the compactness of ${\mathbb{D}}_{N}$ (see (2.24)), the laws of
$(h^{\epsilon},X^{\epsilon},W)$ in
${\mathbb{D}}_{N}\times{\mathbb{C}}_{T}({\mathbb{X}})\times{\mathbb{C}}_{T}({\mathbb{U}})$
is tight. By Skorohod’s embedding theorem (cf. [39]), the conclusions (i) and
(ii) hold.
We now prove (iii). Note that by (i) (cf. [54, Section 8])
$\displaystyle\tilde{X}^{k}(t)$ $\displaystyle=$ $\displaystyle
g(t)+\int^{t}_{0}A(t,s,\tilde{X}^{k}(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}B(t,s,\tilde{X}^{k}(s))\dot{\tilde{h}}^{k}(s){\mathord{{\rm
d}}}s$
$\displaystyle+\sqrt{\epsilon_{k}}\int^{t}_{0}B(t,s,\tilde{X}^{k}(s)){\mathord{{\rm
d}}}\tilde{W}^{k}(s)$ $\displaystyle=:$ $\displaystyle
g(t)+J^{k}_{1}(t)+J^{k}_{2}(t)+J^{k}_{3}(t),\ \ \tilde{P}-a.s..$
Set
$\tilde{\tau}^{k}_{n}:=\inf\\{t\geqslant
0:\|\tilde{X}^{k}(t)\|_{\mathbb{X}}>n\\}.$
Then for any $\delta>0$, by (i) and (C2) there exists an $n$ large enough such
that
$\displaystyle\sup_{k\in{\mathbb{N}}}\tilde{P}\\{\tilde{\tau}^{k}_{n}<T\\}$
$\displaystyle=$
$\displaystyle\sup_{k\in{\mathbb{N}}}\tilde{P}\left\\{\sup_{s\in[0,T)}\|\tilde{X}^{k}(s)\|_{\mathbb{X}}>n\right\\}$
$\displaystyle=$
$\displaystyle\sup_{k\in{\mathbb{N}}}P\left\\{\sup_{s\in[0,T)}\|X^{\epsilon_{k}}(s)\|_{\mathbb{X}}>n\right\\}$
$\displaystyle=$
$\displaystyle\sup_{k\in{\mathbb{N}}}P\\{\tau^{\epsilon_{k}}_{n}<T\\}\leqslant\delta.$
Hence, for any $\delta^{\prime}>0$, by the BDG inequality (2.17) and (H2)′ we
have
$\displaystyle\tilde{P}\Big{\\{}\|J^{k}_{3}(t)\|_{\mathbb{X}}\geqslant\delta^{\prime}\Big{\\}}$
$\displaystyle\leqslant$
$\displaystyle\tilde{P}\Big{\\{}J^{k}_{3}(t)\geqslant\delta^{\prime};\tilde{\tau}^{k}_{n}\geqslant
T\Big{\\}}+\tilde{P}\Big{\\{}\tilde{\tau}^{k}_{n}<T\Big{\\}}$
$\displaystyle\leqslant$
$\displaystyle\frac{{\mathbb{E}}^{\tilde{P}}\|J^{k}_{3}(t)\cdot
1_{\\{t\leqslant\tilde{\tau}^{k}_{n}\\}}\|^{2}_{\mathbb{X}}}{\delta^{\prime
2}}+\delta$ $\displaystyle\leqslant$ $\displaystyle\frac{\epsilon_{k}\cdot
C_{n}{\mathbb{E}}^{\tilde{P}}\left(\int^{t\wedge\tilde{\tau}^{k}_{n}}_{0}\kappa_{1,n}(t,s){\mathord{{\rm
d}}}s\right)}{\delta^{\prime 2}}+\delta$ $\displaystyle\leqslant$
$\displaystyle\frac{\epsilon_{k}\cdot C_{n,t}}{\delta^{\prime 2}}+\delta.$
Thus, we get
$\lim_{k\to\infty}\tilde{P}\Big{\\{}\|J^{k}_{3}(t)\|_{\mathbb{X}}\geqslant\delta^{\prime}\Big{\\}}=0.$
Let $J_{i}(t),i=1,2$ be the corresponding terms in Eq.(4.5). In order to prove
that $X^{h}$ solves Eq.(4.5), it is now enough to show that for any
$t\in[0,T]$ and $y\in{\mathbb{X}}^{*}$
$\lim_{k\to\infty}{}_{\mathbb{X}}{\langle}J^{k}_{i}(t)-J_{i}(t),y{\rangle}_{{\mathbb{X}}^{*}}=0,\
\ i=1,2,\ \ \tilde{P}-a.s..$
Observe that
$\displaystyle|{}_{\mathbb{X}}{\langle}J^{k}_{2}(t)-J_{2}(t),y{\rangle}_{{\mathbb{X}}^{*}}|$
$\displaystyle\leqslant$
$\displaystyle\|y\|_{{\mathbb{X}}^{*}}\cdot\int^{t}_{0}\|[B(t,s,\tilde{X}^{k}(s))-B(t,s,X^{h}(s))]\dot{\tilde{h}}^{k}(s)\|_{\mathbb{X}}{\mathord{{\rm
d}}}s$
$\displaystyle+\left|\int^{t}_{0}{}_{\mathbb{X}}{\langle}B(t,s,X^{h}(s))[\dot{\tilde{h}}^{k}(s)-\dot{h}(s)],y{\rangle}_{{\mathbb{X}}^{*}}{\mathord{{\rm
d}}}s\right|$ $\displaystyle=:$ $\displaystyle\|y\|_{{\mathbb{X}}^{*}}\cdot
J^{k}_{21}(t)+J^{k}_{22}(t).$
By the weak convergence of $\tilde{h}^{k}$ to $h$ in ${\mathbb{D}}_{N}$, we
have
$\lim_{k\to\infty}J^{k}_{22}(t)=0.$
Noting that by (ii), for almost all $\tilde{\omega}\in\tilde{\Omega}$ and some
$K(\tilde{\omega})\in{\mathbb{N}}$
$n(\tilde{\omega}):=\sup_{s\in[0,T]}\|X^{h}(s,\tilde{\omega})\|_{\mathbb{X}}\vee\sup_{k\geqslant
K(\tilde{\omega})}\sup_{s\in[0,T]}\|\tilde{X}^{k}(s,\tilde{\omega})\|_{\mathbb{X}}<+\infty,$
we have, by Hölder’s inequality and (H3)′
$\displaystyle J^{k}_{21}(t,\tilde{\omega})$ $\displaystyle\leqslant$
$\displaystyle\|\tilde{h}^{k}(\tilde{\omega})\|_{\ell^{2}_{T}}\cdot\left(\int^{t}_{0}\big{\|}B(t,s,\tilde{X}^{k}(s,\tilde{\omega}))-B(t,s,X^{h}(s,\tilde{\omega}))\big{\|}^{2}_{L_{2}(l^{2};{\mathbb{X}})}{\mathord{{\rm
d}}}s\right)^{1/2}$ $\displaystyle\leqslant$ $\displaystyle
N\cdot\left(\int^{t}_{0}\kappa_{2,n(\tilde{\omega})}(t,s)\cdot\|\tilde{X}^{k}(s,\tilde{\omega})-X^{h}(s,\tilde{\omega})\|^{2}_{\mathbb{X}}{\mathord{{\rm
d}}}s\right)^{1/2}$ $\displaystyle\stackrel{{\scriptstyle(ii)}}{{\to}}$
$\displaystyle 0,\ \ \mbox{ as $k\to\infty$},$
where we have used $\tilde{h}^{k}(\tilde{\omega})\in{\mathbb{D}}_{N}$.
Similarly, we have
$\lim_{k\to\infty}\|J^{k}_{1}(t)-J_{1}(t)\|_{{\mathbb{X}}}=0,\
\tilde{P}-a.s..$
Combining the above estimates, we find that $X^{h}$ solves Eq.(4.5). ∎
Let $I(f)$ be defined by
$\displaystyle
I(f):=\frac{1}{2}\inf_{\\{h\in\ell^{2}_{T}:~{}f=X^{h}\\}}\|h\|^{2}_{\ell^{2}_{T}},\
\ f\in{\mathbb{C}}_{T}({\mathbb{X}}),$ (4.6)
where $X^{h}$ is defined by Eq.(4.5). In order to identify $I(f)$, we assume
that
1. (C3)
For any $N\in{\mathbb{N}}$
$\sup_{h\in{\mathbb{D}}_{N}}\sup_{t\in[0,T]}\|X^{h}(t)\|_{\mathbb{X}}<+\infty.$
Similar to the proof of Lemma 4.3, we can prove that:
###### Lemma 4.4.
Under (C3), (LD)2 in Subsection 2.4 holds.
Thus, by Theorem 2.15 we have proven:
###### Theorem 4.5.
Assume that (H1)′′-(H2)′′, (H2)′-(H4)′ and (C1)-(C3) hold. Then,
$\\{X_{\epsilon},\epsilon\in(0,1)\\}$ satisfies the large deviation principle
in ${\mathbb{C}}_{T}({\mathbb{X}})$ with the rate function $I(f)$ given by
(4.6).
###### Remark 4.6.
The conditions (C2) and (C3) are satisfied if (H1)′′, (H2) and (H4) hold, and
$\kappa_{1}$ in (H2) belongs to ${\mathscr{K}}_{>1}$. In fact, we can prove as
the proof of Theorem 3.8
$\sup_{n\in{\mathbb{N}}}\sup_{\epsilon\in(0,1)}{\mathbb{E}}\left(\sup_{t\in[0,T\wedge\tau^{\epsilon}_{n}]}\|X^{\epsilon}(t)\|^{p}_{{\mathbb{X}}}\right)\leqslant
C_{T,p,\kappa_{1}},$
which then implies (C2). The condition (C3) is more direct in this case.
## 5\. Semilinear stochastic evolutionary integral equations
In this section, we consider the following semilinear stochastic evolutionary
integral equation:
$\displaystyle X(t)=x_{0}-\int^{t}_{0}a(t-s){\mathfrak{L}}X(s){\mathord{{\rm
d}}}s+\int^{t}_{0}\Phi(s,X(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}\Psi(s,X(s)){\mathord{{\rm d}}}W(s),$ (5.1)
where $a:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ is a measurable function, and
$\Phi:{\mathbb{R}}_{+}\times\Omega\times{\mathbb{X}}\to{\mathbb{X}}\in{\mathcal{M}}\times{\mathcal{B}}({\mathbb{X}})/{\mathcal{B}}({\mathbb{X}})$
and
$\Psi:{\mathbb{R}}_{+}\times\Omega\times{\mathbb{X}}\to
L_{2}(l^{2};{\mathbb{X}})\in{\mathcal{M}}\times{\mathcal{B}}({\mathbb{X}})/{\mathcal{B}}(L_{2}(l^{2};{\mathbb{X}})).$
Here and below, ${\mathcal{M}}$ stands for the progressively measurable
$\sigma$-algebra over ${\mathbb{R}}_{+}\times\Omega$.
Consider first the following deterministic integral equation:
$\displaystyle x(t)=x_{0}-\int^{t}_{0}a(t-s){\mathfrak{L}}x(s){\mathord{{\rm
d}}}s.$ (5.2)
The solution of this equation is called the resolvent of $(a,{\mathfrak{L}})$,
and denoted by ${\mathfrak{S}}_{t}x_{0}=x(t)$. Note that in general
${\mathfrak{S}}_{t+s}\not={\mathfrak{S}}_{t}\circ{\mathfrak{S}}_{s}.$
We make the following assumptions:
1. (S1)
The resolvent $\\{{\mathfrak{S}}_{t}:t\geqslant 0\\}$ is of analyticity type
$(\omega_{0},\theta_{0})$ in the sense of [63, Definition 2.1], where
$\omega_{0}\in{\mathbb{R}}$ and $\theta_{0}\in(0,\pi/2]$.
2. (S2)
For any $R>0$, there exist $C_{R}>0$ and $\beta\in[0,1)$ such that for all
$s>0$, $\omega\in\Omega$ and $x,y\in{\mathbb{X}}$ with
$\|x\|_{\mathbb{X}},\|y\|_{\mathbb{X}}\leqslant R$
$\|\Phi(s,\omega,x)\|_{{\mathbb{X}}}+\|\Psi(s,\omega,x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}\leqslant\frac{C_{R}}{(s\wedge
1)^{\beta}},$
and
$\displaystyle\|\Phi(s,\omega,x)-\Phi(s,\omega,y)\|_{{\mathbb{X}}}$
$\displaystyle\leqslant$ $\displaystyle\frac{C_{R}}{(s\wedge
1)^{\beta}}\|x-y\|_{{\mathbb{X}}},$
$\displaystyle\|\Psi(s,\omega,x)-\Psi(s,\omega,y)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}$
$\displaystyle\leqslant$ $\displaystyle\frac{C_{R}}{(s\wedge
1)^{\beta}}\|x-y\|_{{\mathbb{X}}}^{2}.$
3. (S3)
For all $s>0$, $\omega\in\Omega$ and $x\in{\mathbb{X}}$, it holds that
$\displaystyle\|\Phi(s,\omega,x)\|_{{\mathbb{X}}}$ $\displaystyle\leqslant$
$\displaystyle\frac{C}{(s\wedge 1)^{\beta}}(1+\|x\|_{\mathbb{X}}),$
$\displaystyle\|\Psi(s,\omega,x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}$
$\displaystyle\leqslant$ $\displaystyle\frac{C}{(s\wedge
1)^{\beta}}(1+\|x\|^{2}_{\mathbb{X}}).$
The following property of analytic resolvent $\\{{\mathfrak{S}}_{t}:t>0\\}$ is
crucial for the proof of Theorem 5.2 below (cf. [63, Corollary 2.1]).
###### Proposition 5.1.
Let ${\mathfrak{S}}_{t}$ be an analytic resolvent of type
$(\omega_{0},\theta_{0})$. Then for any $T>0$
$\displaystyle\sup_{t\in[0,T]}\|{\mathfrak{S}}_{t}\|_{L({\mathbb{X}};{\mathbb{X}})}\leqslant
C_{T}$ (5.3)
and for any $t\in(0,T]$
$\displaystyle\|\dot{\mathfrak{S}}_{t}\|_{L({\mathbb{X}};{\mathbb{X}})}\leqslant
C_{T}t^{-1},$ (5.4)
where the dot denotes the operator derivative and
$\|\cdot\|_{L({\mathbb{X}};{\mathbb{X}})}$ denotes the norm of bounded linear
operators.
By a solution of Eq.(5.1) we mean that $X(t)$ satisfies the following
stochastic Volterra equation:
$\displaystyle
X(t)={\mathfrak{S}}_{t}x_{0}+\int^{t}_{0}{\mathfrak{S}}_{t-s}\Phi(s,X(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}{\mathfrak{S}}_{t-s}\Psi(s,X(s)){\mathord{{\rm d}}}W(s).$
(5.5)
Let us define
$A(t,s,\omega,x):={\mathfrak{S}}_{t-s}\Phi(s,\omega,x),\ \
B(t,s,\omega,x):={\mathfrak{S}}_{t-s}\Psi(s,\omega,x).$
We have:
###### Theorem 5.2.
Under (S1) and (S2), there exists a unique maximal solution $(X,\tau)$ for Eq.
(5.5) in the sense of Definition 3.5. Moreover, if (S3) holds, then
$\tau=+\infty$, a.s..
###### Proof.
First of all, it is easy to see by (5.3) that (H2)′ and (H3)′ hold with
$\kappa_{1,R}(t,s)=\kappa_{2,R}(t,s)=\frac{C_{R}}{(s\wedge
1)^{\beta}}\in{\mathscr{K}}_{>1}.$
For $0\leqslant s<t<t^{\prime}$, $\omega\in\Omega$ and $x\in{\mathbb{X}}$ with
$\|x\|_{\mathbb{X}}\leqslant R$, we have
$\displaystyle\|A(t^{\prime},s,\omega,x)-A(t,s,\omega,x)\|_{\mathbb{X}}$
$\displaystyle=$
$\displaystyle\|({\mathfrak{S}}_{t^{\prime}-s}-{\mathfrak{S}}_{t-s})\Phi(s,\omega,x)\|_{\mathbb{X}}$
$\displaystyle\leqslant$ $\displaystyle\frac{C_{R}}{(s\wedge
1)^{\beta}}\|{\mathfrak{S}}_{t^{\prime}-s}-{\mathfrak{S}}_{t-s}\|_{L({\mathbb{X}};{\mathbb{X}})}$
$\displaystyle\leqslant$ $\displaystyle\frac{C_{R}}{(s\wedge
1)^{\beta}}\int^{t^{\prime}-s}_{t-s}\|\dot{\mathfrak{S}}_{r}\|_{L({\mathbb{X}};{\mathbb{X}})}{\mathord{{\rm
d}}}r$ $\displaystyle\stackrel{{\scriptstyle(\ref{Po})}}{{\leqslant}}$
$\displaystyle\frac{C_{R}}{(s\wedge
1)^{\beta}}\int^{t^{\prime}-s}_{t-s}\frac{1}{r}{\mathord{{\rm d}}}r$
$\displaystyle=$ $\displaystyle\frac{C_{R}}{(s\wedge
1)^{\beta}}\log\left(\frac{t^{\prime}-s}{t-s}\right)$
and
$\|B(t^{\prime},s,\omega,x)-B(t,s,\omega,x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}\leqslant\frac{C_{R}}{(s\wedge
1)^{\beta}}\log^{2}\left(\frac{t^{\prime}-s}{t-s}\right).$
Note that the following elementary inequality holds for any $\gamma\in(0,1)$
$\log(1+s)\leqslant Cs^{\gamma},\ \ \forall s>0.$
Therefore, for $0\leqslant s<t<t^{\prime}$, $\omega\in\Omega$ and
$x\in{\mathbb{X}}$ with $\|x\|_{\mathbb{X}}\leqslant R$
$\displaystyle\|A(t^{\prime},s,\omega,x)-A(t,s,\omega,x)\|_{\mathbb{X}}+\|B(t^{\prime},s,\omega,x)-B(t,s,\omega,x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}$
$\displaystyle\qquad\leqslant\frac{C_{R}(t^{\prime}-t)^{\gamma}}{(s\wedge
1)^{\beta}(t-s)^{\gamma}}\left[1+\frac{(t^{\prime}-t)^{\gamma}}{(t-s)^{\gamma}}\right]=:\lambda_{R}(t^{\prime},t,s).$
Thus, we find that (H4)′ holds if $\gamma\in(0,(1-\beta)/2)$.
Lastly, if (S3) is satisfied, it is clear that (H2) holds with
$\kappa_{1}(t,s)=\frac{C}{(s\wedge 1)^{\beta}}\in{\mathscr{K}}_{>1}$, and (H4)
also holds from the above calculations. The non-explosion now follows from
Theorem 3.8. ∎
We now turn to the small perturbation of Eq.(5.5) and assume that $\Phi$ and
$\Psi$ are non-random. Consider
$X_{\epsilon}(t)={\mathfrak{S}}_{t}x_{0}+\int^{t}_{0}{\mathfrak{S}}_{t-s}\Phi(s,X_{\epsilon}(s)){\mathord{{\rm
d}}}s+\sqrt{\epsilon}\int^{t}_{0}{\mathfrak{S}}_{t-s}\Psi(s,X_{\epsilon}(s)){\mathord{{\rm
d}}}W(s).$
In order to use Theorem 4.5 to get the LDP for
$\\{X_{\epsilon},\epsilon\in(0,1)\\}$, we also assume
1. (S4)
Let $\\{{\mathfrak{S}}_{t}:t\geqslant 0\\}$ be an analytic resolvent of type
$(\omega_{0},\theta_{0})$. Assume that for some $\omega_{1}>\omega_{0}$,
$0<\theta_{1}<\theta_{0}$, $C>0$ and $\alpha_{1}>0$
$\displaystyle|\hat{a}(\lambda)|\geqslant
C(|\lambda-\omega_{1}|^{\alpha_{1}}+1)^{-1},\ \
\forall\lambda\in{\mathbb{C}}\mbox{ with
}|\mathrm{arg}(\lambda-\omega)|<\theta_{1},$ (5.6)
where $\hat{a}$ denotes the Laplace transform of $a$. Moreover, we also assume
that
$\displaystyle\int^{r}_{0}a(s){\mathord{{\rm
d}}}s+\int^{t}_{0}|a(r+s)-a(s)|{\mathord{{\rm d}}}s\leqslant
C_{T}|r|^{\delta},$ (5.7)
where $r,t\in[0,T]$ and $T,\delta>0$.
We have
###### Theorem 5.3.
Under (S1)-(S4) and (C1), for any $x_{0}\in{\mathscr{D}}({\mathfrak{L}})$,
$\\{X_{\epsilon},\epsilon\in(0,1)\\}$ satisfies the large deviation principle
in ${\mathbb{C}}_{T}({\mathbb{X}})$ with the rate function $I(f)$ given by
(4.6).
###### Proof.
From the proof of Theorem 5.2, it is enough to check (H1)′′ and (H2)′′. By
(5.6) and [63, p.57, Theorem 2.2 (ii)], we have
$\|{\mathfrak{L}}{\mathfrak{S}}_{t}\|_{L({\mathbb{X}};{\mathbb{X}})}\leqslant
Ce^{\omega_{1}t}(1+t^{-\alpha_{1}}),\ \ \forall t>0,$
which together with (v) of Proposition 2.11 yields that for any
$\alpha\in(0,1)$ and $T>0$
$\|{\mathfrak{L}}^{\alpha}{\mathfrak{S}}_{t}\|_{L({\mathbb{X}};{\mathbb{X}})}\leqslant
C_{T}(1+t^{-\alpha_{1}\cdot\alpha}),\ \ \forall t\in(0,T].$
Thus, (H2)′′ holds by choosing $\alpha<\frac{1-\beta}{\alpha_{1}}$, where
$\beta$ is from (S3).
For (H1)′′, since $x_{0}\in{\mathscr{D}}({\mathfrak{L}})={\mathbb{X}}_{1}$, by
(5.3) we have
$\|{\mathfrak{L}}{\mathfrak{S}}_{t}x_{0}\|_{\mathbb{X}}=\|{\mathfrak{S}}_{t}{\mathfrak{L}}x_{0}\|_{\mathbb{X}}\leqslant
C\|{\mathfrak{L}}x_{0}\|_{\mathbb{X}}.$
On the other hand, by the resolvent equation (5.2) and (5.7) we have, for any
$0\leqslant t<t^{\prime}\leqslant T$
$\displaystyle\|{\mathfrak{S}}_{t^{\prime}}x_{0}-{\mathfrak{S}}_{t}x_{0}\|_{\mathbb{X}}$
$\displaystyle\leqslant$
$\displaystyle\int^{t}_{0}|a(t^{\prime}-s)-a(t-s)|\cdot\|{\mathfrak{L}}{\mathfrak{S}}_{s}x_{0}\|_{\mathbb{X}}{\mathord{{\rm
d}}}s$
$\displaystyle+\int^{t^{\prime}}_{t}|a(t^{\prime}-s)|\cdot\|{\mathfrak{L}}{\mathfrak{S}}_{s}x_{0}\|_{\mathbb{X}}{\mathord{{\rm
d}}}s$ $\displaystyle\leqslant$ $\displaystyle
C_{T}\|{\mathfrak{L}}x_{0}\|_{\mathbb{X}}\cdot|t^{\prime}-t|^{\delta}.$
The proof is thus completed by Theorem 4.5 and Remark 4.6. ∎
###### Example 5.4.
Let $a$ be a completely monotonic kernel function, i.e.,
$\displaystyle a(t)=\int^{\infty}_{0}e^{-st}{\mathord{{\rm d}}}\rho(s),\ \
t>0,$ (5.8)
where $s\mapsto\rho(s)$ is nondecreasing, and such that
$\int^{\infty}_{1}{\mathord{{\rm d}}}\rho(s)/s<\infty$. Then the resolvent
$\\{{\mathfrak{S}}_{t}:t\geqslant 0\\}$ associated with $a$ is of analyticity
type $(0,\theta)$ for some $\theta\in(0,\pi/2)$ (cf. [63, p.55, Example 2.2]),
i.e., (S1) holds. For (S4), besides (5.8) and (5.7), we also assume that for
some $C,\alpha_{1}>0$
$\displaystyle C(1+\lambda)^{-\alpha_{1}}\leqslant\int^{\infty}_{0}e^{-\lambda
t}\cdot a(t){\mathord{{\rm d}}}t<+\infty,\ \ \forall\lambda>0,$ (5.9)
which implies by [63, p.221, Lemma 8.1 (v)] that (5.6) holds. In particular,
$a_{\alpha}(t)=\frac{t^{\alpha-1}}{\Gamma(\alpha)},\ \ \alpha\in(0,1]$
is completely monotonic, and satisfies (5.7) and (5.9), where $\Gamma$ denotes
the usual Gamma function.
Moreover, for the kernel function $a_{\alpha}$, if
$1<\alpha<2-\frac{2\phi}{\pi}<2,$
where $\phi$ comes from (2.19), then ${\mathfrak{S}}_{t}$ is analytic (cf.
[63, p.55, Example 2.1]). Notice that in [63], $-{\mathfrak{L}}$ is
considered. In this case, (5.6) and (5.7) clearly hold since
$\hat{a}_{\alpha}(\lambda)=\lambda^{-\alpha}$, $\mathrm{Re}\lambda>0$.
## 6\. Semilinear stochastic partial differential equations
When $a=1$ in Eq.(5.1), one sees that Eq.(5.1) contains a class of semilinear
SPDEs. However, it cannot deal with the equation like stochastic Navier-Stokes
equation. In this section, we shall discuss strong solutions of a large class
of semilinear SPDEs by using the properties of analytic semigroups.
### 6.1. Mild solutions of SPDEs driven by Brownian motions
Consider the following semilinear stochastic partial differential equation:
$\displaystyle{\mathord{{\rm
d}}}X(t)=[-{\mathfrak{L}}X(t)+\Phi(t,X(t))]{\mathord{{\rm
d}}}t+\Psi(t,X(t)){\mathord{{\rm d}}}W(t),\ \ X(0)=x_{0}.$ (6.1)
We study two cases, in application, which correspond to different types of
SPDEs. First of all, we introduce the following assumptions on the
coefficients:
1. (M1)
For some $\alpha\in(0,1)$
$\Phi:{\mathbb{R}}_{+}\times\Omega\times{\mathbb{X}}_{\alpha}\to{\mathbb{X}}\in{\mathcal{M}}\times{\mathcal{B}}({\mathbb{X}}_{\alpha})/{\mathcal{B}}({\mathbb{X}})$
and
$\Psi:{\mathbb{R}}_{+}\times\Omega\times{\mathbb{X}}_{\alpha}\to
L_{2}(l^{2};{\mathbb{X}}_{\frac{\alpha}{2}})\in{\mathcal{M}}\times{\mathcal{B}}({\mathbb{X}}_{\alpha})/{\mathcal{B}}(L_{2}(l^{2};{\mathbb{X}}_{\frac{\alpha}{2}})).$
2. (M2)
For any $R>0$, there exist $C_{R}>0$ and $\beta\in[0,1)$ with
$\alpha+\beta<1$
such that for all $s>0$, $\omega\in\Omega$ and $x,y\in{\mathbb{X}}_{\alpha}$
with $\|x\|_{{\mathbb{X}}_{\alpha}},\|y\|_{{\mathbb{X}}_{\alpha}}\leqslant R$,
$\|\Phi(s,\omega,x)\|_{{\mathbb{X}}}+\|\Psi(s,\omega,x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}}_{\frac{\alpha}{2}})}\leqslant\frac{C_{R}}{(s\wedge
1)^{\beta}}$
and
$\displaystyle\|\Phi(s,\omega,x)-\Phi(s,\omega,y)\|_{{\mathbb{X}}}$
$\displaystyle\leqslant$ $\displaystyle\frac{C_{R}}{(s\wedge
1)^{\beta}}\|x-y\|_{{\mathbb{X}}_{\alpha}},$
$\displaystyle\|\Psi(s,\omega,x)-\Psi(s,\omega,y)\|^{2}_{L_{2}(l^{2};{\mathbb{X}}_{\frac{\alpha}{2}})}$
$\displaystyle\leqslant$ $\displaystyle\frac{C_{R}}{(s\wedge
1)^{\beta}}\|x-y\|_{{\mathbb{X}}_{\alpha}}^{2}.$
3. (M3)
For all $s>0$, $\omega\in\Omega$ and $x\in{\mathbb{X}}_{\alpha}$, it holds
that
$\displaystyle\|\Phi(s,\omega,x)\|_{{\mathbb{X}}}$ $\displaystyle\leqslant$
$\displaystyle\frac{C}{(s\wedge 1)^{\beta}}(1+\|x\|_{{\mathbb{X}}_{\alpha}}),$
$\displaystyle\|\Psi(s,\omega,x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}}_{\frac{\alpha}{2}})}$
$\displaystyle\leqslant$ $\displaystyle\frac{C}{(s\wedge
1)^{\beta}}(1+\|x\|_{{\mathbb{X}}_{\alpha}}^{2}).$
By a mild solution of equation (6.1) we mean that $X(t)$ solves the following
stochastic Volterra integral equation:
$\displaystyle
X(t)={\mathfrak{T}}_{t}x_{0}+\int^{t}_{0}{\mathfrak{T}}_{t-s}\Phi(s,X(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}{\mathfrak{T}}_{t-s}\Psi(s,X(s)){\mathord{{\rm d}}}W(s).$
(6.2)
###### Theorem 6.1.
Under (M1) and (M2), for any $x_{0}\in{\mathbb{X}}_{\alpha}$ ($\alpha$ is from
(M1)), there exists a unique maximal solution $(X,\tau)$ for Eq.(6.2) so that
1. (i)
$t\mapsto X(t)\in{\mathbb{X}}_{\alpha}$ is continuous on $[0,\tau)$ almost
surely;
2. (ii)
$\lim_{t\uparrow\tau}\|X(t)\|_{{\mathbb{X}}_{\alpha}}=+\infty$ on
$\\{\omega:\tau(\omega)<+\infty\\}$;
3. (iii)
it holds that, $P$-a.s, on $[0,\tau)$
$\displaystyle
X(t)={\mathfrak{T}}_{t}x_{0}+\int^{t}_{0}{\mathfrak{T}}_{t-s}\Phi(s,X(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}{\mathfrak{T}}_{t-s}\Psi(s,X(s)){\mathord{{\rm d}}}W(s).$
Moreover, if (M3) holds, then $\tau=+\infty$, a.s..
###### Proof.
We first consider the following stochastic Volterra integral equation
$\displaystyle Y(t)$ $\displaystyle=$
$\displaystyle{\mathfrak{L}}^{\alpha}{\mathfrak{T}}_{t}x_{0}+\int^{t}_{0}{\mathfrak{L}}^{\alpha}{\mathfrak{T}}_{t-s}\Phi(s,{\mathfrak{L}}^{-\alpha}Y(s)){\mathord{{\rm
d}}}s$ (6.3)
$\displaystyle+\int^{t}_{0}{\mathfrak{L}}^{\alpha}{\mathfrak{T}}_{t-s}\Psi(s,{\mathfrak{L}}^{-\alpha}Y(s)){\mathord{{\rm
d}}}W(s).$
Define
$\displaystyle g(t)$ $\displaystyle:=$
$\displaystyle{\mathfrak{L}}^{\alpha}{\mathfrak{T}}_{t}x_{0},$ $\displaystyle
A(t,s,\omega,y)$ $\displaystyle:=$
$\displaystyle{\mathfrak{L}}^{\alpha}{\mathfrak{T}}_{t-s}\Phi(s,\omega,{\mathfrak{L}}^{-\alpha}y),$
$\displaystyle B(t,s,\omega,y)$ $\displaystyle:=$
$\displaystyle{\mathfrak{L}}^{\alpha}{\mathfrak{T}}_{t-s}\Psi(s,\omega,{\mathfrak{L}}^{-\alpha}y).$
Let us verify (H1)′-(H4)′. Clearly, (H1)′ holds since
$x_{0}\in{\mathbb{X}}_{\alpha}$.
By (iii) of Proposition 2.11 and (M2), for all $t>s>0$, $\omega\in\Omega$ and
$x,y\in{\mathbb{X}}$ with $\|x\|_{\mathbb{X}},\|y\|_{\mathbb{X}}\leqslant R$
we have
$\displaystyle\|A(t,s,\omega,x)\|_{\mathbb{X}}+\|B(t,s,\omega,x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}$
$\displaystyle\qquad\preceq\frac{1}{(t-s)^{\alpha}}\Big{(}\|\Phi(s,\omega,{\mathfrak{L}}^{-\alpha}x)\|_{{\mathbb{X}}}+\|\Psi(s,\omega,{\mathfrak{L}}^{-\alpha}x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}}_{\frac{\alpha}{2}})}\Big{)}$
$\displaystyle\qquad\leqslant\frac{C_{R}}{(t-s)^{\alpha}(s\wedge 1)^{\beta}},$
(6.4)
and
$\displaystyle\|A(t,s,\omega,x)-A(t,s,\omega,y)\|_{\mathbb{X}}$
$\displaystyle\preceq$
$\displaystyle\frac{1}{(t-s)^{\alpha}}\|\Phi(s,\omega,{\mathfrak{L}}^{-\alpha}x)-\Phi(s,\omega,{\mathfrak{L}}^{-\alpha}y)\|_{{\mathbb{X}}}$
$\displaystyle\leqslant$ $\displaystyle\frac{C_{R}}{(t-s)^{\alpha}(s\wedge
1)^{\beta}}\|{\mathfrak{L}}^{-\alpha}x-{\mathfrak{L}}^{-\alpha}y\|_{{\mathbb{X}}_{\alpha}}$
$\displaystyle=$ $\displaystyle\frac{C_{R}}{(t-s)^{\alpha}(s\wedge
1)^{\beta}}\|x-y\|_{{\mathbb{X}}}$
as well as
$\displaystyle\|B(t,s,\omega,x)-B(t,s,\omega,y)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}$
$\displaystyle\qquad\preceq\frac{1}{(t-s)^{\alpha}}\|\Psi(s,\omega,{\mathfrak{L}}^{-\alpha}x)-\Psi(s,\omega,{\mathfrak{L}}^{-\alpha}y)\|_{L_{2}(l^{2};{\mathbb{X}}_{\frac{\alpha}{2}})}^{2}$
$\displaystyle\qquad\leqslant\frac{C_{R}}{(t-s)^{\alpha}(s\wedge
1)^{\beta}}\|x-y\|^{2}_{{\mathbb{X}}}.$
Hence, if we take
$\kappa_{1,R}(t,s)=\kappa_{2,R}(t,s):=\frac{C_{R}}{(t-s)^{\alpha}(s\wedge
1)^{\beta}}\in{\mathscr{K}}_{>1},$
then (H2)′ and (H3)′ hold.
Let $0<\gamma<1-(\alpha+\beta)$. By (iv) of Proposition 2.11 and (M2) we have
$\displaystyle\|A(t^{\prime},s,\omega,x)-A(t,s,\omega,x)\|_{\mathbb{X}}$
$\displaystyle=$
$\displaystyle\|({\mathfrak{T}}_{t^{\prime}-t}-1){\mathfrak{L}}^{\alpha}{\mathfrak{T}}_{t-s}\Phi(s,\omega,{\mathfrak{L}}^{-\alpha}x)\|_{\mathbb{X}}$
$\displaystyle\preceq$
$\displaystyle(t^{\prime}-t)^{\gamma}\|{\mathfrak{L}}^{\alpha+\gamma}{\mathfrak{T}}_{t-s}\Phi(s,\omega,{\mathfrak{L}}^{-\alpha}x)\|_{{\mathbb{X}}}$
$\displaystyle\leqslant$
$\displaystyle\frac{C_{R}(t^{\prime}-t)^{\gamma}}{(t-s)^{\alpha+\gamma}(s\wedge
1)^{\beta}}$
and
$\displaystyle\|B(t^{\prime},s,\omega,x)-B(t,s,\omega,x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}$
$\displaystyle\qquad\preceq\|({\mathfrak{T}}_{t^{\prime}-t}-1){\mathfrak{L}}^{\alpha}{\mathfrak{T}}_{t-s}\Psi(s,\omega,{\mathfrak{L}}^{-\alpha}x)\|_{L_{2}(l^{2};{\mathbb{X}})}^{2}$
$\displaystyle\qquad\preceq(t^{\prime}-t)^{\gamma}\|{\mathfrak{L}}^{\alpha+\gamma/2}{\mathfrak{T}}_{t-s}\Psi(s,{\mathfrak{L}}^{-\alpha}x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}$
$\displaystyle\qquad\preceq\frac{(t^{\prime}-t)^{\gamma}}{(t-s)^{\alpha+\gamma}}\|{\mathfrak{L}}^{\frac{\alpha}{2}}\Psi(s,{\mathfrak{L}}^{-\alpha}x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}$
$\displaystyle\qquad\leqslant\frac{C_{R}(t^{\prime}-t)^{\gamma}}{(t-s)^{\alpha+\gamma}(s\wedge
1)^{\beta}}.$
So, if we take
$\lambda_{R}(t^{\prime},t,s):=\frac{C_{R}(t^{\prime}-t)^{\gamma}}{(t-s)^{\alpha+\gamma}(s\wedge
1)^{\beta}},$
then (H4)′ holds.
Hence, by Theorem 3.7 there is a unique maximal solution $(Y,\tau)$ for
Eq.(6.3) in the sense of Definition 3.5. Set
$X(t)={\mathfrak{L}}^{-\alpha}Y(t).$
It is easy to see that $(X,\tau)$ a unique maximal solution for Eq.(6.2),
which satisfies (i), (ii) and (iii) in the theorem.
Lastly, if (M3) is satisfied, then as estimating (6.4), for the above $A$ and
$B$, (H2) holds with some $\kappa_{1}\in{\mathscr{K}}_{>1}$, and also (H4)
holds. So, by Theorem 3.8 we have $\tau=\infty$ a.s.. ∎
###### Remark 6.2.
The solution $(X,\tau)$ in Theorem 6.1 is clearly a local solution of Eq.(6.2)
in ${\mathbb{X}}$. However, it may be not a maximal solution in ${\mathbb{X}}$
because it may happen that
$\lim_{t\uparrow\tau(\omega)}\|X(t,\omega)\|_{\mathbb{X}}<+\infty\mbox{ on
$\\{\omega:\tau(\omega)<+\infty\\}$}.$
Next, we study the large deviation estimate for Eq.(6.1), and assume that
$\Phi$ and $\Psi$ are non-random. Consider the following small perturbation of
Eq.(6.1):
$\displaystyle{\mathord{{\rm
d}}}X_{\epsilon}(t)=[-{\mathfrak{L}}X_{\epsilon}(t)+\Phi(t,X_{\epsilon}(t))]{\mathord{{\rm
d}}}t+\sqrt{\epsilon}\Psi(t,X_{\epsilon}(t)){\mathord{{\rm d}}}W(t),\ \
X_{\epsilon}(0)=x_{0}.$ (6.5)
In order to apply Theorem 4.5 to this situation, we need the non-explosion
assumptions as (C2) and (C3). For a family of processes
$\\{h^{\epsilon},\epsilon\in(0,1)\\}$ in ${\mathcal{A}}^{T}_{N}$ (see (2.23)
for the definition of ${\mathcal{A}}^{T}_{N}$), consider
$\displaystyle X^{\epsilon}(t)$ $\displaystyle=$
$\displaystyle{\mathfrak{T}}_{t}x_{0}+\int^{t}_{0}{\mathfrak{T}}_{t-s}\Phi(s,X^{\epsilon}(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}{\mathfrak{T}}_{t-s}\Psi(s,X^{\epsilon}(s))\dot{h}^{\epsilon}(s){\mathord{{\rm
d}}}s$
$\displaystyle+\sqrt{\epsilon}\int^{t}_{0}{\mathfrak{T}}_{t-s}\Psi(s,X^{\epsilon}(s)){\mathord{{\rm
d}}}W(s),$
and for $h\in\ell^{2}_{T}$ (see (2.22))
$X^{h}(t)={\mathfrak{T}}_{t}x_{0}+\int^{t}_{0}{\mathfrak{T}}_{t-s}\Phi(s,X^{h}(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}{\mathfrak{T}}_{t-s}\Psi(s,X^{h}(s))\dot{h}(s){\mathord{{\rm
d}}}s.$
Below, for $n\in{\mathbb{N}}$ we define
$\tau^{\epsilon}_{n}:=\inf\\{t>0:\|X^{\epsilon}(t)\|_{{\mathbb{X}}_{\alpha}}>n\\}.$
Our large deviation principle can be stated as follows:
###### Theorem 6.3.
Assume (M1) and (M2). Let $x_{0}\in{\mathbb{X}}_{\delta}$ for some
$1\geqslant\delta>\alpha$, where $\alpha$ is from (M1). We also assume that
${\mathscr{D}}({\mathfrak{L}})={\mathbb{X}}_{1}\subset{\mathbb{X}}$ is
compact, and
$\displaystyle\lim_{n\rightarrow\infty}\sup_{\epsilon\in(0,1)}P\big{\\{}\omega:\tau^{\epsilon}_{n}(\omega)<T\big{\\}}=0$
(6.6)
and for any $N>0$
$\displaystyle\sup_{h\in{\mathbb{D}}_{N}}\sup_{t\in[0,T]}\|X^{h}(t)\|_{{\mathbb{X}}_{\alpha}}<+\infty.$
(6.7)
Then, $\\{X_{\epsilon},\epsilon\in(0,1)\\}$ satisfies the large deviation
principle in ${\mathbb{C}}_{T}({\mathbb{X}}_{\alpha})$ with the rate function
$I(f)$ given by
$\displaystyle
I(f):=\frac{1}{2}\inf_{\\{h\in\ell^{2}_{T}:~{}f=X^{h}\\}}\|h\|^{2}_{\ell^{2}_{T}},\
\ f\in{\mathbb{C}}_{T}({\mathbb{X}}_{\alpha}).$ (6.8)
###### Proof.
By Theorem 4.5, it only need to check (H1)′′ and (H2)′′ for Eq.(6.3). Since
$x_{0}\in{\mathbb{X}}^{\delta}$ with $\delta>\alpha$, by (iv) of Proposition
2.11, (H1)′′ holds with $\delta^{\prime}=\delta-\alpha$ and
$\alpha^{\prime}\in(0,\delta-\alpha)$. As the calculations given in (6.4), one
finds that (H2)′′ holds with $\alpha^{\prime}\in(0,1-\alpha-\beta)$. ∎
###### Remark 6.4.
If (M3) is satisfied, one can see that (6.6) and (6.7) hold by Remark 4.6.
We now consider another group of assumptions on the coefficients:
1. (M1)′
For some $\alpha\in(0,1)$
$\Phi:{\mathbb{R}}_{+}\times\Omega\times{\mathbb{X}}\to{\mathbb{X}}_{-\alpha}\in{\mathcal{M}}\times{\mathcal{B}}({\mathbb{X}})/{\mathcal{B}}({\mathbb{X}}_{-\alpha})$
and
$\Psi:{\mathbb{R}}_{+}\times\Omega\times{\mathbb{X}}\to
L_{2}(l^{2};{\mathbb{X}}_{-\frac{\alpha}{2}})\in{\mathcal{M}}\times{\mathcal{B}}({\mathbb{X}})/{\mathcal{B}}(L_{2}(l^{2};{\mathbb{X}}_{-\frac{\alpha}{2}})).$
2. (M2)′
For any $R>0$, there exist $C_{R}>0$ and $\beta\in(0,1)$ with
$\alpha+\beta<1$
such that for all $s>0$, $\omega\in\Omega$ and $x,y\in{\mathbb{X}}$ with
$\|x\|_{\mathbb{X}},\|y\|_{\mathbb{X}}\leqslant R$,
$\|\Phi(s,\omega,x)\|_{{\mathbb{X}}_{-\alpha}}+\|\Psi(s,\omega,x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}}_{-\frac{\alpha}{2}})}\leqslant\frac{C_{R}}{(s\wedge
1)^{\beta}}$
and
$\displaystyle\|\Phi(s,\omega,x)-\Phi(s,\omega,y)\|_{{\mathbb{X}}_{-\alpha}}$
$\displaystyle\leqslant$ $\displaystyle\frac{C_{R}}{(s\wedge
1)^{\beta}}\|x-y\|_{{\mathbb{X}}},$
$\displaystyle\|\Psi(s,\omega,x)-\Psi(s,\omega,y)\|^{2}_{L_{2}(l^{2};{\mathbb{X}}_{-\frac{\alpha}{2}})}$
$\displaystyle\leqslant$ $\displaystyle\frac{C_{R}}{(s\wedge
1)^{\beta}}\|x-y\|_{{\mathbb{X}}}^{2}.$
3. (M3)′
For all $s>0$, $\omega\in\Omega$ and $x\in{\mathbb{X}}$, it holds that
$\displaystyle\|\Phi(s,\omega,x)\|_{{\mathbb{X}}_{-\alpha}}$
$\displaystyle\leqslant$ $\displaystyle\frac{C}{(s\wedge
1)^{\beta}}(1+\|x\|_{\mathbb{X}}),$
$\displaystyle\|\Psi(s,\omega,x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}}_{-\frac{\alpha}{2}})}$
$\displaystyle\leqslant$ $\displaystyle\frac{C}{(s\wedge
1)^{\beta}}(1+\|x\|^{2}_{\mathbb{X}}).$
The following two results are parallel to Theorems 6.1 and 6.3, we omit the
details.
###### Theorem 6.5.
Under (M1)′ and (M2)′, for any $x_{0}\in{\mathbb{X}}$, there exists a unique
maximal mild solution $(X,\tau)$ for Eq. (6.2) in the sense of Definition 3.5.
Moreover, if (M3)′ holds, then $\tau=+\infty$, a.s..
###### Theorem 6.6.
Assume that (M1)′, (M2)′ and (C1)-(C3) hold. Let
$x_{0}\in{\mathbb{X}}_{\delta}$ for some $\delta>0$ Then,
$\\{X_{\epsilon},\epsilon\in(0,1)\\}$ satisfies the large deviation principle
in ${\mathbb{C}}_{T}({\mathbb{X}})$ with the rate function $I(f)$ given by
(4.6).
###### Remark 6.7.
Theorem 6.5 is due to Brzeźniak [11]. Compared with Theorem 6.1, the solution
in Theorem 6.1 has better regularity, and is in fact a strong solution under a
slightly stronger assumption (M4) below.
### 6.2. Strong solutions of SPDEs driven by Brownian motions
In this subsection, following the method used in the deterministic case (cf.
[37, 58]), we prove the existence of strong solutions for Eq.(6.1). For this
aim, in addition to (M1) and (M2) with $\beta=0$, we also assume
1. (M4)
For any $R,T>0$, there exist $\delta>0$ and $\alpha^{\prime}>1$ such that for
all $s,s^{\prime}\in[0,T]$, $\omega\in\Omega$ and $x\in{\mathbb{X}}_{\alpha}$
with $\|x\|_{{\mathbb{X}}_{\alpha}}\leqslant R$
$\displaystyle\|\Phi(s^{\prime},\omega,x)-\Phi(s,\omega,x)\|_{{\mathbb{X}}}$
$\displaystyle\leqslant$ $\displaystyle C_{T,R}|s^{\prime}-s|^{\delta},$ (6.9)
$\displaystyle\|\Psi(s,\omega,x)\|^{2}_{L_{2}(l^{2};{\mathbb{X}}_{\frac{\alpha^{\prime}}{2}})}$
$\displaystyle\leqslant$ $\displaystyle C_{T,R}.$ (6.10)
Let us first recall the following result (cf. [37, Theorem 3.2.2] or [58,
p.114, Theorem 3.5]).
###### Lemma 6.8.
Let $[0,T]\ni s\mapsto f(s)\in{\mathbb{X}}$ be a Hölder continuous function.
Then
$t\mapsto\int^{t}_{0}{\mathfrak{T}}_{t-s}f(s){\mathord{{\rm d}}}s\in
C([0,T];{\mathbb{X}}_{1}).$
Using this lemma, we can prove the existence of strong solutions for Eq.(6.1).
###### Theorem 6.9.
Assume that (M1), (M2) and (M4) hold. For any $x_{0}\in{\mathbb{X}}_{1}$, let
$(X,\tau)$ be the unique maximal solution of Eq.(6.2) in Theorem 6.1. Then
1. (i)
$t\mapsto X(t)\in{\mathbb{X}}_{1}$ is continuous on $[0,\tau)$ a.s.;
2. (ii)
it holds that in ${\mathbb{X}}$
$\displaystyle X(t)=x_{0}-\int^{t}_{0}{\mathfrak{L}}X(s){\mathord{{\rm
d}}}s+\int^{t}_{0}\Phi(s,X(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}\Psi(s,X(s)){\mathord{{\rm d}}}W(s)$
for all $t\in[0,\tau)$, $P$-a.s..
We shall call $(X,\tau)$ the unique maximal strong solution of Eq.(6.1).
###### Proof.
For $n\in{\mathbb{N}}$, set
$\tau_{n}:=\inf\\{t>0:\|X(t)\|_{{\mathbb{X}}_{\alpha}}>n\\}$
and
$G(t,s):={\mathfrak{T}}_{t-s}\Psi(s,X(s)).$
Then by (iii) and (iv) of Proposition 2.11 we have
$\|G(t,s)\|_{L_{2}(l^{2};{\mathbb{X}}_{1})}^{2}\preceq\frac{1}{(t-s)^{2-\alpha^{\prime}}}\|\Psi(s,X(s))\|^{2}_{L_{2}(l^{2};{\mathbb{X}}_{\alpha^{\prime}/2})},$
and in view of $\alpha^{\prime}>1$
$\|G(t^{\prime},s)-G(t,s)\|_{L_{2}(l^{2};{\mathbb{X}}_{1})}^{2}\preceq\frac{(t^{\prime}-t)^{(\alpha^{\prime}-1)/2}}{(t-s)^{(3-\alpha^{\prime})/2}}\|\Psi(s,X(s))\|^{2}_{L_{2}(l^{2};{\mathbb{X}}_{\alpha^{\prime}/2})}.$
Hence, by Lemma 3.4 and (6.10),
$t\mapsto\int^{t}_{0}{\mathfrak{T}}_{t-s}\Psi(s,X(s)){\mathord{{\rm
d}}}W(s)\in{\mathbb{X}}_{1}$
admits a continuous modification on $[0,\tau_{n})$.
Moreover, starting from (6.3), as in the proof of Theorem 3.3, there exists an
$a>0$ such that for $p$ sufficiently large
${\mathbb{E}}\left(\sup_{t\not=t^{\prime}\in[0,T\wedge\tau_{n}]}\frac{\|X(t^{\prime})-X(t)\|^{p}_{{\mathbb{X}}^{\alpha}}}{|t^{\prime}-t|^{ap}}\right)\leqslant
C_{n,T,p}.$
Thus, by (M2) and (M4) we know that
$s\mapsto\Phi(s,X(s))\in{\mathbb{X}}\mbox{ is H\"{o}lder continuous on
$[0,T\wedge\tau_{n}]$ $P$-a.s.}.$
Therefore, by Lemma 6.8 we have
$t\mapsto\int^{t}_{0}{\mathfrak{T}}_{t-s}\Phi(s,X(s)){\mathord{{\rm d}}}s\in
C([0,T\wedge\tau_{n}],{\mathbb{X}}_{1}),\ \ P-a.s..$
Noting that $x_{0}\in{\mathbb{X}}_{1}$ and
$\displaystyle 1_{\\{t\leqslant\tau_{n}\\}}\cdot X(t)$ $\displaystyle=$
$\displaystyle
1_{\\{t\leqslant\tau_{n}\\}}\cdot{\mathfrak{T}}_{t}x_{0}+1_{\\{t\leqslant\tau_{n}\\}}\cdot\int^{t}_{0}{\mathfrak{T}}_{t-s}\Phi(s,X(s)){\mathord{{\rm
d}}}s$
$\displaystyle+1_{\\{t\leqslant\tau_{n}\\}}\cdot\int^{t}_{0}{\mathfrak{T}}_{t-s}\Psi(s,X(s)){\mathord{{\rm
d}}}W(s),\ \ \forall t\geqslant 0,\ \ P-a.s,$
by $\tau_{n}\nearrow\tau$, we therefore have that $t\mapsto
X(t)\in{\mathbb{X}}_{1}$ is continuous on $[0,\tau)$ $P$-a.s..
Lastly, by stochastic Fubini’s theorem (cf. [54, Section 6]) we have
$\displaystyle\int^{t}_{0}{\mathfrak{L}}X(s){\mathord{{\rm d}}}s$
$\displaystyle=$
$\displaystyle\int^{t}_{0}{\mathfrak{L}}{\mathfrak{T}}_{s}x_{0}{\mathord{{\rm
d}}}s+\int^{t}_{0}\int^{s}_{0}{\mathfrak{L}}{\mathfrak{T}}_{s-r}\Phi(r,X(r)){\mathord{{\rm
d}}}r{\mathord{{\rm d}}}s$
$\displaystyle+\int^{t}_{0}\int^{s}_{0}{\mathfrak{L}}{\mathfrak{T}}_{s-r}\Psi(r,X(r)){\mathord{{\rm
d}}}W(r){\mathord{{\rm d}}}s$ $\displaystyle=$ $\displaystyle
x_{0}-{\mathfrak{T}}_{t}x_{0}+\int^{t}_{0}\int^{t}_{r}{\mathfrak{L}}{\mathfrak{T}}_{s-r}\Phi(r,X(r)){\mathord{{\rm
d}}}s{\mathord{{\rm d}}}r$
$\displaystyle+\int^{t}_{0}\int^{t}_{r}{\mathfrak{L}}{\mathfrak{T}}_{s-r}\Psi(r,X(r)){\mathord{{\rm
d}}}s{\mathord{{\rm d}}}W(r)$ $\displaystyle=$ $\displaystyle
x_{0}-{\mathfrak{T}}_{t}x_{0}+\int^{t}_{0}\Big{[}\Phi(r,X(r))-{\mathfrak{T}}_{t-r}\Phi(r,X(r))\Big{]}{\mathord{{\rm
d}}}r$
$\displaystyle+\int^{t}_{0}\Big{[}\Psi(r,X(r))-{\mathfrak{T}}_{t-r}\Psi(r,X(r))\Big{]}{\mathord{{\rm
d}}}W(r)$ $\displaystyle=$ $\displaystyle
x_{0}-X(t)+\int^{t}_{0}\Phi(s,X(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}\Psi(s,X(s)){\mathord{{\rm d}}}W(s)$
on $\\{t\leqslant\tau_{n}\\}$. The proof is completed by letting $n\to\infty$.
∎
### 6.3. SPDEs driven by fractional Brownian motions
In this subsection, we study the existence-uniqueness and large deviation for
SPDEs driven by additive fractional Brownian motions. Let for $H\in(0,1)$
$K_{H}(t,s):=\left(c_{H}(t-s)^{H-\frac{1}{2}}+s^{H-\frac{1}{2}}F(t/s)\right)1_{\\{s<t\\}},\quad
s,t\in[0,1],$
where
$c_{H}:=\left(\frac{2H\Gamma(3/2-H)}{\Gamma(H+1/2)\Gamma(2-2H)}\right)^{1/2}$,
$\Gamma$ denotes the usual Gamma function, and
$F(u):=c_{H}(\frac{1}{2}-H)\int^{u}_{1}(r-1)^{H-\frac{3}{2}}(1-r^{H-\frac{1}{2}}){\mathord{{\rm
d}}}r.$
The sequence of independent fractional Brownian motions with Hurst parameter
$H\in(0,1)$ may be defined by (cf. [21])
$W^{k}_{H}(t):=\int^{t}_{0}K_{H}(t,s){\mathord{{\rm d}}}W^{k}(s),\ \
k=1,2,\cdots,$
which has the covariance function
$R_{H}(t,s)={\mathbb{E}}(W^{k}_{H}(t)W^{k}_{H}(s))=\frac{1}{2}(s^{2H}+t^{2H}-|t-s|^{2H}).$
Consider the following stochastic partial differential equation driven by
$\\{W^{k}_{H},k\in{\mathbb{N}}\\}$
$\displaystyle{\mathord{{\rm
d}}}X(t)=[-{\mathfrak{L}}X(t)+\Phi(t,X(t))]{\mathord{{\rm
d}}}t+\Psi(t){\mathord{{\rm d}}}W_{H}(t),\ \ X(0)=x_{0}\in{\mathbb{X}},$
(6.11)
where $\Psi(t)$ is a deterministic function and will be specified in Theorem
6.10.
As above, we consider the mild solution:
$\displaystyle
X(t)={\mathfrak{T}}_{t}x_{0}+\int^{t}_{0}{\mathfrak{T}}_{t-s}\Phi(s,X(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}{\mathfrak{T}}_{t-s}\Psi(s){\mathord{{\rm d}}}W_{H}(s).$
(6.12)
Here the stochastic integral is defined by the integration by parts formula as
$\displaystyle\int^{t}_{0}{\mathfrak{T}}_{t-s}\Psi(s){\mathord{{\rm
d}}}W_{H}(s)$ $\displaystyle:=$
$\displaystyle\Psi(t)W_{H}(t)+\int^{t}_{0}W_{H}(s)[{\mathfrak{L}}{\mathfrak{T}}_{t-s}\Psi(s)-{\mathfrak{T}}_{t-s}\dot{\Psi}(s)]{\mathord{{\rm
d}}}s$ $\displaystyle=$ $\displaystyle\int^{t}_{0}B(t,s){\mathord{{\rm
d}}}W(s),$
where
$B(t,s):=\Psi(t)K_{H}(t,s)+\int^{t}_{s}K_{H}(u,s)\big{[}{\mathfrak{L}}{\mathfrak{T}}_{t-u}\Psi(u)-{\mathfrak{T}}_{t-s}\dot{\Psi}(s)\big{]}{\mathord{{\rm
d}}}u.$
We also define
$A(t,s,x):={\mathfrak{T}}_{t-s}\Phi(s,x).$
Then we have:
###### Theorem 6.10.
Assume that $\Phi$ satisfies (M1)′ and (M2)′ and $\Psi$ satisfies for some
$\gamma>0$
$\displaystyle\|\Psi(t^{\prime})-\Psi(t)\|_{L_{2}(l^{2};{\mathbb{X}})}\leqslant
C|t^{\prime}-t|^{\gamma},\ \ t,t^{\prime}\in[0,1]$ (6.13)
and for some $\delta\in(0,1)$
$\displaystyle\sup_{t\in[0,1]}\Big{(}\|\Psi(t)\|_{L_{2}(l^{2};{\mathbb{X}}_{\delta})}+\|\dot{\Psi}(t)\|_{L_{2}(l^{2};{\mathbb{X}}_{\delta-1})}\Big{)}<+\infty,\
\ t\in[0,1].$ (6.14)
Then for any $x_{0}\in{\mathbb{X}}$, there exists a unique maximal solution
for Eq.(6.12) in the sense of Definition 3.5. In particular, if $\Phi$ also
satisfies (M3)′, then there is no explosion for Eq.(6.12).
###### Proof.
As the proof of Theorem 6.1, one can check that $A$ satisfies (H2)′-(H4)′. In
order to finish the proof by Theorem 3.7, we need to verify that $B$ also
satisfies (H2) and (H4). We first check that for some $\gamma^{\prime}>0$
$\displaystyle\int^{t}_{0}\|B(t^{\prime},s)-B(t,s)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}{\mathord{{\rm
d}}}s\preceq|t-t^{\prime}|^{\gamma^{\prime}},\ \ 0\leqslant
t<t^{\prime}\leqslant 1.$ (6.15)
Noting that
$\displaystyle K_{H}(t,s)\leqslant
C|t-s|^{H-\frac{1}{2}}+Cs^{-|H-\frac{1}{2}|}$ (6.16)
and
$\displaystyle\int^{t}_{0}[K_{H}(t,s)-K_{H}(t^{\prime},s)]^{2}{\mathord{{\rm
d}}}s$ $\displaystyle\leqslant$ $\displaystyle
R_{H}(t,t)-2R_{H}(t,t^{\prime})+R_{H}(t^{\prime},t^{\prime})$ $\displaystyle=$
$\displaystyle
t^{2H}-(t^{2H}+(t^{\prime})^{2H}-|t^{\prime}-t|^{2H})+(t^{\prime})^{2H}$
$\displaystyle=$ $\displaystyle|t^{\prime}-t|^{2H},$
by (6.13) we have
$\int^{t}_{0}\|\Psi(t^{\prime})K_{H}(t^{\prime},s)-\Psi(t)K_{H}(t,s)\|^{2}_{L_{2}(l^{2};{\mathbb{X}})}{\mathord{{\rm
d}}}s\preceq|t^{\prime}-t|^{2H\wedge\gamma}.$
Observe that
$\displaystyle\int^{t}_{0}\left\|\int^{t^{\prime}}_{s}K_{H}(u,s){\mathfrak{L}}{\mathfrak{T}}_{t^{\prime}-u}\Psi(u){\mathord{{\rm
d}}}u-\int^{t}_{s}K_{H}(u,s){\mathfrak{L}}{\mathfrak{T}}_{t-u}\Psi(u){\mathord{{\rm
d}}}u\right\|_{L_{2}(l^{2};{\mathbb{X}})}^{2}{\mathord{{\rm d}}}s$
$\displaystyle\quad\preceq\int^{t}_{0}\left|\int^{t^{\prime}}_{t}K_{H}(u,s)\cdot\|{\mathfrak{L}}{\mathfrak{T}}_{t^{\prime}-u}\Psi(u)\|_{L_{2}(l^{2};{\mathbb{X}})}{\mathord{{\rm
d}}}u\right|^{2}{\mathord{{\rm d}}}s$
$\displaystyle\quad\quad+\int^{t}_{0}\left|\int^{t}_{s}K_{H}(u,s)\cdot\|{\mathfrak{L}}({\mathfrak{T}}_{t^{\prime}-u}-{\mathfrak{T}}_{t-u})\Psi(u)\|_{L_{2}(l^{2};{\mathbb{X}})}{\mathord{{\rm
d}}}u\right|^{2}{\mathord{{\rm d}}}s$ $\displaystyle\quad=:J_{1}+J_{2}.$
By (6.16), (iii) (iv) of Proposition 2.11 and (6.14), we have
$\displaystyle J_{1}$ $\displaystyle\preceq$
$\displaystyle\int^{t}_{0}\left|\int^{t^{\prime}}_{t}\Big{[}(u-s)^{H-\frac{1}{2}}+s^{-|H-\frac{1}{2}|}\Big{]}\cdot(t^{\prime}-u)^{\delta-1}{\mathord{{\rm
d}}}u\right|^{2}{\mathord{{\rm d}}}s$ $\displaystyle\preceq$
$\displaystyle\int^{t}_{0}\Big{[}(t-s)^{H-\frac{1}{2}}+s^{-|H-\frac{1}{2}|}\Big{]}^{2}\cdot\left|\int^{t^{\prime}}_{t}(t^{\prime}-u)^{\delta-1}{\mathord{{\rm
d}}}u\right|^{2}{\mathord{{\rm d}}}s$ $\displaystyle\preceq$
$\displaystyle|t^{\prime}-t|^{2\delta}$
and
$\displaystyle J_{2}$ $\displaystyle\preceq$
$\displaystyle\int^{t}_{0}\left|\int^{t}_{s}\Big{[}(u-s)^{H-\frac{1}{2}}+s^{-|H-\frac{1}{2}|}\Big{]}\cdot(t^{\prime}-t)^{\frac{\delta}{2}}\cdot(t-u)^{\frac{\delta}{2}-1}{\mathord{{\rm
d}}}u\right|^{2}{\mathord{{\rm d}}}s$ $\displaystyle\preceq$
$\displaystyle(t^{\prime}-t)^{\delta}\int^{t}_{0}\left|\int^{t}_{s}(u-s)^{H-\frac{1}{2}}(t-u)^{\frac{\delta}{2}-1}{\mathord{{\rm
d}}}u\right|^{2}{\mathord{{\rm d}}}s$
$\displaystyle+(t^{\prime}-t)^{\delta}\int^{t}_{0}s^{-2|H-\frac{1}{2}|}\left|\int^{t}_{s}(t-u)^{\frac{\delta}{2}-1}{\mathord{{\rm
d}}}u\right|^{2}{\mathord{{\rm d}}}s$ $\displaystyle=:$ $\displaystyle
J_{21}+J_{22}.$
It is clear that
$J_{22}\preceq(t^{\prime}-t)^{\delta}\int^{t}_{0}s^{-2|H-\frac{1}{2}|}(t-s)^{\delta}{\mathord{{\rm
d}}}s\preceq(t^{\prime}-t)^{\delta}.$
For $J_{21}$, let us make the following elementary estimation:
$\displaystyle\int^{t}_{0}\left|\int^{t}_{s}(u-s)^{H-\frac{1}{2}}(t-u)^{\frac{\delta}{2}-1}{\mathord{{\rm
d}}}u\right|^{2}{\mathord{{\rm d}}}s$
$\displaystyle\quad\preceq\int^{t}_{0}\left|\int^{\frac{t+s}{2}}_{s}(u-s)^{H-\frac{1}{2}}(t-u)^{\frac{\delta}{2}-1}{\mathord{{\rm
d}}}u\right|^{2}{\mathord{{\rm d}}}s$
$\displaystyle\quad\quad+\int^{t}_{0}\left|\int^{t}_{\frac{t+s}{2}}(u-s)^{H-\frac{1}{2}}(t-u)^{\frac{\delta}{2}-1}{\mathord{{\rm
d}}}u\right|^{2}{\mathord{{\rm d}}}s$
$\displaystyle\quad\preceq\int^{t}_{0}(t-s)^{\delta-2}\left|\int^{\frac{t+s}{2}}_{s}(u-s)^{H-\frac{1}{2}}{\mathord{{\rm
d}}}u\right|^{2}{\mathord{{\rm d}}}s$
$\displaystyle\quad\quad+\int^{t}_{0}(t-s)^{2H-1}\left|\int^{t}_{\frac{t+s}{2}}(t-u)^{\frac{\delta}{2}-1}{\mathord{{\rm
d}}}u\right|^{2}{\mathord{{\rm d}}}s$
$\displaystyle\quad\preceq\int^{t}_{0}(t-s)^{2H+\delta-1}{\mathord{{\rm
d}}}s\leqslant C.$
Hence
$J_{1}+J_{2}\leqslant C(t^{\prime}-t)^{\delta}.$
Similarly, we have
$\displaystyle\int^{t}_{0}\left\|\int^{t^{\prime}}_{s}K_{H}(u,s){\mathfrak{T}}_{t^{\prime}-u}\dot{\Psi}(u){\mathord{{\rm
d}}}u-\int^{t}_{s}K_{H}(u,s){\mathfrak{T}}_{t-u}\dot{\Psi}(u){\mathord{{\rm
d}}}u\right\|_{L_{2}(l^{2};{\mathbb{X}})}^{2}{\mathord{{\rm d}}}s$
$\displaystyle\qquad\qquad\leqslant C(t^{\prime}-t)^{\delta}.$
Summing up the above calculations, we get (6.15). Thus, $B$ satisfies (H4)′.
Moreover, from the above calculations, one can see that
$\|B(t,s)\|_{L_{2}(l^{2};{\mathbb{X}})}^{2}\leqslant
C(|t-s|^{2H-1}+s^{-|2H-1|}+|t-s|^{2H+\delta-1})=:\kappa(t,s).$
So, $B$ satisfies (H2)′ with $\kappa\in{\mathscr{K}}_{>1}$.
Lastly, if $\Phi$ also satisfies (M3)′, then the non-explosion follows from
Theorem 3.8. ∎
Consider the following small perturbation of Eq.(6.12):
$X_{\epsilon}(t)={\mathfrak{T}}_{t}x_{0}+\int^{t}_{0}{\mathfrak{T}}_{t-s}\Phi(s,X_{\epsilon}(s)){\mathord{{\rm
d}}}s+\sqrt{\epsilon}\int^{t}_{0}{\mathfrak{T}}_{t-s}\Psi(s){\mathord{{\rm
d}}}W_{H}(s).$
A direct application of Theorem 4.5 yields that
###### Theorem 6.11.
Keep the same assumptions as Theorem 6.10, where $\Phi$ satisfies (M1)′-(M3)′.
Then for any $x_{0}\in{\mathbb{X}}_{\alpha}$ ($\alpha>0$),
$\\{X_{\epsilon},\epsilon\in(0,1)\\}$ satisfies the large deviation principle
in ${\mathbb{C}}_{1}({\mathbb{X}})$ with the rate function $I(f)$ given by
$I(f):=\frac{1}{2}\inf_{\\{h\in\ell^{2}_{1}:~{}f=X^{h}\\}}\|h\|^{2}_{\ell^{2}_{1}},\
\ f\in{\mathbb{C}}_{1}({\mathbb{X}}),$
where $X^{h}$ solves the following integral equation
$X^{h}(t)={\mathfrak{T}}_{t}x_{0}+\int^{t}_{0}{\mathfrak{T}}_{t-s}\Phi(s,X^{h}(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}B(t,s)\dot{h}(s){\mathord{{\rm d}}}s.$
###### Remark 6.12.
Let $\Psi_{0}\in L_{2}(l^{2};{\mathbb{X}}_{\delta})$ for some
$\delta\in(0,1)$. Then $\Psi(t):={\mathfrak{T}}_{t}\Psi_{0}$ satisfies (6.13)
and (6.14) by Proposition 2.11. Moreover, under stronger assumptions on
$\Psi(t)$, we can also prove the existence of strong solutions for Eq.(6.12)
as Theorem 6.9.
## 7\. Application to SPDEs in bounded domains of ${\mathbb{R}}^{d}$
Let ${\mathcal{O}}$ be an open bounded domain of ${\mathbb{R}}^{d}$ with
smooth boundary. For $m\in{\mathbb{N}}$, by $C^{m}({\mathcal{O}})$ (resp.
$C^{m}_{0}({\mathcal{O}})$) we denote the set of all $m$-times continuously
differentiable functions in ${\mathcal{O}}$ (resp. with compact support in
${\mathcal{O}}$). For $u\in C^{m}({\mathcal{O}})$ and $p\geqslant 1$ we define
$\|u\|_{m,p}:=\left(\sum_{j=0}^{m}\int_{\mathcal{O}}|D^{j}u(x)|^{p}{\mathord{{\rm
d}}}x\right)^{1/p},$
where $D^{j}$ is the usual derivative operator. The Sobolev spaces
$W^{m,p}({\mathcal{O}})$ and $W^{m,p}_{0}({\mathcal{O}})$ are defined
respectively as the completions of $C^{m}({\mathcal{O}})$ and
$C^{m}_{0}({\mathcal{O}})$ with respect to the norm $\|\cdot\|_{m,p}$.
Let ${\mathscr{A}}(x,D)$ be a strongly elliptic differential operator in
${\mathcal{O}}$ of the form (cf. [27, 58]):
${\mathscr{A}}(x,D)u:=\sum_{k=0}^{2m}\sum_{\alpha_{1}+\cdots\alpha_{d}=k}a_{\alpha_{1}\cdots\alpha_{d}}(x)D^{\alpha_{1}}_{1}\cdots
D^{\alpha_{d}}_{d}u,\ \ m\geqslant 1,$
where $a_{\alpha_{1}\cdots\alpha_{d}}(x)\in C^{\infty}(\bar{\mathcal{O}})$,
and $D^{\alpha_{j}}_{j}$ is the $\alpha_{j}$-order derivative with respect to
the $j$-th variable. We consider the following stochastic partial differential
equation:
$\displaystyle\left\\{\begin{aligned} {\mathord{{\rm
d}}}u(t,x)=&\Big{[}{\mathscr{A}}(x,D)u(t)+\varphi(t,x,u,Du,\cdots,D^{2m-1}u)\Big{]}{\mathord{{\rm
d}}}t\\\ &+\psi(t,x,u,Du,\cdots,D^{m-1}u){\mathord{{\rm d}}}W(t),\\\
\frac{\partial^{j}u(t,x)}{\partial\nu^{j}}=&0,\ \ j=0,1,\cdots,m-1,\
x\in\partial{\mathcal{O}},\\\ u(0,x)=&u_{0}(x),\end{aligned}\right.$ (7.1)
where $\frac{\partial^{j}}{\partial\nu^{j}}$ denotes the $j$-th outward normal
derivative, $\varphi$ and $\psi$ are two measurable functions with the
entries:
$\displaystyle\varphi:{\mathbb{R}}_{+}\times{\mathcal{O}}\times{\mathbb{R}}\times{\mathbb{R}}^{d}\times\cdots\times{\mathbb{R}}^{(2m-1)d}$
$\displaystyle\to$ $\displaystyle{\mathbb{R}},$
$\displaystyle\psi:{\mathbb{R}}_{+}\times{\mathcal{O}}\times{\mathbb{R}}\times{\mathbb{R}}^{d}\times\cdots\times{\mathbb{R}}^{(m-1)d}$
$\displaystyle\to$ $\displaystyle l^{2}.$
Define for $p>1$ and $\lambda>0$
${\mathfrak{L}}_{p}u:=\lambda u-{\mathscr{A}}(x,D)u$
with
$u\in{\mathscr{D}}({\mathfrak{L}}_{p}):=W^{2m,p}({\mathcal{O}})\cap
W^{m,p}_{0}({\mathcal{O}}).$
It is well known that for $u\in{\mathscr{D}}({\mathfrak{L}}_{p})$ (cf. [58, p.
212, Theorem 3.1] or [76])
$\displaystyle\|u\|_{2m,p}\preceq\|{\mathfrak{L}}_{p}u\|_{L^{p}}+\|u\|_{L^{p}},$
(7.2)
and $({\mathfrak{L}}_{p},{\mathscr{D}}({\mathfrak{L}}_{p}))$ is a sectorial
operator on ${\mathbb{X}}^{p}_{0}=L^{p}({\mathcal{O}})$ with
$0\in\rho({\mathfrak{L}}_{p})$ for $\lambda$ large enough (cf. [58, p. 213,
Theorem 3.5]). Below we shall write for $p>1$ and $\alpha\geqslant 0$
${\mathbb{X}}_{\alpha}^{p}:={\mathscr{D}}({\mathfrak{L}}^{\alpha}_{p}).$
We first recall the following well known result (cf. [58, p.243]).
###### Lemma 7.1.
For any $p>1$, $j<2m$ and any
$0\leqslant\alpha^{\prime}<\frac{j}{2m}<\alpha\leqslant 1$ we have
$\displaystyle\|{\mathfrak{L}}^{\alpha^{\prime}}_{p}u\|_{L^{p}}\preceq\|u\|_{j,p}\preceq\|{\mathfrak{L}}_{p}^{\alpha}u\|_{L^{p}},\
\ u\in{\mathscr{D}}({\mathfrak{L}}_{p}^{\alpha}).$ (7.3)
Moreover,
${\mathbb{X}}_{\alpha}^{p}\hookrightarrow W^{k,q}\ \ \mbox{ for
$k-\frac{d}{q}<2m\alpha-\frac{d}{p}$,\ $q\geqslant p$}$
and
$\displaystyle{\mathbb{X}}_{\alpha}^{p}\hookrightarrow
C^{\nu}(\bar{\mathcal{O}})\ \ \mbox{ for
$0\leqslant\nu<2m\alpha-\frac{d}{p}$},$ (7.4)
where $C^{\nu}(\bar{\mathcal{O}})$ is the usual Hölder space (cf. [1, 58]).
In this section, we fix
$\displaystyle p>d\mbox{ and }\frac{2m-1+\frac{d}{p}}{2m}<\alpha_{0}<\alpha<1$
(7.5)
so that
$\displaystyle m(1-\alpha)^{2}<(\alpha-\alpha_{0}).$ (7.6)
Suppose that
1. (F1)
For any $T,R>0$, there exist $\delta>0$ and $C_{R,T}>0$ such that for all
$s,t\in[0,T]$, $x\in{\mathcal{O}}$ and $U,V\in{\mathbb{R}}^{m(2m-1)d+1}$ with
$|U|,|V|\leqslant R$
$|\varphi(t,x,U)-\varphi(s,x,V)|\leqslant C_{R,T}(|t-s|^{\delta}+|U-V|).$
Moreover, $\sup_{x\in{\mathcal{O}}}|\varphi(0,x,0)|<+\infty$.
2. (F2)
For each $t\in{\mathbb{R}}_{+}$, $\psi(t,*)\in
C^{m+1}(\bar{\mathcal{O}}\times{\mathbb{R}}^{m(m+1)d/2+d+1};l^{2})$. Here and
below, the asterisk stands for the rest variables.
3. (F3)
For each $u\in{\mathbb{X}}_{\alpha_{0}}^{p}$,
$\psi(t,\cdot,u,Du,\cdots,D^{m-1}u)\in{\mathbb{X}}_{\frac{\alpha}{2}}^{p}.$
4. (F4)
For any $T>0$, there exist constant $C_{T}>0$ and $\lambda_{0}\in
L^{p}({\mathcal{O}})$ such that for all $t\in[0,T]$, $x\in{\mathcal{O}}$ and
$U\in{\mathbb{R}}^{m(2m-1)d+1}$
$\displaystyle|\varphi(t,x,U)|\leqslant C_{T}(\lambda_{0}(x)+|U|),$ (7.7)
and
$\displaystyle\left\\{\begin{aligned}
&\psi(t,x,u,Du,\cdots,D^{m-1}u)=\sum_{j=0}^{m-1}g_{j}(t)\cdot
D^{j}u+\psi_{0}(t,x),\ \ &m\geqslant 2,\\\ &\left.\begin{aligned} &\mbox{for
some $\delta>0$ and each $r\in{\mathbb{R}}$},\
\mbox{supp}(\psi(t,\cdot,r))\subset{\mathcal{O}}_{\delta},\\\ &\psi(t,*)\in
C^{2}(\bar{\mathcal{O}}\times{\mathbb{R}}^{d+1};l^{2}),\
\|\partial_{r}\psi(t,x,r)\|_{l^{2}}\leqslant C_{T},\\\
&\|D_{x}\psi(t,x,r)\|_{l^{2}\times{\mathbb{R}}^{d}}+\|\psi(t,x,r)\|_{l^{2}}\leqslant
C(f_{0}(t,x)+|r|)\end{aligned}\right\\},&m=1,\end{aligned}\right.$ (7.8)
where
${\mathcal{O}}_{\delta}\subset\bar{\mathcal{O}}_{\delta}\subset{\mathcal{O}}$
is an open subset, and for each $j=0,\cdots,m-1$,
$\displaystyle t\mapsto g_{j}(t)\in l^{2}\times{\mathbb{R}}^{jd},$
$\displaystyle t\mapsto\psi_{0}(t,\cdot)\in
l^{2}\times{\mathbb{X}}^{p}_{\alpha},$ $\displaystyle t\mapsto
f_{0}(t,\cdot)\in L^{p}({\mathcal{O}})$
are bounded measurable functions.
We remark that (F3) is related to the boundary conditions, e.g.,
$\Psi\equiv$constant does not satisfy (F3). It is easy to see that (7.8)
implies (F3).
Set
$\displaystyle\Phi(t,u)(x)$ $\displaystyle:=$
$\displaystyle\varphi(t,x,u,Du,\cdots,D^{2m-1}u)+\lambda u,$ (7.9)
$\displaystyle\Psi(t,u)(x)$ $\displaystyle:=$
$\displaystyle\psi(t,x,u,Du,\cdots,D^{m-1}u).$ (7.10)
Then the system (7.1) can be written as the following abstract form:
$\displaystyle{\mathord{{\rm
d}}}u(t)=[-{\mathfrak{L}}_{p}u+\Phi(t,u(t))]{\mathord{{\rm
d}}}t+\Psi(t,u(t)){\mathord{{\rm d}}}W(t),\ \ u(0)=u_{0}.$ (7.11)
Using Theorem 6.9, we have the following result.
###### Theorem 7.2.
Let $p>d$ and $\alpha,\alpha_{0}$ satisfy (7.5) and (7.6). Assume that
(F1)-(F3) hold. For any $u_{0}\in{\mathbb{X}}_{1}^{p}$, there exists a unique
maximal strong solution $(u,\tau)$ for Eq. (7.11) so that
1. (i)
$t\mapsto u(t)\in{\mathbb{X}}^{p}_{1}$ is continuous on $[0,\tau)$ almost
surely;
2. (ii)
$\lim_{t\uparrow\tau}\|u(t)\|_{{\mathbb{X}}^{p}_{\alpha}}=+\infty$ on
$\\{\omega:\tau(\omega)<+\infty\\}$;
3. (iii)
it holds that in $L^{p}({\mathcal{O}})$
$\displaystyle u(t)$ $\displaystyle=$ $\displaystyle
u_{0}-\int^{t}_{0}{\mathfrak{L}}_{p}u(s){\mathord{{\rm
d}}}s+\int^{t}_{0}\Phi(s,u(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}\Psi(s,u(s)){\mathord{{\rm d}}}W(s)$ $\displaystyle=$
$\displaystyle u_{0}+\int^{t}_{0}{\mathscr{A}}(x,D)u(s){\mathord{{\rm
d}}}s+\int^{t}_{0}\varphi(s,x,u(s),Du(s),\cdots,D^{2m-1}u(s)){\mathord{{\rm
d}}}s$
$\displaystyle+\int^{t}_{0}\psi(s,x,u(s),Du(s),\cdots,D^{m-1}u(s)){\mathord{{\rm
d}}}W(s)$
for all $t<\tau$, $P$-a.s..
Moreover, if (F4) holds, then
$\tau=+\infty,\ \ a.s..$
###### Proof.
We only need to verify that (M1)-(M4) hold for $\Phi$ and $\Psi$ defined by
(7.9) and (7.10). In virtue of (7.5), by (7.4) we have
$\displaystyle\|D^{j}u\|_{C(\bar{\mathcal{O}})}\preceq\|u\|_{{\mathbb{X}}_{\alpha_{0}}^{p}},\
\ j=0,1,\cdots,2m-1.$ (7.12)
It is easy to see by (F1) that $\Phi$ given by (7.9) is locally Lipschitz
continuous and locally bounded with respect to $u$ on
${\mathbb{X}}_{\alpha}^{p}$, and is $\delta$-order Hölder continuous with
respect to $t$.
Note that by the chain rule, for $j=1,\cdots,m+1$
$\displaystyle D^{j}\Psi(t,u)$ $\displaystyle=$
$\displaystyle(\partial_{D^{m-1}u}\psi)(t,x,u,Du,\cdots,D^{m-1}u)\cdot
D^{m-1+j}u$ (7.13) $\displaystyle+\psi_{j}(t,x,u,Du,\cdots,D^{m-2+k}u),$
where $\psi_{j}$ is an $l^{2}$-valued continuously differentiable function of
all its variables with the exception of the $t$-variable. For any
$u,v\in{\mathbb{X}}_{\alpha_{0}}^{p}$ with
$\|u\|_{{\mathbb{X}}_{\alpha_{0}}^{p}},\|v\|_{{\mathbb{X}}_{\alpha_{0}}^{p}}\leqslant
R$, by (F2) and (F3) we have
$\displaystyle\|{\mathfrak{L}}^{\frac{\alpha}{2}}(\Psi(t,u)-\Psi(t,v))\|^{2}_{L_{2}(l^{2};{\mathbb{X}}^{p}_{0})}$
$\displaystyle\stackrel{{\scriptstyle(\ref{Exa})}}{{\preceq}}$
$\displaystyle\|{\mathfrak{L}}^{\frac{\alpha}{2}}(\Psi(t,u)-\Psi(t,v))\|^{2}_{L^{p}({\mathcal{O}})}$
$\displaystyle\stackrel{{\scriptstyle(\ref{PP4})}}{{\preceq}}$
$\displaystyle\sum_{k=1}^{\infty}\sum_{j=0}^{m}\|D^{j}(\Psi_{k}(t,u)-\Psi_{k}(t,v))\|^{2}_{L^{p}({\mathcal{O}};l^{2})}$
$\displaystyle\preceq$
$\displaystyle\sum_{k=1}^{\infty}\sum_{j=0}^{m}\|D^{j}(\Psi_{k}(t,u)-\Psi_{k}(t,v))\|^{2}_{C(\bar{\mathcal{O}})}$
$\displaystyle\stackrel{{\scriptstyle(\ref{PL1},\ref{PO7})}}{{\leqslant}}$
$\displaystyle
C_{R}\sum_{j=0}^{2m-1}\|D^{j}(u-v)\|_{C(\bar{\mathcal{O}})}^{2}$
$\displaystyle\stackrel{{\scriptstyle(\ref{PO7})}}{{\preceq}}$ $\displaystyle
C_{R}\|u-v\|^{2}_{{\mathbb{X}}_{\alpha_{0}}^{p}}\leqslant
C_{R}\|u-v\|_{{\mathbb{X}}_{\alpha}^{p}}^{2}.$
Thus, (M2) holds.
We next look at (M4). As above, by (7.13) and (7.12) we have
$\displaystyle\|D^{m+1}\Psi(t,u)\|_{L^{p}({\mathcal{O}};l^{2})}\leqslant
C_{R}(1+\|D^{2m}u\|_{L^{p}})\stackrel{{\scriptstyle(\ref{Po7})}}{{\leqslant}}C_{R}(1+\|u\|_{{\mathbb{X}}^{p}_{1}})$
(7.14)
for all $u\in{\mathbb{X}}_{1}^{p}$ with
$\|u\|_{{\mathbb{X}}_{\alpha_{0}}^{p}}\leqslant R$. By (7.6), we may choose
$1<\alpha^{\prime}<\alpha^{\prime\prime}<\frac{m+1}{m}$
such that
$\theta:=\frac{\alpha-\alpha_{0}}{1-\alpha_{0}}>\frac{\alpha^{\prime}-\alpha}{\alpha^{\prime\prime}-\alpha}=:\theta^{\prime}.$
Thus, for all $u\in{\mathbb{X}}_{1}^{p}$ with
$\|u\|_{{\mathbb{X}}_{\alpha_{0}}^{p}}\leqslant R$, we have
$\displaystyle\|\Psi(t,u)\|^{2}_{L_{2}(l^{2};{\mathbb{X}}_{\alpha^{\prime\prime}/2}^{p})}$
$\displaystyle\stackrel{{\scriptstyle(\ref{Exa})}}{{\preceq}}$
$\displaystyle\|{\mathfrak{L}}^{\frac{\alpha^{\prime\prime}}{2}}\Psi(t,u)\|^{2}_{L^{p}({\mathcal{O}};l^{2})}$
$\displaystyle\stackrel{{\scriptstyle(\ref{PP4})}}{{\preceq}}$
$\displaystyle\sum_{j=0}^{m+1}\|D^{j}\Psi(t,u)\|^{2}_{L^{p}({\mathcal{O}};l^{2})}$
$\displaystyle\preceq$
$\displaystyle\sum_{j=0}^{m}\sum_{k=1}^{\infty}\|D^{j}\Psi_{k}(t,u)\|^{2}_{C(\bar{\mathcal{O}})}$
$\displaystyle+\|D^{m+1}\Psi(t,u)\|_{L^{p}({\mathcal{O}};l^{2})}^{2}$
$\displaystyle\stackrel{{\scriptstyle(\ref{PO8})}}{{\leqslant}}$
$\displaystyle C_{R}(1+\|u\|^{2}_{{\mathbb{X}}_{1}^{p}}).$
Using Lemma 2.13 with the data $\alpha_{0}$, $\theta$ and $\theta^{\prime}$ as
above and
$\alpha_{1}=1,\ \ \alpha_{2}=\frac{\alpha}{2},\ \
\alpha_{3}=\frac{\alpha^{\prime\prime}}{2},$
we obtain that for all $u\in{\mathbb{X}}^{p}_{\alpha}$ with
$\|u\|_{{\mathbb{X}}_{\alpha}^{p}}\leqslant R$
$\displaystyle\|\Psi(t,u)\|_{L_{2}(l^{2};{\mathbb{X}}^{p}_{\alpha^{\prime}/2})}=\|\Psi(t,u)\|_{L_{2}(l^{2};{\mathbb{X}}^{p}_{\theta^{\prime}(\alpha_{3}-\alpha_{2})+\alpha_{2}})}\leqslant
C_{R}.$
Thus, (M4) holds.
We now verify (M3) under (F4). First of all, by the linear growth of
$\varphi(t,x,*)$ with respect to $*$, we have
$\displaystyle\|\Phi(t,u)\|_{{\mathbb{X}}^{p}_{0}}$ $\displaystyle=$
$\displaystyle\|\varphi(t,\cdot,u,Du,D^{2}u,\cdots,D^{2m-1}u)\|_{L^{p}}$
$\displaystyle\preceq$ $\displaystyle 1+\sum_{j=0}^{2m-1}\|D^{j}u\|_{L^{p}}$
$\displaystyle\preceq$ $\displaystyle 1+\|u\|_{2m-1,p}$
$\displaystyle\stackrel{{\scriptstyle(\ref{PP4})}}{{\preceq}}$ $\displaystyle
1+\|u\|_{{\mathbb{X}}^{p}_{\alpha}}.$
For $\Psi$, we only consider the case of $m=1$, and have
$\displaystyle\|\Psi(t,u)\|^{2}_{L_{2}(l^{2};{\mathbb{X}}^{p}_{\frac{\alpha}{2}})}$
$\displaystyle\stackrel{{\scriptstyle(\ref{Exa})}}{{\preceq}}$
$\displaystyle\|{\mathfrak{L}}_{p}^{\frac{\alpha}{2}}\Psi(t,u)\|^{2}_{L^{p}({\mathcal{O}};l^{2})}$
$\displaystyle\stackrel{{\scriptstyle(\ref{PP4})}}{{\preceq}}$
$\displaystyle\|\Psi(t,u)\|^{2}_{L^{p}({\mathcal{O}};l^{2})}+\|D\Psi(t,u)\|^{2}_{L^{p}({\mathcal{O}};l^{2})}.$
Noting that
$D\Psi(t,u)=(D_{x}\psi)(t,x,u)+(\partial_{u}\psi)(t,x,u)Du$
by (7.8) we have
$\|D\Psi(t,u)\|^{2}_{L^{p}({\mathcal{O}};l^{2})}\preceq
1+\|u\|_{L^{p}}+\|Du\|_{L^{p}}\preceq 1+\|u\|_{{\mathbb{X}}^{p}_{\alpha}}.$
So
$\|\Psi(t,u)\|^{2}_{L_{2}(l^{2};{\mathbb{X}}^{p}_{\frac{\alpha}{2}})}\preceq
1+\|u\|_{{\mathbb{X}}^{p}_{\alpha}}.$
Thus, (M3) holds. ∎
Consider the small perturbation of equation (7.11):
$\displaystyle{\mathord{{\rm
d}}}u_{\epsilon}(t)=[-{\mathfrak{L}}_{p}u_{\epsilon}(s)+\Phi(t,u_{\epsilon}(t))]{\mathord{{\rm
d}}}t+\sqrt{\epsilon}\Psi(t,u_{\epsilon}(t)){\mathord{{\rm d}}}W(t),\ \
u_{\epsilon}(0)=u_{0}.$ (7.15)
Using Theorem 6.3, we have
###### Theorem 7.3.
Let $p>d$ and $\alpha,\alpha_{0}$ satisfy (7.5) and (7.6). Assume that (F1)
and (F4) hold. Let $u_{0}\in{\mathbb{X}}_{1}^{p}$. Then
$\\{u_{\epsilon},\epsilon\in(0,1)\\}$ satisfies the large deviation principle
in ${\mathbb{C}}_{T}({\mathbb{X}}_{\alpha}^{p})$ with the rate function $I(f)$
given by (6.8).
## 8\. Application to SPDEs on complete Riemannian manifolds
Let $(M,{\mathfrak{g}})$ be a $d$-dimensional complete Riemannian manifold
without boundary. The Riemannian volume is denoted by ${\mathord{{\rm
d}}}_{\mathfrak{g}}x$. Let $\nabla$ denote the gradient or covariant
derivative associated with ${\mathfrak{g}}$, $\Delta$ the Laplace Beltrami
operator, $T(M)$ the tangent bundle. Let $L^{p}(M,{\mathord{{\rm
d}}}_{\mathfrak{g}}x)$ be the usual real $L^{p}$-space on $M$ with respect to
${\mathord{{\rm d}}}_{\mathfrak{g}}x$. It is well known that the symmetric
heat semigroup $({\mathfrak{T}}_{t})_{t\geqslant 0}$ associated with $\Delta$
is strongly continuous and contracted on $L^{p}(M,{\mathord{{\rm
d}}}_{\mathfrak{g}}x)$ for $1\leqslant p<+\infty$, which is also contracted on
$L^{\infty}(M,{\mathord{{\rm d}}}_{\mathfrak{g}}x)$ (cf. Strichartz [73,
Theorem 3.5]). Therefore, for each $1<p<+\infty$,
$({\mathfrak{T}}_{t})_{t\geqslant 0}$ forms an analytic semigroup on
$L^{p}(M,{\mathord{{\rm d}}}_{\mathfrak{g}}x)$ (cf. Stein [72, p.67 Theorem
1]). The Bessel spaces over $M$ are defined by
${\mathbb{H}}^{p}_{\alpha}:=(I-\Delta)^{-\alpha/2}(L^{p}(M,{\mathord{{\rm
d}}}_{\mathfrak{g}}x)).$
In this section, we make the following geometric assumptions:
(G)n: The Ricci curvature ${\rm Ricci}_{\mathfrak{g}}$ and curvature
$R_{\mathfrak{g}}$ tensors together with their covariant derivatives up to
$n$-th order are bounded.
(G)inj: The injectivity radius of $(M,{\mathfrak{g}})$ is strictly positive.
It was proved by Yoshida [79] that under (G)n, an equivalent norm of
${\mathbb{H}}^{p}_{n}$ is given by the covariant derivatives up to $n$-th
order, i.e., there are two positive constants $C_{1}$ and $C_{2}$ such that
for any $u\in C_{0}^{\infty}(M)$
$\displaystyle
C_{1}\sum_{k=0}^{n}\|\nabla^{k}u\|_{L^{p}}\leqslant\|(I-\Delta)^{n/2}u\|_{L^{p}}\leqslant
C_{2}\sum_{k=0}^{n}\|\nabla^{k}u\|_{L^{p}},$ (8.1)
where $\nabla^{k}$ denotes the $k$-th covariant derivative. As an example, the
components of $\nabla u$ in local coordinates are given by $(\nabla
u)_{i}=\partial_{i}u$, while the components of $\nabla^{2}u$ in local
coordinates are given by
$(\nabla^{2}u)_{ij}=\partial_{ij}u-\Gamma^{k}_{ij}\partial_{k}u,$
where $\Gamma^{k}_{ij}$ are Christoffel symbols. By definition one has that
$|\nabla^{k}u|^{2}=g^{i_{1}j_{1}}\cdots
g^{i_{k}j_{k}}(\nabla^{k})_{i_{1}\cdots i_{k}}(\nabla^{k})_{j_{1}\cdots
j_{k}},$
where $g_{ij}={\mathfrak{g}}(\partial_{i},\partial_{j})$ and $(g^{ij})$
denotes the inverse matrix of $(g_{ij})$.
We remark that when $n=1$, (8.1) was first proved by Bakry [4] under the
assumption that Ricci curvature is bounded from below.
The following embedding result was proved in [83]. We refer to [2, 35, 36] for
a detailed study of integer order Sobolev spaces over $M$.
###### Theorem 8.1.
Under (G)n+1 and (G)inj, for $\alpha\in(0,1)$ and $p>d/\alpha$ we have
${\mathbb{H}}^{p}_{n+\alpha}\hookrightarrow C^{n}_{b}(M),$
where $C^{n}_{b}(M)$ denotes the Banach space of all $n$-times continuously
differentiable functions on $M$ with
$\|u\|_{C^{n}_{b}}:=\sup_{x\in M}\sum_{k=0}^{n}|\nabla^{k}u(x)|<+\infty.$
Consider the following SPDE:
$\displaystyle\left\\{\begin{aligned} {\mathord{{\rm d}}}u(t,x)=&\big{[}\Delta
u(t,x)+\varphi(t,x,u(t),{\mathfrak{g}}(Y(x),\nabla u(t)))\big{]}{\mathord{{\rm
d}}}t\\\ &+\psi(t,x,u(t,x)){\mathord{{\rm d}}}W(t),\\\
u(0,x)=&u_{0}(x),\end{aligned}\right.$ (8.2)
where $Y:M\to T(M)$ is a measurable vector field with
$\displaystyle\sup_{x\in M}{\mathfrak{g}}(Y(x),Y(x))<+\infty$ (8.3)
and
$\displaystyle\varphi:{\mathbb{R}}_{+}\times\Omega\times
M\times{\mathbb{R}}^{2}\to{\mathbb{R}}$ $\displaystyle\in$
$\displaystyle{\mathcal{M}}\times{\mathcal{B}}(M)\times{\mathcal{B}}({\mathbb{R}}^{2})/{\mathcal{B}}({\mathbb{R}}),$
$\displaystyle\psi:{\mathbb{R}}_{+}\times\Omega\times M\times{\mathbb{R}}\to
l^{2}$ $\displaystyle\in$
$\displaystyle{\mathcal{M}}\times{\mathcal{B}}(M)\times{\mathcal{B}}({\mathbb{R}})/{\mathcal{B}}(l^{2}).$
In this section, we fix
$\displaystyle p>d\mbox{ and
}\frac{3}{2}-\sqrt{\frac{3}{4}-\frac{d}{2p}}<\alpha<1.$ (8.4)
Assume that
1. (R1)
For each $T,R>0$, there exist constants $C_{R,T},\delta>0$ and
$\lambda^{\varphi,1}_{R,T},\lambda^{\varphi,2}_{R,T}\in L^{p}(M,{\mathord{{\rm
d}}}_{\mathfrak{g}}x)$ such that for all $s,t\in[0,T],\omega\in\Omega$, $x\in
M$ and $|u_{1}|,|v_{1}|,|u_{2}|,|v_{2}|\leqslant R$
$\displaystyle|\varphi(t,\omega,x,u_{1},u_{2})-\varphi(t,\omega,x,v_{1},v_{2})|$
$\displaystyle\qquad\leqslant
C_{R,T}\big{(}\lambda^{\varphi,1}_{R,T}(x)\cdot|t-s|^{\delta}+|u_{1}-v_{1}|+|u_{2}-v_{2}|\big{)}$
and
$|\varphi(t,\omega,x,u_{1},u_{2})|\leqslant\lambda^{\varphi,2}_{R,T}(x).$
2. (R2)
For each $(t,\omega)\in{\mathbb{R}}_{+}\times\Omega$,
$\psi(t,\omega,\cdot,\cdot)\in C^{2}(M\times{\mathbb{R}};l^{2})$. For each
$T,R>0$, there exist constant $C_{R,T}>0$ and $\lambda^{\psi}_{R,T}\in
L^{p}(M,{\mathord{{\rm d}}}_{\mathfrak{g}}x)$ such that for all
$t\in[0,T],\omega\in\Omega$, $x\in M$ and $|u|\leqslant R$
$\displaystyle\|\nabla_{x}\partial_{u}\psi(t,\omega,\cdot,u)\|_{l^{2}}+\|\partial^{j}_{u}\psi(t,\omega,\cdot,u)\|_{l^{2}}\leqslant
C_{R,T},\ \ j=1,2$
and
$\displaystyle\|\psi(t,\omega,\cdot,u)\|_{l^{2}}+\|\nabla^{j}_{x}\psi(t,\omega,\cdot,u)\|_{l^{2}}\leqslant\lambda^{\psi}_{R,T}(x),\
\ j=1,2.$
###### Theorem 8.2.
Let $p>d$ and $\alpha$ satisfy (8.4). Under (G)2-(G)inj and (R1)-(R2), for
each $u_{0}\in{\mathbb{H}}^{p}_{2}$, there exists a unique maximal strong
solution $(u,\tau)$ for Eq.(8.2) so that
1. (i)
$t\mapsto u(t)\in{\mathbb{H}}^{p}_{2}$ is continuous on $[0,\tau)$ almost
surely;
2. (ii)
$\lim_{t\uparrow\tau}\|u(t)\|_{{\mathbb{H}}^{p}_{2\alpha}}=+\infty$ on
$\\{\omega:\tau(\omega)<+\infty\\}$;
3. (iii)
it holds that, $P$-a.s., on $[0,\tau)$
$\displaystyle u(t)$ $\displaystyle=$ $\displaystyle
u_{0}+\int^{t}_{0}\big{[}\Delta
u(s)+\varphi(s,\cdot,u(s),{\mathfrak{g}}(Y(\cdot),\nabla
u(s)))\big{]}{\mathord{{\rm d}}}s$
$\displaystyle+\int^{t}_{0}\psi(s,\cdot,u(s)){\mathord{{\rm d}}}W(s)\ \
\mbox{in $L^{p}(M,{\mathord{{\rm d}}}_{\mathfrak{g}}x)$.}$
###### Proof.
Choose $\alpha_{0}$ such that
$\displaystyle\frac{1}{2}+\frac{d}{2p}<\alpha_{0}<3\alpha-\alpha^{2}-1<\alpha<1.$
(8.5)
Let $u,v\in{\mathbb{H}}^{p}_{2\alpha_{0}}$ with
$\|u\|_{{\mathbb{H}}^{p}_{2\alpha_{0}}},\|v\|_{{\mathbb{H}}^{p}_{2\alpha_{0}}}\leqslant
R$. By Theorem 8.1 we have
$\displaystyle\|u\|_{C^{1}_{b}}+\|v\|_{C^{1}_{b}}\leqslant C_{R}.$ (8.6)
Set
$\displaystyle\Phi(t,\omega,u)$ $\displaystyle:=$
$\displaystyle\varphi(t,\omega,\cdot,u,{\mathfrak{g}}(Y(\cdot),\nabla u)),$
$\displaystyle\Psi(t,\omega,u)$ $\displaystyle:=$
$\displaystyle\psi(t,\omega,\cdot,u).$
By (R1) and (8.1) (8.3) (8.6), we have
$\displaystyle\|\Phi(t,\omega,u)-\Phi(s,\omega,v)\|_{L^{p}}$
$\displaystyle\preceq$
$\displaystyle|t-s|^{\delta}+\|u-v\|_{L^{p}}+\|\nabla(u-v)\|_{L^{p}}$
$\displaystyle\preceq$
$\displaystyle|t-s|^{\delta}+\|u-v\|_{{\mathbb{H}}^{p}_{1}}$
$\displaystyle\preceq$
$\displaystyle|t-s|^{\delta}+\|u-v\|_{{\mathbb{H}}^{p}_{2\alpha}}$
and
$\|\Phi(t,\omega,u)\|_{L^{p}}\leqslant C_{R}.$
Note that
$\displaystyle\nabla_{x}\Psi(t,\omega,u)=(\nabla_{x}\psi)(t,\omega,\cdot,u)+(\partial_{u}\psi)(t,\omega,x,u)\nabla_{x}u$
(8.7)
and
$\displaystyle\nabla^{2}_{x}\Psi(t,\omega,u)$ $\displaystyle=$
$\displaystyle(\nabla^{2}_{x}\psi)(t,\omega,\cdot,u)+2(\nabla_{x}\partial_{u}\psi)(t,\omega,\cdot,u)\otimes\nabla_{x}u$
(8.8)
$\displaystyle+(\partial_{u}\psi)(t,\omega,x,u)\nabla_{x}u\otimes\nabla_{x}u+(\partial_{u}\psi)(t,\omega,x,u)\nabla^{2}_{x}u.$
By (R2) and (8.6) we have
$\|\Psi(t,\omega,u)-\Psi(t,\omega,v)\|_{L^{p}}\preceq\|u-v\|_{L^{p}}$
and by (8.1) and (8.7)
$\|\nabla(\Psi(t,\omega,u)-\Psi(t,\omega,v))\|_{L^{p}}\preceq\|u-v\|_{{\mathbb{H}}^{p}_{1}}.$
Hence,
$\|\Psi(t,\omega,u)-\Psi(t,\omega,v)\|_{{\mathbb{H}}^{p}_{\alpha}}\preceq\|\Psi(t,\omega,u)-\Psi(t,\omega,v)\|_{{\mathbb{H}}^{p}_{1}}\preceq\|u-v\|_{{\mathbb{H}}^{p}_{2\alpha_{0}}}.$
Moreover, by (R2) and (8.1) (8.7) (8.8),
$\|\Psi(t,\omega,u)\|_{{\mathbb{H}}^{p}_{1}}\leqslant C_{R,T}$
and
$\|\Psi(t,\omega,u)\|_{{\mathbb{H}}^{p}_{2}}\leqslant
C_{R,T}(1+\|u\|_{{\mathbb{H}}^{p}_{2}}).$
Using Lemma 2.13 with the data $\alpha_{0}$ as above and
$\alpha_{3}=\alpha_{1}=1,\ \ \alpha_{2}=\frac{\alpha}{2},\ \
\frac{\alpha-\alpha_{0}}{1-\alpha_{0}}=:\theta\stackrel{{\scriptstyle(\ref{Lp1})}}{{>}}\theta^{\prime}>\frac{1-\alpha}{2-\alpha},\
\ $
we find that for all $u\in{\mathbb{X}}^{p}_{\alpha}$ with
$\|u\|_{{\mathbb{X}}_{\alpha}^{p}}\leqslant R$
$\displaystyle\|\Psi(t,\omega,u)\|_{L_{2}(l^{2};{\mathbb{H}}^{p}_{\alpha^{\prime}})}=\|\Psi(t,\omega,u)\|_{L_{2}(l^{2};{\mathbb{H}}^{p}_{2\theta^{\prime}(\alpha_{3}-\alpha_{2})+2\alpha_{2}})}\leqslant
C_{R},$
where $\alpha^{\prime}=2\theta^{\prime}(\alpha_{3}-\alpha_{2})+2\alpha_{2}>1$.
Thus, (M2) and (M4) hold, and the theorem follows from Theorem 6.9. ∎
For the non-explosion, we assume that
1. (R3)
For each $T>0$, there exist $\lambda_{i}\in L^{p}(M,{\mathord{{\rm
d}}}_{\mathfrak{g}}x),i=0,1,2$ and $k\in{\mathbb{N}}$ such that for all
$(t,\omega)\in[0,T]\times\Omega,u,v\in{\mathbb{R}}$ such that
$\displaystyle u\cdot\varphi(t,\omega,x,u,v)$ $\displaystyle\leqslant$
$\displaystyle C_{T}|u|\cdot(|u|+|v|+\lambda_{0}(x)),$ (8.9)
$\displaystyle|\varphi(t,\omega,x,u,v)|$ $\displaystyle\leqslant$
$\displaystyle C_{T}(|u|^{k}+|v|+\lambda_{1}(x))$ (8.10)
and
$\displaystyle\|\partial_{u}\psi(t,\omega,\cdot,u)\|_{l^{2}}$
$\displaystyle\leqslant$ $\displaystyle C_{T},$ (8.11)
$\displaystyle\|\psi(t,\omega,\cdot,u)\|_{l^{2}}+\|\nabla_{x}\psi(t,\omega,\cdot,u)\|_{l^{2}}$
$\displaystyle\leqslant$ $\displaystyle C_{T}(|u|+\lambda_{2}(x)).$ (8.12)
The following theorem will follow from the proof of Lemma 8.4 below.
###### Theorem 8.3.
Keep the same assumptions as in Theorem 8.2, and also assume (R3). Let
$(u,\tau)$ be the unique maximal strong solution of Eq.(8.2) in Theorem 8.2.
Then $\tau=+\infty$ a.s..
Let $\varphi$ and $\psi$ be independent of $\omega$. Consider now the small
perturbation of Eq.(8.2):
$\displaystyle\left\\{\begin{aligned} {\mathord{{\rm
d}}}u_{\epsilon}(t,x)=&\big{[}\Delta
u_{\epsilon}(t,x)+\varphi(t,x,u_{\epsilon}(t),{\mathfrak{g}}(Y(x),\nabla
u_{\epsilon}(t)))\big{]}{\mathord{{\rm d}}}t\\\
&+\sqrt{\epsilon}\psi(t,x,u_{\epsilon}(t,x)){\mathord{{\rm d}}}W(t),\\\
u_{\epsilon}(0,x)=&u_{0}(x)\in{\mathbb{H}}^{p}_{2},\end{aligned}\right.$
as well as the control equation
$\displaystyle\left\\{\begin{aligned} {\mathord{{\rm
d}}}u^{\epsilon}(t,x)=&\big{[}\Delta
u^{\epsilon}(t,x)+\varphi(t,x,u^{\epsilon}(t),{\mathfrak{g}}(Y(x),\nabla
u^{\epsilon}(t)))\big{]}{\mathord{{\rm d}}}t\\\
&+\psi(t,x,u^{\epsilon}(t,x))\dot{h}^{\epsilon}(s){\mathord{{\rm
d}}}s+\sqrt{\epsilon}\psi(t,x,u^{\epsilon}(t,x)){\mathord{{\rm d}}}W(t),\\\
u^{\epsilon}(0,x)=&u_{0}(x)\in{\mathbb{H}}^{p}_{2},\end{aligned}\right.$
(8.13)
where $h^{\epsilon}\in{\mathcal{A}}^{T}_{N}$ (see (2.23) for the definition of
${\mathcal{A}}^{T}_{N}$), and $T>0$ is fixed below.
Let $(u^{\epsilon},\tau^{\epsilon})$ be the unique maximal strong solution of
Eq.(8.13). Define
$\tau^{\epsilon}_{n}:=\inf\big{\\{}t:\|u^{\epsilon}(t)\|_{{\mathbb{H}}^{p}_{2\alpha}}>n\big{\\}}.$
Then we have:
###### Lemma 8.4.
Assume (R3). Then
$\displaystyle\lim_{n\rightarrow\infty}\sup_{\epsilon\in(0,1)}P\big{\\{}\omega:\tau^{\epsilon}_{n}(\omega)<T\big{\\}}=0.$
###### Proof.
For the simplicity of notations, we drop the superscript $\epsilon$ in
$u^{\epsilon}$ in the following. First of all, note that (cf. [2])
$u_{0}\in{\mathbb{H}}^{p}_{2}\subset\cap_{q>1}L^{q}(M,{\mathord{{\rm
d}}}_{\mathfrak{g}}x).$
For $q,r\geqslant 2$, by the usual Itô formula (cf. [12, Theorem A.2]), we
have
$\|u(t)\|_{L^{q}}^{rq}=\|u_{0}\|_{L^{q}}^{rq}+J_{1}(t)+J_{2}(t)+J_{3}(t)+J_{4}(t)+J_{5}(t)+J_{6}(t)$
on $[0,\tau^{\epsilon}_{n}]$, where
$\displaystyle J_{1}(t)$ $\displaystyle:=$ $\displaystyle
rq\int^{t}_{0}\|u(s)\|_{L^{q}}^{(r-1)q}{\langle}|u(s)|^{q-2}u(s),\Delta
u(s){\rangle}_{L^{2}}{\mathord{{\rm d}}}s,$ $\displaystyle J_{2}(t)$
$\displaystyle:=$ $\displaystyle
rq\int^{t}_{0}\|u(s)\|_{L^{q}}^{(r-1)q}{\langle}|u(s)|^{q-2}u(s),\varphi(s,\cdot,u(s),{\mathfrak{g}}(Y(\cdot),\nabla
u(s))){\rangle}_{L^{2}}{\mathord{{\rm d}}}s,$ $\displaystyle J_{3}(t)$
$\displaystyle:=$ $\displaystyle
rq\sum_{k}\int^{t}_{0}\|u(s)\|_{L^{q}}^{(r-1)q}{\langle}|u(s)|^{q-2}u(s),\psi_{k}(s,\cdot,u(s)){\rangle}_{L^{2}}{\mathord{{\rm
d}}}W^{k}_{s},$ $\displaystyle J_{4}(t)$ $\displaystyle:=$
$\displaystyle\frac{rq(q-1)}{2}\sum_{k}\int^{t}_{0}\|u(s)\|_{L^{q}}^{(r-1)q}{\langle}|u(s)|^{q-2},|\psi_{k}(s,\cdot,u(s))|^{2}{\rangle}_{L^{2}}{\mathord{{\rm
d}}}s,$ $\displaystyle J_{5}(t)$ $\displaystyle:=$
$\displaystyle\frac{q^{2}r(r-1)}{2}\sum_{k}\int^{t}_{0}\|u(s)\|_{L^{q}}^{(r-2)q}{\langle}|u(s)|^{q-2}u(s),\psi_{k}(s,\cdot,u(s)){\rangle}_{L^{2}}^{2}{\mathord{{\rm
d}}}s,$ $\displaystyle J_{6}(t)$ $\displaystyle:=$ $\displaystyle
rq\int^{t}_{0}\|u(s)\|_{L^{q}}^{(r-1)q}{\langle}|u(s)|^{q-2}u(s),\psi(s,\cdot,u(s))\dot{h}^{\epsilon}(s){\rangle}_{L^{2}}{\mathord{{\rm
d}}}s.$
For $J_{1}(t)$ we have
$\displaystyle
J_{1}(t)=-rq(q-1)\int^{t}_{0}\|u(s)\|_{L^{q}}^{(r-1)q}\int_{M}|u(s)|^{q-2}|\nabla
u(s)|^{2}{\mathord{{\rm d}}}_{\mathfrak{g}}x{\mathord{{\rm d}}}s.$
For $J_{2}(t)$, by (8.9) (8.3) and Young’s inequality we have
$\displaystyle J_{2}(t)$ $\displaystyle\leqslant$ $\displaystyle
rq\int^{t}_{0}\|u(s)\|_{L^{q}}^{(r-1)q}\int_{M}|u(s)|^{q-1}\big{(}|u(s)|+|{\mathfrak{g}}(Y,\nabla
u(s))|+\lambda_{0}\big{)}{\mathord{{\rm d}}}_{\mathfrak{g}}x{\mathord{{\rm
d}}}s$ $\displaystyle\leqslant$
$\displaystyle-\frac{J_{1}(t)}{2}+C\int^{t}_{0}(\|u(s)\|^{rq}_{L^{q}}+1){\mathord{{\rm
d}}}s.$
Similarly, by (8.12) we have
$J_{4}(t)+J_{5}(t)\leqslant
C\int^{t}_{0}(\|u(s)\|^{rq}_{L^{q}}+1){\mathord{{\rm d}}}s,$
and by Young’s inequality
$\displaystyle J_{6}(t)$ $\displaystyle\preceq$
$\displaystyle\int^{t}_{0}\|u(s)\|_{L^{q}}^{(r-1)q}(\|u(s)\|_{L^{q}}^{q}+1)\cdot\|\dot{h}^{\epsilon}(s)\|_{l^{2}}{\mathord{{\rm
d}}}s$ (8.14) $\displaystyle\preceq$ $\displaystyle
N\left(\int^{t}_{0}\|u(s)\|_{L^{q}}^{2(r-1)q}(\|u(s)\|_{L^{q}}^{q}+1)^{2}{\mathord{{\rm
d}}}s\right)^{1/2}$ $\displaystyle\preceq$ $\displaystyle
N\sup_{s\in[0,t]}\|u(s)\|_{L^{q}}^{rq/2}\cdot\left(\int^{t}_{0}(\|u(s)\|_{L^{q}}^{rq}+1){\mathord{{\rm
d}}}s\right)^{1/2}$ $\displaystyle\leqslant$
$\displaystyle\frac{1}{2}\sup_{s\in[0,t]}\|u(s)\|_{L^{q}}^{rq}+C_{N}\cdot\int^{t}_{0}(\|u(s)\|_{L^{q}}^{rq}+1){\mathord{{\rm
d}}}s.$
Combining the above calculations, we obtain
$\displaystyle\sup_{s\in[0,t]}\|u(s\wedge\tau^{\epsilon}_{n})\|^{rq}_{L^{q}}\leqslant
2\|u_{0}\|_{L^{q}}^{rq}+2\sup_{s\in[0,t]}J_{3}(s\wedge\tau^{\epsilon}_{n})+C_{N}\int^{t\wedge\tau^{\epsilon}_{n}}_{0}(\|u(s)\|^{rq}_{L^{q}}+1){\mathord{{\rm
d}}}s.$
Set
$f_{1}(t):={\mathbb{E}}\left(\sup_{s\in[0,t]}\|u(s\wedge\tau^{\epsilon}_{n})\|^{rq}_{L^{q}}\right).$
By BDG’s inequality and as (8.14) we have
$\displaystyle{\mathbb{E}}\left(\sup_{s\in[0,t]}|J_{3}(s\wedge\tau^{\epsilon}_{n})|\right)$
$\displaystyle\preceq$
$\displaystyle{\mathbb{E}}\left(\int^{t\wedge\tau^{\epsilon}_{n}}_{0}\|u(s)\|_{L^{q}}^{2(r-1)q}(\|u(s)\|^{q}_{L^{q}}+1)^{2}{\mathord{{\rm
d}}}s\right)^{1/2}$ $\displaystyle\leqslant$
$\displaystyle\frac{1}{2}f_{1}(t)+C{\mathbb{E}}\left(\int^{t\wedge\tau^{\epsilon}_{n}}_{0}(\|u(s)\|^{rq}_{L^{q}}+1){\mathord{{\rm
d}}}s\right).$
Therefore,
$\displaystyle f_{1}(t)$ $\displaystyle\leqslant$ $\displaystyle
4\|u_{0}\|_{L^{q}}^{rq}+C_{N}{\mathbb{E}}\int^{t\wedge\tau^{\epsilon}_{n}}_{0}(\|u(s)\|^{rq}_{L^{q}}+1){\mathord{{\rm
d}}}s$ $\displaystyle\leqslant$ $\displaystyle
4\|u_{0}\|_{L^{q}}^{rq}+C_{N}\int^{t}_{0}(f_{1}(s)+1){\mathord{{\rm d}}}s,$
which yields by Gronwall’s inequality that
$\displaystyle{\mathbb{E}}\left(\sup_{t\in[0,T]}\|u(t\wedge\tau^{\epsilon}_{n})\|_{L^{q}}^{rq}\right)\leqslant
C_{T,N}.$ (8.15)
Here and below, the constant $C_{T,N}$ is independent of $n$ and $\epsilon$.
Set
$\xi^{\epsilon}_{n}(t):=t\wedge\tau^{\epsilon}_{n}$
and for $q\geqslant 2$
$f_{2}(t):={\mathbb{E}}\left(\sup_{t^{\prime}\leqslant\xi^{\epsilon}_{n}(t)}\|u(t^{\prime})\|^{q}_{{\mathbb{H}}^{p}_{2\alpha}}\right).$
Note that
$\displaystyle u(t)$ $\displaystyle=$
$\displaystyle{\mathfrak{T}}_{t}u_{0}+\int^{t}_{0}{\mathfrak{T}}_{t-s}\varphi(s,\cdot,u(s),{\mathfrak{g}}(Y(\cdot),\nabla
u(s))){\mathord{{\rm d}}}s$
$\displaystyle+\int^{t}_{0}{\mathfrak{T}}_{t-s}\psi(s,\cdot,u(s))\dot{h}^{\epsilon}(s){\mathord{{\rm
d}}}s+\sqrt{\epsilon}\int^{t}_{0}{\mathfrak{T}}_{t-s}\psi(s,\cdot,u(s)){\mathord{{\rm
d}}}W(s)$ $\displaystyle=:$
$\displaystyle{\mathfrak{T}}_{t}u_{0}+{\mathcal{J}}_{1}(t)+{\mathcal{J}}_{2}(t)+{\mathcal{J}}_{3}(t).$
By (iii) of Proposition 2.11 and Hölder’s inequality we have, for
$q>\frac{1}{1-\alpha}$
$\displaystyle{\mathbb{E}}\left(\sup_{t^{\prime}\in[0,\xi^{\epsilon}_{n}(t)]}\|{\mathcal{J}}_{1}(t)\|^{q}_{{\mathbb{H}}^{p}_{2\alpha}}\right)$
$\displaystyle\quad\preceq{\mathbb{E}}\left(\sup_{t^{\prime}\in[0,\xi^{\epsilon}_{n}(t)]}\int^{t^{\prime}}_{0}\frac{1}{(t^{\prime}-s)^{\alpha}}\|\varphi(s,\cdot,u(s),{\mathfrak{g}}(Y(\cdot),\nabla
u(s)))\|_{L^{p}}{\mathord{{\rm d}}}s\right)^{q}$
$\displaystyle\quad\preceq{\mathbb{E}}\left(\int^{\xi^{\epsilon}_{n}(t)}_{0}\|\varphi(s,\cdot,u(s),{\mathfrak{g}}(Y(\cdot),\nabla
u(s)))\|^{q}_{L^{p}}{\mathord{{\rm d}}}s\right)$
$\displaystyle\quad\stackrel{{\scriptstyle(\ref{V9},\ref{YY})}}{{\preceq}}{\mathbb{E}}\left(\int^{\xi^{\epsilon}_{n}(t)}_{0}(1+\|u(s)\|^{kq}_{L^{pk}}+\|\nabla
u(s)\|^{q}_{L^{p}}){\mathord{{\rm d}}}s\right)$
$\displaystyle\quad\stackrel{{\scriptstyle(\ref{re})}}{{\preceq}}\int^{t}_{0}(\|u(s)\|^{q}_{{\mathbb{H}}^{p}_{1}}+1){\mathord{{\rm
d}}}s\stackrel{{\scriptstyle(\ref{V11})}}{{\preceq}}\int^{t}_{0}(f_{2}(s)+1){\mathord{{\rm
d}}}s.$
On the other hand, by (8.1), (2.18) and (R3) we have, for
$u\in{\mathbb{H}}^{p}_{1}$
$\displaystyle\|\psi(s,\cdot,u)\|^{2}_{L_{2}(l^{2};{\mathbb{H}}^{p}_{\alpha})}$
$\displaystyle\preceq$
$\displaystyle\|\psi(s,\cdot,u)\|^{2}_{L^{p}(M;l^{2})}+\|\nabla\psi(s,\cdot,u)\|^{2}_{L^{p}(M;l^{2})}$
(8.16) $\displaystyle\preceq$ $\displaystyle\|u\|^{2}_{L^{p}}+\|\nabla
u\|^{2}_{L^{p}}+1\preceq\|u\|^{2}_{{\mathbb{H}}^{p}_{1}}+1.$
Thus, as above, by (iii) of Proposition 2.11 and Hölder’s inequality we have,
for $q>\frac{2}{1-\alpha}$
$\displaystyle{\mathbb{E}}\left(\sup_{t^{\prime}\in[0,\xi^{\epsilon}_{n}(t)]}\|{\mathcal{J}}_{2}(t)\|^{q}_{{\mathbb{H}}^{p}_{2\alpha}}\right)$
$\displaystyle\qquad\leqslant{\mathbb{E}}\left(\sup_{t^{\prime}\in[0,\xi^{\epsilon}_{n}(t)]}\int^{t}_{0}\|{\mathfrak{T}}_{t-s}\psi(s,\cdot,u(s))\dot{h}^{\epsilon}(s)\|_{{\mathbb{H}}^{p}_{2\alpha}}{\mathord{{\rm
d}}}s\right)^{q}$ $\displaystyle\qquad\leqslant
N{\mathbb{E}}\left(\sup_{t^{\prime}\in[0,\xi^{\epsilon}_{n}(t)]}\int^{t}_{0}\|{\mathfrak{T}}_{t-s}\psi(s,\cdot,u(s))\|^{2}_{L_{2}(l^{2};{\mathbb{H}}^{p}_{2\alpha})}{\mathord{{\rm
d}}}s\right)^{q/2}$ $\displaystyle\qquad\leqslant
C_{N}{\mathbb{E}}\left(\sup_{t^{\prime}\in[0,\xi^{\epsilon}_{n}(t)]}\int^{t}_{0}\frac{1}{(t-s)^{\alpha}}\|\psi(s,\cdot,u(s))\|^{2}_{L_{2}(l^{2};{\mathbb{H}}^{p}_{\alpha})}{\mathord{{\rm
d}}}s\right)^{q/2}$
$\displaystyle\qquad\stackrel{{\scriptstyle(\ref{YY0})}}{{\leqslant}}C_{T,N}{\mathbb{E}}\left(\int^{\xi^{\epsilon}_{n}(t)}_{0}(\|u(s)\|^{q}_{{\mathbb{H}}^{p}_{1}}+1){\mathord{{\rm
d}}}s\right)$ $\displaystyle\qquad\leqslant
C_{T,N}\int^{t}_{0}(f_{2}(s)+1){\mathord{{\rm d}}}s.$
Set
$G(t,s):=\sqrt{\epsilon}{\mathfrak{T}}_{t-s}\psi(s,\cdot,u(s)).$
Then by (iii) and (iv) of Proposition 2.11 we have
$\|G(t,s)\|^{2}_{{\mathbb{H}}^{p}_{2\alpha}}\leqslant\frac{C}{(t-s)^{\alpha}}\|\psi(s,\cdot,u(s))\|^{2}_{L_{2}(l^{2};{\mathbb{H}}^{p}_{\alpha})}$
and for $\gamma\in(0,(1-\alpha)/2)$
$\|G(t^{\prime},s)-G(t,s)\|^{2}_{{\mathbb{H}}^{p}_{2\alpha}}\leqslant\frac{|t^{\prime}-t|^{\gamma}}{(t-s)^{\alpha+2\gamma}}\|\psi(s,\cdot,u(s))\|^{2}_{L_{2}(l^{2};{\mathbb{H}}^{p}_{\alpha})}.$
Therefore, using Lemma 3.4, for $q$ large enough, we have
$\displaystyle{\mathbb{E}}\left(\sup_{t^{\prime}\in[0,\xi^{\epsilon}_{n}(t)]}\|{\mathcal{J}}_{3}(t)\|^{q}_{{\mathbb{H}}^{p}_{2\alpha}}\right)$
$\displaystyle=$
$\displaystyle{\mathbb{E}}\left(\sup_{t^{\prime}\in[0,T\wedge\xi_{n}^{\epsilon}(t)]}\left\|\int^{t^{\prime}}_{0}G(t^{\prime},s){\mathord{{\rm
d}}}W(s)\right\|^{q}_{{\mathbb{H}}^{p}_{2\alpha}}\right)$
$\displaystyle\leqslant$ $\displaystyle
C_{T}{\mathbb{E}}\left(\int^{T\wedge\xi_{n}^{\epsilon}(t)}_{0}\|\psi(s,\cdot,u(s))\|_{L_{2}(l^{2};{\mathbb{H}}^{p}_{\alpha})}^{q}{\mathord{{\rm
d}}}s\right)$ $\displaystyle\stackrel{{\scriptstyle(\ref{YY0})}}{{\leqslant}}$
$\displaystyle C_{T}\int^{t}_{0}(f_{2}(s)+1){\mathord{{\rm d}}}s.$
Combining the above estimates, we get
$f_{2}(t)\leqslant
C\|u_{0}\|^{q}_{{\mathbb{H}}^{p}_{2}}+C_{T,N}\int^{t}_{0}(f_{2}(s)+1){\mathord{{\rm
d}}}s.$
By Gronwall’s inequality again, we find
$\displaystyle{\mathbb{E}}\left(\sup_{t\leqslant
T\wedge\tau^{\epsilon}_{n}}\|u(t)\|^{q}_{{\mathbb{H}}^{p}_{2\alpha}}\right)=f_{2}(T)\leqslant
C_{T,N},$
which in turn implies that
$\lim_{n\to\infty}\sup_{\epsilon\in(0,1)}P\\{\tau^{\epsilon}_{n}<T\\}=0.$
The proof is complete. ∎
Moreover, under (R3), similar to the above lemma, we can check that (6.7)
holds. Thus, using Theorem 6.3 we obtain
###### Theorem 8.5.
Let $(M,{\mathfrak{g}})$ be a compact Riemannian manifold, and $p>d$, $\alpha$
satisfy (8.4). Let $u_{0}\in{\mathbb{H}}_{2}^{p}$. Under (R1)-(R3),
$\\{u_{\epsilon},\epsilon\in(0,1)\\}$ satisfies the large deviation principle
in ${\mathbb{C}}_{T}({\mathbb{H}}_{2\alpha}^{p})$ with the rate function
$I(f)$ given by
$I(f):=\frac{1}{2}\inf_{\\{h\in\ell^{2}_{T}:~{}f=u^{h}\\}}\|h\|^{2}_{\ell^{2}_{1}},\
\ f\in{\mathbb{C}}_{T}({\mathbb{H}}^{p}_{2\alpha}),$
where $u^{h}$ solves the following equation:
$\displaystyle u^{h}(t)$ $\displaystyle=$ $\displaystyle
u_{0}+\int^{t}_{0}\big{[}\Delta
u^{h}(s)+\varphi(s,\cdot,u^{h}(s),{\mathfrak{g}}(Y(\cdot),\nabla
u^{h}(s)))\big{]}{\mathord{{\rm d}}}s$
$\displaystyle+\int^{t}_{0}\psi(s,\cdot,u^{h}(s))\dot{h}(s){\mathord{{\rm
d}}}s.$
## 9\. Application to stochastic Navier-Stokes equations
### 9.1. Unique maximal strong solution for SNSEs
Let ${\mathcal{O}}$ be a bounded smooth domain in
${\mathbb{R}}^{d}$($d\geqslant 2$), or the whole space ${\mathbb{R}}^{d}$, or
$d$-dimensional torus ${\mathbb{T}}^{d}$. Let
${\bf W}^{m,p}({\mathcal{O}}):=(W^{m,p}({\mathcal{O}}))^{d},\ \ {\bf
W}^{m,p}_{0}({\mathcal{O}}):=(W^{m,p}_{0}({\mathcal{O}}))^{d}$
and
$\mathord{{\bf C}}^{\infty}_{0,\sigma}({\mathcal{O}}):=\\{\mathord{{\bf
u}}\in(C^{\infty}_{0}({\mathcal{O}}))^{d}:{\mathord{{\rm div}}}(\mathord{{\bf
u}})=0\\}.$
Notice that ${\bf W}^{m,p}({\mathbb{R}}^{d})={\bf
W}^{m,p}_{0}({\mathbb{R}}^{d})$ and ${\bf W}^{m,p}({\mathbb{T}}^{d})={\bf
W}^{m,p}_{0}({\mathbb{T}}^{d})$.
Let $\mathord{{\bf L}}^{p}_{\sigma}({\mathcal{O}})$ be the closure of
$\mathord{{\bf C}}^{\infty}_{0,\sigma}({\mathcal{O}})$ with respect to the
norm in $\mathord{{\bf L}}^{p}({\mathcal{O}}):=(L^{p}({\mathcal{O}}))^{d}$.
Let ${\mathscr{P}}_{2}$ be the orthonormal projection from $\mathord{{\bf
L}}^{2}({\mathcal{O}})$ to $\mathord{{\bf L}}^{2}_{\sigma}({\mathcal{O}})$. It
is well known that ${\mathscr{P}}_{2}$ can be extended to a bounded linear
operator from $\mathord{{\bf L}}^{p}({\mathcal{O}})$ to $\mathord{{\bf
L}}^{p}_{\sigma}({\mathcal{O}})$ (cf. [28]) so that for every $\mathord{{\bf
u}}\in\mathord{{\bf L}}^{p}({\mathcal{O}})$
$\mathord{{\bf u}}={\mathscr{P}}_{p}\mathord{{\bf u}}+\nabla\pi,\ \
\pi\in(L^{p}_{loc}({\mathcal{O}}))^{d}.$
The stokes operator is defined by
$\displaystyle A_{p}\mathord{{\bf u}}:=-{\mathscr{P}}_{p}\Delta\mathord{{\bf
u}},\ \ {\mathscr{D}}(A_{p}):={\mathbb{H}}^{p}_{2}\cap\mathord{{\bf
L}}^{p}_{\sigma}({\mathcal{O}}),$ (9.1)
where
${\mathbb{H}}^{p}_{2}:={\bf W}^{2,p}({\mathcal{O}})\cap{\bf
W}^{1,p}_{0}({\mathcal{O}})={\mathscr{D}}(I-\Delta_{p})$
and $\Delta_{p}$ is the Laplace operator on $\mathord{{\bf
L}}^{p}({\mathcal{O}})$.
It is well known that $(A_{p},{\mathscr{D}}(A_{p}))$ is a sectorial operator
on $\mathord{{\bf L}}^{p}_{\sigma}({\mathcal{O}})$ (cf. [29]). It should be
noticed that when ${\mathcal{O}}={\mathbb{R}}^{d}$ or ${\mathbb{T}}^{d}$,
since the projection ${\mathscr{P}}_{p}$ can commute with $\nabla$ (cf. [46,
p.84]), we have
$A_{p}\mathord{{\bf u}}=-\Delta{\mathscr{P}}_{p}\mathord{{\bf
u}}=-\Delta\mathord{{\bf u}},\ \ \mathord{{\bf u}}\in{\mathscr{D}}(A_{p}).$
That is, the stokes operator is just the restriction of $-\Delta_{p}$ on ${\bf
W}^{2,p}({\mathcal{O}})\cap\mathord{{\bf L}}^{p}_{\sigma}({\mathcal{O}})$,
where ${\mathcal{O}}={\mathbb{R}}^{d}$ or ${\mathbb{T}}^{d}$.
Below, we write
${\mathfrak{L}}_{p}:=I+A_{p}$
and
${\mathbf{H}}_{\alpha}^{p}:={\mathscr{D}}({\mathfrak{L}}^{\alpha/2}_{p}).$
Giga [30] proved that for $\alpha\in[0,1]$
$\displaystyle{\mathbf{H}}_{\alpha}^{p}=[\mathord{{\bf
L}}^{p}_{\sigma}({\mathcal{O}}),{\mathscr{D}}(A_{p})]_{\alpha}={\mathbb{H}}^{p}_{\alpha}\cap\mathord{{\bf
L}}^{p}_{\sigma}({\mathcal{O}}),$ (9.2)
where ${\mathbb{H}}^{p}_{\alpha}=[\mathord{{\bf
L}}^{p}({\mathcal{O}}),{\mathbb{H}}^{p}_{2}]_{\alpha}$ and
$[\cdot,\cdot]_{\alpha}$ stands for the complex interpolation space between
two Banach spaces. In particular, the following embedding results hold (see
(7.3) and (7.4)): for $p>1$ and
$0\leqslant\alpha^{\prime}<\frac{1}{2}<\alpha\leqslant 1$
$\displaystyle\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{p}_{2\alpha^{\prime}}}\preceq\|\mathord{{\bf
u}}\|_{1,p}\preceq\|\mathord{{\bf u}}\|_{{\mathbf{H}}^{p}_{2\alpha}},\ \
\mathord{{\bf u}}\in{\mathbf{H}}^{p}_{\alpha},$ (9.3)
and for $q\geqslant p$, $k-\frac{d}{q}<2\alpha-\frac{d}{p}$
$\displaystyle{\mathbf{H}}^{p}_{2\alpha}\hookrightarrow{\bf
W}^{k,q}({\mathcal{O}}),$ (9.4)
and for $\alpha>\frac{d}{p}$
$\displaystyle{\mathbf{H}}_{\alpha}^{p}\hookrightarrow C_{b}({\mathcal{O}}).$
(9.5)
In what follows, we fix
$\displaystyle p>d,\ \ \ \frac{1}{2}<\alpha<1,$ (9.6)
and consider the following stochastic Navier-Stokes equation with Dirichlet
boundary (only for bounded smooth domain):
$\displaystyle\left\\{\begin{aligned} &{\mathord{{\rm d}}}\mathord{{\bf
u}}(t)=\big{[}\Delta\mathord{{\bf u}}(t)+(\mathord{{\bf
u}}(t)\cdot\nabla)\mathord{{\bf u}}(t)+\nabla\pi(t)\big{]}{\mathord{{\rm
d}}}t\\\ &\qquad\qquad+F(t,\mathord{{\bf u}}(t)){\mathord{{\rm
d}}}t+\Psi(t,\mathord{{\bf u}}(t)){\mathord{{\rm d}}}W(t)\\\ &\mathord{{\bf
u}}(t,\cdot)|_{\partial{\mathcal{O}}}=0,\ \ \ \ \ \ {\mathord{{\rm
div}}}\mathord{{\bf u}}(t)=0,\\\ &\mathord{{\bf u}}(0,x)=\mathord{{\bf
u}}_{0}(x),\end{aligned}\right.$ (9.7)
where $\mathord{{\bf u}}$ and $\pi$ are unknown functions, and
$F:{\mathbb{R}}_{+}\times{\mathbf{H}}^{p}_{2\alpha}\to{\mathbf{H}}^{p}_{0}\mbox{
and
}\Psi:{\mathbb{R}}_{+}\times{\mathbf{H}}^{p}_{2\alpha}\to{\mathbf{H}}^{p}_{\alpha}$
are two measurable functions.
We assume that
1. (N1)
For each $T,R>0$, there exist $\delta>0$ and $C_{T,R,\delta}>0$ such that for
all $t,s\in[0,T]$ and $\mathord{{\bf u}},\mathord{{\bf
v}}\in{\mathbf{H}}^{p}_{2\alpha}$ with $\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{p}_{2\alpha}},\|\mathord{{\bf
v}}\|_{{\mathbf{H}}^{p}_{2\alpha}}\leqslant R$
$\|F(t,\mathord{{\bf u}})-F(s,\mathord{{\bf
v}})\|_{{\mathbf{H}}^{p}_{0}}\leqslant
C_{T,R,\delta}\Big{(}|t-s|^{\delta}+\|\mathord{{\bf u}}-\mathord{{\bf
v}}\|_{{\mathbf{H}}^{p}_{2\alpha}}\Big{)}.$
2. (N2)
For each $T,R>0$, there exist $\alpha^{\prime}>1$ and $C_{T,R}>0$ such that
for all $t\in[0,T]$ and $\mathord{{\bf u}},\mathord{{\bf
v}}\in{\mathbf{H}}^{p}_{2\alpha}$ with $\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{p}_{2\alpha}},\|\mathord{{\bf
v}}\|_{{\mathbf{H}}^{p}_{2\alpha}}\leqslant R$
$\|\Psi(t,\mathord{{\bf u}})-\Psi(t,\mathord{{\bf
v}})\|_{L_{2}(l^{2};{\mathbf{H}}^{p}_{\alpha})}\leqslant
C_{T,R}\|\mathord{{\bf u}}-\mathord{{\bf v}}\|_{{\mathbf{H}}^{p}_{2\alpha}}$
and
$\displaystyle\|\Psi(t,\mathord{{\bf
u}})\|_{L_{2}(l^{2};{\mathbf{H}}^{p}_{\alpha^{\prime}})}\leqslant C_{T,R}.$
(9.8)
Set
$\displaystyle\Phi(t,\mathord{{\bf u}}):=\mathord{{\bf
u}}+{\mathscr{P}}_{p}[(\mathord{{\bf u}}\cdot\nabla)\mathord{{\bf
u}}]+F(t,\mathord{{\bf u}}).$ (9.9)
Then Eq.(9.7) can be written as the following abstract form:
$\displaystyle{\mathord{{\rm d}}}\mathord{{\bf
u}}(t)=[-{\mathfrak{L}}_{p}\mathord{{\bf u}}(t)+\Phi(t,\mathord{{\bf
u}})]{\mathord{{\rm d}}}t+\Psi(t,\mathord{{\bf u}}){\mathord{{\rm d}}}W(s),\ \
\mathord{{\bf u}}(0)=\mathord{{\bf u}}_{0}.$ (9.10)
###### Theorem 9.1.
Let $p>d$ and $\frac{1}{2}<\alpha<1$. Under (N1) and (N2), for any
$\mathord{{\bf u}}_{0}\in{\mathbf{H}}^{p}_{2}$, there exists a unique maximal
strong solution $(\mathord{{\bf u}},\tau)$ for Eq.(9.10) so that
1. (i)
$t\mapsto\mathord{{\bf u}}(t)\in{\mathbf{H}}^{p}_{2}$ is continuous on
$[0,\tau)$ a.s.;
2. (ii)
$\lim_{t\uparrow\tau}\|\mathord{{\bf
u}}(t)\|_{{\mathbf{H}}^{p}_{2\alpha}}=\infty$ on $\\{\tau<+\infty\\}$;
3. (iii)
it holds that in $\mathord{{\bf
L}}^{p}_{\sigma}({\mathcal{O}})={\mathbf{H}}^{p}_{0}$
$\displaystyle\mathord{{\bf u}}(t)$ $\displaystyle=$
$\displaystyle\mathord{{\bf
u}}_{0}+\int^{t}_{0}[-{\mathfrak{L}}_{p}\mathord{{\bf
u}}(s)+\Phi(s,\mathord{{\bf u}}(s))]{\mathord{{\rm
d}}}s+\int^{t}_{0}\Psi(s,\mathord{{\bf u}}(s)){\mathord{{\rm d}}}W(s)$
$\displaystyle=$ $\displaystyle\mathord{{\bf
u}}_{0}+\int^{t}_{0}[A_{p}\mathord{{\bf
u}}(s)+{\mathscr{P}}_{p}((\mathord{{\bf u}}(s)\cdot\nabla)\mathord{{\bf
u}}(s))]{\mathord{{\rm d}}}s$ $\displaystyle+\int^{t}_{0}F(s,\mathord{{\bf
u}}(s)){\mathord{{\rm d}}}s+\int^{t}_{0}\Psi(s,\mathord{{\bf
u}}(s)){\mathord{{\rm d}}}W(s),$
for all $t\in[0,\tau)$, $P$-a.s..
###### Proof.
In view of (9.6), (9.3) and (9.5), for any $\mathord{{\bf u}},\mathord{{\bf
v}}\in{\mathbf{H}}^{p}_{2\alpha}$ we have
$\displaystyle\|{\mathscr{P}}_{p}[(\mathord{{\bf u}}\cdot\nabla)\mathord{{\bf
u}}-(\mathord{{\bf v}}\cdot\nabla)\mathord{{\bf v}}]\|_{\mathord{{\bf
L}}^{p}_{\sigma}}$ $\displaystyle\preceq$ $\displaystyle\|(\mathord{{\bf
u}}\cdot\nabla)\mathord{{\bf u}}-(\mathord{{\bf v}}\cdot\nabla)\mathord{{\bf
v}}\|_{\mathord{{\bf L}}^{p}}$ $\displaystyle\preceq$
$\displaystyle\|\mathord{{\bf u}}-\mathord{{\bf v}}\|_{\mathord{{\bf
L}}^{\infty}}\cdot\|\nabla\mathord{{\bf u}}\|_{\mathord{{\bf L}}^{p}}$
$\displaystyle+\|\mathord{{\bf v}}\|_{\mathord{{\bf
L}}^{\infty}}\cdot\|\nabla(\mathord{{\bf u}}-\mathord{{\bf
v}})\|_{\mathord{{\bf L}}^{p}}$ $\displaystyle\preceq$
$\displaystyle\|\mathord{{\bf u}}-\mathord{{\bf
v}}\|_{{\mathbf{H}}^{p}_{2\alpha}}\cdot\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{p}_{2\alpha}}$ $\displaystyle+\|\mathord{{\bf
v}}\|_{{\mathbf{H}}^{p}_{2\alpha}}\cdot\|\mathord{{\bf u}}-\mathord{{\bf
v}}\|_{{\mathbf{H}}^{p}_{2\alpha}},$
Thus, by (N1) and (N2), it is easy to see that (M2) and (M4) hold for the
above $\Phi$ and $\Psi.$ The result now follows by Theorem 6.9. ∎
We now give two concrete functionals so that (N1) and (N2) are satisfied. Let
$\mathord{{\bf
f}}:{\mathbb{R}}_{+}\times{\mathcal{O}}\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$
be a measurable function, and satisfy that: for any $T,R>0$, there exist
constants $\delta,C_{T,R}>0$ and $\lambda^{\mathord{{\bf f}}}_{R,T}\in
L^{p}({\mathcal{O}})$ such that for all $t,s\in[0,T],x\in{\mathcal{O}}$ and
$\mathord{{\bf u}},\mathord{{\bf v}}\in{\mathbb{R}}^{d}$ with $|\mathord{{\bf
u}}|,|\mathord{{\bf v}}|\leqslant R$
$|\mathord{{\bf f}}(t,x,\mathord{{\bf u}})-\mathord{{\bf f}}(s,x,\mathord{{\bf
v}})|\leqslant C_{T,R}\big{(}\lambda^{\mathord{{\bf
f}}}_{R,T}(x)\cdot|t-s|^{\delta}+|\mathord{{\bf u}}-\mathord{{\bf
v}}|\big{)}.$
Let $\mathord{{\bf
g}}:{\mathbb{R}}_{+}\times{\mathcal{O}}\times{\mathbb{R}}^{d}\to
l^{2}\times{\mathbb{R}}^{d}$ be a measurable function, and satisfy that:
$\displaystyle\left\\{\begin{aligned} \left.\begin{aligned} &\mathord{{\bf
g}}(t,x,\mathord{{\bf u}})=c(t)\mathord{{\bf u}}+\mathord{{\bf g}}_{2}(t,x),\
\\\ &\exists\alpha^{\prime}>1\ \ s.t.\ \
\sup_{t\in[0,T]}(|c(t)|+\|\mathord{{\bf
g}}_{2}(t,\cdot)\|_{{\mathbf{H}}^{p}_{\alpha^{\prime}}})\leqslant
C_{T},\end{aligned}\right\\},&\mbox{ ${\mathcal{O}}$ bounded;}\\\
\left.\begin{aligned} &\mathord{{\bf g}}(t,\cdot,\cdot)\in
C^{2}({\mathcal{O}}\times{\mathbb{R}}^{d};l^{2}),\mbox{ and for
$|\mathord{{\bf u}}|\leqslant R$},\\\ &\|\nabla_{x}\partial_{\mathord{{\bf
u}}}\mathord{{\bf g}}(t,x,\mathord{{\bf
u}})\|_{l^{2}}+\|\partial^{j}_{\mathord{{\bf u}}}\mathord{{\bf
g}}(t,x,\mathord{{\bf u}})\|_{l^{2}}\leqslant C_{R,T},j=1,2,\\\
&\sup_{t\in[0,T]}\|\nabla^{j}_{x}\mathord{{\bf
g}}(t,\cdot,0)\|_{{\mathbb{R}}^{d}\times l^{2}}\leqslant\lambda^{\mathord{{\bf
g}}}_{R,T}(x),\ j=0,1,2\end{aligned}\right\\},&\mbox{
${\mathcal{O}}={\mathbb{R}}^{d}$ or ${\mathbb{T}}^{d}$},\end{aligned}\right.$
where $\lambda^{\mathord{{\bf g}}}_{R,T}\in L^{p}({\mathcal{O}})$.
We define
$\displaystyle F(t,\mathord{{\bf u}}):={\mathscr{P}}_{p}(\mathord{{\bf
f}}(t,\cdot,\mathord{{\bf u}}))$ (9.11)
and
$\displaystyle\Psi(t,\mathord{{\bf u}}):={\mathscr{P}}_{p}(\mathord{{\bf
g}}(t,\cdot,\mathord{{\bf u}})).$ (9.12)
One can see that (N1) and (N2) hold. Indeed, for $\mathord{{\bf
u}},\mathord{{\bf v}}\in{\mathbf{H}}^{p}_{2\alpha}$ with $\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{p}_{2\alpha}},\|\mathord{{\bf
v}}\|_{{\mathbf{H}}^{p}_{2\alpha}}\leqslant R$, we have
$\displaystyle\|F(t,\mathord{{\bf u}})-F(s,\mathord{{\bf v}})\|_{\mathord{{\bf
L}}^{p}_{\sigma}}$ $\displaystyle\preceq$ $\displaystyle\|\mathord{{\bf
f}}(t,\cdot,\mathord{{\bf u}})-\mathord{{\bf f}}(s,\cdot,\mathord{{\bf
v}})\|_{\mathord{{\bf L}}^{p}}$
$\displaystyle\stackrel{{\scriptstyle(\ref{PL02})}}{{\leqslant}}$
$\displaystyle C_{T,R}(|t-s|^{\delta}+\|\mathord{{\bf u}}-\mathord{{\bf
v}}\|_{\mathord{{\bf L}}^{p}})$ $\displaystyle\leqslant$ $\displaystyle
C_{T,R}(|t-s|^{\delta}+\|\mathord{{\bf u}}-\mathord{{\bf
v}}\|_{{\mathbf{H}}^{p}_{2\alpha}}).$
Thus, (N1) holds. For (N2), let us look at the case of
${\mathcal{O}}={\mathbb{R}}^{d}$ or ${\mathbb{T}}^{d}$. Since $\Delta_{p}$ can
commute with ${\mathscr{P}}_{p}$, we have, for $\mathord{{\bf
u}},\mathord{{\bf v}}\in{\mathbf{H}}^{p}_{2\alpha}$ with $\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{p}_{2\alpha}},\|\mathord{{\bf
v}}\|_{{\mathbf{H}}^{p}_{2\alpha}}\leqslant R$
$\displaystyle\|\Psi(t,\mathord{{\bf u}})-\Psi(t,\mathord{{\bf
v}})\|_{L_{2}(l^{2};{\mathbf{H}}^{p}_{\alpha})}^{2}$
$\displaystyle\stackrel{{\scriptstyle(\ref{Exa})}}{{\preceq}}$
$\displaystyle\|{\mathfrak{L}}^{\frac{\alpha}{2}}{\mathscr{P}}_{p}[\mathord{{\bf
g}}(t,\mathord{{\bf u}})-\mathord{{\bf g}}(t,\mathord{{\bf
v}})]\|_{\mathord{{\bf L}}^{p}_{\sigma}({\mathcal{O}};l^{2})}^{2}$
$\displaystyle\preceq$
$\displaystyle\|(I-\Delta_{p})^{\frac{\alpha}{2}}[\mathord{{\bf
g}}(t,\mathord{{\bf u}})-\mathord{{\bf g}}(t,\mathord{{\bf
v}})]\|_{\mathord{{\bf L}}^{p}({\mathcal{O}};l^{2})}^{2}$
$\displaystyle\stackrel{{\scriptstyle(\ref{PP04})}}{{\preceq}}$
$\displaystyle\sum_{k}\|\mathord{{\bf g}}_{k}(t,\mathord{{\bf
u}})-\mathord{{\bf g}}_{k}(t,\mathord{{\bf v}})\|_{1,p}^{2}$
$\displaystyle\preceq$ $\displaystyle C_{R}\|\mathord{{\bf u}}-\mathord{{\bf
v}}\|^{2}_{{\mathbf{H}}^{p}_{2\alpha}}.$
Using Lemma 2.13, as the calculations given in Theorem 8.2, one can verify
that (9.8) holds under (8.4). Thus, (N2) holds.
### 9.2. Non-explosion and large deviation for 2D SNSEs
In this subsection, we study the non-explosion and large deviation for SNSE in
the case of two dimension. For this aim, in addition to (N1) and (N2), we also
suppose that
1. (N3)
For any $T>0$, there exists $C_{T}>0$ such that for all $t\in[0,T]$ and
$\mathord{{\bf u}}\in{\mathbf{H}}^{p}_{2}$
$\displaystyle\|F(t,\mathord{{\bf u}})\|_{{\mathbf{H}}^{2}_{0}}$
$\displaystyle\leqslant$ $\displaystyle C_{T}(\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{2}_{1}}+1),$ $\displaystyle\|F(t,\mathord{{\bf
u}})\|_{{\mathbf{H}}^{p}_{0}}$ $\displaystyle\leqslant$ $\displaystyle
C_{T}(\|\mathord{{\bf u}}\|_{{\mathbf{H}}^{p}_{2\alpha}}+1)$
and for $i=0,1$
$\displaystyle\|\Psi(s,\mathord{{\bf
u}})\|_{L_{2}(l^{2};{\mathbf{H}}^{2}_{i})}$ $\displaystyle\leqslant$
$\displaystyle C_{T}(1+\|\mathord{{\bf u}}\|_{{\mathbf{H}}^{2}_{i}}),$
$\displaystyle\|\Psi(s,\mathord{{\bf
u}})\|_{L_{2}(l^{2};{\mathbf{H}}^{p}_{\alpha})}$ $\displaystyle\leqslant$
$\displaystyle C_{T}(1+\|\mathord{{\bf u}}\|_{{\mathbf{H}}^{p}_{2\alpha}}),$
where $p$ and $\alpha$ satisfy (9.6).
We remark that $F$ and $\Psi$ defined by (9.11) and (9.12) satisfy (N3) when
$\mathord{{\bf f}}$ satisfies
$|\mathord{{\bf f}}(t,x,\mathord{{\bf u}})|\leqslant C_{T}(|\mathord{{\bf
u}}|+\lambda_{0}(x))$
and $\mathord{{\bf g}}$ satisfies (${\mathcal{O}}={\mathbb{R}}^{2}$ or
${\mathbb{T}}^{2}$)
$\displaystyle\|\partial_{\mathord{{\bf u}}}\mathord{{\bf
g}}(t,x,\mathord{{\bf u}})\|_{l^{2}}$ $\displaystyle\leqslant$ $\displaystyle
C_{T},$ $\displaystyle\|\mathord{{\bf g}}(t,x,\mathord{{\bf
u}})\|_{l^{2}}+\|\nabla_{x}\mathord{{\bf g}}(t,x,\mathord{{\bf u}})\|_{l^{2}}$
$\displaystyle\leqslant$ $\displaystyle C_{T}(|\mathord{{\bf
u}}|+\lambda_{1}(x)),$
where $\lambda_{0},\lambda_{1}\in L^{p}({\mathcal{O}})$.
We have the following result, the proof will be given in Lemma 9.7 below.
###### Theorem 9.2.
Let $p>d$ and $\frac{1}{2}<\alpha<1$. Assume that (N1)-(N3) hold. Let
$(\mathord{{\bf u}},\tau)$ be the unique maximal solution of Eq.(9.13) in
Theorem 9.1. Then $\tau=+\infty$ a.s..
We now consider the small perturbation for 2D stochastic Navier-Stokes
equation:
$\displaystyle{\mathord{{\rm d}}}\mathord{{\bf
u}}_{\epsilon}(t)=\big{[}-{\mathfrak{L}}_{p}\mathord{{\bf
u}}_{\epsilon}(t)+\Phi(t,\mathord{{\bf u}}_{\epsilon}(t))\big{]}{\mathord{{\rm
d}}}t+\sqrt{\epsilon}\Psi(t,\mathord{{\bf u}}_{\epsilon}(t)){\mathord{{\rm
d}}}W(t),\ \ \mathord{{\bf u}}_{\epsilon}(0)=\mathord{{\bf u}}_{0}$
as well as the control equation:
$\displaystyle{\mathord{{\rm d}}}\mathord{{\bf u}}^{\epsilon}(t)$
$\displaystyle=$ $\displaystyle\big{[}-{\mathfrak{L}}_{p}\mathord{{\bf
u}}^{\epsilon}(t)+\Phi(t,\mathord{{\bf u}}^{\epsilon}(t))+\Psi(t,\mathord{{\bf
u}}^{\epsilon}(t))\dot{h}^{\epsilon}(t)\big{]}{\mathord{{\rm d}}}t$ (9.13)
$\displaystyle+\sqrt{\epsilon}\Psi(t,\mathord{{\bf
u}}^{\epsilon}(t)){\mathord{{\rm d}}}W(t),\ \ \mathord{{\bf
u}}^{\epsilon}(0)=\mathord{{\bf u}}_{0},$
where $h^{\epsilon}\in{\mathcal{A}}^{T}_{N}$ (see (2.23) for the definition of
${\mathcal{A}}^{T}_{N}$), and $T>0$ is fixed below.
Let $(\mathord{{\bf u}}^{\epsilon},\tau^{\epsilon})$ be the unique maximal
strong solution of Eq. (9.13) with the properties:
$\lim_{t\uparrow\tau^{\epsilon}}\|\mathord{{\bf
u}}^{\epsilon}(t)\|_{{\mathbf{H}}^{p}_{2\alpha}}=+\infty\mbox{ on
$\\{\tau^{\epsilon}<\infty\\}$},$
and $t\mapsto\mathord{{\bf u}}^{\epsilon}(t)\in{\mathbf{H}}^{p}_{2}$ is
continuous on $[0,\tau^{\epsilon})$.
Before proving the non-explosion result (Lemma 9.7), we first prepare a series
of lemmas.
###### Lemma 9.3.
There exists a constant $C_{T}>0$ such that for any $t\in[0,T]$ and
$\mathord{{\bf u}}\in{\mathbf{H}}^{2}_{2}$
$\displaystyle{\langle}\mathord{{\bf u}},-{\mathfrak{L}}_{2}\mathord{{\bf
u}}+\Phi(s,\mathord{{\bf u}}){\rangle}_{{\mathbf{H}}^{2}_{0}}$
$\displaystyle\leqslant$ $\displaystyle-\frac{1}{2}\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{2}_{1}}^{2}+C_{T}(\|\mathord{{\bf
u}}\|^{2}_{{\mathbf{H}}^{2}_{0}}+1),$ (9.14)
$\displaystyle{\langle}{\mathfrak{L}}_{2}\mathord{{\bf
u}},-{\mathfrak{L}}_{2}\mathord{{\bf u}}+\Phi(s,\mathord{{\bf
u}}){\rangle}_{{\mathbf{H}}^{2}_{0}}$ $\displaystyle\leqslant$ $\displaystyle
C\|\mathord{{\bf u}}\|^{2}_{{\mathbf{H}}^{2}_{0}}\|\mathord{{\bf
u}}\|^{4}_{{\mathbf{H}}^{2}_{1}}+C_{T}\big{(}1+\|\mathord{{\bf
u}}\|^{2}_{{\mathbf{H}}^{2}_{1}}\big{)}$ (9.15)
and
$\displaystyle\|\Phi(t,\mathord{{\bf u}})\|_{{\mathbf{H}}^{p}_{0}}\leqslant
C_{T}\big{(}1+\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{2}_{1}}\big{)}\cdot\big{(}1+\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{p}_{2\alpha}}\big{)}.$ (9.16)
###### Proof.
Let $\mathord{{\bf u}}\in{\mathbf{H}}^{2}_{2}$. Noting that
${\langle}\mathord{{\bf u}},{\mathscr{P}}_{2}((\mathord{{\bf
u}}\cdot\nabla)\mathord{{\bf
u}}){\rangle}_{{\mathbf{H}}^{2}_{0}}={\langle}\mathord{{\bf u}},(\mathord{{\bf
u}}\cdot\nabla)\mathord{{\bf u}}{\rangle}_{\mathord{{\bf
L}}^{2}}=\frac{1}{2}\int_{\mathcal{O}}\mathord{{\bf
u}}(x)\cdot\nabla|\mathord{{\bf u}}(x)|^{2}{\mathord{{\rm d}}}x=0,$
by (N3) and Young’s inequality we have
$\displaystyle{\langle}\mathord{{\bf u}},-{\mathfrak{L}}_{2}\mathord{{\bf
u}}+\Phi(s,\mathord{{\bf u}}){\rangle}_{{\mathbf{H}}^{2}_{0}}$
$\displaystyle=$ $\displaystyle-\|\mathord{{\bf
u}}\|^{2}_{{\mathbf{H}}^{2}_{1}}+{\langle}\mathord{{\bf u}},\mathord{{\bf
u}}+F(t,\mathord{{\bf u}})){\rangle}_{{\mathbf{H}}^{2}_{0}}$
$\displaystyle\leqslant$ $\displaystyle-\frac{1}{2}\|\mathord{{\bf
u}}\|^{2}_{{\mathbf{H}}^{2}_{1}}+C_{T}(\|\mathord{{\bf
u}}\|^{2}_{{\mathbf{H}}^{2}_{0}}+1).$
Thus, (9.14) is proved.
For (9.15), noting that by Gagliado-Nirenberge’s inequality (cf. [27, p.24
Theoerem 9.3]) and (9.2)
$\|\mathord{{\bf u}}\|^{2}_{\mathord{{\bf L}}^{\infty}}\preceq\|\mathord{{\bf
u}}\|_{{\mathbb{H}}^{2}_{2}}\cdot\|\mathord{{\bf
u}}\|_{{\mathbb{H}}^{2}_{0}}\preceq\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{2}_{2}}\cdot\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{2}_{0}},$
by Young’s inequality we have
$\displaystyle{\langle}{\mathfrak{L}}_{2}\mathord{{\bf
u}},{\mathscr{P}}_{2}((\mathord{{\bf u}}\cdot\nabla)\mathord{{\bf
u}}){\rangle}_{{\mathbf{H}}^{2}_{0}}$ $\displaystyle\leqslant$
$\displaystyle\frac{1}{4}\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{2}_{2}}^{2}+\|{\mathscr{P}}_{2}((\mathord{{\bf
u}}\cdot\nabla)\mathord{{\bf u}})\|^{2}_{{\mathbf{H}}^{2}_{0}}$
$\displaystyle\leqslant$ $\displaystyle\frac{1}{4}\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{2}_{2}}^{2}+C\|(\mathord{{\bf
u}}\cdot\nabla)\mathord{{\bf u}})\|^{2}_{\mathord{{\bf L}}^{2}}$
$\displaystyle\leqslant$ $\displaystyle\frac{1}{4}\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{2}_{2}}^{2}+C\|\mathord{{\bf u}}\|_{\mathord{{\bf
L}}^{\infty}}^{2}\cdot\|\nabla\mathord{{\bf u}}\|^{2}_{\mathord{{\bf L}}^{2}}$
$\displaystyle\leqslant$ $\displaystyle\frac{1}{4}\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{2}_{2}}^{2}+C\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{2}_{0}}\cdot\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{2}_{2}}\cdot\|\mathord{{\bf
u}}\|^{2}_{{\mathbf{H}}^{2}_{1}}$ $\displaystyle\leqslant$
$\displaystyle\frac{1}{2}\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{2}_{2}}^{2}+C\|\mathord{{\bf
u}}\|^{2}_{{\mathbf{H}}^{2}_{0}}\cdot\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{2}_{1}}^{4}$
and by (N3)
${\langle}{\mathfrak{L}}_{2}\mathord{{\bf u}},F(s,\mathord{{\bf
u}}){\rangle}_{{\mathbf{H}}^{2}_{0}}\leqslant\frac{1}{2}\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{2}_{2}}^{2}+C_{T}(1+\|\mathord{{\bf
u}}\|^{2}_{{\mathbf{H}}^{2}_{1}}).$
Thus, (9.15) holds.
Let
$p<q<\frac{d}{1+\frac{d}{p}-2\alpha},\ \ q^{*}=\frac{qp}{q-p}.$
By Hölder’s inequality we have
$\displaystyle\|{\mathscr{P}}_{p}(\mathord{{\bf u}}\cdot\nabla)\mathord{{\bf
u}}\|_{{\mathbf{H}}^{p}_{0}}$ $\displaystyle\preceq$
$\displaystyle\|\mathord{{\bf u}}\cdot\nabla\mathord{{\bf u}}\|_{\mathord{{\bf
L}}^{p}}$ $\displaystyle\preceq$ $\displaystyle\|\mathord{{\bf
u}}\|_{\mathord{{\bf L}}^{q^{*}}}\cdot\|\nabla\mathord{{\bf
u}}\|_{\mathord{{\bf L}}^{q}}$
$\displaystyle\stackrel{{\scriptstyle(\ref{Em})}}{{\preceq}}$
$\displaystyle\|\mathord{{\bf u}}\|_{{\mathbf{H}}^{2}_{1}}\cdot\|\mathord{{\bf
u}}\|_{{\mathbf{H}}^{p}_{2\alpha}}.$
The estimate (9.16) now follows by (N3). ∎
Below, set for $n\in{\mathbb{N}}$
$\displaystyle\tau^{\epsilon}_{n}:=\inf\Big{\\{}t\geqslant 0:\|\mathord{{\bf
u}}^{\epsilon}(t)\|_{{\mathbf{H}}_{2\alpha}^{p}}>n\Big{\\}}.$
###### Lemma 9.4.
There exists a constant $C_{T}>0$ such that for all $\epsilon\in(0,1)$ and
$n\in{\mathbb{N}}$
$\displaystyle{\mathbb{E}}\left(\sup_{s\in[0,T\wedge\tau^{\epsilon}_{n}]}\|\mathord{{\bf
u}}^{\epsilon}(s)\|^{2}_{{\mathbf{H}}^{2}_{0}}\right)+{\mathbb{E}}\left(\int^{T\wedge\tau^{\epsilon}_{n}}_{0}\|\mathord{{\bf
u}}^{\epsilon}(s)\|^{2}_{{\mathbf{H}}^{2}_{1}}{\mathord{{\rm
d}}}s\right)\leqslant C_{T}.$
###### Proof.
By Ito’s formula we have
$\displaystyle\|\mathord{{\bf u}}^{\epsilon}(t)\|_{{\mathbf{H}}^{2}_{0}}^{2}$
$\displaystyle=$ $\displaystyle\|\mathord{{\bf
u}}_{0}\|_{{\mathbf{H}}^{2}_{0}}^{2}+2\int^{t}_{0}{\langle}\mathord{{\bf
u}}^{\epsilon}(s),-{\mathfrak{L}}_{2}\mathord{{\bf
u}}^{\epsilon}(s)+\Phi(s,\mathord{{\bf
u}}^{\epsilon}(s)){\rangle}_{{\mathbf{H}}^{2}_{0}}{\mathord{{\rm d}}}s$
$\displaystyle+2\int^{t}_{0}{\langle}\mathord{{\bf
u}}^{\epsilon}(s),\Psi(s,\mathord{{\bf
u}}^{\epsilon}(s))\dot{h}^{\epsilon}(s){\rangle}_{{\mathbf{H}}^{2}_{0}}{\mathord{{\rm
d}}}s$
$\displaystyle+2\sqrt{\epsilon}\sum_{k}\int^{t}_{0}{\langle}\mathord{{\bf
u}}^{\epsilon}(s),\Psi_{k}(s,\mathord{{\bf
u}}^{\epsilon}(s)){\rangle}_{{\mathbf{H}}^{2}_{0}}{\mathord{{\rm d}}}W^{k}(s)$
$\displaystyle+\epsilon\sum_{k}\int^{t}_{0}\|\Psi_{k}(s,\mathord{{\bf
u}}^{\epsilon}(s))\|^{2}_{{\mathbf{H}}^{2}_{0}}{\mathord{{\rm d}}}s$
$\displaystyle=:$ $\displaystyle\|\mathord{{\bf
u}}_{0}\|_{{\mathbf{H}}^{2}_{1}}^{2}+J_{1}(t)+J_{2}(t)+J_{3}(t)+J_{4}(t).$
Set
$f(t):={\mathbb{E}}\left(\sup_{s\in[0,t\wedge\tau^{\epsilon}_{n}]}\|\mathord{{\bf
u}}^{\epsilon}(s)\|^{2}_{{\mathbf{H}}^{2}_{0}}\right).$
First of all, noting that by (9.14)
$J_{1}(t)\leqslant-\int^{t}_{0}\|\mathord{{\bf
u}}^{\epsilon}(s)\|_{{\mathbf{H}}^{2}_{1}}^{2}+C_{T}\int^{t}_{0}(\|\mathord{{\bf
u}}^{\epsilon}(s)\|^{2}_{{\mathbf{H}}^{2}_{0}}+1){\mathord{{\rm d}}}s,$
we have
$\displaystyle{\mathbb{E}}\left(\sup_{s\in[0,t\wedge\tau^{\epsilon}_{n}]}J_{1}(s)\right)+{\mathbb{E}}\left(\int^{t\wedge\tau^{\epsilon}_{n}}_{0}\|\mathord{{\bf
u}}^{\epsilon}(s)\|^{2}_{{\mathbf{H}}^{2}_{1}}{\mathord{{\rm
d}}}s\right)\leqslant C_{T}\int^{t}_{0}(f(s)+1){\mathord{{\rm d}}}s.$
By (N3) and Young’s inequality we have
$\displaystyle{\mathbb{E}}\left(\sup_{s\in[0,t\wedge\tau^{\epsilon}_{n}]}J_{2}(s)\right)$
$\displaystyle\leqslant$ $\displaystyle
2{\mathbb{E}}\left(\int^{t\wedge\tau^{\epsilon}_{n}}_{0}\|\mathord{{\bf
u}}^{\epsilon}(s)\|_{{\mathbf{H}}^{2}_{0}}\cdot\|\Psi(s,\mathord{{\bf
u}}^{\epsilon}(s))\|_{L_{2}(l^{2};{\mathbf{H}}^{2}_{0})}\cdot\|\dot{h}^{\epsilon}(s)\|_{l^{2}}{\mathord{{\rm
d}}}s\right)$ $\displaystyle\leqslant$ $\displaystyle
2N{\mathbb{E}}\left(\int^{t\wedge\tau^{\epsilon}_{n}}_{0}\|\mathord{{\bf
u}}^{\epsilon}(s)\|^{2}_{{\mathbf{H}}^{2}_{0}}\cdot\|\Psi(s,\mathord{{\bf
u}}^{\epsilon}(s))\|^{2}_{L_{2}(l^{2};{\mathbf{H}}^{2}_{0})}{\mathord{{\rm
d}}}s\right)^{1/2}$ $\displaystyle\leqslant$
$\displaystyle\frac{1}{4}f(t)+C_{N}{\mathbb{E}}\left(\int^{t\wedge\tau^{\epsilon}_{n}}_{0}(1+\|\mathord{{\bf
u}}^{\epsilon}(s)\|^{2}_{{\mathbf{H}}^{2}_{0}}){\mathord{{\rm d}}}s\right)$
$\displaystyle\leqslant$
$\displaystyle\frac{1}{4}f(t)+C_{N}\int^{t}_{0}(1+f(s)){\mathord{{\rm d}}}s.$
Similarly, we also have
${\mathbb{E}}\left(\sup_{s\in[0,t\wedge\tau^{\epsilon}_{n}]}J_{3}(s)\right)\leqslant\frac{1}{4}f(t)+C\int^{t}_{0}(1+f(s)){\mathord{{\rm
d}}}s$
and
${\mathbb{E}}\left(\sup_{s\in[0,t\wedge\tau^{\epsilon}_{n}]}J_{4}(s)\right)\leqslant
C\int^{t}_{0}(1+f(s)){\mathord{{\rm d}}}s.$
Combining the above calculations we get
$f(t)+2{\mathbb{E}}\int^{t\wedge\tau^{\epsilon}_{n}}_{0}\|\mathord{{\bf
u}}^{\epsilon}(s)\|^{2}_{{\mathbf{H}}^{2}_{1}}{\mathord{{\rm d}}}s\leqslant
2\|\mathord{{\bf
u}}_{0}\|_{{\mathbf{H}}^{2}_{0}}^{2}+C_{N}+C_{N}\int^{t}_{0}(1+f(s)){\mathord{{\rm
d}}}s.$
The desired estimate follows by Gronwall’s inequality. ∎
Set for $n\in{\mathbb{N}}$
$\displaystyle\eta^{\epsilon}_{n}(t)$ $\displaystyle:=$
$\displaystyle\int^{t\wedge\tau^{\epsilon}_{n}}_{0}\|\mathord{{\bf
u}}^{\epsilon}(s)\|^{2}_{{\mathbf{H}}^{2}_{1}}\cdot\|\mathord{{\bf
u}}^{\epsilon}(s)\|^{2}_{{\mathbf{H}}^{2}_{0}}{\mathord{{\rm d}}}s+t$
$\displaystyle=$ $\displaystyle\int^{t}_{0}\|\mathord{{\bf
u}}^{\epsilon}(s)\|^{2}_{{\mathbf{H}}^{2}_{1}}\cdot\|\mathord{{\bf
u}}^{\epsilon}(s)\|^{2}_{{\mathbf{H}}^{2}_{0}}\cdot
1_{[0,\tau^{\epsilon}_{n}]}(s){\mathord{{\rm d}}}s+t$
and
$\theta^{\epsilon}_{n}(t):=\inf\left\\{s\geqslant
0:\eta^{\epsilon}_{n}(s)\geqslant t\right\\}.$
Clearly, $t\mapsto\eta^{\epsilon}_{n}(t)$ is a continuous and strictly
increasing function, and the inverse function of
$t\mapsto\theta^{\epsilon}_{n}(t)$ is just given by $\eta^{\epsilon}_{n}$.
Moreover, since $\eta^{\epsilon}_{n}(t)>t$, we have
$\theta^{\epsilon}_{n}(t)<t.$
###### Lemma 9.5.
For any $K>0$, there exists a constant $C_{K,N}>0$ such that for all
$\epsilon\in(0,1)$ and $n\in{\mathbb{N}}$
${\mathbb{E}}\left(\sup_{s\in[0,\theta^{\epsilon}_{n}(K)\wedge\tau^{\epsilon}_{n}]}\|\mathord{{\bf
u}}^{\epsilon}(s))\|^{2}_{{\mathbf{H}}^{2}_{1}}\right)\leqslant C_{K,N}.$
###### Proof.
Consider the following evolution triple
${\mathbf{H}}^{2}_{2}\subset{\mathbf{H}}^{2}_{1}\subset{\mathbf{H}}^{2}_{0}.$
By Ito’s formula (cf. [68]), we have
$\displaystyle\|\mathord{{\bf u}}^{\epsilon}(t)\|_{{\mathbf{H}}^{2}_{1}}^{2}$
$\displaystyle=$ $\displaystyle\|\mathord{{\bf
u}}_{0}\|_{{\mathbf{H}}^{2}_{1}}^{2}+2\int^{t}_{0}{\langle}{\mathfrak{L}}_{2}\mathord{{\bf
u}}^{\epsilon}(s),-{\mathfrak{L}}_{2}\mathord{{\bf
u}}^{\epsilon}(s)+\Phi(s,\mathord{{\bf
u}}^{\epsilon}(s)){\rangle}_{{\mathbf{H}}^{2}_{0}}{\mathord{{\rm d}}}s$
$\displaystyle+2\int^{t}_{0}{\langle}{\mathfrak{L}}_{2}\mathord{{\bf
u}}^{\epsilon}(s),\Psi(s,\mathord{{\bf
u}}^{\epsilon}(s))\dot{h}^{\epsilon}(s){\rangle}_{{\mathbf{H}}^{2}_{0}}{\mathord{{\rm
d}}}s$
$\displaystyle+2\sqrt{\epsilon}\sum_{k}\int^{t}_{0}{\langle}\mathord{{\bf
u}}^{\epsilon}(s),\Psi_{k}(s,\mathord{{\bf
u}}^{\epsilon}(s)){\rangle}_{{\mathbf{H}}^{2}_{1}}{\mathord{{\rm d}}}W^{k}(s)$
$\displaystyle+\epsilon\sum_{k}\int^{t}_{0}\|\Psi_{k}(s,\mathord{{\bf
u}}^{\epsilon}(s))\|^{2}_{{\mathbf{H}}^{2}_{1}}{\mathord{{\rm d}}}s$
$\displaystyle=:$ $\displaystyle\|\mathord{{\bf
u}}_{0}\|_{{\mathbf{H}}^{2}_{1}}^{2}+J_{1}(t)+J_{2}(t)+J_{3}(t)+J_{4}(t).$
Set
$\displaystyle f(t)$ $\displaystyle:=$
$\displaystyle{\mathbb{E}}\left(\sup_{s\in[0,t]}\|\mathord{{\bf
u}}^{\epsilon}(\theta^{\epsilon}_{n}(s)\wedge\tau^{\epsilon}_{n})\|^{2}_{{\mathbf{H}}^{2}_{1}}\right)$
$\displaystyle=$
$\displaystyle{\mathbb{E}}\left(\sup_{s\in[0,\theta^{\epsilon}_{n}(t)\wedge\tau^{\epsilon}_{n}]}\|\mathord{{\bf
u}}^{\epsilon}(s)\|^{2}_{{\mathbf{H}}^{2}_{1}}\right).$
For $J_{1}(t)$, by (9.15) we have, for $t\in[0,K]$
$\displaystyle J_{1}(\theta^{\epsilon}_{n}(t)\wedge\tau^{\epsilon}_{n})$
$\displaystyle\leqslant$
$\displaystyle\int^{\theta^{\epsilon}_{n}(t)\wedge\tau^{\epsilon}_{n}}_{0}\Big{[}C\|\mathord{{\bf
u}}^{\epsilon}(s)\|^{2}_{{\mathbf{H}}^{2}_{0}}\cdot\|\mathord{{\bf
u}}^{\epsilon}(s)\|_{{\mathbf{H}}^{2}_{1}}^{4}+C_{K}(1+\|\mathord{{\bf
u}}^{\epsilon}(s)\|_{{\mathbf{H}}^{2}_{1}}^{2})\Big{]}{\mathord{{\rm d}}}s$
$\displaystyle\leqslant$ $\displaystyle
C\int^{\theta^{\epsilon}_{n}(t)}_{0}\|\mathord{{\bf
u}}^{\epsilon}(s\wedge\tau^{\epsilon}_{n})\|_{{\mathbf{H}}^{2}_{1}}^{2}{\mathord{{\rm
d}}}\eta_{n}^{\epsilon}(s)+C_{K}$ $\displaystyle=$ $\displaystyle
C\int^{t}_{0}\|\mathord{{\bf
u}}^{\epsilon}(\theta^{\epsilon}_{n}(s)\wedge\tau^{\epsilon}_{n})\|_{{\mathbf{H}}^{2}_{1}}^{2}{\mathord{{\rm
d}}}s+C_{K},$
where the last step is due to the substitution of variable formula . So,
${\mathbb{E}}\left(\sup_{s\in[0,t]}J_{1}(\theta^{\epsilon}_{n}(s)\wedge\tau^{\epsilon}_{n})\right)\leqslant
C\int^{t}_{0}f(s){\mathord{{\rm d}}}s+C_{K}.$
Using the same trick as used in Lemma 9.4 and by (N3), we also have
${\mathbb{E}}\left(\sup_{s\in[0,t]}J_{i}(\theta^{\epsilon}_{n}(s)\wedge\tau^{\epsilon}_{n})\right)\leqslant\frac{1}{2}f(t)+C_{N,K}\int^{t}_{0}(f(s)+1){\mathord{{\rm
d}}}s,\ \ \ i=2,3,4.$
Thus, we get
$f(t)\leqslant 2\|\mathord{{\bf
u}}_{0}\|_{{\mathbf{H}}^{2}_{1}}^{2}+C_{N,K}\int^{t}_{0}(f(s)+1){\mathord{{\rm
d}}}s,$
which yields the desired estimate by Gronwall’s inequality. ∎
Set for $M>0$
$\zeta^{\epsilon}_{n}(M):=\inf\Big{\\{}t\geqslant 0:\|\mathord{{\bf
u}}^{\epsilon}(t\wedge\tau^{\epsilon}_{n})\|_{{\mathbf{H}}^{2}_{1}}\geqslant
M\Big{\\}}.$
###### Lemma 9.6.
For any $M>0$ and $q\geqslant 2$, there exists a constant $C_{T,M,N}>0$ such
that for all $\epsilon\in(0,1)$ and $n\in{\mathbb{N}}$
${\mathbb{E}}\left[\sup_{t\in[0,T\wedge\tau^{\epsilon}_{n}\wedge\zeta^{\epsilon}_{n}(M)]}\|\mathord{{\bf
u}}^{\epsilon}(t)\|_{{\mathbf{H}}_{2\alpha}^{p}}^{q}\right]\leqslant
C_{T,M,N}.$
###### Proof.
Set for $t\in[0,T]$
$\xi_{n}^{\epsilon}(t):=t\wedge\tau^{\epsilon}_{n}\wedge\zeta^{\epsilon}_{n}(M)$
and for $q\geqslant 2$
$f(t):={\mathbb{E}}\left[\sup_{t^{\prime}\in[0,\xi^{\epsilon}_{n}(t)]}\|\mathord{{\bf
u}}^{\epsilon}(t)\|_{{\mathbf{H}}_{2\alpha}^{p}}^{q}\right].$
Note that
$\displaystyle\mathord{{\bf u}}^{\epsilon}(t)$ $\displaystyle=$
$\displaystyle{\mathfrak{T}}_{t}\mathord{{\bf
u}}_{0}+\int^{t}_{0}{\mathfrak{T}}_{t-s}\Phi(s,\mathord{{\bf
u}}^{\epsilon}(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}{\mathfrak{T}}_{t-s}\Psi(s,\mathord{{\bf
u}}^{\epsilon}(s))\dot{h}^{\epsilon}(s){\mathord{{\rm d}}}s$
$\displaystyle+\sqrt{\epsilon}\int^{t}_{0}{\mathfrak{T}}_{t-s}\Psi(s,\mathord{{\bf
u}}^{\epsilon}(s)){\mathord{{\rm d}}}W(s).$
By (iii) of Proposition 2.11, Hölder’s inequality and Lemma 9.16, we have, for
$q>\frac{1}{1-\alpha}$
$\displaystyle{\mathbb{E}}\left[\sup_{t^{\prime}\in[0,\xi^{\epsilon}_{n}(t)]}\left\|\int^{t^{\prime}}_{0}{\mathfrak{T}}_{t^{\prime}-s}\Phi(s,\mathord{{\bf
u}}^{\epsilon}(s)){\mathord{{\rm
d}}}s\right\|^{q}_{{\mathbf{H}}^{p}_{2\alpha}}\right]$
$\displaystyle\qquad\preceq{\mathbb{E}}\left[\sup_{t^{\prime}\in[0,\xi^{\epsilon}_{n}(t)]}\left(\int^{t^{\prime}}_{0}\frac{1}{(t^{\prime}-s)^{\alpha}}\|\Phi(s,\mathord{{\bf
u}}^{\epsilon}(s))\|_{{\mathbf{H}}_{0}^{p}}{\mathord{{\rm
d}}}s\right)^{q}\right]$
$\displaystyle\qquad\preceq{\mathbb{E}}\left[\int^{\xi^{\epsilon}_{n}(t)}_{0}\|\Phi(s,\mathord{{\bf
u}}^{\epsilon}(s))\|_{{\mathbf{H}}_{0}^{p}}^{q}{\mathord{{\rm d}}}s\right]$
$\displaystyle\qquad\stackrel{{\scriptstyle(\ref{LL4})}}{{\preceq}}{\mathbb{E}}\left[\int^{\xi^{\epsilon}_{n}(t)}_{0}\Big{[}\big{(}1+\|\mathord{{\bf
u}}^{\epsilon}(s)\|_{{\mathbf{H}}^{2}_{1}}^{q}\big{)}\cdot\big{(}1+\|\mathord{{\bf
u}}^{\epsilon}(s)\|^{q}_{{\mathbf{H}}^{p}_{2\alpha}}\big{)}\Big{]}{\mathord{{\rm
d}}}s\right]$ $\displaystyle\qquad\leqslant
C_{M}\int^{t}_{0}(f(s)+1){\mathord{{\rm d}}}s.$
On the other hand, set
$G(t,s):={\mathfrak{T}}_{t-s}\Psi(s,\mathord{{\bf u}}^{\epsilon}(s)).$
Then by (iii) and (iv) of Proposition 2.11, we have
$\|G(t,s)\|^{2}_{{\mathbf{H}}^{p}_{2\alpha}}\leqslant\frac{C}{(t-s)^{\alpha}}\|\Psi(s,\mathord{{\bf
u}}^{\epsilon}(s))\|^{2}_{L_{2}(l^{2};{\mathbf{H}}^{p}_{\alpha})}$
and for $\gamma\in(0,(1-\alpha)/2)$
$\|G(t^{\prime},s)-G(t,s)\|^{2}_{{\mathbf{H}}^{p}_{2\alpha}}\leqslant\frac{|t^{\prime}-t|^{\gamma}}{(t-s)^{\alpha+2\gamma}}\|\Psi(s,\mathord{{\bf
u}}^{\epsilon}(s))\|^{2}_{L_{2}(l^{2};{\mathbf{H}}^{p}_{\alpha})}.$
Therefore, using Lemma 3.4 for $q$ large enough, we get
$\displaystyle{\mathbb{E}}\left(\sup_{t^{\prime}\in[0,T\wedge\xi_{n}^{\epsilon}(t)]}\left\|\int^{t^{\prime}}_{0}G(t^{\prime},s){\mathord{{\rm
d}}}W(s)\right\|^{q}_{{\mathbf{H}}^{p}_{2\alpha}}\right)$
$\displaystyle\qquad\leqslant
C_{T}{\mathbb{E}}\left(\int^{T\wedge\xi_{n}^{\epsilon}(t)}_{0}\|\Psi(s,\mathord{{\bf
u}}^{\epsilon}(s))\|_{L_{2}(l^{2};{\mathbf{H}}^{p}_{\alpha})}^{q}{\mathord{{\rm
d}}}s\right)$
$\displaystyle\qquad\stackrel{{\scriptstyle{\bf(N3)}}}{{\leqslant}}C_{T}\int^{t}_{0}(f(s)+1){\mathord{{\rm
d}}}s.$
Similarly, we have
$\displaystyle{\mathbb{E}}\left(\sup_{t^{\prime}\in[0,T\wedge\xi_{n}^{\epsilon}(t)]}\left\|\int^{t}_{0}{\mathfrak{T}}_{t-s}\Psi(s,\mathord{{\bf
u}}^{\epsilon}(s))\dot{h}^{\epsilon}(s){\mathord{{\rm
d}}}s\right\|^{q}_{{\mathbf{H}}^{p}_{2\alpha}}\right)$
$\displaystyle\qquad\leqslant C_{T,N}\int^{t}_{0}(f(s)+1){\mathord{{\rm
d}}}s.$
Combining the above calculations, we obtain
$f(t)\leqslant C_{T,M,N}\int^{t}_{0}f(s){\mathord{{\rm d}}}s+C_{T,M,N},$
which yields the desired estimate by Gronwall’s inequality. ∎
###### Lemma 9.7.
It holds that
$\displaystyle\lim_{n\rightarrow\infty}\sup_{\epsilon\in(0,1)}P\Big{\\{}\omega:\tau^{\epsilon}_{n}(\omega)\leqslant
T\Big{\\}}=0.$ (9.17)
###### Proof.
First of all, for any $M,K>0$ we have
$\displaystyle P\\{\zeta^{\epsilon}_{n}(M)<T\\}$ $\displaystyle\leqslant$
$\displaystyle P\\{\zeta^{\epsilon}_{n}(M)<T;\theta^{\epsilon}_{n}(K)\geqslant
T\\}+P\\{\theta^{\epsilon}_{n}(K)<T\\}$ $\displaystyle=$ $\displaystyle
P\left\\{\sup_{t\in[0,T)}\|\mathord{{\bf
u}}^{\epsilon}(t\wedge\tau^{\epsilon}_{n})\|_{{\mathbf{H}}^{2}_{1}}>M;\theta^{\epsilon}_{n}(K)\geqslant
T\right\\}$
$\displaystyle+P\left\\{\sup_{s\in[0,T)}\eta^{\epsilon}_{n}(s)>K\right\\}$
$\displaystyle\leqslant$ $\displaystyle
P\left\\{\sup_{t\in[0,\theta^{\epsilon}_{n}(K)\wedge\tau^{\epsilon}_{n}]}\|\mathord{{\bf
u}}^{\epsilon}(t)\|_{{\mathbf{H}}^{2}_{1}}>M\right\\}+P\big{\\{}\eta^{\epsilon}_{n}(T)>K\big{\\}}$
$\displaystyle\leqslant$
$\displaystyle{\mathbb{E}}\left(\sup_{t\in[0,\theta^{\epsilon}_{n}(K)\wedge\tau^{\epsilon}_{n})}\|\mathord{{\bf
u}}^{\epsilon}(t)\|_{{\mathbf{H}}^{2}_{1}}^{2}\right)/M^{2}+{\mathbb{E}}\left(\eta^{\epsilon}_{n}(T)\right)/K.$
Hence, by Lemmas 9.4 and 9.5 we have
$\lim_{M\to\infty}\sup_{n,\epsilon}P\\{\zeta^{\epsilon}_{n}(M)<T\\}=0.$
Secondly, we also have
$\displaystyle P\\{\tau^{\epsilon}_{n}<T\\}\leqslant
P\\{\tau^{\epsilon}_{n}<T;\zeta^{\epsilon}_{n}(M)\geqslant
T\\}+P\\{\zeta^{\epsilon}_{n}(M)<T\\}.$ (9.18)
For the first term, by Lemma 9.6 we have
$\displaystyle P\\{\tau^{\epsilon}_{n}<T;\zeta^{\epsilon}_{n}(M)\geqslant
T\\}$ $\displaystyle=$ $\displaystyle P\left\\{\sup_{t\in[0,T)}\|\mathord{{\bf
u}}^{\epsilon}(t)\|_{{\mathbf{H}}^{p}_{2\alpha}}>n;\zeta^{\epsilon}_{n}(M)\geqslant
T\right\\}$ $\displaystyle\leqslant$ $\displaystyle
P\left\\{\sup_{t\in[0,T\wedge\tau^{\epsilon}_{n}]}\|\mathord{{\bf
u}}^{\epsilon}(t)\|_{{\mathbf{H}}^{p}_{2\alpha}}\geqslant
n;\zeta^{\epsilon}_{n}(M)\geqslant T\right\\}$ $\displaystyle\leqslant$
$\displaystyle
P\left\\{\sup_{s\in[0,T\wedge\zeta^{\epsilon}_{n}(M)\wedge\tau^{\epsilon}_{n}]}\|\mathord{{\bf
u}}^{\epsilon}(t)\|_{{\mathbf{H}}^{p}_{2\alpha}}\geqslant n\right\\}$
$\displaystyle\leqslant$
$\displaystyle{\mathbb{E}}\left(\sup_{s\in[0,T\wedge\zeta^{\epsilon}_{n}(M)\wedge\tau^{\epsilon}_{n}]}\|\mathord{{\bf
u}}^{\epsilon}(t)\|^{q}_{{\mathbf{H}}^{p}_{2\alpha}}\right)/n^{q}$
$\displaystyle\leqslant$ $\displaystyle\frac{C_{T,M,N}}{n^{q}},$
where $C_{T,M,N}$ is independent of $\epsilon$ and $n$. The desired limit now
follows by taking limits for (9.18), first $n\to\infty$, then $M\to\infty$. ∎
Thus, using Theorem 6.3 we get:
###### Theorem 9.8.
Let ${\mathcal{O}}={\mathbb{T}}^{2}$ or a bounded smooth domain in
${\mathbb{R}}^{2}$. Under (N1)-(N3), for $\mathord{{\bf
u}}_{0}\in{\mathbf{H}}^{p}_{2}$, $\\{\mathord{{\bf
u}}_{\epsilon},\epsilon\in(0,1)\\}$ satisfies the large deviation principle in
${\mathbb{C}}_{T}({\mathbf{H}}_{2\alpha}^{p})$ with the rate function $I(f)$
given by
$I(f):=\frac{1}{2}\inf_{\\{h\in\ell^{2}_{T}:~{}f=\mathord{{\bf
u}}^{h}\\}}\|h\|^{2}_{\ell^{2}_{T}},\ \
f\in{\mathbb{C}}_{T}({\mathbf{H}}_{2\alpha}^{p}),$
where $\mathord{{\bf u}}^{h}$ solves the following equation:
$\displaystyle\mathord{{\bf u}}^{h}(t)$ $\displaystyle=$
$\displaystyle\mathord{{\bf u}}_{0}+\int^{t}_{0}\Delta\mathord{{\bf
u}}^{h}(s){\mathord{{\rm d}}}s+\int^{t}_{0}{\mathscr{P}}_{p}((\mathord{{\bf
u}}^{h}(s)\cdot\nabla)\mathord{{\bf u}}^{h}(s)){\mathord{{\rm d}}}s$
$\displaystyle+\int^{t}_{0}F(s,\mathord{{\bf u}}^{h}(s)){\mathord{{\rm
d}}}s+\int^{t}_{0}\Psi(s,\mathord{{\bf u}}^{h}(s))\dot{h}(s){\mathord{{\rm
d}}}s.$
Acknowledgements:
The author would like to thank Professor Benjamin Goldys for providing him an
excellent environment to work in the University of New South Wales. His work
is supported by ARC Discovery grant DP0663153 of Australia and NSF of China
(No. 10871215).
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|
arxiv-papers
| 2008-12-04T00:13:48 |
2024-09-04T02:48:59.177966
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xicheng Zhang",
"submitter": "Xicheng Zhang",
"url": "https://arxiv.org/abs/0812.0834"
}
|
0812.0841
|
# Genetic noise control via protein oligomerization
Cheol-Min Ghim and Eivind Almaas C.-M. Ghim - cmghim@llnl.gov E. Almaas -
almaas@llnl.gov Microbial Systems Biology Group, Biosciences and Biotechnology
Division, Lawrence Livermore National Laboratory, 7000 East Avenue Livermore,
CA 94550, USA
###### Abstract
#### Background:
Gene expression in a cell entails random reaction events occurring over
disparate time scales. Thus, molecular noise that often results in phenotypic
and population-dynamic consequences sets a fundamental limit to biochemical
signaling. While there have been numerous studies correlating the architecture
of cellular reaction networks with noise tolerance, only a limited effort has
been made to understand the dynamic role of protein-protein interactions.
#### Results:
We have developed a fully stochastic model for the positive feedback control
of a single gene, as well as a pair of genes (toggle switch), integrating
quantitative results from previous in vivo and in vitro studies. In
particular, we explicitly account for the fast binding-unbinding kinetics
among proteins, RNA polymerases, and the promoter/operator sequences of DNA.
We find that the overall noise-level is reduced and the frequency content of
the noise is dramatically shifted to the physiologically irrelevant high-
frequency regime in the presence of protein dimerization. This is independent
of the choice of monomer or dimer as transcription factor and persists
throughout the multiple model topologies considered. For the toggle switch, we
additionally find that the presence of a protein dimer, either homodimer or
heterodimer, may significantly reduce its random switching rate. Hence, the
dimer promotes the robust function of bistable switches by preventing the
uninduced (induced) state from randomly being induced (uninduced).
#### Conclusions:
The specific binding between regulatory proteins provides a buffer that may
prevent the propagation of fluctuations in genetic activity. The capacity of
the buffer is a non-monotonic function of association-dissociation rates.
Since the protein oligomerization per se does not require extra protein
components to be expressed, it provides a basis for the rapid control of
intrinsic or extrinsic noise. The stabilization of regulatory circuits and
epigenetic memory in general is of direct implications to organism fitness.
Our results also suggest possible avenues for the design of synthetic gene
circuits with tunable robustness for a wide range of engineering purposes.
## Background
Recent experiments on isogenic populations of microbes with single-cell
resolution [pedraza05, rosenfeld05, newman06] have demonstrated that
stochastic fluctuations, or noise, can override genetic and environmental
determinism. In fact, the presence of noise may significantly affect the
fitness of an organism [raser05]. The traditional approach for modeling the
process of molecular synthesis and degradation inside a cell is by
deterministic rate equations, where the continuous change of arbitrarily small
fractions of molecules is controlled instantaneously and frequently
represented through sigmoidal dose-response relations. However, the rate-
equation approaches can not explain the observed phenotypic variability in an
isogenic population in stable environments. In particular, when molecules
involved in feedback control exist in low copy numbers, noise may give rise to
significant cell-to-cell variation as many regulatory events are triggered by
molecules with very low copy numbers $\lesssim 100$ [guptasarma95]. A well
known example is the regulation of inorganic trace elements [nelson99], such
as iron, copper, and zinc. While these trace elements are essential for the
activity of multiple enzymes, their presence may quickly turn cytotoxic unless
their concentrations are carefully controlled.
Although the presence of phenotypic variation due to stochastic fluctuations
need not be detrimental for a population of cells [kussell05], elaborate
regulatory mechanisms have evolved to attenuate noise [chen06]. Several
systems-biology studies have recently focused on a select set gene-regulatory
circuits, in particular those with feedback control. Feedback control circuits
have been identified as important for multiple species and proven responsible
for noise reduction and increased functional stability in many housekeeping
genes through negative autoregulation [thieffry98], long cascades of
ultrasensitive signaling [thattai02], bacterial chemotaxis [yi00], and the
circadian clock [vilar02]. Additionally, recent studies on iron homeostasis
[semsey06, levine07] in E. coli highlight the noise-reducing capability
mediated by small RNAs.
Here, we study reversible protein-protein binding as a novel source for
genetic noise control. In particular, we have quantitatively analyzed the
effects of protein oligomerization on noise in positive autoregulatory
circuits as well as a simple toggle-switch [gardner00]. The all-or-none
threshold behavior of positive-feedback circuits typically improves robustness
against “leaky” switching. However, due to their functional purposes, gene
circuits involved in developmental processes or stress responses that often
accompany genome-wide changes in gene expression are intrinsically noisier
than negative feedback circuits.
It is frequently observed that transcription factors exist in oligomeric form
[beckett01], and protein oligomerization is an important subset of protein-
protein interactions, constituting a recurring theme in enzymatic proteins as
well as regulatory proteins. Well studied examples include the $\lambda$-phage
repressor, ${\lambda}$CI (dimer), the TrpR (dimer), LacR (tetramer), and Lrp
(hexadecamer or octamer). While many of the RNA-binding proteins dimerize
exclusively in the cytosol, the LexA repressor [kim92], the leucine-zipper
activator [berger98, kohler99], and the Arc repressor [rentzeperis99] have
been shown to form an oligomer either in the cytosol (“dimer path”) or on the
DNA by sequential binding (“monomer path”). Previously, the efficacy of
monomer and dimer transcription-regulation paths to reduce noise was
separately studied for a negative-feedback autoregulatory circuit
[bundschuh03jtb]. In contrast, we have focused on oligomerization in positive-
feedback autoregulatory circuits, as well as genetic toggle switches based on
the mutual repression of genes[gardner00]. We find that cytosolic
transcription-factor oligomerization acts as a significant buffer for
abundance-fluctuations in the monomer, overall reducing noise in the circuit.
Additionally, the noise-power spectral density is shifted from the low- to the
high-frequency regime. In the toggle switch, cytosolic oligomerization may
significantly stabilize the functional state of the circuit. This is
especially evident for heterodimerization.
Yet another interesting case of ligand-binding-mediated receptor
oligomerization has been reported [alarcon06, macnamara07], where the
formation of various structures of oligomers may act to buffer the
intracellular signaling against noise. Although our modeling and analysis is
based on prokaryotic cells, we expect our main findings to be organism
independent since protein oligomers, especially homodimers, is such a common
occurrence across the species [ispolatov05], with homodimers comprising 12.6%
of the high-fidelity human proteome [ramirez07, mcdermott05].
## Results and Discussion
### Dimerization breaks long-time noise correlations in autogenous circuit
To evaluate the dynamic effects of protein-protein binding in positive-
autoregulation gene circuits, we construct several alternative models of
positive autogenous circuits. Each model emphasizes a different combination of
possible feedback mechanisms, and the network topologies considered can be
grouped into the two classes of monomer-only (MO) and dimer-allowed (DA)
circuits, according to the availability of a protein-dimer state (color coding
in Fig. 1). We further group the DA circuits into three variations, DA1
through DA3, depending on which form of the protein is the functional
transcription factor (TF) and where the dimerization occurs. For DA1, we only
allow the dimer to bind with the DNA-operator sequence (dimeric transcription
factor, DTF), while for DA2 dimerization occurs through sequential binding of
monomers on the DNA. In DA3, the protein-DNA binding kinetics is the same as
in the MO circuit, hence monomeric transcription factor (MTF), with the
addition of a cytosolic protein dimer state. While we will only present
results for DA1 in this paper, there is no significant difference for DA2 and
DA3 [Additional file 1].
Note that the feedback loop is not explicit in Fig. 1 but implicitly included
through the dependence of RNAp-promoter binding equilibrium on the binding
status of the TF-operator pair. The sign (positive or negative) and strength
of the feedback control is determined by the relative magnitude of the
dissociation constants between RNAp and DNA which is either free or TF-bound.
For instance, topology DA1 has positive feedback control if
$K_{30}=k_{30}/q_{30}>K_{32}=k_{32}/q_{32}$, and $K_{30}$ corresponds to the
level of constitutive transcription (transcription initiation in the absence
of bound transcription factor). For each topology, we study the dependence of
noise characteristics on the kinetic rates by varying the dimer lifetime,
binding affinity, and the individual association/dissociation rates (see Table
Acknowledgments and Fig. 1). While we only discuss positive feedback control
of the autogenous circuit in this paper, we have obtained corresponding
results for negative feedback control [Additional file 1].
Fig. 2 shows a sample of ten representative time courses for the protein
abundance. The effect of stochastic fluctuations is marked in the MO circuit.
However, in all the DA circuits where the protein may form a cytosolic dimer
we observe a significantly reduced level of noise in the monomer abundance.
The suppression of fluctuations persists throughout the range of kinetic
parameters that (so far) is known to be physiologically relevant (see Table
1).
Calculating the steady-state distribution for the monomer and dimer abundances
(Fig. 3) we observe a clear trend that the monomer Fano factor (variance-to-
mean ratio) is reduced as the binding equilibrium is shifted towards the
dimer. This trend is conserved for all the investigated DA topologies (see
Supplementary Information). As long as dimerization is allowed in the cytosol,
the fast-binding equilibrium absorbs long-time fluctuations stemming from
bursty synthesis or decay of the monomer. When a random fluctuation brings
about a sudden change in the monomer copy number, dimerization provides a
buffering pool that absorbs the sudden change. Otherwise, random bursts in the
monomer abundance will propagate to the transcriptional activity of the
promoter, leading to erratic control of protein expression. It should be
emphasized that this has nothing to do with the sign of regulation and is in
agreement with the observations of Ref. [bundschuh03jtb] for negative
autoregulation. Surprisingly, the magnitude of noise reduction in the positive
autoregulatory circuit is nearly the same as that for negative autoregulation
which is typically considered a highly stable construct [Additional file 1].
A heuristic explanation can be found from Jacobian analysis of a deterministic
dynamical system, which is justified for small perturbations around a steady
state. When a random fluctuation shifts the monomer copy number away from its
steady-state value, the decay toward the steady state can be described by the
system Jacobian. The disparity in the magnitude of the (negative) eigenvalues
of the Jacobian matrix for the MO versus the DA circuits signifies that the
perturbed state is buffered by fast settlement of the monomer-dimer
equilibrium. This buffering occurs before random fluctuation can accumulate,
possibly with catastrophic physiological effects, explaining the coarse long-
time patterns observed in the MO model in contrast with the DA circuits (Fig.
2).
### Frequency-selective whitening of Brownian noise
The dimerization process itself generates stochastic fluctuations on a short
time scale. However, since this time scale is essentially separated from that
of monomer synthesis and decay (orders of magnitude faster), dimerization
effectively mitigates monomer-level fluctuations. The frequency content of the
fluctuations is best studied by an analysis of the power spectral density
(PSD), which is defined as the Fourier transform of the autocorrelation
function [numrec], originally introduced for signal processing. Fig. 4 shows
the noise power spectra of DA1, and the distinction between the MO circuit and
the DA topology is immediately evident. In particular, we note the following
two features. (i) A power-law decay with increasing frequency and (ii) a
horizontal plateau for the DA circuits. The power-law feature is explained by
the “random walk” nature of protein synthesis and decay: The power-law
exponent is approximately 2, which is reminiscent of Brownian motion (a Wiener
process) in the limit of large molecular copy numbers. Compared to other
commonly observed signals, such as white (uncorrelated) noise or $1/f$ noise,
protein synthesis/decay has a longer correlation time. If the autocorrelation
function of a time course is characterized by a single exponential decay, as
is the case for Brownian noise, the PSD is given by a Lorentzian profile, and
thus, well approximated by an inverse-square law in the low-frequency regime.
We do not observe a saturation value for the MO circuit, and it is likely not
in the frequency window of physiological interest. This may especially be the
case for circuits where the correlation times are long.
The noise reduction is in the physiologically relevant low-frequency regime,
and in Fig. 4 we have indicated the typical values for a cell cycle and mRNA
lifetime. Although stochastic fluctuations impose a fundamental limit in
cellular information processing, multiple noise sources may affect cellular
physiology non-additively. For a living cell, fluctuations are especially
relevant when their correlation time is comparable to, or longer than, the
cell cycle. At the same time, short-time scale fluctuations (relative to the
cell cycle) are more easily attenuated or do not propagate [tan07].
Additionally, the observed flat region in the PSD of the DA circuits implies
that as far as mid-range frequency fluctuations are concerned, we can safely
approximate them as a white noise. This insight may shed light on the
reliability of approximation schemes for effective stochastic dynamics in
protein-only models.
### Increased lifetime of dimer plays an important role
The virtue of the cytosolic dimer state is also directly related to the
extended lifetime of proteins when in a complex. Except for the degradation
tagging for active proteolysis, a much slower turnover of protein oligomers is
the norm. This is partly explained by the common observation that monomers
have largely unfolded structures, which are prone to be target of proteolysis
[herman03]. It has also been pointed out that the prolonged lifetime of the
oligomeric form is a critical factor for enhancing the feasible parameter
ranges of gene circuits [buchler05]. As seen from Fig. 3 (also Table 2), the
fold change of the noise reduction, while still significant, is not as strong
for the (hypothetical) case of dimer lifetime being the same as that of the
monomer ($\gamma_{2}/\gamma_{1}=1/2$). However, the low-frequency power
spectra still exhibit almost an order-of-magnitude smaller noise power than in
the MO circuit with the same rate parameters (Fig. 4). Hence, the noise
reduction capability holds good as long as the dimer lifetime is kept
sufficiently long compared with the monomer-dimer transition.
### Effects of homo-dimerization in genetic toggle switch
The exceptionally stable lysogeny of the phage $\lambda$, for which the
spontaneous loss rate is $\lesssim 10^{-7}$ per cell per generation
[rozanov98, little99], has motivated the synthesis of a genetic toggle switch
[gardner00]. Toggle switch is constructed from a pair of genes, which we will
denote as gene $A$ and $B$, that transcriptionally repress each other’s
expression. This mutual negative regulation can be considered an effective
positive feedback loop and provides the basis for the multiple steady states.
The existence of multistability, in turn, may be exploited as a device for
epigenetic memory or for decision making [losick08].
As the general attributes of positive feedback with cooperativity suggest, a
genetic toggle switch responds to external cues in an ultrasensitive way: When
the strength of a signal approaches a threshold value, the gene expression
state can be flipped by a small change in the signal. For example, the
concentration of protein $A$ ($B$) may rapidly switch from high to low and
vice versa. However, previous studies of a synthetic toggle switch have shown
that the noise-induced state switching is a rare event [kobayashi04, tian06,
gardner00]. In the ensuing analysis, we aim to delineate the origin of this
exceptional stability.
In a simple model, the monomer-only (MO) toggle, regulatory proteins only
exist in monomeric form. Although an external signal is not explicitly
included, random fluctuations in the abundance of the circuit’s molecular
components will occasionally flip the toggle-state for the two protein
species. Drawing on the results from our analysis of positive autoregulatory
gene circuits, we hypothesize that dimerization in the regulatory proteins of
the toggle switch will serve to stabilize its performance against noise. We
allow the protein products of each gene to form a homodimer, being either $AA$
or $BB$, which is similar to the cI-cro system in phage $\lambda$ [ptashne].
The dissociation constant for the dimers is defined as $K_{1}=q_{1}/k_{1}$,
where $k_{1}$ is the rate of two monomers forming a complex, and $q_{1}$ the
rate of the complex breaking up into its two constituents.
We evaluate the effect of the fast protein binding-unbinding dynamics on the
toggle switch performance by using either (i) the monomers or (ii) the
homodimers as the functional form of the repressor. Fig. 5 shows, for selected
values of the dissociation constant $K_{1}$, representative time series of the
protein monomer (left) and dimer (right) abundances for the case of (a)
monomeric or (b) dimeric transcription factors, respectively. A careful
analysis of the phase space (in presence of noise) for our chosen set of
parameters confirms that the studied toggle-switch systems are in the bistable
region [ghim08].
When monomer is the functional form of the repressor molecule (Fig. 5(a)) and
$K_{1}$ is large (limit of low dimer affinity), the protein populations are
dominated by monomers. Hence, the circuit effectively behaves as an MO toggle.
As $K_{1}$ decreases, we see that the level of random switching is suppressed:
Analogous to the autogenous circuit, the dimer pool stabilizes the protein
monomer population. However, the noise suppression is not monotonic with
increasing dimer binding affinity. Indeed, for very large binding affinities
(small $K_{1}$), the number of random switching events is increased since the
monomer is only available in low copy numbers. Consequently in this limit, it
becomes more likely that a small fluctuation in the monomer abundance can
cause a dramatic change in the overall gene expression profile. The noise-
stabilizing effect of dimerization is also reflected in the corresponding PSDs
[Additional file 1]. For instance, we observe a marked suppression of low-
frequency fluctuations in the monomer abundance with increasing $K_{1}$.
In Fig. 5(b) we show corresponding sample time series for the case of a
dimeric repressor, all other properties being the same as in (a). While the
overall trends are similar, we do note the following difference. Contrary to
the monomeric repressor case, there are very few toggle events in the strong
binding limit: Since the signaling molecules (dimers) of the dominant gene
(the “on”-gene) tend to exist in large copy numbers, a significant fluctuation
is needed to flip the state of the toggle switch. In the case of monomeric
repression, the signaling molecule exists in low abundance in this limit.
Thus, the dominant protein species in the dimeric-repressor system is able to
maintain much better control over the state of the toggle switch.
In Fig. 6, we show the distribution $(N_{A}-N_{B})$, the difference in
molecule abundance for the two protein species in the case of monomeric (left)
and dimeric (right) transcription factor. The asymmetry with respect to the
zero axis is caused by our choice of initial conditions (protein species $A$
in high concentration and species $B$ in low concentration), as well as the
finite length of the time series. For monomeric transcription, the presence of
dimers with moderate binding affinity sharpens the monomer abundance
distribution while accentuating its bimodal character. This is in agreement
with the qualitative observation from Fig. 5 on switching stability. For
dimeric transcription, we clearly observe that the symmetry of the system is
broken for small values of $K_{1}$, indicating that the state of the toggle
switch is extremely stable, and hence, likely determined by the choice of the
initial conditions.
To systematically quantify our observations on the interplay between dimer-
binding affinity and the functional stability of the toggle switch, we
generated long time series ($\approx 3\cdot 10^{7}$ sec) to measure the
average spontaneous switching rate. In Fig. 7, we show the average toggle
frequency relative to that of the MO toggle for the binding affinities
$K_{1}/\textrm{nM}=\\{2,20,100,1000\\}$, and the average MO switching rate is
$7.5\times 10^{-6}$/hour. As expected, we find that intermediate values of
$K_{1}$ are able to stabilize the toggle switch. Fig. 7 also highlights the
increased stability of the toggle switch for a dimeric versus monomeric
transcription factor, the dimeric switching rates always being lower and
approaching zero for strong dimer binding.
### Heterodimerization in genetic toggle switch
We have also considered the case of heterodimerization in the toggle switch,
since the noise- and functional stabilization of the switch may be directly
affected by the composition and source of the dimers. Note that, the gene-
regulation activity is conferred by the two monomer proteins $A$ and $B$ and
not the heterodimer $AB$. However, we find that the presence of (inactive)
heterodimers gives rise to very similar noise-stabilizing effects as that of
homodimers (Fig. 7). In fact, the existence of heterodimer state allows the
dominant protein species to effectively suppress the (active) monomers of the
minority species. Thus the heterodimer circuit shows dramatically enhanced
functional stability as compared to the case of homodimeric repressors, not
sharing the discussed vulnerability of MO circuit to intrinsic noise.
Although, to our knowledge, this is a purely hypothetical toggle-switch
design, it provides a general strategy for noise control in synthetic gene
circuits, along with previously proposed approach of overlapping upstream
regulatory domains [warren04].
## Conclusions
Cells have evolved distinct strategies to combat the fundamental limits
imposed by intrinsic and environmental fluctuations. We investigated the role
of protein oligomerization on noise originating from the random occurrence of
reaction events and the discrete nature of molecules. Recent efforts to
correlate network structure with functional aspects may provide valuable
insights into approaches for network-level noise control [barabasi04]. While
negative feedback is one of the most abundantly observed patterns to achieve
the goal of stability, it begs the question of how cells reliably change the
expression of genes from one state to another. The ultrasensitive response
circuit, exemplified by the ubiquitous signal transduction cascades in
eukaryotic cells, has been proposed as an answer to this question
[goldbeter81, huang96].
In addition to the combinatorial expansion of functional specificity, we argue
that the availability of oligomeric states contributes to the attenuation of
stochastic fluctuations in protein abundance. In positive autoregulatory gene
circuits, where the abundance of an expressed protein controls its own
synthesis rate, dimerization provides a buffer serving to mitigate random
fluctuations associated with the bursty transcription-translation process. We
find that short-time binding-unbinding dynamics reduce the overall noise level
by converting potentially pathological low-frequency noise to physiologically
unimportant, and easily attenuated, high-frequency noise [tan07].
Noise-induced switching generally signals a defect in cellular information
processing. Untimely exit from latency in the lambda-phage system directly
implies, as the immediate consequence to viruses, increased chance of being
targeted by a host immune system. In the case of a bacterium, the expression
of a specific set of sugar uptake genes when the sugar is absent from the
external medium is a considerable waste of cellular resources. For example,
lac operon of E. coli can be considered to have the circuitry of mutual
antagonism between the lacI gene and lactose uptake-catabolic genes
[lacoperon]. A difference lies in the non-transcriptional deactivation of the
allosteric transcription factor LacI. LacY, lactose permease, indirectly
regulates LacI by increasing lactose uptake, which in turn catalytically
deactivates LacI. Likewise, many pili operons of Gram-negative bacteria are
also known to utilize heritable expression states, which are of crucial role
in pathogenesis [hernday02, sauer00].
We expect that the random flipping of gene expression states in the examples
of positive-feedback-based genetic switches may very well be closely coupled
with the fitness of an organism. Phenomenological models relating the fitness
of an organism to random phenotypic switching in fluctuating environments have
provided important insights into the role of noise [thattai04], but still many
questions remain unanswered.
Applying these insights to the design of a synthetic gene switch demonstrates
the potential use of affinity-manipulation for synthetic biology, where the
construction of genetic circuits with tunable noise-resistance is of central
importance. In particular, our analysis highlights the potential utility of
heterodimerization to stabilize ultrasensitive switches against random
fluctuations. In practice, small ligand molecules may be employed to regulate
and tune the binding affinity of regulatory proteins, being either monomers or
dimers. Our results further suggest that the structure of the protein-
interaction network [losick08] may provide important insights on methods for
genome-level noise control in synthetic and natural systems.
## Methods
### Model construction
To evaluate the general role of protein oligomerization in a broad functional
context, we studied the two most common motifs found in genetic regulatory
circuits: positive autoregulation and the bistable switch. The reaction scheme
studied is summarized in Fig. 1, where the binding/unbinding reactions between
RNAp and promoter or between TF and operator are made explicit. Each distinct
binding status of DNA is associated with a unique transcription initiation
rate, and then the overall rate of mRNA synthesis is a weighted average of the
initiation rates for distinct binding status, where the weights are given by
the relative abundance of each configuration at equilibrium, determined by the
calculation of binding energy [gilman02]. Note that, neither binding
equilibrium nor empirical Hill-type cooperativity is assumed ad hoc. In
particular, we split the lumped transcription process into two separate
events, (i) isomerization of closed RNAp-promoter complex to its open form and
(ii) transcription elongation followed by termination. This is to reflect the
availability of the free promoter while the transcription machinery proceeds
along the coding sequence of a gene as soon as the promoter region is cleared
of the RNAp holoenzyme. Otherwise, the promoter would be inaccessible during a
whole transcription event, altering the random mRNA synthesis dynamics.
To realize the genetic toggle switch in a stochastic setting, we keep track of
the microscopic origin of cooperativity that gives rise to bistability. Among
various strategies, we employ multiple operator sites which have the same
binding affinity with the repressor. The resultant circuitry is, in essence,
two autogenous circuits, $A$ and $B$, which are connected through the active
form of their expressed proteins (the active form being either monomer or
dimer). The connection is implemented by allowing the active form of proteins
$A$ ($B$) to bind the operator sites of gene $B$ ($A$). In order to make the
interaction between the two genes repressive, unlike positive autogenous
circuits, $K_{31}$ and $K_{32}$ in Fig. 1 are now greater than $K_{30}$,
making the protein transcriptional repressors. For reasons of analytical
simplicity, we have studied the symmetric toggle switch, where the reaction
descriptions of each component follow those of the autogenous circuit. Again,
the quantitative characteristics of macromolecular binding-unbinding are
chosen based on the phage lambda-E. coli system. The only exception is related
to the multiple operator sites, where the second repressor binds an operator
site with higher binding affinity when the first site is already occupied by
the repressor protein [tian04]. We introduce three different dimerization
schemes. Three different dimerization scheme have been introduced: (i)
homodimerization with monomeric repressor, (ii) homodimerization with dimeric
repressor, and (iii) heterodimerization with monomeric repressor. By solving
for the stationary states of the deterministic rate equations, we could
identify the bistability region in parameter space to which all the model
systems under consideration belong.
### Stochastic simulation
While the deterministic rate equation approach or Langevin dynamics explicitly
gives the time-evolution of molecular concentration in the form of ordinary
differential equations, chemical master equation (CME) describes the evolution
of a molecular number state as a continuous-time jump Markov process. To
generate the statistically correct trajectories dictated by CME, we used the
Gillespie direct [gillespie77] and Next Reaction (Gibson-Bruck) [gibson00]
algorithms, both based on the exact chemical master equation. The Dizzy
package [dizzy] were used as the core engine of the simulations. To ensure
that calculations were undertaken in a steady state, we solved the
deterministic set of equations for steady state using every combination of
parameters investigated. We employed these deterministic steady-state
solutions as initial conditions for the stochastic simulations. For each model
system, we generated $10^{5}$ ensemble runs with identical initial conditions
and used the instantaneous protein copy number at a fixed time point $t=5000$
sec. To achieve high-quality power spectra in the low- and high-frequency
limits, we ran time courses ($\sim 10^{5}$ sec) with higher sampling frequency
(20 measure points per sec).
To calculate the average switching rate, we generated time series of minimum
length $3\cdot 10^{7}$ sec (approximately corresponding to 1 year). We
identify a state change in the toggle switch by monitoring the ratio of the
monomer and dimer abundance for the two protein species. In order to avoid
counting short-time fluctuations that do not correspond to a prolonged change
of the toggle state, we a applied sliding-window average to the time series,
using a window size of $1000$ sec.
## Authors’ contributions
CMG and EA designed the study. CMG performed the computations. CMG and EA
analyzed the results and wrote the paper. Both authors have read and approved
the final version of the paper.
## Acknowledgments
The authors thank Dr. Navid for thoughtful discussion and suggestions. This
work was performed under the auspices of the U. S. Department of Energy by
Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and
funded by the Laboratory Directed Research and Development Program (project
06-ERD-061) at LLNL.
Tables
Table 1 - Probability rates for positive autogenous circuit.
Category | Symbol | Reaction | Value (s-1) | Ref.
---|---|---|---|---
protein dimerization | $k_{1}$ | P1 $+$ P${}_{1}\rightarrow$ P2 | 0.001-0.1 | [sauer79, arkin98]
$q_{1}$ | P${}_{2}\rightarrow$ P1 $+$ P1 | 0.1-1
TF-operator int | $k_{20}$ | P2 $+$ D00 $\rightarrow$ D20 | 0.012 | [ackers82, hawley82, hawley85]
$q_{20}$ | D20 $\rightarrow$ P2 $+$ D00 | 0.9
$k_{21}$ | P1 $+$ D00 $\rightarrow$ D10 | 0.038
$q_{21}$ | D10 $\rightarrow$ P1 $+$ D00 | 0.3
$k_{22}$ | P1 $+$ D10 $\rightarrow$ D20 | 0.011
$q_{22}$ | D20 $\rightarrow$ P1 $+$ D10 | 0.9
RNAp-promoter int | $k_{30}$ | R $+$ D00 $\rightarrow$ D01 | 0.038 | [dehaseth98, ujvari96, shea85]
$q_{30}$ | D01 $\rightarrow$ R $+$ D00 | 0.3
$k_{31}$ | R $+$ D10 $\rightarrow$ D11 | 0.038†, 0.38‡
$q_{31}$ | D11 $\rightarrow$ R $+$ D10 | 0.3†, 0.03‡
$k_{32}$ | R $+$ D20 $\rightarrow$ D21 | 0.38∗†
$q_{32}$ | D21 $\rightarrow$ R $+$ D20 | 0.03∗†
Isomerization | $v$ | Dx1 $\rightarrow$ C $+$ Dx0 | 0.0078 | [hawley85]
tsx-tsl elongation & decay | $\alpha$ | C $\rightarrow$ M $+$ R | 0.03 | [alberts02, lewin04]
$\beta$ | M $\rightarrow$ P1 $+$ M | 0.044
$\gamma_{0}$ | M $\rightarrow\varnothing$ | 0.0039
$\gamma_{1}$ | P${}_{1}\rightarrow\varnothing$ | 7$\times$10-4
$\gamma_{2}$ | P${}_{2}\rightarrow\varnothing$ | 0.7-3.5$\times$10-4
Kinetic rates for the positive autogenous circuit. Experimentally available
rates are all taken from lambda phage-E. coli complex. The values with
superscript correspond to the circuit topologies DA1 (*), DA2 ($\dagger$), and
DA3 ($\ddagger$) in Fig. 1.
Table 2 - Relative Fano factors of protein abundance distributions
$K_{1}$ (nM) | $\gamma_{2}=\gamma_{1}/10$ | $\gamma_{2}=\gamma_{1}/2$
---|---|---
monomer | dimer | monomer | dimer
1 | 0.127 | 0.809 | 0.132 | 0.679
20 | 0.209 | 0.936 | 0.230 | 0.716
500 | 0.866 | 0.478 | 0.826 | 0.426
The Fano factor of protein abundance distribution for the autogenous circuits
(topology DA1), relative to that of the monomer-only (MO) circuit, $8.729$.
Figures
Figure 1. Schematic of model autoregulation gene circuit.
The DNA binding status is indicated by Dxy, where x corresponds to the
operator region (empty=0, monomer=1, dimer=2), and y to the promoter region
(empty=0, RNA polymerase bound=1). C represents the open complex of DNA-RNAp
holoenzyme with the promoter sequence just cleared of RNAp and is subject to
transcription elongation. Finally, M, P1 and P2 correspond to mRNA, protein
monomer, and dimer, respectively. The network topologies can be grouped into
two classes, monomer-only (MO) or dimer-allowed (DA) circuits. We have studied
DA1 (red lines), which only allows the dimer to bind with the DNA-operator
sequence, DA2 (green) with sequential binding of monomers on the DNA, and DA3
(blue), which shares protein-DNA binding kinetics with MO while allowing
dimerization in the cytosol. Note that for topology DA2, we have chosen
$K_{31}=K_{30}$ (see text for details) We have assumed cells to be in the
exponential growth phase and the number of RNAp (R) constant.
Figure 2. Ten independent time courses of the abundance of protein monomers in
the (positive) autoregulatory circuit.
The availability of a cytosolic dimer state (red, using circuit topology DA1)
significantly reduces the copy-number fluctuations of the monomer compared to
the monomer-only (MO) circuit (blue). All corresponding MO and DA1 parameters
have the same values. In the ensuing simulations initial conditions are chosen
to be the steady state solution of the corresponding deterministic rate
equation so that the transient behavior should be minimized.
Figure 3. Stationary state distribution of monomer (black) and dimer (orange)
protein abundance in the positive autogenous circuits.
The left (right) column corresponds to a ratio of the dimer and monomer decay
rates of $\gamma_{2}/\gamma_{1}=1/10$ ($\gamma_{2}/\gamma_{1}=1/2$). The
molecular copy numbers are collected at a fixed time interval ($5\cdot 10^{3}$
sec) after the steady state has been reached. Here $K_{1}\equiv q_{1}/k_{1}$
is the dissociation constant of the protein dimer. As the binding equilibrium
is shifted towards the dimer state (decreasing $K_{1}$), the noise level is
monotonically reduced (see Table 2). Note that the prolonged protein lifetime
due to the complex formation (left column) affects the noise level.
Figure 4. Power spectral density (PSD) of fluctuations in protein abundance.
The PSD of the MO circuit clearly displays a power-law behavior. All other
model systems with an available cytosolic protein dimer state (DA1 shown here)
develop a plateau in the mid-frequency region regardless of the model details
(see Supplementary Information). As the dimer binding affinity increases, the
noise level is further reduced. We have included the MO result in the dimer
panel (right) for reference. Datasets with solid (empty) symbols correspond to
$\gamma_{2}/\gamma_{1}=1/10$ ($\gamma_{2}/\gamma_{1}=1/2$).
Figure 5. Sample time series of monomer and dimer copy numbers in genetic
toggle switch.
(a) MTF circuit, where monomer is the functional form of the repressor. (b)
DTF circuit, where dimer is the functional form of the repressor. The left
(right) column shows the number of the two monomer molecules $A$ and $B$
(dimers $AA$ and $BB$), and the initial state is always with species $A$ (red)
in high abundance. Note that the switching frequency depends on the binding
affinity of protein dimer.
Figure 6. Distribution of monomer abundance differences between protein
species $A$ and $B$.
The asymmetry with respect to the zero axis is due to the choice of initial
state (species $A$ high) and the finite time span of simulations.
Figure 7. Random switching rates of genetic toggle switches.
Ordinate is the ratio of the random switching rates of various toggle switches
to that of the monomer-only (MO) circuit, $7.5\times 10^{-6}$/hour. MTF,
monomeric transcription factor; DTF, dimeric transcription factor; Het-MTF,
monomeric transcription factor with deactivated heterodimer state.
Additional Files
Additional file 1. Supplementary results for the positive autoregulatory
circuits with various topology.
Protein abundance distribution and power spectral density of autogenous DA2
and DA3 circuits are presented.
|
arxiv-papers
| 2008-12-04T01:02:15 |
2024-09-04T02:48:59.202428
|
{
"license": "Public Domain",
"authors": "C.-M. Ghim, E. Almaas",
"submitter": "C.-M. Ghim",
"url": "https://arxiv.org/abs/0812.0841"
}
|
0812.0890
|
# Wakimoto realization of Drinfeld current for
the elliptic quantum algebra $U_{q,p}(\widehat{sl}_{3})$
###### Abstract
We study a free field realization of the elliptic quantum algebra
$U_{q,p}(\widehat{sl_{3}})$ for arbitrary level $k$. We give the free field
realization of elliptic analogue of Drinfeld current associated with
$U_{q,p}(\widehat{sl_{3}})$ for arbitrary level $k$. In the limit $p\to 0,q\to
1$ our realization reproduces Wakimoto realization for the affine Lie algebra
$\widehat{sl_{3}}$.
PACS numbers : 02.20.Sv, 02.20.Uw, 02.30.Ik
Takeo KOJIMA
Department of Mathematics, College of Science and Technology, Nihon
University,
Surugadai, Chiyoda-ku, Tokyo 101-0062, JAPAN
## 1 Introduction
The elliptic quantum group has been proposed in papers [1, 2, 3, 4, 5]. There
are two types of elliptic quantum groups, the vertex type ${\cal
A}_{q,p}(\widehat{sl_{N}})$ and the face type ${\cal B}_{q,\lambda}({g})$,
where ${g}$ is a Kac-Moody algebra associated with a symmetrizable Cartan
matrix. The elliptic quantum groups have the structure of quasi-triangular
quasi-Hopf algebras introduced by V.Drinfeld [6]. H.Konno [7] introduced the
elliptic quantum algebra $U_{q,p}(\widehat{sl_{2}})$ as an algebra of the
screening currents of the extended deformed Virasoro algebra in terms of the
fusion SOS model [8]. M.Jimbo, H.Konno, S.Odake, J.Shiraishi [9] continued to
study the elliptic quantum algebra $U_{q,p}(\widehat{sl_{2}})$. They
constructed the elliptic alnalogue of Drinfeld currents and identified
$U_{q,p}(\widehat{sl_{2}})$ with the tensor product of ${\cal
B}_{q,\lambda}(\widehat{sl_{2}})$ and a Heisenberg algebra ${\cal H}$. The
elliptic quantum group ${\cal B}_{q,\lambda}(\widehat{sl_{2}})$ is a quasi-
Hopf algebra while the elliptic algebra $U_{q,p}(\widehat{sl_{2}})$ is not.
The intertwining relation of the vertex operator of ${\cal
B}_{q,\lambda}(\widehat{sl_{2}})$ is based on the quasi-Hopf structure of
${\cal B}_{q,\lambda}(\widehat{sl_{2}})$. By the above isomorphism
$U_{q,p}(\widehat{sl_{2}})\simeq{\cal
B}_{q,\lambda}(\widehat{sl_{2}})\otimes{\cal H}$, we can understand
”intertwining relation” of the vertex operator for the elliptic algebra
$U_{q,p}(\widehat{sl_{2}})$. Along the above scheme the elliptic analogue of
Drinfeld current of $U_{q,p}(\widehat{sl_{2}})$ is extended to those of
$U_{q,p}({g})$ for non-twisted affine Lie algebra ${g}$ [9, 10]. In this paper
we are interested in higher-rank generalization of level $k$ free field
realization of the elliptic quantum algebra. For the elliptic algebra
$U_{q,p}(\widehat{sl_{2}})$, there exist two kind of free field realizations
for arbitrary level $k$, the one is parafermion realization [7, 9], the other
is Wakimoto realization [16]. In this paper we are interested in the higher-
rank generalization of Wakimoto realization of $U_{q,p}(\widehat{sl_{2}})$. We
construct level $k$ free field realization of Drinfeld current associated with
the elliptic algebra $U_{q,p}(\widehat{sl_{3}})$. This gives the first example
of arbitrary level free field realization of the higher-rank elliptic algebra.
This free field realization can be applied for construction of the integrals
of motion for the elliptic algebra $U_{q,p}(\widehat{sl_{3}})$. For this
purpose, see references [17, 18, 19].
The organization of this paper is as follows. In section 2 we set the notation
and introduce bosons. In section 3 we review the level $k$ free field
realization of the quantum group $U_{q}(\widehat{sl_{3}})$ [15]. In section 4
we give the level $k$ free field realization of the elliptic quantum algebra
$U_{q,p}(\widehat{sl_{3}})$. In appendix we summarize the normal ordering of
the basic operators.
## 2 Boson
The purpose of this section is to set up the basic notation and to introduce
the boson. In this paper we fix three parameters $q,k,r\in{\mathbb{C}}$. Let
us set $r^{*}=r-k$. We assume $k\neq 0,-3$ and ${\rm Re}(r)>0$, ${\rm
Re}(r^{*})>0$. We assume $q$ is a generic with $|q|<1,q\neq 0$. Let us set a
pair of parameters $p$ and $p^{*}$ by
$\displaystyle p=q^{2r},~{}~{}p^{*}=q^{2r^{*}}.$
We use the standard symbol of $q$-integer $[n]$ by
$\displaystyle[n]=\frac{q^{n}-q^{-n}}{q-q^{-1}}.$
Let us set the elliptic theta function $\Theta_{p}(z)$ by
$\displaystyle\Theta_{p}(z)=(z;p)_{\infty}(p/z;p)_{\infty}(p;p)_{\infty},$
$\displaystyle(z;p)_{\infty}=\prod_{n=0}^{\infty}(1-p^{n}z).$
It is convenient to work with the additive notation. We use the
parametrization
$\displaystyle q$ $\displaystyle=$ $\displaystyle e^{-\pi\sqrt{-1}/r\tau},$
$\displaystyle p$ $\displaystyle=$ $\displaystyle
e^{-2\pi\sqrt{-1}/\tau},~{}~{}~{}p^{*}=e^{-2\pi\sqrt{-1}/\tau^{*}},~{}~{}(r\tau=r^{*}\tau^{*}),$
$\displaystyle z$ $\displaystyle=$ $\displaystyle q^{2u}.$
Let us set Jacobi elliptic theta function $[u]_{r},[u]_{r^{*}}$ by
$\displaystyle~{}[u]_{r}$ $\displaystyle=$ $\displaystyle
q^{\frac{u^{2}}{r}-u}\frac{\Theta_{p}(z)}{(p;p)_{\infty}^{3}},~{}~{}~{}[u]_{r^{*}}=q^{\frac{u^{2}}{r^{*}}-u}\frac{\Theta_{p^{*}}(z)}{(p^{*};p^{*})_{\infty}^{3}}.$
The function $[u]_{r}$ has a zero at $u=0$, enjoys the quasi-periodicity
property
$\displaystyle~{}[u+r]_{r}=-[u]_{r},~{}~{}~{}~{}[u+r\tau]_{r}=-e^{-\pi\sqrt{-1}\tau-\frac{2\pi\sqrt{-1}u}{r}}[u]_{r}.$
Let us set the delta-function $\delta(z)$ as formal power series.
$\displaystyle\delta(z)=\sum_{n\in{\mathbb{Z}}}z^{n}.$
Following [15] we introduce free bosons
$a_{n}^{1},a_{n}^{2},b_{n}^{1},b_{n}^{2},b_{n}^{3},c_{n}^{1},c_{n}^{2},c_{n}^{3},(n\in{\mathbb{Z}}_{\neq
0})$.
$\displaystyle~{}[a_{n}^{i},a_{m}^{j}]$ $\displaystyle=$
$\displaystyle\frac{[(k+3)n][A_{i,j}n]}{n}\delta_{n+m,0},~{}~{}[p_{a}^{i},q_{a}^{j}]=(k+3)A_{i,j},~{}~{}(i,j=1,2),$
(2.1) $\displaystyle~{}[b_{n}^{i},b_{m}^{j}]$ $\displaystyle=$
$\displaystyle-\frac{[n]^{2}}{n}\delta_{i,j}\delta_{n+m,0},~{}~{}[p_{b}^{i},q_{b}^{j}]=-\delta_{i,j},~{}~{}(i,j=1,2,3),$
(2.2) $\displaystyle~{}[c_{n}^{i},c_{m}^{j}]$ $\displaystyle=$
$\displaystyle\frac{[n]^{2}}{n}\delta_{i,j}\delta_{n+m,0},~{}~{}[p_{c}^{i},q_{c}^{j}]=\delta_{i,j},~{}~{}(i,j=1,2,3).$
(2.3)
Here we have used Cartan matrix $\left(\begin{array}[]{cc}A_{11}&A_{12}\\\
A_{21}&A_{22}\end{array}\right)=\left(\begin{array}[]{cc}2&-1\\\
-1&2\end{array}\right)$.
For parameters
$a_{1},a_{2},b_{1},b_{2},b_{3},c_{1},c_{2},c_{3}\in{\mathbb{R}}$, we set the
vacuum vector $|a,b,c\rangle$ of the Fock space ${\cal
F}_{a_{1}a_{2}b_{1}b_{2}b_{3}c_{1}c_{2}c_{3}}$ as following.
$\displaystyle
a_{n}^{i}|a,b,c\rangle=b_{n}^{j}|a,b,c\rangle=c_{n}^{j}|a,b,c\rangle=0,~{}~{}(i=1,2;j=1,2,3),$
(2.4) $\displaystyle
p_{a}^{i}|a,b,c\rangle=a_{i}|a,b,c\rangle,~{}p_{b}^{j}|a,b,c\rangle=b_{j}|a,b,c\rangle,~{}p_{c}^{j}|a,b,c\rangle=c_{j}|a,b,c\rangle,$
$\displaystyle~{}~{}(i=1,2;j=1,2,3;n>0).$ (2.5)
The Fock space ${\cal F}_{a_{1}a_{2}b_{1}b_{2}b_{3}c_{1}c_{2}c_{3}}$ is
generated by bosons
$a_{-n}^{1},a_{-n}^{2},b_{-n}^{1},b_{-n}^{2},b_{-n}^{3},c_{-n}^{1},c_{-n}^{2},c_{-n}^{3}$
for $n\in{\mathbb{N}}_{\neq 0}$. The dual Fock space ${\cal
F}_{a_{1}a_{2}b_{1}b_{2}b_{3}c_{1}c_{2}c_{3}}^{*}$ is defined as the same
manner. In this paper we construct the elliptic analogue of Drinfeld current
for $U_{q,p}(\widehat{sl_{3}})$ by these bosons
$a_{n}^{i},b_{n}^{j},c_{n}^{j}$ acting on the Fock space.
## 3 Free Field Realization of $U_{q}(\widehat{sl_{3}})$
The purpose of this section is to give the free field realization of the
quantum affine algebra $U_{q}(\widehat{sl_{3}})$. We give a review of Wakimoto
realization of $U_{q}(\widehat{sl_{3}})$ [15]. Let us set the bosonic
operators $a_{\pm}^{i}(z),b_{\pm}^{i}(z)$, $\gamma^{i}(z),\beta_{s}^{i}(z)$ by
$\displaystyle a_{\pm}^{i}(z)$ $\displaystyle=$
$\displaystyle\pm(q-q^{-1})\sum_{n>0}a_{\pm n}^{i}z^{\mp n}\pm p_{a}^{i}{\rm
log}q,~{}~{}(i=1,2),$ (3.1) $\displaystyle b_{\pm}^{i}(z)$ $\displaystyle=$
$\displaystyle\pm(q-q^{-1})\sum_{n>0}b_{\pm n}^{i}z^{\mp n}\pm p_{b}^{i}{\rm
log}q,~{}~{}(i=1,2,3),$ (3.2) $\displaystyle b^{i}(z)$ $\displaystyle=$
$\displaystyle-\sum_{n\neq
0}\frac{b_{n}^{i}}{[n]}z^{-n}+q_{b}^{i}+p_{b}^{i}{\rm log}z,~{}~{}(i=1,2,3),$
(3.3) $\displaystyle c^{i}(z)$ $\displaystyle=$ $\displaystyle-\sum_{n\neq
0}\frac{c_{n}^{i}}{[n]}z^{-n}+q_{c}^{i}+p_{c}^{i}{\rm log}z,~{}~{}(i=1,2,3),$
(3.4) $\displaystyle\gamma^{i}(z)$ $\displaystyle=$ $\displaystyle-\sum_{n\neq
0}\frac{(b+c)_{n}^{i}}{[n]}z^{-n}+(q_{b}^{i}+q_{c}^{i})+(p_{b}^{i}+p_{c}^{i}){\rm
log}(-z),~{}~{}(i=1,2,3),$ (3.5) $\displaystyle\beta_{1}^{i}(z)$
$\displaystyle=$ $\displaystyle
b_{+}^{i}(z)-(b^{i}+c^{i})(qz),~{}\beta_{2}^{i}(z)=b_{-}^{i}(z)-(b^{i}+c^{i})(q^{-1}z),~{}~{}(i=1,2,3),$
(3.6) $\displaystyle\beta_{1}^{i}(z)$ $\displaystyle=$ $\displaystyle
b_{+}^{i}(z)+(b^{i}+c^{i})(qz),~{}\beta_{2}^{i}(z)=b_{-}^{i}(z)+(b^{i}+c^{i})(q^{-1}z),~{}~{}(i=1,2,3).$
(3.7)
We give a free field realiztaion of Drinfeld current for
$U_{q}(\widehat{sl_{3}})$.
###### Definition 3.1
We define the bosonic operators
$e_{1}^{+}(z),e_{2}^{+}(z),e_{1}^{-}(z),e_{2}^{-}(z)$ by
$\displaystyle e_{1}^{+}(z)$ $\displaystyle=$
$\displaystyle\frac{-1}{(q-q^{-1})z}(e_{1}^{+,1}(z)-e_{1}^{+,2}(z)),$ (3.8)
$\displaystyle e_{2}^{+}(z)$ $\displaystyle=$
$\displaystyle\frac{-1}{(q-q^{-1})z}(e_{2}^{+,1}(z)-e_{2}^{+,2}(z)+e_{2}^{+,3}(z)-e_{2}^{+,4}(z)),$
(3.9) $\displaystyle e_{1}^{-}(z)$ $\displaystyle=$
$\displaystyle\frac{-1}{(q-q^{-1})z}(e_{1}^{-,1}(z)-e_{1}^{-,2}(z)-e_{1}^{-,3}(z)+e_{1}^{-,4}(z)),$
(3.10) $\displaystyle e_{2}^{-}(z)$ $\displaystyle=$
$\displaystyle\frac{-1}{(q-q^{-1})z}(e_{2}^{-,1}(z)-e_{2}^{-,2}(z)+e_{2}^{-,3}(z)-e_{2}^{-,4}(z)).$
(3.11) $\displaystyle\psi^{\pm}_{1}(z)$ $\displaystyle=$
$\displaystyle:\exp\left(b_{\pm}^{1}(q^{\pm
k}z)+b_{\pm}^{1}(q^{\pm(k+2)}z)+b_{\pm}^{2}(q^{\pm(k+3)}z)-b_{\pm}^{3}(q^{\pm(k+2)}z)+a_{\pm}^{1}(q^{\pm\frac{k+3}{2}}z)\right):,$
$\displaystyle\psi^{\pm}_{2}(z)$ $\displaystyle=$
$\displaystyle:\exp\left(-b_{\pm}^{1}(q^{\pm(k+1)}z)+b_{\pm}^{2}(q^{\pm
k}z)+b_{\pm}^{3}(q^{\pm(k+1)}z)+b_{\pm}^{3}(q^{\pm(k+3)}z)+a_{\pm}^{2}(q^{\pm\frac{k+3}{2}}z)\right):,$
Here we have set
$\displaystyle e_{1}^{+,1}(z)$ $\displaystyle=$
$\displaystyle:\exp\left(\beta_{1}^{1}(z)\right):,$ (3.14) $\displaystyle
e_{1}^{+,2}(z)$ $\displaystyle=$
$\displaystyle:\exp\left(\beta_{2}^{1}(z)\right):,$ (3.15) $\displaystyle
e_{2}^{+,1}(z)$ $\displaystyle=$
$\displaystyle:\exp\left(\gamma^{1}(z)+\beta_{1}^{2}(z)\right):,$ (3.16)
$\displaystyle e_{2}^{+,2}(z)$ $\displaystyle=$
$\displaystyle:\exp\left(\gamma^{1}(z)+\beta_{2}^{2}(z)\right):,$ (3.17)
$\displaystyle e_{2}^{+,3}(z)$ $\displaystyle=$
$\displaystyle:\exp\left(\beta_{1}^{3}(qz)+b_{+}^{2}(z)-b_{+}^{1}(qz)\right):,$
(3.18) $\displaystyle e_{2}^{+,4}(z)$ $\displaystyle=$
$\displaystyle:\exp\left(\beta_{2}^{3}(qz)+b_{+}^{2}(z)-b_{+}^{1}(qz)\right):,$
(3.19) $\displaystyle e_{1}^{-,1}(z)$ $\displaystyle=$
$\displaystyle:\exp\left(\beta_{4}^{1}(q^{-k-2}z)+b_{-}^{2}(q^{-k-3}z)-b_{-}^{3}(q^{-k-2}z)+a_{-}^{1}(q^{-\frac{k+3}{2}}z)\right):,$
(3.20) $\displaystyle e_{1}^{-,2}(z)$ $\displaystyle=$
$\displaystyle:\exp\left(\beta_{3}^{1}(q^{k+2}z)+b_{+}^{2}(q^{k+3}z)-b_{+}^{3}(q^{k+2}z)+a_{+}^{1}(q^{\frac{k+3}{2}}z)\right):,$
(3.21) $\displaystyle e_{1}^{-,3}(z)$ $\displaystyle=$
$\displaystyle:\exp\left(\gamma^{2}(q^{k+2}z)+\beta_{1}^{3}(q^{k+2}z)+b_{+}^{2}(q^{k+3}z)-b_{+}^{3}(q^{k+2}z)+a_{+}^{1}(q^{\frac{k+3}{2}}z\right):,$
(3.22) $\displaystyle e_{1}^{-,4}(z)$ $\displaystyle=$
$\displaystyle:\exp\left(\gamma^{2}(q^{k+2}z)+\beta_{2}^{3}(q^{k+2}z)+b_{+}^{2}(q^{k+3}z)-b_{+}^{3}(q^{k+2}z)+a_{+}^{1}(q^{\frac{k+3}{2}}z\right):,$
(3.23) $\displaystyle e_{2}^{-,1}(z)$ $\displaystyle=$
$\displaystyle:\exp\left(\gamma^{2}(q^{-k-1}z)-\beta_{3}^{1}(q^{-k-1}z)+2b_{-}^{3}(q^{-k-1}z)+a_{-}^{2}(q^{-\frac{k+3}{2}}z)\right):,$
(3.24) $\displaystyle e_{2}^{-,2}(z)$ $\displaystyle=$
$\displaystyle:\exp\left(\gamma^{2}(q^{-k-1}z)-\beta_{4}^{1}(q^{-k-1}z)+2b_{-}^{3}(q^{-k-1}z)+a_{-}^{2}(q^{-\frac{k+3}{2}}z)\right):,$
(3.25) $\displaystyle e_{2}^{-,3}(z)$ $\displaystyle=$
$\displaystyle:\exp\left(\beta_{4}^{3}(q^{-k-3}z)+a_{-}^{2}(q^{-\frac{k+3}{2}}z)\right):,$
(3.26) $\displaystyle e_{2}^{-,4}(z)$ $\displaystyle=$
$\displaystyle:\exp\left(\beta_{3}^{3}(q^{k+3}z)+a_{+}^{2}(q^{\frac{k+3}{2}}z)\right):.$
(3.27)
Here the symbol $:{\cal O}:$ represents the normal ordering of ${\cal O}$. For
example we have
$\displaystyle:b_{k}b_{l}:=\left\\{\begin{array}[]{cc}b_{k}^{i}b_{l}^{i},&k<0\\\
b_{l}^{i}b_{k}^{i},&k>0.\end{array}\right.~{}~{}~{}:p_{b}^{i}q_{b}^{i}:=:q_{b}^{i}p_{b}^{i}:=q_{b}^{i}p_{b}^{i}.$
(3.30)
###### Theorem 3.1
[15] The bosonic operators $e_{i}^{\pm}(z)$, $\psi_{i}^{\pm}(z)$, $(i=1,2)$
satisfy the following commutation relations.
$\displaystyle(z_{1}-q^{A_{i,j}}z_{2})e_{i}^{+}(z_{1})e_{j}^{+}(z_{2})$
$\displaystyle=$
$\displaystyle(q^{A_{i,j}}z_{1}-z_{2})e_{j}^{+}(z_{2})e_{i}^{+}(z_{1}),$
(3.31)
$\displaystyle(z_{1}-q^{-A_{i,j}}z_{2})e_{i}^{-}(z_{1})e_{j}^{-}(z_{2})$
$\displaystyle=$
$\displaystyle(q^{-A_{i,j}}z_{1}-z_{2})e_{j}^{-}(z_{2})e_{i}^{-}(z_{1}),$
(3.32) $\displaystyle~{}[\psi_{i}^{\pm}(z_{1}),\psi_{j}^{\pm}(z_{2})]$
$\displaystyle=$ $\displaystyle 0,$ (3.33)
$\displaystyle(z_{1}-q^{A_{i,j}-k}z_{2})(z_{1}-q^{-A_{i,j}+k}z_{2})\psi_{i}^{\pm}(z_{1})\psi_{j}^{\mp}(z_{2})$
(3.34) $\displaystyle=$
$\displaystyle(z_{1}-q^{A_{i,j}+k}z_{2})(z_{1}-q^{-A_{i,j}-k}z_{2})\psi_{j}^{\mp}(z_{2})\psi_{i}^{\pm}(z_{1}),$
$\displaystyle(z_{1}-q^{\pm(A_{i,j}-\frac{k}{2})}z_{2})\psi_{i}^{+}(z_{1})e^{\pm}_{j}(z_{2})$
$\displaystyle=$ $\displaystyle(q^{\pm
A_{i,j}}z_{1}-q^{\mp\frac{k}{2}}z_{2})e^{\pm}_{j}(z_{2})\psi_{i}^{+}(z_{1}),$
(3.35)
$\displaystyle(z_{1}-q^{\pm(A_{i,j}-\frac{k}{2})}z_{2})e^{\pm}_{i}(z_{1})\psi_{j}^{-}(z_{2})$
$\displaystyle=$ $\displaystyle(q^{\pm
A_{i,j}}z_{1}-q^{\mp\frac{k}{2}}z_{2})\psi_{j}^{-}(z_{2})e^{\pm}_{i}(z_{1}),$
(3.36)
$\displaystyle\left\\{e_{i}^{\pm}(z_{1})e_{i}^{\pm}(z_{2})e_{j}^{\pm}(z_{3})-(q+q^{-1})e_{i}^{\pm}(z_{1})e_{j}^{\pm}(z_{3})e_{j}^{\pm}(z_{2})+e_{i}^{\pm}(z_{3})e_{i}^{\pm}(z_{1})e_{j}^{\pm}(z_{2})\right\\}$
$\displaystyle+\left\\{z_{1}\leftrightarrow z_{2}\right\\}=0,~{}~{}{\rm
for}~{}~{}(i\neq j),$ (3.37)
$\displaystyle[e_{i}^{+}(z_{1}),e_{j}^{-}(z_{2})]=\frac{\delta_{i,j}}{(q-q^{-1})z_{1}z_{2}}\left(\delta\left(q^{-k}\frac{z_{1}}{z_{2}}\right)\psi_{i}^{+}(q^{-\frac{k}{2}}z_{1})-\delta\left(q^{k}\frac{z_{1}}{z_{2}}\right)\psi_{i}^{-}(q^{-\frac{k}{2}}z_{2})\right).$
Hence $e_{i}^{\pm}(z),\psi_{i}^{\pm}(z)$ give level $k$ free field realization
of $U_{q}(\widehat{sl_{3}})$.
## 4 Free Field Realization of $U_{q,p}(\widehat{sl_{3}})$
The purpose of this section is to give a free field realization of the
elliptic analogue of Drinfeld current for $U_{q,p}(\widehat{sl_{3}})$ with
arbitrary level $k\neq 0,-3$. Let us set the bosonic operators ${\cal
B}_{\pm}^{*i}(z),{\cal B}_{\pm}^{i}(z),(i=1,2,3)$, ${\cal A}^{*i}(z),{\cal
A}^{i}(z),(i=1,2)$ by
$\displaystyle{\cal B}_{\pm}^{*i}(z)$ $\displaystyle=$
$\displaystyle\exp\left(\pm\sum_{n>0}\frac{b_{-n}^{i}}{[r^{*}n]}z^{n}\right),~{}~{}(i=1,2,3),$
(4.1) $\displaystyle{\cal B}_{\pm}^{i}(z)$ $\displaystyle=$
$\displaystyle\exp\left(\pm\sum_{n>0}\frac{b_{n}^{i}}{[rn]}z^{-n}\right),~{}~{}(i=1,2,3),$
(4.2) $\displaystyle{\cal A}^{i*}(z)$ $\displaystyle=$
$\displaystyle\exp\left(\sum_{n>0}\frac{a_{-n}^{i}}{[r^{*}n]}z^{n}\right),~{}~{}(i=1,2),$
(4.3) $\displaystyle{\cal A}^{i}(z)$ $\displaystyle=$
$\displaystyle\exp\left(-\sum_{n>0}\frac{a_{n}^{i}}{[rn]}z^{-n}\right),~{}~{}(i=1,2).$
(4.4)
###### Definition 4.1
Let us set the bosonic operators $e_{i}(z),f_{i}(z),\Psi_{i}^{\pm}(z),(i=1,2)$
by
$\displaystyle e_{i}(z)={U}^{*i}(z)e_{i}^{+}(z),~{}~{}(i=1,2),$ (4.5)
$\displaystyle f_{i}(z)=e_{i}^{-}(z){U}^{i}(z),~{}~{}(i=1,2),$ (4.6)
$\displaystyle\Psi_{i}^{+}(z)=U^{*i}(q^{\frac{k}{2}}z)\psi_{i}^{+}(z)U^{i}(q^{-\frac{k}{2}}z),~{}~{}(i=1,2),$
(4.7)
$\displaystyle\Psi_{i}^{+}(z)=U^{*i}(q^{-\frac{k}{2}}z)\psi_{i}^{-}(z)U^{i}(q^{\frac{k}{2}}z),~{}~{}(i=1,2).$
(4.8)
Here we have set
$\displaystyle{U}^{*1}(z)$ $\displaystyle=$ $\displaystyle{\cal
B}_{+}^{*1}(q^{r^{*}}z){\cal B}_{+}^{*1}(q^{r^{*}-2}z){\cal
B}_{+}^{*2}(q^{r^{*}-3}z){\cal B}_{-}^{*3}(q^{r^{*}-2}z){\cal
A}^{*1}(q^{r^{*}+\frac{k-3}{2}}z),$ (4.9) $\displaystyle{U}^{*2}(z)$
$\displaystyle=$ $\displaystyle{\cal B}_{+}^{*3}(q^{r^{*}-3}z){\cal
B}_{+}^{*3}(q^{r^{*}-1}z){\cal B}_{+}^{*2}(q^{r^{*}}z){\cal
B}_{-}^{*1}(q^{r^{*}-1}z){\cal A}^{*2}(q^{r^{*}+\frac{k-3}{2}}z),$ (4.10)
$\displaystyle{U}^{1}(z)$ $\displaystyle=$ $\displaystyle{\cal
B}_{-}^{1}(q^{-r^{*}}z){\cal B}_{-}^{1}(q^{-r^{*}+2}z){\cal
B}_{-}^{2}(q^{-r^{*}+3}z){\cal B}_{+}^{3}(q^{-r^{*}+2}z){\cal
A}^{1}(q^{-r^{*}-\frac{k-3}{2}}z),$ (4.11) $\displaystyle{U}^{2}(z)$
$\displaystyle=$ $\displaystyle{\cal B}_{-}^{3}(q^{-r^{*}+1}z){\cal
B}_{-}^{3}(q^{-r^{*}+1}z){\cal B}_{-}^{2}(q^{-r^{*}}z){\cal
B}_{+}^{1}(q^{-r^{*}+1}z){\cal A}^{2}(q^{-r^{*}-\frac{k-3}{2}}z).$ (4.12)
The above free field realization of the twistors $U^{*i}(z),U^{i}(z)$,
$(i=1,2)$ is the main result of this paper.
###### Proposition 4.1
The bosonic operators $e_{i}(z),f_{i}(z),\Psi_{i}^{\pm}(z)$, $(i=1,2)$ satisfy
the following commutation relations.
$\displaystyle e_{i}(z_{1})e_{j}(z_{2})$ $\displaystyle=$ $\displaystyle
q^{-A_{i,j}}\frac{\Theta_{p^{*}}(q^{A_{i,j}}z_{1}/z_{2})}{\Theta_{p^{*}}(q^{-A_{i,j}}z_{1}/z_{2})}e_{j}(z_{2})e_{i}(z_{1}),$
(4.13) $\displaystyle f_{i}(z_{1})f_{j}(z_{2})$ $\displaystyle=$
$\displaystyle
q^{A_{i,j}}\frac{\Theta_{p}(q^{-A_{i,j}}z_{1}/z_{2})}{\Theta_{p}(q^{A_{i,j}}z_{1}/z_{2})}f_{j}(z_{2})f_{i}(z_{1}),$
(4.14) $\displaystyle\Psi_{i}^{\pm}(z_{1})\Psi_{j}^{\pm}(z_{2})$
$\displaystyle=$
$\displaystyle\frac{\Theta_{p}(q^{-A_{i,j}}z_{1}/z_{2})\Theta_{p^{*}}(q^{A_{i,j}}z_{1}/z_{2})}{\Theta_{p}(q^{A_{i,j}}z_{1}/z_{2})\Theta_{p^{*}}(q^{-A_{i,j}}z_{1}/z_{2})}\Psi_{j}^{\pm}(z_{2})\Psi_{i}^{\pm}(z_{1}),$
(4.15) $\displaystyle\Psi_{i}^{\pm}(z_{1})\Psi_{j}^{\mp}(z_{2})$
$\displaystyle=$
$\displaystyle\frac{\Theta_{p}(pq^{-A_{i,j}-k}z_{1}/z_{2})\Theta_{p^{*}}(p^{*}q^{A_{i,j}+k}z_{1}/z_{2})}{\Theta_{p}(pq^{A_{i,j}-k}z_{1}/z_{2})\Theta_{p^{*}}(p^{*}q^{-A_{i,j}+k}z_{1}/z_{2})}\Psi_{j}^{\mp}(z_{2})\Psi_{i}^{\pm}(z_{1}),$
(4.16) $\displaystyle\Psi_{i}^{\pm}(z_{1})e_{j}(z_{2})$ $\displaystyle=$
$\displaystyle\frac{\Theta_{p^{*}}(q^{A_{i,j}\pm\frac{k}{2}}z_{1}/z_{2})}{\Theta_{p^{*}}(q^{-A_{i,j}\pm\frac{k}{2}}z_{1}/z_{2})}e_{j}(z_{2})\Psi_{i}^{\pm}(z_{1}),$
(4.17) $\displaystyle\Psi_{i}^{\pm}(z_{1})f_{j}(z_{2})$ $\displaystyle=$
$\displaystyle\frac{\Theta_{p^{*}}(q^{-A_{i,j}\mp\frac{k}{2}}z_{1}/z_{2})}{\Theta_{p^{*}}(q^{A_{i,j}\mp\frac{k}{2}}z_{1}/z_{2})}e_{j}(z_{2})\Psi_{i}^{\pm}(z_{1}),$
(4.18)
$\displaystyle~{}[e_{i}(z_{1}),f_{j}(z_{2})]=\frac{\delta_{i,j}}{(q-q^{-1})z_{1}z_{2}}\left(\delta\left(q^{-k}\frac{z_{1}}{z_{2}}\right)\Psi_{i}^{+}(q^{-k/2}z_{1})-\delta\left(q^{k}\frac{z_{1}}{z_{2}}\right)\Psi_{i}^{-}(q^{-k/2}z_{2})\right),$
$\displaystyle(i\neq j).~{}~{}~{}~{}$ (4.19)
We introduce the Heisenberg algebra ${\cal H}$ generated by the following
$P_{i},Q_{i}$, $(i=1,2)$.
$\displaystyle~{}[P_{i},Q_{j}]=\frac{A_{i,j}}{2},~{}~{}(i,j=1,2).$ (4.20)
###### Definition 4.2
Let us define the bosonic operators $E_{i}(z),F_{i}(z),H_{i}^{\pm}(z)\in
U_{q}(\widehat{sl_{3}}){\otimes}{\cal H}$, $(i=1,2)$ by
$\displaystyle E_{1}(z)$ $\displaystyle=$ $\displaystyle
e_{1}(z)e^{2Q_{1}}z^{-\frac{P_{1}-1}{r^{*}}},~{}~{}E_{2}(z)=e_{2}(z)e^{2Q_{2}}z^{-\frac{P_{2}-1}{r^{*}}},$
(4.21) $\displaystyle F_{1}(z)$ $\displaystyle=$ $\displaystyle
f_{1}(z)z^{\frac{2p_{b}^{1}+p_{b}^{2}-p_{b}^{3}+p_{a}^{1}}{r}}z^{\frac{P_{1}-1}{r}},~{}~{}F_{2}(z)=f_{2}(z)z^{\frac{2p_{b}^{3}+p_{b}^{2}-p_{b}^{1}+p_{a}^{2}}{r}}z^{\frac{P_{2}-1}{r}},$
(4.22) $\displaystyle H_{1}^{\pm}(z)$ $\displaystyle=$
$\displaystyle\Psi_{1}^{\pm}(z)e^{2Q_{1}}(q^{\mp\frac{k}{2}}z)^{\frac{2p_{b}^{1}+p_{b}^{2}-p_{b}^{3}+p_{a}^{1}}{r}}(q^{\pm(r-\frac{k}{2})}z)^{\frac{P_{1}-1}{r}-\frac{P_{1}-1}{r^{*}}},$
(4.23) $\displaystyle H_{2}^{\pm}(z)$ $\displaystyle=$
$\displaystyle\Psi_{2}^{\pm}(z)e^{2Q_{2}}(q^{\mp\frac{k}{2}}z)^{\frac{2p_{b}^{3}+p_{b}^{2}-p_{b}^{1}+p_{a}^{2}}{r}}(q^{\pm(r-\frac{k}{2})}z)^{\frac{P_{2}-1}{r}-\frac{P_{2}-1}{r^{*}}}.$
(4.24)
###### Theorem 4.2
The bosonic operators $E_{i}(z),F_{i}(z),H_{i}^{\pm}(z)$, $(i=1,2)$ satisfy
the following commutation relations.
$\displaystyle E_{i}(z_{1})E_{j}(z_{2})$ $\displaystyle=$
$\displaystyle\frac{\displaystyle\left[u_{1}-u_{2}+\frac{A_{i,j}}{2}\right]_{r^{*}}}{\displaystyle\left[u_{1}-u_{2}-\frac{A_{i,j}}{2}\right]_{r^{*}}}E_{j}(z_{2})E_{i}(z_{1}),$
(4.25) $\displaystyle F_{i}(z_{1})F_{j}(z_{2})$ $\displaystyle=$
$\displaystyle\frac{\displaystyle\left[u_{1}-u_{2}-\frac{A_{i,j}}{2}\right]_{r}}{\displaystyle\left[u_{1}-u_{2}+\frac{A_{i,j}}{2}\right]_{r}}F_{j}(z_{2})F_{i}(z_{1}),$
(4.26) $\displaystyle H^{\pm}_{i}(z_{1})H^{\pm}_{j}(z_{2})$ $\displaystyle=$
$\displaystyle\frac{\displaystyle\left[u_{1}-u_{2}-\frac{A_{i,j}}{2}\right]_{r}\left[u_{1}-u_{2}+\frac{A_{i,j}}{2}\right]_{r^{*}}}{\displaystyle\left[u_{1}-u_{2}+\frac{A_{i,j}}{2}\right]_{r}\left[u_{1}-u_{2}-\frac{A_{i,j}}{2}\right]_{r^{*}}}H^{\pm}_{j}(z_{2})H^{\pm}_{i}(z_{1}),$
(4.27) $\displaystyle H^{+}_{i}(z_{1})H^{-}_{j}(z_{2})$ $\displaystyle=$
$\displaystyle\frac{\displaystyle\left[u_{1}-u_{2}-\frac{A_{i,j}}{2}-\frac{k}{2}\right]_{r}\left[u_{1}-u_{2}+\frac{A_{i,j}}{2}+\frac{k}{2}\right]_{r^{*}}}{\displaystyle\left[u_{1}-u_{2}+\frac{A_{i,j}}{2}-\frac{k}{2}\right]_{r}\left[u_{1}-u_{2}-\frac{A_{i,j}}{2}+\frac{k}{2}\right]_{r^{*}}}H^{-}_{j}(z_{2})H^{+}_{i}(z_{1}),$
$\displaystyle H^{\pm}_{i}(z_{1})E_{j}(z_{2})$ $\displaystyle=$
$\displaystyle\frac{\displaystyle\left[u_{1}-u_{2}\pm\frac{k}{4}+\frac{A_{i,j}}{2}\right]_{r^{*}}}{\displaystyle\left[u_{1}-u_{2}\pm\frac{k}{4}-\frac{A_{i,j}}{2}\right]_{r^{*}}}E_{j}(z_{2})H^{\pm}_{i}(z_{1}),$
(4.29) $\displaystyle H^{\pm}_{i}(z_{1})F_{j}(z_{2})$ $\displaystyle=$
$\displaystyle\frac{\displaystyle\left[u_{1}-u_{2}\mp\frac{k}{4}-\frac{A_{i,j}}{2}\right]_{r}}{\displaystyle\left[u_{1}-u_{2}\mp\frac{k}{4}+\frac{A_{i,j}}{2}\right]_{r}}F_{j}(z_{2})H^{\pm}_{i}(z_{1}),$
(4.30)
$\displaystyle~{}[E_{i}(z_{1}),F_{j}(z_{2})]=\frac{\delta_{i,j}}{(q-q^{-1})z_{1}z_{2}}\left(\delta\left(q^{-k}\frac{z_{1}}{z_{2}}\right)H_{i}^{+}(q^{-\frac{k}{2}}z_{1})-\delta\left(q^{k}\frac{z_{1}}{z_{2}}\right)H_{i}^{-}(q^{-\frac{k}{2}}z_{2})\right).$
(4.31)
Now we have costructed level $k$ free field realization of Drinfeld current
$E_{i}(z),F_{i}(z),H_{i}^{\pm}(z)$ for the elliptic algebra
$U_{q,p}(\widehat{sl_{3}})$. This gives the first example of arbitrary-level
free field realization of higher-rank elliptic algebra.
## Acknowledgement
The author would like to thank the organizing committee of the 27-th
International Colloquium of the Group Theoretical Method in Physics held at
Yerevan, Armenia 2008. The author would like to thank Prof.A.Kluemper for his
kindness at Armenia. This work is partly supported by the Grant-in Aid for
Young Scientist B(18740092) from Japan Society for the Promotion of Science.
## Appendix
In appendix we summarize the normal ordering of the basic operators.
$\displaystyle:e^{\gamma^{i}(z_{1})}:{\cal B}_{+}^{*i}(z_{2})$
$\displaystyle=$ $\displaystyle:e^{\gamma^{i}(z_{1})}{\cal
B}_{+}^{*i}(z_{2}):\frac{(q^{r^{*}+1}z_{2}/z_{1};p^{*})_{\infty}}{(q^{r^{*}-1}z_{2}/z_{1};p^{*})_{\infty}},$
$\displaystyle:e^{\beta_{1}^{i}(z_{1})}:{\cal B}_{+}^{*i}(z_{2})$
$\displaystyle=$ $\displaystyle:e^{\beta_{1}^{i}(z_{1})}{\cal
B}_{+}^{*i}(z_{2}):\frac{(q^{r^{*}}z_{2}/z_{1};p^{*})_{\infty}}{(q^{r^{*}+2}z_{2}/z_{1};p^{*})_{\infty}},$
$\displaystyle:e^{\beta_{2}^{i}(z_{1})}:{\cal B}_{+}^{*i}(z_{2})$
$\displaystyle=$ $\displaystyle:e^{\beta_{2}^{i}(z_{1})}{\cal
B}_{+}^{*i}(z_{2}):\frac{(q^{r^{*}}z_{2}/z_{1};p^{*})_{\infty}}{(q^{r^{*}+2}z_{2}/z_{1};p^{*})_{\infty}},$
$\displaystyle:e^{\beta_{3}^{i}(z_{1})}:{\cal B}_{+}^{*i}(z_{2})$
$\displaystyle=$ $\displaystyle:e^{\beta_{3}^{i}(z_{1})}{\cal
B}_{+}^{*i}(z_{2}):\frac{(q^{r^{*}}z_{2}/z_{1};p^{*})_{\infty}}{(q^{r^{*}-2}z_{2}/z_{1};p^{*})_{\infty}},$
$\displaystyle:e^{\beta_{4}^{i}(z_{1})}:{\cal B}_{+}^{*i}(z_{2})$
$\displaystyle=$ $\displaystyle:e^{\beta_{4}^{i}(z_{1})}{\cal
B}_{+}^{*i}(z_{2}):\frac{(q^{r^{*}}z_{2}/z_{1};p^{*})_{\infty}}{(q^{r^{*}+2}z_{2}/z_{1};p^{*})_{\infty}},$
$\displaystyle{\cal B}_{-}^{i}(z_{1}):e^{\gamma^{i}(z_{2})}:$ $\displaystyle=$
$\displaystyle:{\cal
B}_{-}^{i}(z_{1})e^{\gamma^{i}(z_{2})}:\frac{(q^{r+1}z_{2}/z_{1};p)_{\infty}}{(q^{r-1}z_{2}/z_{1};p)_{\infty}},$
$\displaystyle{\cal B}_{-}^{i}(z_{1}):e^{\beta_{1}^{i}(z_{2})}:$
$\displaystyle=$ $\displaystyle:{\cal
B}_{-}^{i}(z_{1})e^{\beta_{1}^{i}(z_{2})}:\frac{(q^{r}z_{2}/z_{1};p)_{\infty}}{(q^{r+2}z_{2}/z_{1};p)_{\infty}},$
$\displaystyle{\cal B}_{-}^{i}(z_{1}):e^{\beta_{2}^{i}(z_{2})}:$
$\displaystyle=$ $\displaystyle:{\cal
B}_{-}^{i}(z_{1})e^{\beta_{2}^{i}(z_{2})}:\frac{(q^{r}z_{2}/z_{1};p)_{\infty}}{(q^{r+2}z_{2}/z_{1};p)_{\infty}},$
$\displaystyle{\cal B}_{-}^{i}(z_{1}):e^{\beta_{3}^{i}(z_{1})}:$
$\displaystyle=$ $\displaystyle:{\cal
B}_{-}^{i}(z_{1})e^{\beta_{3}^{i}(z_{1})}:\frac{(q^{r}z_{2}/z_{1};p)_{\infty}}{(q^{r-2}z_{2}/z_{1};p)_{\infty}},$
$\displaystyle{\cal B}_{-}^{i}(z_{1}):e^{\beta_{4}^{i}(z_{2})}:$
$\displaystyle=$ $\displaystyle:{\cal
B}_{-}^{i}(z_{1})e^{\beta_{4}^{i}(z_{2})}:\frac{(q^{r}z_{2}/z_{1};p)_{\infty}}{(q^{r-2}z_{2}/z_{1};p)_{\infty}},$
$\displaystyle e^{b_{+}^{i}(z_{1})}{\cal B}_{+}^{*i}(z_{2})$ $\displaystyle=$
$\displaystyle:e^{b_{+}^{i}(z_{1})}{\cal
B}_{+}^{*i}(z_{2}):\frac{(q^{r^{*}}z_{2}/z_{1};p^{*})_{\infty}^{2}}{(q^{r^{*}+2}z_{2}/z_{1};p^{*})_{\infty}(q^{r^{*}-2}z_{2}/z_{1};p^{*})_{\infty}},$
$\displaystyle{\cal B}_{-}^{i}(z_{1})e^{b_{-}^{i}(z_{2})}$ $\displaystyle=$
$\displaystyle:{\cal
B}_{-}^{i}(z_{1})e^{b_{-}^{i}(z_{2})}:\frac{(q^{r}z_{2}/z_{1};p)_{\infty}^{2}}{(q^{r+2}z_{2}/z_{1};p)_{\infty}(q^{r-2}z_{2}/z_{1};p)_{\infty}},$
$\displaystyle e^{a_{+}^{i}(z_{1})}{\cal A}^{*i}(z_{2})$ $\displaystyle=$
$\displaystyle:e^{a_{+}^{i}(z_{1})}{\cal
A}^{*i}(z_{2}):\frac{(q^{r^{*}+k+5}z_{2}/z_{1};p^{*})_{\infty}(q^{r^{*}-k-5}z_{2}/z_{1};p^{*})_{\infty}}{(q^{r^{*}+k+1}z_{2}/z_{1};p^{*})_{\infty}(q^{r^{*}-k-1}z_{2}/z_{1};p^{*})_{\infty}},$
$\displaystyle e^{a_{+}^{1}(z_{1})}{\cal A}^{*2}(z_{2})$ $\displaystyle=$
$\displaystyle:e^{a_{+}^{1}(z_{1})}{\cal
A}^{*2}(z_{2}):\frac{(q^{r^{*}+k+2}z_{2}/z_{1};p^{*})_{\infty}(q^{r^{*}-k-2}z_{2}/z_{1};p^{*})_{\infty}}{(q^{r^{*}+k+4}z_{2}/z_{1};p^{*})_{\infty}(q^{r^{*}-k-4}z_{2}/z_{1};p^{*})_{\infty}},$
$\displaystyle e^{a_{+}^{2}(z_{1})}{\cal A}^{*1}(z_{2})$ $\displaystyle=$
$\displaystyle:e^{a_{+}^{2}(z_{1})}{\cal
A}^{*1}(z_{2}):\frac{(q^{r^{*}+k+2}z_{2}/z_{1};p^{*})_{\infty}(q^{r^{*}-k-2}z_{2}/z_{1};p^{*})_{\infty}}{(q^{r^{*}+k+4}z_{2}/z_{1};p^{*})_{\infty}(q^{r^{*}-k-4}z_{2}/z_{1};p^{*})_{\infty}},$
$\displaystyle{\cal A}^{i}(z_{1})e^{a_{-}^{i}(z_{2})}$ $\displaystyle=$
$\displaystyle:{\cal
A}^{i}(z_{1})e^{a_{-}^{i}(z_{2})}:\frac{(q^{r+k+5}z_{2}/z_{1};p)_{\infty}(q^{r-k-5}z_{2}/z_{1};p)_{\infty}}{(q^{r+k+1}z_{2}/z_{1};p)_{\infty}(q^{r-k-1}z_{2}/z_{1};p)_{\infty}},$
$\displaystyle{\cal A}^{1}(z_{1})e^{a_{-}^{2}(z_{2})}$ $\displaystyle=$
$\displaystyle:{\cal
A}^{1}(z_{1})e^{a_{-}^{2}(z_{2})}:\frac{(q^{r+k+2}z_{2}/z_{1};p)_{\infty}(q^{r-k-2}z_{2}/z_{1};p)_{\infty}}{(q^{r+k+4}z_{2}/z_{1};p)_{\infty}(q^{r-k-4}z_{2}/z_{1};p)_{\infty}},$
$\displaystyle{\cal A}^{2}(z_{1})e^{a_{-}^{1}(z_{2})}$ $\displaystyle=$
$\displaystyle:{\cal
A}^{2}(z_{1})e^{a_{-}^{1}(z_{2})}:\frac{(q^{r+k+2}z_{2}/z_{1};p)_{\infty}(q^{r-k-2}z_{2}/z_{1};p)_{\infty}}{(q^{r+k+4}z_{2}/z_{1};p)_{\infty}(q^{r-k-4}z_{2}/z_{1};p)_{\infty}},$
$\displaystyle{\cal B}_{-}^{i}(z_{1}){\cal B}_{+}^{*i}(z_{2})$
$\displaystyle=$ $\displaystyle:{\cal B}_{-}^{i}(z_{1}){\cal
B}_{+}^{*i}(z_{2}):\frac{(q^{k}z_{2}/z_{1};q^{2k},p^{*})_{\infty}^{2}}{(q^{k+2}z_{2}/z_{1};q^{2k},p^{*})_{\infty}(q^{k-2}z_{2}/z_{1};q^{2k},p^{*})_{\infty}}$
$\displaystyle\times$
$\displaystyle\frac{(q^{k+2}z_{2}/z_{1};q^{2k},p)_{\infty}(q^{k-2}z_{2}/z_{1};q^{2k},p)_{\infty}}{(q^{k}z_{2}/z_{1};q^{2k},p)_{\infty}^{2}},$
$\displaystyle{\cal A}^{i}(z_{1}){\cal A}^{*i}(z_{2})$ $\displaystyle=$
$\displaystyle:{\cal A}^{i}(z_{1}){\cal
A}^{*i}(z_{2}):\frac{(q^{2k+5}z_{2}/z_{1};q^{2k},p^{*})_{\infty}(q^{-5}z_{2}/z_{1};q^{2k},p^{*})_{\infty}}{(q^{2k+1}z_{2}/z_{1};q^{2k},p^{*})_{\infty}(q^{-1}z_{2}/z_{1};q^{2k},p^{*})_{\infty}}$
$\displaystyle\times$
$\displaystyle\frac{(q^{2k+1}z_{2}/z_{1};q^{2k},p)_{\infty}(q^{-1}z_{2}/z_{1};q^{2k},p)_{\infty}}{(q^{2k+5}z_{2}/z_{1};q^{2k},p)_{\infty}(q^{-5}z_{2}/z_{1};q^{2k},p)_{\infty}},$
$\displaystyle{\cal A}^{1}(z_{1}){\cal A}^{*2}(z_{2})$ $\displaystyle=$
$\displaystyle:{\cal A}^{1}(z_{1}){\cal
A}^{*2}(z_{2}):\frac{(q^{2k+2}z_{2}/z_{1};q^{2k},p^{*})_{\infty}(q^{-2}z_{2}/z_{1};q^{2k},p^{*})_{\infty}}{(q^{2k+4}z_{2}/z_{1};q^{2k},p^{*})_{\infty}(q^{-4}z_{2}/z_{1};q^{2k},p^{*})_{\infty}}$
$\displaystyle\times$
$\displaystyle\frac{(q^{2k+4}z_{2}/z_{1};q^{2k},p)_{\infty}(q^{-4}z_{2}/z_{1};q^{2k},p)_{\infty}}{(q^{2k+2}z_{2}/z_{1};q^{2k},p)_{\infty}(q^{-2}z_{2}/z_{1};q^{2k},p)_{\infty}},$
$\displaystyle{\cal A}^{2}(z_{1}){\cal A}^{*1}(z_{2})$ $\displaystyle=$
$\displaystyle:{\cal A}^{2}(z_{1}){\cal
A}^{*1}(z_{2}):\frac{(q^{2k+2}z_{2}/z_{1};q^{2k},p^{*})_{\infty}(q^{-2}z_{2}/z_{1};q^{2k},p^{*})_{\infty}}{(q^{2k+4}z_{2}/z_{1};q^{2k},p^{*})_{\infty}(q^{-4}z_{2}/z_{1};q^{2k},p^{*})_{\infty}}$
$\displaystyle\times$
$\displaystyle\frac{(q^{2k+4}z_{2}/z_{1};q^{2k},p)_{\infty}(q^{-4}z_{2}/z_{1};q^{2k},p)_{\infty}}{(q^{2k+2}z_{2}/z_{1};q^{2k},p)_{\infty}(q^{-2}z_{2}/z_{1};q^{2k},p)_{\infty}}.$
Here we have used the notation
$(z;p_{1},p_{2})_{\infty}=\prod_{n_{1},n_{2}=0}^{\infty}(1-p_{1}^{n_{1}}p_{2}^{n_{2}}z).$
## References
* [1] O.Foda, K.Iohara, M.Jimbo, R.Kedem, T.Miwa, H.Yan: An elliptic quantum algebra for $\widehat{sl_{2}}$, Lett.Math.Phys.32, 259-268 (1994).
* [2] G.Felder: Elliptic quantum groups, Proceedings for International Conference of Mathematical Physics 1994, Paris, Cambridge-Hong Kong : International Press, 1995, pp.211-218.
* [3] C.Fr$\phi$nsdal : Quasi-Hopf deformation of quantum group, Lett.Math.Phys.40, 117-134 (1997).
* [4] B.Enriquez and G.Felder : Elliptic quantum group $E_{\tau,\eta}(sl_{2})$ and quasi-Hopf algebra, Commun.Math.Phys.195, 651-689, (1998).
* [5] M.Jimbo, H.Konno, S.Odake and J.Shiraishi : Quasi-Hopf twistors for elliptic quantum groups, Transformation Groups 4, 303-327, (1999).
* [6] V.Drinfeld : Quasi-Hopf algebras, Leningrad Math.J. 1, 1419-1457, (1990).
* [7] H.Konno : An Elliptic Algebra $U_{q,p}(\widehat{sl_{2}})$ and the fusion RSOS model, Commun.Math.Phys.195, 373-403, (1998).
* [8] E.Date, M.Jimbo, A.Kuniba, T.Miwa and M.Okado : Exactly solvable SOS model: II. proof of the star triangle relation and combinatrial identities, Adv.Stu.Pure.Math.16,17-122, (1998).
* [9] M.Jimbo, H.Konno, S.Odake and J.Shiraishi : Elliptic Algebra $U_{q,p}(\widehat{sl_{2}})$ : Drinfeld currents and vertex operators, Commun.Math.Phys.199, 605-647, (1999).
* [10] T.Kojima and H.Konno : The Elliptic Algebra $U_{q,p}(\widehat{sl_{N}})$ and the Drinfeld Realization of the Elliptic Quantum Group ${\cal B}_{q,\lambda}(\widehat{sl_{N}})$, Commun.Math.Phys.237, 405-447, (2003).
* [11] M.Wakimoto : Fock representation of the Affine Lie Algebra $A_{1}^{(1)}$, Commun.Math.Phys.104, 605-609, (1986).
* [12] B.Feigin and E.Frenkel : Representation of Affine Kac-Moody algebra and bosonization, Physics and Mathematics of Strings, World Scientific, 1990, 271-316.
* [13] A.Matsuo : A q-deformation of Wakimoto modules, primary fields and screening operators, Commun.Math.Phys.160,33-48,(1994).
* [14] J.Shiraishi : Free Boson Realization of $U_{q}(\widehat{sl_{2}})$, Phys.Lett.A171, 243-248, (1992).
* [15] H.Awata, S.Odake and J.Shiraishi : Free Boson Representation of $U_{q}(\widehat{sl_{3}})$, Lett.Math.Phys.30, 207-216, (1994).
* [16] W.Chang and X.Ding : On the vertex operators of the elliptic quantum algebra $U_{q,p}(\widehat{sl_{2}})_{k}$ J.Math.Phys.49, 043513, (2008).
* [17] T.Kojima and J.Shiraishi : The Integrals of Motion for the Deformed $W$-Algebra $W_{q,t}(\widehat{sl_{N}})$.II.Proof of the Commutation Relations, Commun.Math.Phys.283, 795-851, (2008).
* [18] T.Kojima and J.Shiraishi : A remark on the integrals of motion associated with level $k$ realization of the elliptic algebra $U_{q,p}(\widehat{sl_{2}})$, Proceedings for 10-th international conference on Geometry, Integrability and Quantization 2008, Bulgaria.
* [19] T.Kojima and J.Shiraishi : The integrals of motion for the elliptic deformation of the Virasoro and $W_{N}$ algebra, Proceedings for 5-th World Congress for Nonlinear Analysts 2008, Florida, USA.
|
arxiv-papers
| 2008-12-04T09:48:33 |
2024-09-04T02:48:59.211860
|
{
"license": "Public Domain",
"authors": "Takeo Kojima",
"submitter": "Takeo Kojima",
"url": "https://arxiv.org/abs/0812.0890"
}
|
0812.1095
|
00footnotetext: Received 29 May 2008, Revised 11 June 2008
# A simulation study on the measurement of $D^{0}-\overline{D^{0}}$ mixing
parameter y at BESIII ††thanks: Supported by National Natural Science
Foundation of China (10491300,10491303,10735080), Research and Development
Project of Important Scientific Equipment of CAS (H7292330S7), 100 Talents
Programme of CAS (U-25, U-54,U-612) and Scientific Research Fund of
GUCAS(110200M202)
HUANG Bin1,2;1) huangb@ihep.ac.cn ZHENG Yang-Heng2;2) zhengyh@gucas.ac.cn LI
Wei-Dong1;3) liwd@ihep.ac.cn BIAN Jian-Ming1,2 CAO Guo-Fu1,2 CAO Xue-Xiang1,2
CHEN Shen-Jian4 DENG Zi-Yan1 Fu Cheng-Dong3,1 GAO Yuan-Ning3 HE Kang-Lin1 HE
Miao1,2 HUA Chun-Fei5 HUANG Xing-Tao10 JI Xiao-Bin1 LI Hai-Bo1 LIANG Yu-Tie6
LIU Chun-Xiu1 LIU Huai-Min1 LIU Qiu-Guang1 LIU Suo7 MA Qiu-Mei1 MA Xiang1,2
MAO Ya-Jun6 MAO Ze-Pu1 MO Xiao-Hu1 PAN Ming-Hua8 PANG Cai-Ying8 PING Rong-
Gang1 QIN Gang1,2 QIN Ya-Hong5 QIU Jin-Fa1 SUN Sheng-Sen1 SUN Yong-Zhao1,2
WANG Ji-Ke1,2 WANG Liang-Liang1,2 WEN Shuo-Pin1 WU Ling-Hui1 XIE Yu-Guang1,2
XU Min9 YAN Liang1,2 YOU Zheng-Yun6 YU Guo-Wei1 YUAN Chang-Zheng1 YUAN Ye1
ZHANG Chang-Chun1 ZHANG Jian-Yong1 ZHANG Xue-Yao10 ZHANG Yao1 ZHU Yong-Sheng1
ZHU Zhi-Li8 ZOU Jia-Heng10 1 (Institute of High Energy Physics, CAS, Beijing
100049, China
2 Graduate University of Chinese Academy of Sciences, Beijing 100049,China)
3 Tsinghua University, Beijing 100084, China
4 (Nanjing University, Nanjing 210093, China)
5 Zhengzhou University, Zhengzhou 450001, China
6 Peking University, Beijing 100871, China
7 Liaoning University, Shenyang 110036, China
8 Guangxi Normal University, Guilin 541004, China
9 Department of Modern Physics, University of Science and Technology of China,
Hefei 230026, China
10 Shandong University, Jinan 250100, China
###### Abstract
We established a method on measuring the $D^{0}-\overline{D^{0}}$ mixing
parameter $y$ for BESIII experiment at the BEPCII $e^{+}e^{-}$ collider. In
this method, the doubly tagged $\psi(3770)\rightarrow D^{0}\overline{D^{0}}$
events, with one $D$ decays to $CP$-eigenstates and the other $D$ decays
semileptonically, are used to reconstruct the signals. Since this analysis
requires good $e/\pi$ separation, a likelihood approach, which combines the
$dE/dx$, time of flight and the electromagnetic shower detectors information,
is used for particle identification. We estimate the sensitivity of the
measurement of $y$ to be $0.007$ based on a $20fb^{-1}$ fully simulated MC
sample.
###### keywords:
likelihood, electron identification, $D^{0}-\overline{D^{0}}$ mixing, mixing
parameter $y$
###### pacs:
1
2.15.Ff, 13.20.Fc, 13.25.Ft, 14.40.Lb, 07.05.Kf
00footnotetext: 0 — LABEL:LastPage
## 1 Introduction
The mixing between a particle and its antiparticle has been observed
experimentally in neutral $K$, $B_{d}$ and $B_{s}$ system. In the Standard
Model, however, the mixing rate of the neutral $D$ system is expected in
general to be small and long-distance contributions make it difficult to be
calculated[2, 3, 4, 5]. Recently, several measurements[6, 7, 8, 9, 10] present
evidences for $D^{0}-\bar{D^{0}}$ mixing with the significance ranging from 3
to 4 standard deviations. This highlights the need for independent
measurements of the mixing parameters. Here, we present a study on measuring
the mixing parameter $y$ at BESIII experiment, which takes advantage of the
correlated threshold production of $D^{0}-\bar{D^{0}}$ pairs in $e^{+}e^{-}$
collisions.
For the neutral $D$ meson system, two mass eigenstates and flavor eigenstates
are not equivalent and can be expressed as the following form of the two
quantum states:
$\displaystyle|D_{A,B}\rangle=p|D^{0}\rangle\pm q|\bar{D^{0}}\rangle,$ (1)
with eigenvalues of masses and widths to be $m_{A,B}$ and $\Gamma_{A,B}$.
Conventionally, the $D^{0}-\bar{D^{0}}$ mixing is described by two small
dimensionless parameters:
$x\equiv\displaystyle\frac{\Delta
m}{\Gamma},y\equiv\displaystyle\frac{\Delta\Gamma}{2\Gamma},$ (2)
where $\Delta m\equiv m_{A}-m_{B}$, $\Delta\Gamma\equiv\Gamma_{A}-\Gamma_{B}$
and $\Gamma\equiv(\Gamma_{A}+\Gamma_{B})/2$. The mixing rate $R_{M}$ is
approximately
$R_{M}\approx\displaystyle\frac{x^{2}+y^{2}}{2}.$ (3)
In the limit of $CP$ conservation, the $|D_{A}\rangle$ and $|D_{B}\rangle$
denote the $CP$ eigenstates.
The mixing parameters can be measured in several ways. The $B$-factories
measured $R_{M}$ with semileptonic $D^{0}$ decay samples[11, 12].
Reference[13, 14] also gave an estimation on the sensitivities of $R_{M}$
measurement at BESIII. Other attempts[6, 7, 8, 9, 10] are based on the proper-
time measurements of the neutral $D$ meason decays. However, the time-
dependent analyse are not possible at symmetric charm factory, which operates
at the $\psi(3770)$ resonance. In this analysis, we utilize the quantum-
coherent threshold production of $D^{0}-\overline{D^{0}}$ pairs in a state of
definite $C=-1$. Applying the kinematics of the process of
$e^{+}e^{-}\rightarrow\psi(3770)\rightarrow D^{0}\bar{D^{0}}$, we can
reconstruct both neutral $D$ mesons (double tagging (DT) technique) to obtain
clean samples to measure the mixing parameters, the strong phase difference
and the $CP$ violation. For the single $D^{0}$ meson decays into a $CP$
eigenstate, the time-integrated decay rate can be written as[15, 16]:
$\Gamma_{CP\pm}\equiv\Gamma\left(D^{0}\rightarrow
f_{CP\pm}\right)=2A^{2}_{CP\pm}\left[1\mp y\right],$ (4)
where $f_{CP\pm}$ is a $CP$ eigenstate with eigenvalue $\pm 1$, and
$A_{CP\pm}\equiv|\langle f_{CP\pm}|{\cal H}|D^{0}\rangle|$ is the magnitude of
decay amplitude. If we consider the coherent $D$-pair decays, in which one $D$
decays into $CP$ eigenstates and the other $D$ decays semileptonically, the
decay rate of
$\left(D^{0}\overline{D^{0}}\right)^{C=-1}\rightarrow\left(l^{\pm}X\right)\left(f_{CP\pm}\right)$
is described as[17, 18, 19]:
$\Gamma_{l;CP}\equiv\Gamma\left[\left(l^{\pm}X\right)\left(f_{CP}\right)\right]\approx
A^{2}_{l^{\pm}X}A^{2}_{CP},$ (5)
where $A_{l^{\pm}X}\equiv|\langle l^{\pm}X|{\cal H}|D^{0}\rangle|$. Here, we
neglect terms to order $y^{2}$ or higher since $y$ is much smaller than unit.
We, thus, can derive:
$y={1\over
4}\left(\frac{\Gamma_{l;CP+}\Gamma_{CP-}}{\Gamma_{l;CP-}\Gamma_{CP+}}-\frac{\Gamma_{l;CP-}\Gamma_{CP+}}{\Gamma_{l;CP+}\Gamma_{CP-}}\right).$
(6)
To measure $y$ at BESIII, only the electron channels are used to reconstruct
the semileptonic $D^{0}$ decays. In the muon channels, the transverse momentum
of muon is too low to be efficiently identified by the BESIII muon detector.
Thus, the $e/\pi$ separation plays an essential role to suppress the
backgrounds. Fig. 1 shows the momentum distribution of the electrons from the
semileptonic $D$ decays. The momentum distribution of the pions from $s$ quark
decays is similar to Fig. 1. As a result, the performance of electron
identification (e-ID) will determine the precision of the measurement of $y$
parameter.
Momentum of the electron from $D^{0}$ semileptonic decays
The designed peak luminosity of BEPCII (Beijing Electron Position Collider) is
$10^{33}cm^{-2}s^{-1}$ at beam energy $E_{beam}$ = 1.89 GeV, which is the
highest in the tau-charm region ever planned and an unprecedented large number
of $\psi(3770)$ events is expected.
This paper is organized as follows: an improved electron identification
technique for BESIII is described in Section 2. In Section 3, we describe the
method on reconstructing the signals with Mente Carlo(MC) simulation samples.
Section 4 presents the estimated sensitivity of $y$ measurement. The summary
is presented in Section 5.
## 2 Electron identification
The BESIII detector operates at BEPCII and consists of a beryllium beam pipe,
a helium-based small-celled drift chamber, Time-Of-Flight (TOF) counters for
particle identification, a CsI(Tl) crystal electromagnetic calorimeter (EMC),
a super-conducting solenoidal magnet with the field of 1 Tesla, and a muon
identifier of Resistive Plate Counters (RPC) interleaved with the magnet yoke
plates. The BESIII Offiline Software System (BOSS)[20] of version 6.1.0 is
used for this analysis. The detector simulation[21] is based on GEANT4[22].
The BESIII detector has four subsystems for particle identification: the dE/dx
of the main drift chamber(MDC), TOF, EMC and the muon counter. Among them, the
dE/dx and the TOF systems are mainly used for hadron separation, the EMC
provides information for electron and photon identification, the MUC has good
performance on muon identification[23].
For electron identification, Refs[24, 25] illustrate the use of dE/dx of MDC
and TOF information. Here, an improved $e/\pi$ separation technique is
introduced in the following sections.
### 2.1 Electromagnetic calorimeter
The BESIII electromagnetic calorimeter[23, 26] is composed of one barrel and
two endcap sections, covering 93% of 4$\pi$. There are a total of 44 rings of
crystals along the z direction in the barrel, each with 120 crystals. And
there are 6 layers in the endcap, with different number of crystals in each
layer. The entire calorimeter has 6240 CsI(Tl) crystals with a total weight of
about 24 tons. The energy resolution is expected to be 2.5% and the spatial
resolution is expected to be 0.6 cm for 1 GeV/c photon.
The primary function of the EMC is to precisely measure the energies and
positions of electron and photon. In order to distinguish electron from
hadron, we make use of significant differences in energy deposition and the
shower shape of different type of the particles.
### 2.2 Variables used in e-ID
The following variables are used to identify the electron from pion:
1)
Ratio of the energy measured by the EMC and the momentum of the charged track
by the MDC ($E/p$).
Ratio of the energy measured by the EMC and the momentum of the charged track
by the MDC ($E/p$). When an electron passes through the calorimeter, the
electron produces electromagnetic shower and loses its energy by pair-
production, Bremsstrahlung and ionizing/exciting atomic electrons. Since the
mass of electron is negligible in the energy range of interest, we expect to
have the ratio $E/p=1$ within the measurement errors. For hadrons, the $E/p$
is typically smaller than one.
2)
Lateral shower shape at the EMC.
In order to enhance the separation between the electrons and the interacting
hadrons, the lateral shower shape can also be utilized. These variables
include: $E_{seed}/E_{3\times 3},E_{3\times 3}/E_{5\times 5}$ and the second-
moment. Here the $E_{seed}$ is the energy deposited in the central crystal,
the $E_{3\times 3}$ and $E_{5\times 5}$ represent the energy deposit in the
$3\times 3$ and $5\times 5$ crystal array, respectively. The second-moment $S$
is defined as
$\displaystyle S=\frac{\sum_{i}{E_{i}\cdot d_{i}^{2}}}{\sum_{i}{E_{i}}},$ (7)
where $E_{i}$ is the energy deposit in the $i$-th crystal, and $d_{i}$ is the
distance between the $i$-th crystal and the center position of reconstructed
shower. Detailed description of $E/p$ and the lateral shower shape can be
found in Ref [24].
3)
Longitudinal shower shape at the EMC.
The longitudinal shower shape provides additional information for electron
identification. The variable $\Delta\phi$, between the polar angles where the
track intersects the EMC and the shower center, can be used. The distributions
of $\Delta\phi$ for electron and pion are drawn in Fig. 2.2. The center of
electron showers is closer to the impact point of track on EMC since the
electron showers reach their maximum earlier than hadrons.
$\Delta\phi$ of (a) electron (b) pion.
### 2.3 The correlation between variables
The $E/p$ ratio, lateral shower shape and longitudinal shower shape are all
depending on the deposited energy in the crystals. Thus, these variables may
be correlated. We calculate the correlation coefficients $\rho_{ij}$ between
the $E/p$, $E_{3\times 3}/E_{5\times 5}$ and $\Delta\phi$ using the function:
$\displaystyle
M_{i,j}=\sum_{i,j}{(x_{i}-\bar{x_{i}})\cdot(x_{j}-\bar{x_{j}})},\quad\rho_{ij}=\frac{M_{ij}}{\sqrt{M_{ii}\times
M_{jj}}},$ (8)
where $i,j$ are the indices of the variable names. Figure. 2.3 shows the
correlation between any two of the variables of electron and pion,
respectively, with the momentum ranging from 0.2 GeV/c to 2.0 GeV/c. Here, the
x-axis represents the particle momentum and the y-axis represents the
correlation coefficient $\rho_{ij}$. The distribution indicates strong
correlation between the variables.
Correlations between (a)E/p and $E_{seed}/E_{3\times 3}$ of electron; (b)E/p
and $E_{3\times 3}/E_{5\times 5}$ of electron; (c)$E_{seed}/E_{3\times 3}$ and
$E_{3\times 3}/E_{5\times 5}$ of electron; (d)E/p and $E_{seed/E3\times 3}$ of
pion; (e)E/p and $E_{3\times 3}/E_{5\times 5}$ of pion;
(f)$E_{seed}/E_{3\times 3}$ and $E_{3\times 3}/E_{5\times 5}$ of pion.
### 2.4 PID Algorithm
Considering the correlations between the variables, the traditional method for
particle identification may be underperforming. In the e-ID, we implement the
artificial neural network (ANN) [27] to provide a general framework for
estimating non-linear functional mapping between the input variables and the
output variable. For the neural network (NN) training, we use the momentum,
traverse momentum and other six discriminants (total deposit energy, Eseed,
E3x3, E5x5, second moment and $\Delta\phi$) as the input variables. The
network we choose has one hidden layer with 16 neurons and one output value.
Figure. 2.4 shows the two-dimension distributions of the output value versus
the momentum of the electrons and pions. It is obvious that the distribution
of the output value depends on the momentum, especially at low momentum
region. Thus, it is unsuitable to apply a single cut on the output value to
separate the electrons from the pions. In practice, we construct probability
density function (PDF) of the NN output value at every 0.1 GeV/c momentum bin.
The PDF is obtained from fitting the nearest 4 bins of the NN output value,
with the third-order polynomial function. Then, the PDF value of the NN output
can be extracted from the fit. Finally, we make the PID decision by comparing
the likelihood values of electron and pion hypothesis.
The NN outputs of (a) pion (b) electron samples with the momentum ranging from
0.2GeV/c to 1.6GeV/c
### 2.5 The performance check
To combine the $dE/dx$, TOF and EMC information, the likelihood approach[28]
is adopted. Firstly, the likelihood value of each subsystem is calculated.
Then, the total likelihood value of each hypothesis is calculated by the
following formula:
$\displaystyle L_{tot}=L_{dE/dx}*L_{TOF}*L_{EMC},$ (9)
where $L_{dE/dx}$ and $L_{TOF}$ represent the likelihood value of $dE/dx$ and
TOF subsystems respectively. Finally, the likelihood ratio of electron
hypothesis is defined as:
$\displaystyle lhf_{{}_{e}}=\frac{L_{e}}{L_{e}+L_{\pi}},$ (10)
where $L_{e}$ and $L_{\pi}$ are the total likelihood value of electron and
$\pi$ hypothesis. To check the performance of the $e/\pi$ separation, both the
electron and pion samples are generated with the momentum ranging from 0.2
Gev/c to 1.6 GeV/c, by using single particle generator. Fig. 2.5(a) shows the
electron likelihood ratio distributions of these samples. For a particle to be
identified as an electron, we require $lhf_{{}_{e}}>0.5$. Fig. 2.5(b) shows
the combined $e/\pi$ separation performance using the $dE/dx$, TOF and EMC
systems.
(a)$lhf_{{}_{e}}$ of electron and $\pi$ samples; (b) performance of $e/\pi$
seperation.
## 3 Simulation and reconstruction
### 3.1 The reconstruction of $CP$ tags
For the neutral $D$ meson decays, the main decay modes of $CP+$ eigenstate are
$K^{+}K^{-}$, $\pi^{+}\pi^{-}$, $K_{S}\pi^{0}\pi^{0}$, $\pi^{0}\pi^{0}$,
$K_{S}K_{S}$ and $\rho^{0}\pi^{0}$. The $CP-$ eigenstates decay through the
modes $K_{S}(\pi^{0},\rho^{0},\eta,\eta^{\prime},\phi,\omega)$. Considering
the branching ratio and the reconstruction efficiency, we only simulated the
$K^{+}K^{-}$, $\pi^{+}\pi^{-}$ for $CP+$ tagging, and the
$K_{S}(\pi^{0},\eta,\eta^{\prime})$ for $CP-$ tagging.
For selecting the charged tracks, the following selection criteria are
adopted:
1)
All charged tracks must have a good helix fit, and are required to be measured
in the fiducial region of MDC;
2)
Their parameters must be corrected for energy loss and multiple scattering
according to the assigned mass hypotheses;
3)
The tracks not associated with $K^{0}_{S}$ reconstruction are required to be
originated from the interaction point(IP).
For reconstructing the $CP+$ eigenstates, two opposite-charged tracks of $K$
or $\pi$ are selected with the requirements that they are from IP and to pass
a common vertex constraint. To identify a track as a $\pi$ or $K$, we use the
likelihood method to combine the information of $dE/dx$ and TOF with the
likelihood fraction of $\pi$ or $K$ greater than 0.5. Then the beam
constrained mass($M_{bc}$) of the $D$ meson is used to distinguish the signal
and background, and it is defined as:
$\displaystyle
M_{bc}\equiv\sqrt{E_{beam}^{2}-(\sum\bm{p}_{i})^{2}}=\sqrt{E_{beam}^{2}-(\bm{p}_{D})^{2}},$
(11)
where the $E_{beam}$ is the beam energy, the $\bm{p}_{i}$ is the momentum of
the $i$-th track and the $\bm{p}_{D}=\sum\bm{p}_{i}$ is the momentum of the
reconstructed $D$ meson.
For tagging the $CP-$ eigenstates, we need to reconstruct the neutral mesons
$K_{S}$, $\pi^{0}$, $\eta$ and $\eta^{{}^{\prime}}$. The $K_{S}$ candidates
are reconstructed through the decay of $K_{S}\rightarrow\pi^{+}\pi^{-}$. The
decay vertex formed by $\pi^{+}\pi^{-}$ pair is required to be away from the
interaction point, and the momentum vector of $\pi^{+}\pi^{-}$ pair must be
aligned with the position vector of the decay vertex to the IP. Here we set
$L_{vtx}/\sigma_{vtx}$ to be greater than 2, where $L_{vtx}$ and
$\sigma_{vtx}$ are the measured decay length and the error of the decay length
of the $K_{S}$. The $\pi^{+}\pi^{-}$ invariant mass is required to be
consistent with the $K_{S}$ nominal mass within $\pm 10$ MeV. To identify the
neutral tracks, one has to address a number of processes which can produce
both real and spurious showers in EMC. The major source of these “fake
photons” arises from hadronic interaction, which can create a “split-off”
shower. This shower does not associate with the main shower and may be
recognized as a photon. Other sources of fake photons include particle decays,
back splash, beam associated background and electronic noise. To reject “fake
photons”, the selection criteria for “good photon” include a deposit energy
cut, and a spatial cut, which requires that the cluster is isolated from the
nearest charged tracks. These “cuts” are set to be $E_{\gamma}>40MeV$ and
$\Delta_{c\gamma}>18^{\circ}$, where $E_{\gamma}$ and $\Delta_{c\gamma}$
represent the deposited energy and the crossing angle of the cluster to the
nearest charged track, respectively. The neutral pions are reconstructed from
$\pi^{0}\rightarrow\gamma\gamma$ decays using the photons observed in the
barrel and endcap regions of EMC. At the energies of interest, a $\pi^{0}$
decays into two isolated photons. In addition, we also reconstruct
$\eta/\eta^{{}^{\prime}}$ candidates in the modes of
$\eta\rightarrow\gamma\gamma,\eta\rightarrow\pi^{+}\pi^{-}\pi^{0},\eta^{{}^{\prime}}\rightarrow\gamma\rho^{0}$
and $\eta^{{}^{\prime}}\rightarrow\eta\pi^{+}\pi^{-}$. For these modes,
3$\sigma$ consistency with the $\pi^{0}/\eta/\eta^{{}^{\prime}}$ mass is
required, followed by a kinematic mass constraint. For $CP-$ eigenstates, the
beam constrained mass is also used to select the signal.
Under the environment of BOSS 6.1.0, we simulated $D^{0}-\overline{D^{0}}$
pairs production at the $\psi(3770)$ peak with one $D$ decayed into $CP$
eigenstates and the other $D$ decayed semileptonically. The $CP+$ eigenstates
are decayed through $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ according to their
branching ratios. For $CP-$ eigenstates, the decay modes $K_{S}\pi^{0}$,
$K_{S}\eta$ and $K_{S}\eta^{{}^{\prime}}$ are included. In the $K_{S}$,
$\pi^{0}$, $\eta$ and $\eta^{{}^{\prime}}$ decays, the decay modes are listed
as follows: $K_{S}\rightarrow\pi^{+}\pi^{-}$,
$\pi^{0}\rightarrow\gamma\gamma$, $\eta\rightarrow\gamma\gamma$,
$\eta\rightarrow\pi^{+}\pi^{-}\pi^{0}$,
$\eta^{{}^{\prime}}\rightarrow\gamma\rho^{0}$, and
$\eta^{{}^{\prime}}\rightarrow\eta\pi^{+}\pi^{-}$. For the $CP+$ and $CP-$
eigenstates, we generated 30,000 events for each MC sample. The distributions
of the beam constrained mass of $D$ meson are shown in Fig. 3.1.
The $M_{bc}$ of (a)the $CP-$ tags; (b)the $CP+$ tags.
### 3.2 The reconstruction of semileptonic tags
For tagging the semileptonic decays, we use the decay mode $D^{0}\rightarrow
K^{-}e^{+}\upsilon_{e}$. To reconstruct the neutral $D$ meson, good tracks for
one electron and one kaon candidate are required. The good track selection
criteria are the same as the $CP$ tagging discussed in Section 3.1. The
electron and kaon candidates are also required to be from the IP, and the
likelihood ratio of the electron and kaon must both be greater than 0.5.
Moreover, the two tracks need to pass a common vertex constraint. After the
electron and pion selections, a standard partial reconstruction technique is
applied to this semileptonic decay channel with one neutrino associated. Here,
we use the “missing mass” ($U_{miss}$) of neutrino to select the signal
candidates. The “missing mass” is defined as follows:
$\displaystyle U_{miss}\equiv E_{miss}-P_{miss},$ (12)
where $E_{miss}=E_{D_{tag}}-E_{K}-E_{e}$ and
$P_{miss}=|\bm{p}_{D_{tag}}-\bm{p}_{K}-\bm{p}_{e}|$ are the missing energy and
momentum of the neutrino. Here, $E_{K,e}$ and $\bm{p}_{K,e}$ are the measured
energy and momentum of the selected kaon and electron track. $E_{D_{tag}}$ is
the energy of the $D$ meson, which is equal to the beam energy.
$\bm{p}_{D_{tag}}=-\bm{p}_{D_{CP}}$ is the 3-momentum of the $D$ meson, which
can be obtained from the reconstructed momentum of the CP tagged $D$ meson.
For neutrino, the energy and momentum are equal. Thus, the distribution of
$U_{miss}$ must have a mean value at zero. The distribution of $U_{miss}$ is
shown in Fig. 3.2. We apply a $3\sigma$ cut on the $U_{miss}$ to select the
semileptonic $D$ decays.
$U_{miss}$ of $\nu_{e}$ for (a) $CP+$ tags and (b) $CP-$ tags.
## 4 The sensitivity of $y$
Table 4 shows the reconstruction efficiency and the number of estimated
doubly-tagged events for different simulated decay channels. For $\sim
20\textrm{fb}^{-1}$ luminosity at $\psi(3770)$ peak, which approximately
corresponds to four years data taking at BESIII, about $8.0\times 10^{7}$
$D^{0}-\overline{D^{0}}$ pairs can be produced. According to the full
simulation, about 11000 doubly tagged $CP+$ decays and 9000 doubly tagged
$CP-$ decays can be reconstructed.
The efficiency and expected events for $8.0\times 10^{7}$
$D^{0}-\overline{D^{0}}$ decays for different decay channels. decay mode
efficiency event estimation $K^{-}e^{+}\upsilon_{e}$
$K^{-}K^{+},\pi^{-}\pi^{+}$ 40% 11701 $K^{-}e^{+}\upsilon_{e}$ $K_{s}\pi^{0}$
16.8% 7345 $K^{-}e^{+}\upsilon_{e}$ $K_{s}\eta$ 7.7% 715
$K^{-}e^{+}\upsilon_{e}$ $K_{s}\eta^{\prime}$ 4.7% 953
For a small y, to calculate the $\sigma_{y}$ of Equation(6), we ignore the
statistical error from single tagged events. Hence, the statistical error of
$y$ parameter can be obtained from the following equation:
$\displaystyle\sigma_{y}=\frac{1}{2}\times\sqrt{\frac{1}{N_{1}}+\frac{1}{N_{2}}},$
(13)
where $N_{1}$ and $N_{2}$ represent the reconstructed doubly-tagged $CP+$ and
$CP-$ events. As a result, the $\sigma_{y}$ is estimated to be $0.007$ with
$\sim 20\textrm{fb}^{-1}$ data at $\psi(3770)$ peak in this analysis. Since
the double tagging technique is adopted here, the background effect can be
ignored comparing to the statistical sensitivity estimated above.
## 5 Summary
In this paper, we presented a MC study on measuring the
$D^{0}-\overline{D^{0}}$ mixing parameter $y$ at the BESIII experiment. Based
on a $20fb^{-1}$ fully simulated MC sample of $\psi(3770)$ resonance decays,
we estimated the sensitivity of the $y$ measurement to be 0.007. In this
analysis, the double tagging technique was used for reconstructing the $D$
meson pairs. Here, the signal is reconstructed such that one $D$ decays to
$CP$ eigenstates and the other $D$ decays semileptonically. The electron
identification is essential for this analysis. We improved the e-ID technique
for BESIII experiment, which can also be applied to many other important
physics topics. Our next step is to include more semileptonic decay modes,
such as $D^{0}\rightarrow K^{*}e\nu_{e}$, into this analysis to improve the
sensitivity of y measurement.
## References
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* [22] Agostinelli S et al.(Geant4 Collaboration). Nucl. Instrum. Methods, 2003, 506: 250
* [23] BESIII Design Report, Interior Document in Institute of High Energy Physics, 2004
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* [26] Harris F A et al.(BES Collab). arXiv:physics/0606059, 2006
* [27] Bishop C M. Neural Networks for Pattern Recognition. Oxford: Clarendon, 1998; Beale R, Jackson T. Neural Computing: An Introduction. New York: Adam Hilger, 1991
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|
arxiv-papers
| 2008-12-05T09:11:02 |
2024-09-04T02:48:59.221696
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bin Huang, Yang-Heng Zheng, Wei-dong Li",
"submitter": "Bin Huang",
"url": "https://arxiv.org/abs/0812.1095"
}
|
0812.1108
|
# LABORATORI NAZIONALI DI FRASCATI SIS-Pubblicazioni
LNF–08/29(P) November 29, 2008
THE CMS RPC GAS GAIN MONITORING SYSTEM:
an Overview and Preliminary Results
L. Benussi1, S. Bianco1, S. Colafranceschi${}^{1},2,3$, D. Colonna1, L.
Daniello1, F. L. Fabbri1, M. Giardoni1
B. Ortenzi1, A. Paolozzi,${}^{1},2$ L. Passamonti1, D. Pierluigi1
B. Ponzio1, C. Pucci1, A. Russo1, G. Roselli5, A. Colaleo4, F. Loddo4, M.
Maggi4
A. Ranieri4, M. Abbrescia${}^{4},5$, G. Iaselli${}^{4},5$, B.
Marangelli${}^{4},5$, S. Natali${}^{4},5$
S. Nuzzo${}^{4},5$, G.Pugliese${}^{4},5$, F. Romano${}^{4},5$, R.
Trentadue${}^{4},5$
S. Tupputi${}^{4},5$, R. Guida3, G. Polese${}^{3},6$, N. Cavallo7 A.
Cimmino${}^{7},8$
D. Lomidze8, P. Noli${}^{7},8$ D. Paolucci8, P. Piccolo8, C. Sciacca${}^{7},8$
P. Baesso9, M. Necchi9
D. Pagano9, S. P. Ratti9, P. Vitulo9, C. Viviani9
1) INFN Laboratori Nazionali di Frascati, Via E. Fermi 40, I-00044 Frascati,
Italy.
2) Università degli Studi di Roma ”La Sapienza”, Piazzale A. Moro.
3) CERN CH-1211 Genéve 23 F-01631 Switzerland.
4) INFN Sezione di Bari, Via Amendola, 173I-70126 Bari, Italy.
5) Dipartimento Interateneo di Fisica, Via Amendola, 173I-70126 Bari, Italy.
6) Lappeenranta University of Technology, P.O. Box 20 FI-538 1 Lappeenranta,
Finland.
7) INFN Sezione di Napoli, Complesso Universitario di Monte Sant’Angelo,
edificio 6, 80126 Napoli, Italy.
8) Università di Napoli Federico II, Complesso Universitario di Monte
Sant’Angelo, edificio 6, 80126 Napoli, Italy.
9) INFN Sezione di Pavia, Via Bassi 6, 27100 Pavia, Italy and Università degli
studi di Pavia, Via Bassi 6, 27100 Pavia, Italy.
The status of the CMS RPC Gas Gain Monitoring (GGM) system developed at the
Frascati Laboratory of INFN (Istituto Nazionale di Fisica Nucleare) is
reported on. The GGM system is a cosmic ray telescope based on small RPC
detectors operated with the same gas mixture used by the CMS RPC system. The
GGM gain and efficiency are continuously monitored on-line, thus providing a
fast and accurate determination of any shift in working point conditions. The
construction details and the first result of GGM commissioning are described.
PACS: 07.77.Ka; 95.55.Vj; 29.40.Cs
Presented by L. Benussi at the RPC07 - February 2008, Mumbai, India
## 1 Introduction
Design parameters, construction, prototyping and preliminary commissioning
results of the CMS RPC Gas Gain Monitoring (GGM) system are presented [1]. The
Resistive Plate Counter (RPC) system is part of the muons detector of the
Compact Solenoid Spectrometer (CMS) experiment[2] at the Large Hadron Collider
(LHC) collider in CERN (Geneva, Switzerland), with the primary task of
providing first level trigger and synchronization. The CMS RPCs are bakelite-
based double-gap RPC with strip readout (for construction details see [3] and
reference therein) operated with 96.2% C2H2F4 \- 3.5% Iso-C4H10 \- 0.3% SF6
gas mixture humidified at about 40%. The large volume of the whole CMS RPC
system and the cost of gas used make mandatory the operation of RPC in a
closed-loop gas system (for a complete description see [4]), in which the gas
fluxing the gaps is reused after being purified by a set of filters[5].
The operation of the CMS RPC system is strictly correlated to the ratio
between the gas mixture components and to the presence of pollution due to
contaminants that can be be produced inside the gaps during discharges (i.e.
HF produced by SF6 or C2H2F4 molecular break-up and further fluorine
recombination), accumulated in the closed-loop or by pollution that can be
present in the gas piping system (tubes, valves, filters, bubblers, etc.) and
flushed into the gaps by the gas flow. To monitor the presence of these
contaminants as well as the gas mixture stability, is therefore mandatory to
avoid RPC damage and to ensure their correct functionality.
A monitoring system of the RPC working point due to changes of gas composition
and pollution must provide a faster and sensitive response than the CMS RPC
system itself in order to avoid irreversible damage of the whole system. Such
a Gas Gain Monitoring system monitors efficiency and signal charge
continuously by means of a small sized cosmic ray telescope based on RPC
detectors. In the following will be briefly described the final setup of the
GGM system, its construction details and the first results obtained during its
commissioning done at the ISR area (CERN).
## 2 The Gas Gain Monitoring System
The GGM system is composed by the same type of RPC used in the CMS detector
but of smaller size (2mm Bakelite gaps, 50$\times$50 cm2). Twelve gaps are
arranged in a stack located in the CMS gas area (SGX5 building) in the
surface, close to CMS assembly hall (LHC-P5). The choice to install the system
in the surface instead of underground allows one to profit from maximum cosmic
muon rates. In order to ensure a fast response to working point shifts with a
precision of 10%, $10^{3}$ events are are required, corresponding to about 30
minutes exposure time on surface, to be compared with a 100-fold lesser rate
underground. The trigger will be provided by four out of twelve gaps of the
stack, while the remaining eight gaps will be used to monitor the working
point stability.
The eight gaps are arranged in three sub-system: one sub-system (two gaps)
will be fluxed with the fresh CMS mixture and its output sent to vent. The
second sub-system (three gaps) will be fluxed with CMS gas coming from the
closed-loop gas system and extracted before the gas purifiers, while the third
sub-system (three gaps) are operated with CMS gas extracted from the closed-
loop extracted after the gas filters. The basic idea is to compare the
operation of the three sub-system and, if some changes are observed, to send a
warning to the experiment. In this way, the gas going to and coming from the
CMS RPC detector is monitored by using the two gaps fluxed with the fresh
mixture as reference gaps. This setup will ensure that pressure, temperature
and humidity changes affecting the gaps behavior do cancel out by comparing
the response of the three sub-system operating in the same ambient condition.
The monitoring is performed by measuring the charge distributions of each
chamber. The eight gaps will be operated at different high voltages, fixed for
each chamber, in order to monitor the total range of operating modes of the
gaps. The operation mode of the RPC changes as a function of the voltage
applied. A fraction of the eight gaps will work in pure avalanche mode, while
the remaining will be operated in avalanche+streamer mode. Comparison of
signal charge distributions and the ratio of the avalanche to streamer
components of the ADC provides a monitoring of the stability of working point
for changes due to gas mixture variations.
## 3 Construction and commissioning
Each chamber of the GGM system consists of a single gap with double sided pad
read-out: two copper pads are glued on the two opposite external side of the
gap. Fig.1 shows a sketch of a chamber whose photos are shown in fig.2; the
two foam planes are used to reduce the capacity coupling between the pad and
the copper shields. The signal is read-out by a transformer based circuit A3
(Fig.3). The circuit
Figure 1: A schematic layout of a GGM chamber
Figure 2: Pictures of a GGM gap and chamber. A) a bare gap with the HV and
signal cables. B) a completed chamber. The gap is sandwiched between two foam
panels and fully covered with a copper shield. C) a section of a chamber with
the two foam panels visible.
allows to algebraically subtract the two signal, which have opposite
polarities, and to obtain an output signal with subtraction of the coherent
noise, with an improvement by about a factor 4 of the signal to noise ratio.
Fig.4 shows the typical operation mode of the GGM double-pad readout with
positive and negative pads pulses, and the output pulse from circuit A3. The
output signals from circuit A3 are sent to a CAEN V965 ADC [7] for charge
analysis. The GGM has been tested with cosmic rays at LNF and then shipped to
CERN for the final commissioning (fig.5 show the final stack assembly).
Figure 3: The electric scheme of the read-out circuit providing the algebraic
sum of the two pad signal (PAD + and PAD -).
Figure 4: An oscilloscope screen-shot of the two pad signals (upper traces)
which are effected by a coherent noise and are barely visible on the screen.
In the lower trace the coherent noise is highly reduced by A3 circuit. The
vertical scale is the same for both cases 5 mV/div.
Figure 5: A picture of the GGM system ready to be shipped to CERN. The stack
is enclosed into an aluminum box for further shielding.
A typical ADC distribution of a GGM gap is shown in fig.6 for two different
effective operating voltage, defined as the high voltage set on the HV power
supply corrected for the local atmospheric pressure and temperature. Fig.6 a)
corresponding to HVeff=9.9kV shows a clean avalanche peak well separated from
the pedestal. Fig.6 b) shows the charge distribution at HVeff=10.7kV with two
signal regions corresponding to the avalanche and to avalanche+streamer mode.
Figure 6: Typical ADC charge distributions of one GGM chamber at two operating
voltages. Distribution (a) correspond to HVeff = 9.9kV while distribution (b)
to HVeff=10.7kV. In (b) is clearly visible the streamer peak around 1900 ADC
channels. The events on the left of the vertical line (1450 ADC channels in
this case) are assumed to be pure avalanche events.
Figure 7: Efficiency plot (full dots) of GGM chambers as a function of HVeff.
The efficiency is defined as the ratio between the number of ADC entries above
3$\sigma_{ped}$ and the number of acquired triggers. Open dot plots correspond
to the streamer fraction of the chamber signal as a function of HVeff.
Fig.7 shows the GGMS single gap efficiency (full dots), and the ratio between
the avalanche and the streamer component (open circles), as a function of the
effective high voltage. Each point corresponds to a total of 10000 entries in
the full ADC spectrum. The efficiency is defined as the ratio between the
number of triggers divided by the number of events above 3$\sigma_{ped}$ over
ADC pedestal, where $\sigma_{ped}$ is the pedestal width. The avalanche to
streamer ratio is defined by counting the number of entries in the avalanche
(below the ADC threshold (fig.6 b) and above the pedestal region) and dividing
it by the number of streamer events above the avalanche threshold. Both
efficiency and avalanche plateau are in good agreement with previous results
[6].
In order to determine the sensitivity of GGM gaps to working point shifts, the
avalanche to streamer transition was studied by two methods, the charge method
and the efficiency method. In the charge method, the mean value of the ADC
charge distribution in the whole ADC range is studied as a function of HVeff
(fig.8). Each point corresponds to 10000 events in the whole ADC spectrum. In
the plot three working point regions are identified
1. 1.
inefficiency (HV${}_{eff}<$ 9.7 kV);
2. 2.
avalanche (9.7 kV $<$ HV${}_{eff}<$ 10.6 kV;
3. 3.
avalanche+streamer mode (HV${}_{eff}>$ 10.6 kV).
The best sensitivity to working point shifts is achieved in the
avalanche+streamer region, estimated to be about 25 ADC ch/10 V or 1.2pC/10V.
Figure 8: Avarege avalanche charge of the eight monitor chamber signal as a
function of HVeff. The slope is about 25 ADC ch/10 or 1.2pC/10V. Each point
corresponds to 10000 triggers.
In the efficiency method, the ADC avalanche event yield is studied as a
function of HVeff (9). The avalanche signal increases by increasing the HV
applied to the gap, until it reaches a maximum value after which the streamer
component starts to increase. The 9.0kV-10.0kV shows a sensitivity to work
point changes of approximately 1.3/%/10V.
Figure 9: Streamer and avalanche yields as a function of HVeff. Each point
corresponds to 10000 collected triggers. The solid line has a slope of
approximately 130 events/10 V corresponding to a sensitivity of 1.3%/10V.
## 4 Conclusions
The status of the Gas Gain Monitoring System for the CMS RPC Detector has been
reported on. The purpose of GGM is to monitor any shift of the working point
of the CMS RPC detector. The GGM is being commissioned at CERN and is planned
to start operation by the end of 2008. Preliminary results show good
sensitivity to working point changes. Further tests are in progress to
determine the sensitivity to gas variations.
## 5 Acknowledgements
We warmly thank F. Hahn and the CERN Gas Group for useful discussions and
cooperation.
## References
* [1] M. Abbrescia et al., “Gas analysis and monitoring systems for the RPC detector of CMS at LHC”, presented by S.Bianco at the IEEE 2006, San Diego (USA), arXiv:physics/0701014.
* [2] The CMS Collab., The CMS experiment at the CERN LHC, to appear on Journal of Instrumentation 2008\.
* [3] A. Colaleo et al., The Compact Muon Solenoid RPC Barrel Detector, these Proceedings.
* [4] R. Guida, these Proceedings.
* [5] G. Saviano, these Proceedings.
* [6] M. Abbrescia et al., Nucl. Instrum. Meth. A 550, 116 (2005).
* [7] C.A.E.N. Costruzioni Apparecchiature Elettroniche Nucleari S.p.A. Via Vetraia 11 - 55049 Viareggio (Italy).
|
arxiv-papers
| 2008-12-05T10:52:58 |
2024-09-04T02:48:59.228138
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L. Benussi (1), S. Bianco (1), S. Colafranceschi (1,2,3), D. Colonna\n (1), L. Daniello (1), F. L. Fabbri (1), M. Giardoni (1), B. Ortenzi (1), A.\n Paolozzi (1,2), L. Passamonti (1), D. Pierluigi (1), B. Ponzio (1), C. Pucci\n (1), A. Russo (1), G. Roselli (5), A. Colaleo (4), F. Loddo (4), M. Maggi\n (4), A. Ranieri (4), M. Abbrescia (4,5), G. Iaselli (4,5), B. Marangelli\n (4,5), S. Natali (4,5), S. Nuzzo (4,5), G. Pugliese (4,5), F. Romano (4,5),\n R. Trentadue (4,5), S. Tupputi (4,5), R. Guida (3), G. Polese (3,6), N.\n Cavallo (7), A. Cimmino (7,8), D. Lomidze (8), P. Noli (7,8), D. Paolucci\n (8), P. Piccolo (8), C. Sciacca (7,8), P. Baesso (9), M. Necchi (9), D.\n Pagano (9), S. P. Ratti (9), P. Vitulo (9), C. Viviani (9)",
"submitter": "Luigi Benussi Dr.",
"url": "https://arxiv.org/abs/0812.1108"
}
|
0812.1119
|
# An analysis of a random algorithm for estimating all the matchings
Jinshan Zhang zjs02@mails.tsinghua.edu.cn Yan Huo
huoy03@mails.tsinghua.edu.cn Fengshan Bai fbai@math.tsinghua.edu.cn
Department of Mathematical Sciences,
Tsinghua University, 100084,
Beijing, PRC.
###### Abstract
Counting the number of all the matchings on a bipartite graph has been
transformed into calculating the permanent of a matrix obtained from the
extended bipartite graph by Yan Huo, and Rasmussen presents a simple approach
(RM) to approximate the permanent, which just yields a critical ratio
O($n\omega(n)$) for almost all the 0-1 matrices, provided it’s a simple
promising practical way to compute this #P-complete problem. In this paper,
the performance of this method will be shown when it’s applied to compute all
the matchings based on that transformation. The critical ratio will be proved
to be very large with a certain probability, owning an increasing factor
larger than any polynomial of $n$ even in the sense for almost all the 0-1
matrices. Hence, RM fails to work well when counting all the matchings via
computing the permanent of the matrix. In other words, we must carefully
utilize the known methods of estimating the permanent to count all the
matchings through that transformation.
matching; permanent; critical ratio; bipartite graph; determinant; Monte-Carlo
algorithm;random algorithm; RM;fpras
###### pacs:
pacs numbers here
## I Introduction
Let $G=(V,E)$ be a bipartite graph, where $V=V_{1}\cup V_{2}$ is the set of
vertices and $E\subset V_{1}\times V_{2}$ is the set of edges. In the
following sections we suppose $\\#V_{1}=\\#V_{2}=n$ if there’s no special
illustration. A set of edges $S\subset E$ is called a matching if no two
distinct edges $e_{1},e_{2}\in S$ contain a common vertex. $S$ is called a
k-matching if $\\#S=k$. In special case, $S$ is called a perfect matching if
$k=n$. Let $S_{k}$ be the set of k-matching in $G$ and $A(G)$ be the set of
all the k-matching, $k=0,1,\ldots,n$. For the convenience of discussion, let
$\\#S_{0}=1$, then the number of all the matchings in $G$ is
$\\#A(G)=\sum\limits_{i=0}^{n}\\#S_{k}$.
The permanent of a 0-1 $A={a_{ij},1\leq i,j\leq n}$ is defined as
$Per(A)=\sum\limits_{\pi}\prod\limits_{i=1}^{n}a_{i,\pi(i)}$ (1)
where the sum is over all the permutations $\pi$ of $[n]=\\{1,\ldots,n\\}$.
It’s well known that the permanent of an adjacent matrix of bipartite graph
equals the number of its perfect matching. Let AM(G) denote the number of all
the matchings in $G$, and $A$ be adjacent matrix of $G$. Yhuo07 has proved
that
$AM(G)=\frac{1}{n!}per\left(\begin{array}[]{cc}A&I_{n\times n}\\\ 1_{n\times
n}&1_{n\times n}\end{array}\right)$ (2)
where $I_{n\times n}$ is $n\times n$ unit matrix, $1_{n\times n}$ denotes
$n\times n$ matrix with all the elements 1. This means in order to count the
number of all the matchings of a bipartite graph with $2n$ vertices we only
need to compute the permanent of a $2n\times 2n$ corresponding matrix
transformed from adjacent matrix. The computation of permanent has a long
history and was shown to be $\\#$P-complete in Val79 . Thus, in the past $20$
years or so, many random algorithms have been developed to approximate the
permanent, which can been divided at least four categoriesChien02 : elementary
recursive algorithms(the original one is Rasmussen method(RM)) Ras94 ;
reductions to determinants Go81 ; Kar93 ; Bar99 ; Bar00 ; iterative balancing
Lin00 ; and Markov chain Monte Carlo Sin89 ; JSV01 ; KRS96 . All these methods
try to find a fully-polynomial randomized approximation scheme fpras for
computing the permanent. fpras is such a scheme which, when given
$\varepsilon$ and inputs matrix $A$, outputs a estimator(usually a unbiased
estimator)$Y$ of the permanent such that
$Pr((1-\varepsilon)per(A)\leq Y\leq(1+\varepsilon)per(A))\geq\frac{3}{4}$ (3)
and runs in polynomial time in $n$ and $\varepsilon^{-1}$, here $3/4$ may be
boosted to $1-\delta$ for any desired $\delta>0$ by running the algorithm
$O(log(\delta^{-1}))$ and taking the median of the trials Che52 . Then a
straightforward application of Chebychev’s inequality shows that running the
algorithm $O(\frac{E(Y^{2})}{E^{2}(Y)}\varepsilon^{-2})$ times and taking the
mean of the results can make the probability more than $3/4$(e.g. running
$4\frac{E(Y^{2})}{E^{2}(Y)}\varepsilon^{-2}$ times). Hence, if the critical
ratio $\frac{E(Y^{2})}{E^{2}(Y)}$ is bounded by a polynomial of inputs $A$,
we’ll get an fpras for the permanent of $A$. Another modified scheme called
fpras for almost all inputs means: choose a matrix from
$\mathcal{A}(n,1/2)$($\mathcal{A}(n,1/2)$ denotes a probability space of
$n\times n$ 0-1 matrices where each entry is chosen to be 1 or 0 with the same
probability 1/2), or equivalently choose a matrix u.a.r. from $\mathcal{A}(n)$
($\mathcal{A}(n)$ represents the set of $n\times n$ 0-1 matrices), and the
following
Pr(critical ratio of $A$ is bounded by a polynomial of the input $A$
)$=1-o(1)$ as $n\longrightarrow\infty$
holds.(Note that this is a much weaker requirement than that of an fpras). If
a proposition $P$ relating to $n$ satisfies Pr(P is true)$=1-o(n)$, we say P
holds whp(whp is the abbreviation of ”with high probability”). Thus, that
there is an fpras for almost all the matrix means the critical ratio of $A$ is
bounded by a polynomial of the input $A$ whp. A exciting result, that Markov
Chain approach led to the first fpras for the permanent of any 0-1
matrix(actually of any matrix with nonnegative entry) was shown byJSV01 .
However, its high exponent of polynomial running time makes it difficult to be
a practical method to approximate the permanent. RM and reductions to
determinants seem to be two practical approaches estimating permanent due to
their simply feasibility, and both of them have been proved to be an fpras for
almost all the 0-1 matrices. besides, Chien02 promises a good prospect on
computing permanent via clifford algebra if some difficulties can be
conquered. RM also has developed to be a kind of approaches called sequential
importance sampling way, which is widely used in statistical physics, seeBS99
.
In this paper, we’ll, by RM, compute the number of all the matchings based on
the above transformation and give its performance theoretically, say, an
analysis of critical ratio in the sense ”for almost all the 0-1 matrix” of
that matrix with a special structure. In section II, A new alternative
estimator operating directly on the adjacent matrix without any transformation
will be presented and proved to be equivalent to approximation performing on
the transformed matrix by RM. In section III, a low bound of the critical
ratio for almost all the matrices will be presented, which is larger than any
polynomial of $n$ with a certain probability. Hence, RM does not perform well
in computing the number of all the matchings as in computing the number of
perfect matching. In section IV we’ll propose some analytic results w.r.t. the
expectation and variance of the number of all the matchings of a matrix
selected u.a.r from $\mathcal{G}(m,n)$($\mathcal{G}(m,n)$ denotes the set of
bipartite graph with $\\#V_{1}=\\#V_{2}=n$ as its vertices and exact $m$
edges). These results seem likely to contribute to the upper bound of critical
ratio for almost all matrices, but the calculations are more arduous and will
be left for latter paper.
## II An equivalent estimator
All the notations have the same meanings as those in the previous section
without special illustration. Let A an $n\times n$ 0-1 matrix be an adjacent
matrix of a bipartite graph $G=(V,E)$, $(V=V_{1}\bigcup V_{2})$. Set $Y_{A}$ a
random variable. Then RM can be stated as follows:
inputs: A an $n\times n$ 0-1 matrix;
outputs: $Y_{A}$ the estimator of permanent A;
if n=0; then
$Y_{A}=1$
else
$W=\\{j:a_{1j}=1\\}$
if $W=\emptyset$ then
$Y_{A}=0$
else
Choose $J$ u.a.r. from W
$Y_{A}=|W|Y_{1J}$
$Y_{1j}$ denotes the submatrix obtained from A by removing the 1st row and the
jth column. Note this heuristic idea comes from the Laplace’s expansion. Our
following algorithm(for easy discussion, call it AMM) is also inspired by
another expansion. we first presents our algorithm for the number of all the
matchings, and then give the explanation and proof of equivalence between AMM
and RM on the transformed matrix:
inputs: A an $n\times n$ 0-1 adjacent matrix of $G$;
outputs: $Y_{A}$ the estimator of the number of all the matchings of $G$;
if n=0; then
$Y_{A}=1$
else
$W=\\{j:a_{1j}=1\\}\bigcup\\{0\\}$
Choose $J$ u.a.r. from W
$Y_{A}=|W|Y_{1J}$
$Y_{10}$ denotes a submatrix of A by removing the 1st row(of course, it’s not
necessarily a square matrix). Define a new terminology AM on the matrix. let
$B=\\{b_{ij},1\leq i\leq m,1\leq j\leq n\\}$ an $m\times n$ matrix, $m\leq n$.
let $AM(\emptyset)=1$, by induction on $m$.
$AM(B):=AM(B_{10})+\sum\limits_{j=1}^{n}b_{1,j}B_{1j}$ (4)
Then we have the following theorem.
Theorem 1. Let A be an $n\times n$ adjacent matrix of a bipartite graph G,
Then AM(A)is the number of all the matchings of G.
Proof: It’s easy to check, when $k\geq 1$, the number of k-matching of G
equals
$\sum\limits_{i_{1},\cdots,i_{k}}\sum\limits_{\pi}a_{i_{1},\pi(i_{1})}\cdots
a_{i_{k},\pi(i_{k})}$, where $i_{1}<i_{2}\cdots<i_{k}$ chosen from
$\\{1,2,\cdots,n\\}$, $\pi$ denotes the permutation
of$\\{i_{1},i_{2},\cdots,i_{k}\\}$. Thus, the number of all the matchings
is$\sum\limits_{k=1}^{n}\sum\limits_{i_{1}<\cdots<i_{k}\\\
\subseteq\\{1,\cdots,n\\}}\sum\limits_{\pi}a_{i_{1},\pi(i_{1})}\cdots
a_{i_{k},\pi(i_{k})}+1$, where 1 denotes the number of 0-matching. Note that
if the AM(A) is written in terms of sum of elements of the matrix A, then it’s
clearly to see $AM(A)=\sum\limits_{k=1}^{n}\sum\limits_{i_{1}<\cdots<i_{k}\\\
\subseteq\\{1,\cdots,n\\}}\sum\limits_{\pi}a_{i_{1},\pi(i_{1})}\cdots
a_{i_{k},\pi(i_{k})}+1$.$\Box$
Corollary1. Let $A=\\{a_{ij}1\leq i,j\leq n\\}$ be an $n\times n$ 0-1 matrix
and $Y_{A}$ is obtained by above AMM. Then $Y_{A}$ is unbiased for AM(A),
$E(Y_{A})=AM(A)$
Proof: We prove for any $m\times n$ 0-1 matrix A, $1\leq i\leq m,1\leq j\leq
n\\}$, which will be widely used in the following proves. AMM is unbiased for
AM(A). For any fixed $n$, by induction on m, k=0,$\forall 1\leq l\leq n$,
$\forall$ a $k\times l$ 0-1 matrix A, the equation $E(Y_{A})=AM(A)$ is
trivial. Now suppose $\forall k\leq m,k\leq l\leq n$, a $k\times l$ 0-1 matrix
A has $E(Y_{A})=AM(A)$. Then when $k=m$, let $|W|=q$, we have
$\begin{split}E(Y_{A})&=\sum\limits_{j\in W}E(Y_{A}|J=j)Pr(J=j)\\\
&=\sum\limits_{j\in W}E(qY_{A_{1j}}|J=j)q^{-1}\\\ &=\sum\limits_{j\in
W}E(Y_{A_{1j}})\\\ &=\sum\limits_{j\in W}AM(A_{1j})\\\ &=AM(A).\end{split}$
$\Box$
Another simple corollary can also be obtained. To estimate the number of all
the matching in G, by RM operating on $B=\left(\begin{array}[]{cc}A&I_{n\times
n}\\\ 1_{n\times n}&1_{n\times n}\end{array}\right)$ divided by $n!$ is
equivalent to operating on A by AMM, in precise words, which can be stated as
follows.
Corollary2. Let $X_{A}$ be the output of RM operating on $A$ , $Y_{B}$ be the
output of AMM operating on transformed matrix $B$ divided by $n!$. Then
$X_{A}$ and $Y_{B}$ has the same distribution.
Proof: Note that by RM after n-th step operating on $B$,
$Y_{B}=S_{n}*Y_{1_{n\times n}}/n!$, where $S_{n}$ is a number obtained from
the first n steps, and obviously $Y_{1_{n\times n}}\equiv n!$. Hence, we have
$Y_{B}=S_{n}$. The same distribution of $S_{n}$ and $X_{A}$ can be verified
step by step.$\Box$
Corollary3. $AM(A)=\frac{1}{n!}per\left(\begin{array}[]{cc}A&I_{n\times n}\\\
1_{n\times n}&1_{n\times n}\end{array}\right)$.
Proof: This is a direct deduction of corollary2. Let $X_{A}$ be the output of
RM operating on $A$ , $Y_{B}$ be the output of AMM operating on transformed
matrix $B$ divided by $n!$.
$AM(A)=E(X_{A})=E(Y_{B})=\frac{1}{n!}per\left(\begin{array}[]{cc}A&I_{n\times
n}\\\ 1_{n\times n}&1_{n\times n}\end{array}\right)$
$\Box$
So in the following section, we’ll use AMM to compute all the matchings
instead of RM since some methodologies similar to Rasmussen can be utilized.
Another small advantage by AMM is that the critical ratio is smaller than that
directly obtained from RM. The critical ratio by RM would be $(2n)!$, see
Theorem 2.2Ras94 , while the critical ratio by AMM would be $(n+1)^{n}$.
Theorem2. Let $A=\\{a_{ij},1\leq i,j\leq n\\}$ be an $n\times n$ adjacent
matrix of a bipartite graph G, and let $X_{A}$ be the output of AMM. Then
$\frac{E(X_{A})^{2}}{E(X^{2}_{A})}\leq(n+1)^{n}$. Generally, Let A be an
$m\times n$ 0-1 matrix, $m\leq n$. $X_{A}$ be the output of AMM. Then
$\frac{E(X_{A})^{2}}{E(X^{2}_{A})}\leq(n+1)^{m}$
Proof: Induction on $m$, For any fixed $n$. $k=0$,$\forall 1\leq l\leq n$,
$\forall$ a $k\times l$ 0-1 matrix $A$, the inequation is trivial. In the case
$k=m$, let $|W|=q$, we have
$\begin{split}E(X^{2}_{A})&=\sum\limits_{j\in W}E(X^{2}_{A}|J=j)Pr(J=j)\\\
&=\sum\limits_{j\in W}E(q^{2}X^{2}_{A_{1j}}|J=j)q^{-1}\\\ &=\sum\limits_{j\in
W}E(X^{2}_{A_{1j}})q\\\ &\leq\sum\limits_{j\in
W}E(X_{A_{1j}})^{2}(n+1)^{m-1}q\\\ &\leq(\sum\limits_{j\in
W}E(X_{A_{1j}}))^{2}(n+1)^{m-1}q\\\ &=E(X_{A})^{2}(n+1)^{m}\end{split}$
$\Box$
## III A lower bound of critical ratio for almost all the matrices
Rasmussen shows that although the critical ratio of RM is factorial in n, it
does indeed provide an fpras for almost all the matrix. However, the similar
result can not be anticipated when computing all the matchings by RM. In fact
the critical ratio for almost all the matrix would be more than
$n^{\sqrt{n}/2-1}$ with a certain probability. To prove this, we need to
define some new denotations. Since there’re two probability spaces, we use the
subscript $\sigma$ denote the calculus w.r.t. the probability space the
algorithm lies in, say, coin-tosses, and subscript $\mathcal{A}$ represent the
calculus w.r.t. the space probability the random matrices lie in.
$\mathcal{A}(m,n,p)$ denotes the probability space of all $m\times n$ 0-1
random matrices where each entry is chosen to be 1 with probability $p$,
and$\mathcal{A}(m,n)$ denotes the set of all $m\times n$ 0-1 matrices .
To obtain the mean and variance of the output of AMM on average under
probability measure $Pr_{\mathcal{A}}$, we need the following lemma.
Lemma1 Let $f(m,n)$ defined as $f(m,n)=a_{n}f(m-1,n)+c_{n}f(m-1,n-1)$, where
$m\leq n$ are two nonnegative integers, $a_{n}$ and $c_{n}$ are two infinite
positive series w.r.t. $n$. And $\forall$ $0\leq l\leq n$, $f(0,l)=1$. Then
$f(m,n)=\sum\limits_{k=1}^{m}\sum\limits_{\begin{subarray}{c}s_{0}+s_{1}+\cdots+s_{k}=m-k\\\
s_{0},\cdots s_{k}\geq 0\end{subarray}}c_{n}\cdots
c_{n-k+1}a_{n}^{s_{0}}\cdots a_{n-k}^{s_{k}}+a_{n}^{m}$
Proof: By induction on m. Obviously, the case p=0 is trivial. Suppose when
$p\leq m-1$ $\forall$ $p\leq l\leq n$,
$f(p,l)=\sum\limits_{k=1}^{p}\sum\limits_{s_{0}+s_{1}+\cdots+s_{k}=p-k}c_{l}\cdots
c_{l-k+1}a_{l}^{s_{0}}\cdots a_{l-k}^{s_{k}}+a_{l}^{p}$ holds, then when p=m,
we have
$\begin{split}a_{n}f(m-1,n)&=\sum\limits_{k=1}^{m-1}\sum\limits_{s_{0}+s_{1}+\cdots+s_{k}=m-1-k}c_{n}\cdots
c_{n-k+1}a_{n}^{s_{0}+1}\cdots a_{n-k}^{s_{k}}+a_{n}^{m}\\\
&=\sum\limits_{k=1}^{m-1}\sum\limits_{\begin{subarray}{c}s_{0}+s_{1}+\cdots+s_{k}=m-k\\\
s_{0}\geq 1\end{subarray}}c_{n}\cdots c_{n-k+1}a_{n}^{s_{0}}\cdots
a_{n-k}^{s_{k}}+a_{n}^{m}\end{split}$
and
$\begin{split}c_{n}f(m-1,n-1)&=\sum\limits_{k=1}^{m-1}\sum\limits_{s_{0}+s_{1}+\cdots+s_{k}=m-1-k}c_{n}\cdots
c_{n-k}a_{n-1}^{s_{0}}\cdots a_{n-1-k}^{s_{k}}+c_{n}a_{n-1}^{m-1}\\\
&=\sum\limits_{k=1}^{m-1}\sum\limits_{s_{1}+s_{2}+\cdots+s_{k+1}=m-1-k}c_{n}\cdots
c_{n-k}a_{n-1}^{s_{1}}\cdots a_{n-1-k}^{s_{k+1}}+c_{n}a_{n-1}^{m-1}\\\
&=\sum\limits_{k=2}^{m}\sum\limits_{s_{1}+s_{2}+\cdots+s_{k}=m-k}c_{n}\cdots
c_{n-k+1}a_{n-1}^{s_{1}}\cdots a_{n-k}^{s_{k}}+c_{n}a_{n-1}^{m-1}\\\
&=\sum\limits_{k=1}^{m}\sum\limits_{s_{1}+s_{2}+\cdots+s_{k}=m-k}c_{n}\cdots
c_{n-k+1}a_{n-1}^{s_{1}}\cdots a_{n-k}^{s_{k}}\\\
&=\sum\limits_{k=1}^{m}\sum\limits_{\begin{subarray}{c}s_{0}+s_{1}+\cdots+s_{k}=m-k\\\
s_{0}=0\end{subarray}}c_{n}\cdots c_{n-k+1}a_{n}^{s_{0}}a_{n-1}^{s_{1}}\cdots
a_{n-k}^{s_{k}}\\\ \end{split}$
From the above two equation, there holds
$\begin{split}f(m,n)&=a_{n}f(m-1,n)+c_{n}f(m-1,n-1)\\\
&=\sum\limits_{k=1}^{m}\sum\limits_{\begin{subarray}{c}s_{0}+s_{1}+\cdots+s_{k}=m-k\\\
s_{0},\cdots s_{k}\geq 0\end{subarray}}c_{n}\cdots
c_{n-k+1}a_{n}^{s_{0}}\cdots a_{n-k}^{s_{k}}+a_{n}^{m}\end{split}$
The previous $n$ can be replaced by any $l$, where $m\leq l\leq n$ $\Box$
Using lemma1 we can easily obtain two following Theorems.
Theorem3. Choose $A_{m,n}$ u.a.r. from $\mathcal{A}(m,n)$, $m\leq n$, or
equivalently let $A_{m,n}$ from $\mathcal{A}(m,n,1/2)$. Then
$E_{\mathcal{A}}(AM(A_{m,n}))=\sum\limits_{k=0}^{m}C_{m}^{k}\frac{P_{n}^{k}}{2^{k}}$
where $C_{m}^{k}=\frac{m!}{k!(m-k)!}$ and $P_{n}^{k}=\frac{n!}{(n-k)!}$
Proof: Induction on m. The case p=0, $E_{\mathcal{A}}(AM(A))=1$ is trivial.
Suppose $\forall$ $p\leq m-1$, $p\leq l\leq n$
$E_{\mathcal{A}}(AM(A_{p,l}))=\sum\limits_{k=0}^{p}C_{p}^{k}\frac{P_{l}^{p-k}}{2^{p-k}}=\sum\limits_{k=0}^{p}C_{p}^{k}\frac{P_{l}^{k}}{2^{k}}$
when p=m, $\forall$ $m\leq l\leq n$, we have
$\begin{split}E_{\mathcal{A}}(AM(A_{m,l}))&=E_{\mathcal{A}}(AM(A_{m,l}^{1,0})+\sum\limits_{j=1}^{n}a_{1,j}AM(A_{m,l}^{1,j}))\\\
&=E_{\mathcal{A}}(AM(A_{m-1,l}))+\sum\limits_{j=1}^{n}E_{\mathcal{A}}(a_{1,j})E_{\mathcal{A}}(AM(A_{m-1,l-1})\\\
&=E_{\mathcal{A}}(AM(A_{m-1,l}))+\frac{n}{2}E_{\mathcal{A}}(AM(A_{m-1,l-1})\\\
\end{split}$
Using lemma1, here $a_{l}\equiv 1$, and$c_{l}=\frac{l}{2}$ then
$\begin{split}E_{\mathcal{A}}(AM(A_{m,l}))&=\sum\limits_{k=1}^{m}\sum\limits_{\begin{subarray}{c}s_{0}+s_{1}+\cdots+s_{k}=m-k\\\
s_{0},\cdots s_{k}\geq 0\end{subarray}}c_{l}\cdots c_{l-k+1}+1\\\
&=\sum\limits_{k=1}^{m}\frac{P_{l}^{k}}{2^{k}}\sum\limits_{\begin{subarray}{c}s_{0}+s_{1}+\cdots+s_{k}=m-k\\\
s_{0},\cdots s_{k}\geq 0\end{subarray}}1+1\\\
&=\sum\limits_{k=1}^{m}\frac{P_{l}^{k}}{2^{k}}C_{m}^{k}+1\\\
&=\sum\limits_{k=0}^{m}\frac{P_{l}^{k}}{2^{k}}C_{m}^{k}\end{split}$
$\Box$
Theorem4 Choose $A_{m,n}$ u.a.r. from $\mathcal{A}(m,n)$, $m\leq n$, and let
$X_{A_{m,n}}$ be the output by AMM. Then
$E_{\mathcal{A}}(E_{\sigma}(X_{A_{m,n}}))=\sum\limits_{k=0}^{m}C_{m}^{k}\frac{P_{n}^{k}}{2^{k}}$
and
$E_{\mathcal{A}}(E_{\sigma}(X_{A_{m,n}}^{2}))=\sum\limits_{k=0}^{m}\frac{P_{n}^{k}P_{n+3}^{k}}{2^{m+k}}\sum\limits_{\begin{subarray}{c}s_{0}+s_{1}+\cdots+s_{k}=m-k\\\
s_{0},\cdots s_{k}\geq
0\end{subarray}}(n+2)^{s_{0}}(n+2-1)^{s_{1}}\cdots(n+2-k)^{s_{k}}$
Proof: The first equation is is trivial since
$E_{\sigma}(X_{A_{m,n}}^{2})=AM(A_{m,l})$. For the second one, we use
induction on m. The case p=0 is obvious. Suppose $\forall A_{p,l}$ where
$0\leq p\leq m-1$, $p\leq l\leq n$ the second equation holds. When $p=m$,
noting the fact $M=|W|-1$ is a binomial variable with parameter $l$ and
$1/2$(recall $W/\\{0\\}$ is the set of column indices with a 1 in the first
row), then
$\begin{split}E_{\mathcal{A}}(E_{\sigma}(X_{A_{m,l}}^{2}))&=\sum\limits_{q=0}^{l}E_{\mathcal{A}}(E_{\sigma}(X_{A_{m,l}}^{2})|M=q)Pr_{\mathcal{A}}(M=q)\\\
&=\sum\limits_{q=0}^{l}E_{\mathcal{A}}((q+1)\sum\limits_{j\in
W}E_{\sigma}(X_{A_{m,l}^{1j}}^{2})|M=q)Pr_{\mathcal{A}}(M=q)\\\
&=\sum\limits_{q=0}^{l}E_{\mathcal{A}}((q+1)E_{\sigma}(X_{A_{m-1,l}}^{2})+q(q+1)E_{\sigma}(X_{A_{m-1,l-1}}^{2}))Pr_{\mathcal{A}}(M=q)\\\
&=(E_{\mathcal{A}}(M)+1)E_{\mathcal{A}}(E_{\sigma}(X_{A_{m-1,l}}^{2}))+(E_{\mathcal{A}}(M^{2})+E_{\mathcal{A}}(M))E_{\mathcal{A}}(E_{\sigma}(X_{A_{m-1,l-1}}^{2}))\\\
&=(\frac{l+2}{2})E_{\mathcal{A}}(E_{\sigma}(X_{A_{m-1,l}}^{2}))+(\frac{l^{2}+3l}{4})E_{\mathcal{A}}(E_{\sigma}(X_{A_{m-1,l-1}}^{2}))\\\
\end{split}$
Using lemma1, here $a_{l}=\frac{l+2}{2}$, and $c_{l}=\frac{l^{2}+3l}{4}$.Then
$\begin{split}E_{\mathcal{A}}(E_{\sigma}(X_{A_{m,l}}^{2}))&=\sum\limits_{k=1}^{m}\frac{P_{l}^{k}P_{l+3}^{k}}{4^{k}}\sum\limits_{\begin{subarray}{c}s_{0}+s_{1}+\cdots+s_{k}=m-k\\\
s_{0},\cdots s_{k}\geq
0\end{subarray}}(\frac{l+2}{2})^{s_{0}}(\frac{l+2-1}{2})^{s_{1}}\cdots(\frac{l+2-k}{2})^{s_{k}}+(\frac{l+2}{2})^{m}\\\
&=\sum\limits_{k=0}^{m}\frac{P_{l}^{k}P_{l+3}^{k}}{2^{k+m}}\sum\limits_{\begin{subarray}{c}s_{0}+s_{1}+\cdots+s_{k}=m-k\\\
s_{0},\cdots s_{k}\geq
0\end{subarray}}(l+2)^{s_{0}}(l+2-1)^{s_{1}}\cdots(l+2-k)^{s_{k}}\\\
\end{split}$
$\Box$
Theorem5 Choose $A_{n,n}$ u.a.r. from $\mathcal{A}(n,n)$, and let
$X_{A_{n,n}}$ be the output by AMM. Then whp $h(n)\leq
E_{\mathcal{A}}(E_{\sigma}(X_{A_{n,n}}))\leq nh(n)$, where
$h(n)=\frac{(n!)^{2}}{2^{n}}\frac{2^{k^{\ast}}}{(n-k^{\ast})!(k^{\ast})^{2}}$,
$k^{\ast}=\lfloor-1+\sqrt{2n+3}\rfloor$. where $\lfloor\ast\rfloor$ denotes
the largest integer no more than $\ast$.
Proof:
$\begin{split}E_{\mathcal{A}}(E_{\sigma}(X_{A_{n,n}}))&=\sum\limits_{k=0}^{n}C_{n}^{k}\frac{P_{n}^{k}}{2^{k}}\\\
&=\sum\limits_{k=0}^{n}C_{n}^{k}\frac{P_{n}^{n-k}}{2^{n-k}}\\\
&=\frac{(n!)^{2}}{2^{n}}\sum\limits_{k=0}^{n}\frac{2^{k}}{(n-k)!(k!)^{2}}\end{split}$
and let $b_{k}=\frac{2^{k}}{(n-k)!(k!)^{2}}$, then
$\frac{b_{k}}{b_{k-1}}=\frac{2(n-k+1)}{k^{2}}$, set $\frac{b_{k}}{b_{k-1}}\geq
1$ we have $k\leq-1+\sqrt{2n+3}$, thus,
$b_{k^{\ast}}=\max\limits_{k=0,\cdots,n}b_{k}$. Thus, obviously
$\frac{(n!)^{2}}{2^{n}}b_{k^{\ast}}\leq
E_{\mathcal{A}}(E_{\sigma}(X_{A_{n,n}}))\leq
n\frac{(n!)^{2}}{2^{n}}b_{k^{\ast}}$
$\Box$
Theorem6 Choose $A_{n,n}$ u.a.r. from $\mathcal{A}(n,n)$, and let
$X_{A_{n,n}}$ be the output by AMM. Then whp
$\frac{E_{\mathcal{A}}(E_{\sigma}(X_{A_{n,n}}^{2}))}{E_{\mathcal{A}}^{2}(E_{\sigma}(X_{A_{n,n}}))}\geq
n^{(\sqrt{n}/2)}$
Proof: Numerical experiment shows the above result. however the theoretical
analysis seems so hard than until now I haven’t thought out the way to show
the comparably tight for $E_{\mathcal{A}}(E_{\sigma}(X_{A_{n,n}}^{2}))$ since
the order of
$\sum\limits_{s_{0}+s_{1}+\cdots+s_{k}=n-k}(n+2)^{s_{0}}(n+2-1)^{s_{1}}\cdots(n+2-k)^{s_{k}}$
is too difficult to gain a good lower bound. The following bound is easy to
check and the best one among methods I thought out,
$E_{\mathcal{A}}(E_{\sigma}(X_{A_{n,n}}^{2}))\geq\sum\limits_{k=0}^{n}\frac{(n!)^{2}(n+3)!}{2^{2n}}\frac{2^{k}(k+2)^{k}}{(k!)^{2}(k+3)!(n-k)!}$
However it still can’t reach the goal. Therefore, the proof of this theorem
will be left for the future.
Even if Theorem6 has been proved, unfortunately, the critical ratio for almost
all the matrices can not obtained from this theorem since two random variables
are not independent. In order to accomplish the ultimate result, we need to
calculate the $E_{\mathcal{A}}(E_{\sigma}^{2}(X_{A_{n,n}}^{2}))$. Using the
induction similar to theorem4, we can obtain the recursion of
$E_{\mathcal{A}}(E_{\sigma}^{2}(X_{A_{m,n}}^{2}))$(recall M is a binomial
variable with parameter n and $\frac{1}{2}$).
$E_{\mathcal{A}}(E_{\sigma}^{2}(X_{A_{m,n}}^{2}))=2(E_{\mathcal{A}}(M^{3})+2E_{\mathcal{A}}(M^{2})+E_{\mathcal{A}}(M))E_{\mathcal{A}}(E_{\sigma}(X_{A_{m-1,n}}^{2})E_{\sigma}(X_{A_{m-1,n-1}}^{2}))\\\
+(E_{\mathcal{A}}(M^{2})+2E_{\mathcal{A}}(M)+1)E_{\mathcal{A}}(E_{\sigma}^{2}(X_{A_{m-1,n}}^{2}))+(E_{\mathcal{A}}(M^{4})+2E_{\mathcal{A}}(M^{3})+E_{\mathcal{A}}(M^{2}))E_{\mathcal{A}}(E_{\sigma}^{2}(X_{A_{m-1,n-1}}^{2}))$
Comparing $E_{\mathcal{A}}(E_{\sigma}^{2}(X_{A_{m,n}}^{2}))$ with
$E_{\mathcal{A}}^{2}(E_{\sigma}(X_{A_{n,n}}^{2}))$ and computing their ratio
have to be done. Our main aim of doing this is to find the matrices satisfying
$E_{\sigma}(X_{A_{m,n}}^{2})\leq
E_{\mathcal{A}}(E_{\sigma}^{2}(X_{A_{m,n}}^{2}))g(n)$, where $g(n)$ is a
polynomial of $n$. However, the ratio of
$\frac{E_{\mathcal{A}}(E_{\sigma}^{2}(X_{A_{m,n}}^{2}))}{E_{\mathcal{A}}^{2}(E_{\sigma}(X_{A_{n,n}}^{2}))}$
is so large that it can’t accomplish our goal. Thus we deduce our requirement
whp to with a certain probability $p>0$, and in our results
$p=\frac{1}{2}-\varepsilon$ where $\varepsilon$ is no more than 0.02. To prove
the theorem, we need the following lemma, which will be proved in section IV.
Lemma2 Let $\mathcal{B}(m,n)$ denote the set of all $n\times n$ 0-1 matrices
with exact m 1’s, $m\gg n$. Choose $B$ u.a.r. from $\mathcal{B}(m,n)$. Then
$E(AM(B))=\sum\limits_{k=0}^{n}(C_{n}^{k})^{2}k!\frac{C_{n^{2}-k}^{m-k}}{C_{n^{2}}^{m}}$
and
$\frac{E(AM^{2}(B))}{E^{2}(AM(B))}=1+o(1),n\rightarrow\infty$
Theorem7 Choose $A_{n,n}$ u.a.r. from $\mathcal{A}(n,n)$, and let
$X_{A_{n,n}}$ be the output by AMM. Then
$Pr(\frac{E_{\sigma}(X_{A_{n,n}}^{2})}{E_{\sigma}^{2}(X_{A_{n,n}})}\geq
n^{\sqrt{n}/2-1})\geq\frac{\sum\limits_{i=(1/2+\varepsilon)n^{2}}^{n^{2}}C_{n^{2}}^{k}}{2^{n^{2}}}$
where c is a constant no more 10, and$\varepsilon\leq 0.02$.
Proof: From lemma2 we know if we set $m=(1/2+\varepsilon)n^{2}$ and
$q=\frac{C_{n^{2}-k}^{m-k}}{C_{n^{2}}^{m}}$. When n goes to infinity, noting
$k\leq n\ll m,n^{2}$, there holds
$q=\frac{C_{n^{2}-k}^{m-k}}{C_{n^{2}}^{m}}=\frac{m(m-1)\cdots(m-k)}{n^{2}(n^{2}-1)\cdots(n^{2}-k)}$
and
$\begin{split}ln(q)&=\sum\limits_{i=0}^{k-1}[ln(m-i)-ln(n^{2}-i)]\\\
&=kln(\frac{m}{n^{2}})+\sum\limits_{i=0}^{k-1}[ln(1-\frac{i}{m})-ln(1-\frac{i}{n^{2}})]\\\
&=kln(\frac{m}{n^{2}})-\sum\limits_{i=0}^{k-1}[\frac{i}{m}-\frac{i}{n^{2}}+O(\frac{i^{2}}{m^{2}})]\\\
&=kln(\frac{m}{n^{2}})-\frac{k(k-1)}{2}(\frac{1}{m}-\frac{1}{n^{2}})+O(\frac{k^{3}}{m^{2}})\end{split}$
Thus, noting that $km^{-1}\leq 2nm^{-1}=O(n^{3}m^{-2})$
$\begin{split}q&=(\frac{m}{n^{2}})^{k}exp[-\frac{k^{2}}{2}(\frac{1}{m}-\frac{1}{n^{2}})+O(\frac{n^{3}}{m^{2}})]\\\
&=(\frac{(1/2+\varepsilon)n^{2}}{n^{2}})^{k}exp[-\frac{k^{2}}{2}(\frac{1}{(1/2+\varepsilon)n^{2}}-\frac{1}{n^{2}})+O(\frac{n^{3}}{((1/2+\varepsilon)n^{2})^{2}})]\\\
&\leq e^{-1}(1/2+\varepsilon)^{k}\end{split}$
Let B selected u.a.r. from $\mathcal{B}(m,n)$ Since
$\frac{E(AM^{2}(B))}{E^{2}(AM(B))}=1+o(1)$, as $n\rightarrow\infty$
then $Pr(AM(B)<\frac{5}{6}E(AM(B)))\rightarrow 0$, as $n\rightarrow\infty$.
So, if $m\geq(1/2+\varepsilon)n^{2}$ and $\varepsilon\leq 0.02$, we have whp
$\begin{split}E_{\sigma}(X_{B}^{2})&\geq E_{\sigma}^{2}(X_{B})\\\
&=AM^{2}(B)\\\ &\geq(\frac{5}{6}E(AM(B)))^{2}\\\
&=(\frac{5}{6}\sum\limits_{k=0}^{n}(C_{n}^{k})^{2}k!\frac{C_{n^{2}-k}^{m-k}}{C_{n^{2}}^{m}})^{2}\\\
&\geq(\sum\limits_{k=0}^{n}(C_{n}^{k})^{2}k!\frac{5e^{-1}}{6}(1/2+\varepsilon)^{k})^{2}\\\
&\geq\sum\limits_{k=0}^{n}\frac{P_{n}^{k}P_{n+3}^{k}}{2^{n+k}}\sum\limits_{\begin{subarray}{c}s_{0}+s_{1}+\cdots+s_{k}=n-k\\\
s_{0},\cdots s_{k}\geq
0\end{subarray}}(n+2)^{s_{0}}(n+2-1)^{s_{1}}\cdots(n+2-k)^{s_{k}}\\\
&=E_{\mathcal{A}}(E_{\sigma}(X_{A_{n,n}})).\end{split}$
Noting
$Pr(A\in\bigcup\limits_{m\geq(1/2+\varepsilon)n^{2}}\mathcal{B}(m,n))=\frac{\sum\limits_{i=(1/2+\varepsilon)n^{2}}^{n^{2}}C_{n^{2}}^{k}}{2^{n^{2}}}$,
thus $Pr(E_{\sigma}(X_{A_{n,n}}^{2})\geq
E_{\mathcal{A}}(E_{\sigma}(X_{A_{n,n}})))\geq\frac{\sum\limits_{i=(1/2+\varepsilon)n^{2}}^{n^{2}}C_{n^{2}}^{k}}{2^{n^{2}}}$.
Using Markov’s inequality,
$Pr(E_{\sigma}(X_{A_{n,n}}^{2})\geq
nE_{\mathcal{A}}(E_{\sigma}(X_{A_{n,n}})))\leq\frac{1}{n}\rightarrow 0$
then whp $E_{\sigma}(X_{A_{n,n}}^{2})\leq
nE_{\mathcal{A}}(E_{\sigma}(X_{A_{n,n}}))$. Finally, we have
$Pr(\frac{E_{\sigma}(X_{A_{n,n}}^{2})}{E_{\sigma}^{2}(X_{A_{n,n}})}\geq\frac{1}{n}\frac{E_{\mathcal{A}}(E_{\sigma}(X_{A_{n,n}}^{2}))}{E_{\mathcal{A}}(E_{\sigma}(X_{A_{n,n}}))})\geq\frac{\sum\limits_{i=(1/2+\varepsilon)n^{2}}^{n^{2}}C_{n^{2}}^{k}}{2^{n^{2}}}$
Apply theorem6 to the above formula, we have
$Pr(\frac{E_{\sigma}(X_{A_{n,n}}^{2})}{E_{\sigma}^{2}(X_{A_{n,n}})}\geq
n^{\sqrt{n}/2-1})\geq\frac{\sum\limits_{i=(1/2+\varepsilon)n^{2}}^{n^{2}}C_{n^{2}}^{k}}{2^{n^{2}}}$
## IV The number of all the matchings on random graph.
In this section, we consider the expectation and variance of the number of all
the matchings on G selected u.a.r. from $\mathcal{G}(m,n)$. We have the
following theorem.
Theorem8 Choose G u.a.r. from $\mathcal{G}(m,n)$, where $\mathcal{G}(m,n)$
denotes the set of bipartite graph with $\\#V_{1}=\\#V_{2}=n$ as its vertices
and exact $m$ edges, $m\gg n$, and let AM(G) denotes the number of all the
matchings in G. Then we have
$E(AM(G))=\sum\limits_{k=0}^{n}(C_{n}^{k})^{2}k!E(X_{M(k)})$
and
$E(AM^{2}(G))=\sum\limits_{k=0}^{n}\sum\limits_{i=0}^{k}(C_{n}^{k})^{2}k!\sum\limits_{p=0}^{min(i,n-k)}C_{n-k}^{p}C_{k}^{i-p}P_{n-i+p}^{p}\sum\limits_{j=0}^{i-p}C_{i-p}^{j}[F_{n-j}(i-p-j)]E(X_{M(k+i-j)})\\\
+\sum\limits_{k=1}^{n}\sum\limits_{i=0}^{k-1}(C_{n}^{k})^{2}k!\sum\limits_{p=0}^{min(i,n-k)}C_{n-k}^{p}C_{k}^{i-p}P_{n-i+p}^{p}\sum\limits_{j=0}^{i-p}C_{i-p}^{j}[F_{n-j}(i-p-j)]E(X_{M(k+i-j)})$
where $E(X_{M(k)})=C_{n^{2}-k}^{m-k}/C_{n^{2}}^{m}$ and
$F_{n}(p)=\sum\limits_{r=0}^{p}(-1)^{r}C_{p}^{r}P_{n-r}^{p-r}$
Proof: we’ll use the methodology in Jerr95 ; Let $M(k)$ be a k-matching on
$V_{1}+V_{2}$, For $G\in\mathcal{G}(m,n)$, define the random variable
$X_{M}(G)$ to be 1 if $M(k)$ is contained in G, and otherwise 0. The
expectation and second moment of $AM(G)$ is as follows.
$E(AM(G))=E(\sum\limits_{k=0}^{n}\sum\limits_{M(k)}X_{M(k)})=\sum\limits_{k=0}^{n}\sum\limits_{M(k)}E(X_{M(k)})$
and
$E(AM^{2}(G))=E((\sum\limits_{k=0}^{n}\sum\limits_{M(k)}X_{M(k)})^{2})=\sum\limits_{k=0}^{n}\sum\limits_{i=0}^{n}\sum\limits_{M(k),M^{{}^{\prime}}(i)}E(X_{M(k)}X_{M(i)}^{{}^{\prime}})$
where $\forall 0\leq k\leq n$, $M(k)$ and $M^{{}^{\prime}}(k)$ range over all
$(C_{n}^{k})^{2}k!$ k-matching’s on $V_{1}+V_{2}$. Note that
$E(X_{M(k)})=\frac{C_{n^{2}-k}^{m-k}}{C_{n^{2}}^{m}}$
The first equation follows quickly. For the second, in order to compute
$E(X_{M(k)}X_{M(i)}^{{}^{\prime}})$, we have to calculate the number of pairs
of $M(k)$ and $M^{{}^{\prime}}(i)$ as a function of the overlap
$j=|M(k)\bigcap M^{{}^{\prime}}(i)|$. For any fixed $k$, suppose $i\leq k$, we
need to compute the number of the pairs of $M(k)$ and $M^{{}^{\prime}}(i)$,
where $i=0,\cdots,k$, and $M^{{}^{\prime}}(i)$ ranges over all
$(C_{n}^{i})^{2}i!$ $i$-matching’s on $V_{1}+V_{2}$. The problem can be
equivalently stated as follows: There’re $n$ different letters and $n$
different envelopes. Among these letters, there’re exact $k(0\leq k\leq n)$
labeled letters, each of which has only one ’mother envelope’ among envelopes.
Different labeled letters have different mother envelopes. We call a $j$-fit
if there’re exact $j$ labeled letters put into its own mother envelope. Now
choose $i$($0\leq i\leq k$)letters from these $n$ letters, then put them into
$i$ envelopes, and each letter can only be put into one envelope. $\forall$
possible $j$, how many circumstances of $j$-fit are there? We can solve this
problem like this: Suppose there’re $p$ letters unlabeled and $i-p$ labeled
letters among the selected letters, obviously, $0\leq p\leq min(n-k,i)$, the
number of ways of choosing letters is $C_{n-k}^{p}C_{k}^{i-p}$. If the labeled
letters has been laid, then the number of the ways of putting $p$ unlabeled
letters is $P_{n-(i-p)}^{p}$. For any $j$($0\leq j\leq i-p$), there’re
$C_{i-p}^{j}$ ways putting exact $j$ labeled letters in its own mother
envelope. The last one we need to deal with is how many ways to put $i-p-j$
labeled letters into $n-j$ envelopes which contain all these $i-p-j$ letters’
mother envelopes, satisfying $0$-fit. By the principle of inclusion-exclusion
seeHal67 , we can easily obtain the number of the ways is $F_{n-j}(i-p-j)$,
where $F_{n}(p)=\sum\limits_{r=0}^{p}(-1)^{r}C_{p}^{r}P_{n-r}^{p-r}$. Noting
that $p$ ranges over $0$ to $min(i,n-k)$, and $j$ ranges over $0$ to $i-p$,
for each $k$ and $i\leq k$. Then
$\begin{split}\sum\limits_{M^{{}^{\prime}}(i)}E(X_{M(k)}X_{M(i)}^{{}^{\prime}})&=\sum\limits_{p=0}^{min(i,n-k)}C_{n-k}^{p}C_{k}^{i-p}P_{n-i+p}^{p}\sum\limits_{j=0}^{i-p}C_{i-p}^{j}[F_{n-j}(i-p-j)]E(X_{M(k+i-j)})\end{split}$
where $E(X_{M(k)})=C_{n^{2}-k}^{m-k}/C_{n^{2}}^{m}$ and
$F_{n}(p)=\sum\limits_{r=0}^{p}(-1)^{r}C_{p}^{r}P_{n-r}^{p-r}$.
Consider,
$\begin{split}\sum\limits_{k=0}^{n}\sum\limits_{i=0}^{n}\sum\limits_{M(k),M^{{}^{\prime}}(i)}E(X_{M(k)}X_{M(i)}^{{}^{\prime}})&=(\sum\limits_{k=0}^{n}\sum\limits_{i=0}^{k}+\sum\limits_{k=0}^{n-1}\sum\limits_{i=k+1}^{n})\sum\limits_{M(k),M^{{}^{\prime}}(i)}E(X_{M(k)}X_{M(i)}^{{}^{\prime}})\\\
&=(\sum\limits_{k=0}^{n}\sum\limits_{i=0}^{k}+\sum\limits_{k=0}^{n-1}\sum\limits_{i=k+1}^{n})\sum\limits_{M(k),M^{{}^{\prime}}(i)}E(X_{M(k)}X_{M(i)}^{{}^{\prime}})\\\
&=(\sum\limits_{k=0}^{n}\sum\limits_{i=0}^{k}+\sum\limits_{i=1}^{n}\sum\limits_{k=0}^{i-1})\sum\limits_{M(k),M^{{}^{\prime}}(i)}E(X_{M(k)}X_{M(i)}^{{}^{\prime}})\\\
&=(\sum\limits_{k=0}^{n}\sum\limits_{i=0}^{k}+\sum\limits_{k=1}^{n}\sum\limits_{i=0}^{k-1})\sum\limits_{M(k),M^{{}^{\prime}}(i)}E(X_{M(k)}X_{M(i)}^{{}^{\prime}})\\\
&=(\sum\limits_{k=0}^{n}\sum\limits_{i=0}^{k}+\sum\limits_{k=1}^{n}\sum\limits_{i=0}^{k-1})\sum\limits_{M(k)}\sum\limits_{M^{{}^{\prime}}(i)}E(X_{M(k)}X_{M(i)}^{{}^{\prime}})\\\
&=(\sum\limits_{k=0}^{n}(C_{n}^{k})^{2}k!\sum\limits_{i=0}^{k}+\sum\limits_{k=1}^{n}(C_{n}^{k})^{2}k!\sum\limits_{i=0}^{k-1})\sum\limits_{M^{{}^{\prime}}(i)}E(X_{M(k)}X_{M(i)}^{{}^{\prime}})\end{split}$
Replace $\sum\limits_{M^{{}^{\prime}}(i)}E(X_{M(k)}X_{M(i)}^{{}^{\prime}})$ by
$\sum\limits_{p=0}^{min(i,n-k)}C_{n-k}^{p}C_{k}^{i-p}P_{n-i+p}^{p}\sum\limits_{j=0}^{i-p}C_{i-p}^{j}[F_{n-j}(i-p-j)]E(X_{M(k+i-j)})$,
then the second equation is achieved.
$\Box$
Remark: To complete the proof of theorem7, we also need to know whether the
ratio $\frac{E(AM^{2}(G))}{E^{2}(AM(G))}$ goes to 1 as n goes to infinity,
adding the condition such as $m^{2}n^{-3}$ $\rightarrow$ $\infty$ as $n$
$\rightarrow$ $\infty$. We guess such a result is right, however the calculus
seems very difficult. And this result also contributes to the upper bound of
critical ratio for almost all the matrices.
###### Acknowledgements.
## References
* (1) M. Hall JR. Combinatorial Theory, Blaisdell, Waltham Massachusetts (1967).
* (2) Valiant. The complexity of computing the permanent, _Theoretical Computer Science_. 8, 189-201 (1979).
* (3) S. Chien, L. Rasmussen and A. Sinclair. Clifford Algebras and approximating the permanent, _Proceedings of the 34th Annual Symposium on Theory of Computing (STOC)_. 222 C231 (2002).
* (4) L. E. Rasmussen. Approximating the Permanent: a Simple Approach, _Random Structures and Algorithms_. 5, 349-361 (1994).
* (5) C. Godsil and I. Gutman. On the matching polynomial of a graph, _Algebraic Methods in Graph Theory_. 241-249 (1981).
* (6) A. Frieze and M. Jerrum. An Analysis of A Monte Carlo Algorithm For Estimating the Permanent, _Combinatorica_. 15(1), 67-83 (1995).
* (7) N. Karmarkar, R. Karp, R. Lipton, L. Lovász, and M. Luby,. A Monte-Carlo algorithm for estimating the permanent, _SIAM Journal on Computing_. 22, 284-293 (1993).
* (8) Yan Huo, Heng Liang, Siqi Liu and Fengshan Bai. Approximating the monomer-dimer constants through matrix permanent, _arXiv:0708.1641v2:cond-mat.stat-mech_. (2007).
* (9) A. Barvinok. Polynomial time algorithms to approximate permanents and mixed discriminants within a simply exponential factor, _Random Struct. Algorithms_. 14, 29-61 (1999).
* (10) H. Chernoff. A measure ofasymptotic efficiencyfor tests ofa hypothesis based on the sum ofobservations, _Ann. Math. Stat._ 23, 493-509 (1952).
* (11) A. Barvinok. New permanent estimators via non-commutative determinants, Preprint. (2000).
* (12) N. Linial, A. Samorodnitsky and A. Wigderson. A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents, _Combinatorica_. 20, 545-568 (2000).
* (13) M. Jerrum and A. Sinclair. Approximating the permanent, , _SIAM Journal on Computing_. 18, 1149-1178 (1989).
* (14) I.Beichl and F. Sullivan. Approximating the Permanent via Importance Sampling with Application to the Dimer Covering Problem, _Journal of Computational Physics_. 149, 128-147 (1999).
* (15) Shmuel Friedland and Daniel Levy. A polynomial-time approximation algorithm for the number of k-matchings in bipartite graphs, _arXiv:cs.CC/0607135 v1_. 392-401 (2006).
* (16) M. Jerrum, A. Sinclair and E. Vigoda. A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries, _Proceedings of the 33rd ACM Symposium on Theory of Computing_. 712 C721 (2001).
* (17) A. Frieze and S. Suen. Counting the number of hamilton cycles in random digraphs, _Random Structrues and Algorithms_.(1992).
* (18) B. Bollobás. Random Graph, _Cambridge University Press_. Second Edition (2001).
* (19) C. Kenyon, D. Randall, and A. Sinclair. Approximating the number of dimer coverings of a lattice, _Journal of Statistical Physics_. 83, 637-659 (1996).
* (20) George E. Andrews, Richard Askey and Ranjan Roy. Special Function, _Cambridge University Press_. (1999).
|
arxiv-papers
| 2008-12-05T12:16:53 |
2024-09-04T02:48:59.234865
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jinshan Zhang",
"submitter": "Jinshan Zhang",
"url": "https://arxiv.org/abs/0812.1119"
}
|
0812.1233
|
# Ultracold molecules from ultracold atoms: a case study with the KRb molecule
Paul S. Julienne
Joint Quantum Institute, National Institute of Standards and
Technology and The University of Maryland
100 Bureau Drive Stop 8423, Gaithersburg, Maryland 20899-8423, USA
E-mail: paul.julienne@nist.gov
Ultracold collisions of cold atoms or molecules make the bound states of the
collision complex formed from the two colliding species accessible for control
and manipulation of the cold species or the complex. Such resonances are best
treated by a resonant scattering theory, which in the ultracold domain can
take advantage of the properties of the long-range potential and the methods
of multichannel quantum defect theory. Coupled channels calculations on the
threshold scattering states and bound states of the 40K87Rb molecule
illustrate the ideas and methodology of quantum defect theory using the long-
range potential and also demonstrate the spin properties of the bound states
throughout the spectrum.
## 1 Introduction
The success of cooling and trapping of neutral atoms in the ultracold regime
on the order of $\mu$K or less has led to a broad range of advances in
multidisciplinary research with quantum degenerate gases of bosons [1, 2, 3]
or fermions [4, 5, 6]. Recent work has emphasized strongly correlated many-
body physics and reduced dimensional physics [7, 8]. The collisional
interaction between cold atoms is a crucial factor in determining the
properties and stability of an ultracold gas. These properties can in many
cases be precisely controlled using tunable scattering resonances known as
Feshbach resonances [9, 10].
Ultracold atom research is especially promising with atoms confined in optical
lattices, which can provide confinement in one, two, or three dimensions [11,
12, 13]. Three dimensional lattices are comprised of a periodic array of
trapping cells and offer the possibility of realizing quantum phase
transitions whereby each cell holds an individual atom, or a pair of atoms.
Cell trapping depths can be on the order of $\mu$K and intercell spacings on
the order of hundreds of nm. Given the ability to control lattice depth and
spacing and the strength of interatomic interactions, lattices of cold atoms
or molecules can realize a rich variety of physical systems for a variety of
applications.
Molecules are more difficult to cool than atoms, because their much more
complex internal structure does not allow simple laser cooling methods [14,
15, 16] to be applied to them [17]. While other methods to cool molecules are
being developed, such as molecular beam deceleration [18], they have not yet
succeeded in reaching the ultracold regime; see the review by Krems [19]. So
far, the only successful route to getting ultracold molecules in the $\mu$K
domain is to associate two already cold atoms to make a dimer molecule. The
first work in this area, reviewed in References [20] and [21], was carried out
by a number of groups that used photoassociation to make molecules in magneto-
optical traps at relatively low phase space density and translational
temperatures on the order of 100 $\mu$K. This early work relied on spontaneous
emission from excited electronic states, which resulted in a distribution of
ground state vibrational and rotational levels and negligible population in
the vibrational ground state of the molecule.
The most effective way to make an ultracold molecular gas with high phase
space density is to convert atom pairs in an already cold and dense atomic gas
to bound molecular states. This can be done, for example, by using a time-
dependent magnetic field to sweep a Feshbach resonance across threshold from
higher to lower energy [22]. Kokkelmans et al. [23] suggested such a sweep
could be combined with optical Raman population transfer to enhance the
production of deeply bound molecules in the vibrational ground state. The
vibrationally excited, weakly bound threshold molecules are in most cases not
collisionally stable but undergo rapid vibrational relaxation upon collision
with an atom or another molecule. On the other hand, $v=0$ levels would not
suffer from collisional vibrational relaxation.
Jaksch et al. [24] suggested that cold ground state 87Rb2 molecules could be
made by photoassociation of pairs of 87Rb atoms in optical lattice cells to
make a weakly bound state, followed by a series of Raman population transfer
steps to transfer population stepwise to the $v=0$ ground state. Upon turning
off the lattice these vibrational ground state molecules could be expected to
form a molecular Bose-Einstein condensate. Damski et al. [25] suggested a
dipolar superfluid could be made in a similar way by associating K and Rb
atoms in optical lattice cells and optically converting them to ground state
KRb polar molecules. The essential step in any of these schemes is the first
step of associating two atoms to make a weakly bound molecular state. Once
associated, well-established optical techniques should be capable of
transferring the population of the highly vibrationally excited molecular
state to the ground state.
Sage et al. [26] showed that a single stimulated Raman step could be used to
transfer population from a weakly bound excited vibrational level to the $v=0$
ground state of the RbCs molecule. Figure 1 schematically illustrates this
kind of process. The initial molecular state in Ref. [26] was formed by
spontaneous emission in a low-phase space density gas. However, it is much
better to use magnetoassociation to convert atom pairs in a high phase space
density gas to very weakly bound molecules known as Feshbach molecules. Köhler
et al. [9] have extensively reviewed research that use magnetically tunable
Feshbach resonances to achieve such association. Molecules formed this way are
just as translationally cold as the atoms that are initially present, and
their density is similar to that of the atoms when the transfer efficiency is
high. It is now apparent that the key to making deeply bound ultracold
molecules is to take advantage of a tunable Feshbach resonance to pair two
unbound atoms into a weakly bound molecular state, which in turn can be
converted by optical methods to the more deeply bound state.
A number of homonulcear alkali dimer Feshbach molecules have been made by
magnetoassociation: 6Li2 [27, 28, 29], 23Na2 [30], 42K2 [31], 87Rb2 [32], and
133Cs2 [33, 34], including molecular Bose-Einstein condensates made of pairs
of fermionic atoms [35, 36, 37, 31]. Thalheimer et al. [38] demonstrated high
efficiency in converting pairs of 87Rb atoms in a optical lattice cells to
make weakly bound 87Rb2 Feshbach molecules, which are also trapped in the
lattice cells. Winkler et al.[39] were also able to demonstrate efficient
coherent optical transfer of population to a lower energy weakly bound 87Rb2
vibrational level using the STImulated Raman Adiabatic Passage (STIRAP)
technique. Danzl et al. [40] succeeded in using the STIRAP method to convert a
weakly bound 133Cs2 Feshbach molecule to a much more deeply bound level with a
binding energy of around 1000 cm-1.
Ni et al. [41] have succeeded in using magnetoassociation plus STIRAP to
convert cold atom pairs in a dense gas of 40K fermions and 87Rb bosons at 350
nK to $v=0$ 40K87Rb ground vibrational state molecules in both the
a${}^{3}\Sigma_{u}^{+}$ state and the X${}^{1}\Sigma_{g}^{+}$ electronic
ground state. Previously Zirbel et al. [42] had characterized the
magnetoassociation to a Feshbach molecule near $54.6$ mT, and Ospelkaus et al.
[43] demonstrated STIRAP to form a dense, ultracold molecular gas of levels
bound by about $E/h=10$ GHz. Since formation of 40K87Rb in lattice cells has
previously been demonstrated by association of two atoms in a lattice cell
[44], it should be straightforward to make a lattice of $v=0$ 40K87Rb
molecules. Polar molecules in lattices have a number of possible applications,
including the realization of exotic condensed matter phases [45, 46, 47] and
quantum computation [48].
Collisions are an essential aspect of understanding and applying ultracold
atomic or molecular gases or lattices. Collisions can be beneficial coherent
ones that allow control of the system or destructive ones that cause loss or
decoherence of trapped atoms or molecules. The goal of this paper is to
illustrate a number of features of ultracold collisions of atoms and molecules
using calculations on the bound and scattering states of the 40K87Rb molecule
as an example system. We use coupled channels calculations with accurate
potential energy curves in calculations that extend from the $E=0$ collision
threshold to the energy of the deeply bound $v=0$ ground state vibrational
level. It is clear that near-threshold scattering resonances and bound states
play a crucial role in ultracold molecule formation. In particular, we would
like to demonstrate the usefulness of a resonant scattering viewpoint,
amplified by the insights of generalized multichannel quantum defect theory
based on the long-range potential between the colliding species.
Section 2 develops the resonant scattering view of an ultracold collision,
whereby a collision allows access to a wide range of bound states of a
collision complex. Section 3 describes the scattering channels of the 40K87Rb
system, the coupled channels method, and the near-threshold scattering
resonances in this system. Section 4 shows how the properties of the van der
Waals long range potential determine the qualitative and semi-quantitative
features of the near-threshold bound and scattering states. Section 5
describes the character of the molecular levels as energy decreases below
threshold to the domain of ”normal” molecular levels, including the
vibrational ground state level. A final section provides a summary of the
results of the paper.
## 2 The Resonant Scattering Viewpoint
### 2.1 Ultracold Collisions for precise spectroscopy and state control
We will begin with some general considerations that are generic to ultracold
atomic or molecular collisions. First, let us assume that the species A and B,
each of which could be an atom or molecule, are prepared in specific internal
states $|p\rangle_{A}$ and $|q\rangle_{B}$ in an ultracold gas at temperature
$T$, where relative collision energies tend to be on the order of
$E/k_{B}\approx 1$ $\mu$K, where $k_{B}$ is the Boltzmann constant; in other
units, 1 $\mu$K corresponds to $E/h=21$ kHz, $E/hc=7.0\times 10^{-7}$ cm-1, or
$E=$ 86 peV. Consequently, the colliding species are prepared in a very
precise energy state relative to the energy scale associated with the
”collision complex” AB, where the fragment A and B interact strongly when they
are close together. The energy scale associated with AB is on the order of
typical chemical bond strengths, say $E=1$ eV, equivalent to $E/h=242$ THz or
$E/hc=8066$ cm-1. Thus, the complex AB can be prepared with a precise energy
spread that may be only 1 part of $10^{10}$ of its ground state binding
energy.
Figure 2 shows a schematic view of an ultracold collision, indicating that the
species A and B have internal structure and could be prepared in one of
several energy states $E_{\alpha}=E_{p}+E_{q}$ of the pair. For simplicity, we
take the energy of the state that is prepared as the zero of the energy scale,
illustrated in the Figure as $E_{\alpha}=E_{1}=0$. Atoms typically have
hyperfine and Zeeman spin structure in their ground states, whereas molecules
will have rotational and vibrational degrees of freedom as well. Collision
channels $\beta$ with $E_{\beta}>E>E_{\alpha}$ are ”closed channels” at the
collision at energy $E$, whereas channels $\beta$ with $E>E_{\alpha}\geq
E_{\beta}$ are ”open channels.” Collision products can exit the collision in
open channels but not closed ones, due to energy conservation. If more than
two atoms are present in the AB species, reactive channels may also be open.
The long-range potential of the AB complex will generally support a series of
bound levels $E_{n\beta}$ leading up to the dissociation limits at $E_{\beta}$
corresponding to the various internal states $\beta$ of the separated species.
Such levels are indicated schematically for three channels in Fig. 2. Levels
in closed channels with $E_{n\beta}>E_{\alpha}$ are quasibound levels that can
decay to channel $\alpha$, depending of the presence of a coupling term in the
Hamiltonian of the AB complex. Such levels give rise to scattering resonances
when collision energy $E$ is near $E_{n\beta}$. Such resonances can be tuned
or coupled to threshold scattering states by external fields. Magnetically
tunable resonances, such as the 54.6 mT 40K87Rb resonance studied in this
paper, can be used to control elastic and inelastic collision rates, and to
form weakly bound molecular states by time-dependent manipulation of the
magnetic field [9, 10]. Optically tunable resonances, or photoassociation
resonances [20], can be tuned or turned on and off by varying the respective
frequency or intensity of the driving electromagnetic radiation. Magnetically
or optically tunable resonances are treated by formally equivalent theory.
If the atoms are assumed to start in a scattering state with $E>0$, Fig. 1
gives examples of optically tunable resonances, namely the one-color
photoassociation process at frequency $\nu_{1}$ or the two-color
photoassociation process involving $\nu_{1}$ and $\nu_{2}$. Photoassociation
is most naturally treated as a resonant scattering process with a decaying
resonance level [49, 50, 51, 52]. The theory of threshold resonant scattering
of a decaying resonance can be readily extended to tightly confining optical
lattices of reduced dimension geometries [53] and can be readily adapted to
magnetically tunable or other kinds of resonances.
Extraordinary success has been achieved with ultracold atoms with high
resolution spectroscopic probing of the states of the AB molecule starting
from the precisely prepared states of the atoms. The collision complex AB can
be prepared in a sharp energy state defined by the spread in energy $k_{B}T$
of the colliding atoms. Tunable magnetic or radiofrequency probes can tune
states within a few GHz of threshold [9, 10]. One-color photoassociation is
especially successful in probing excited states of the AB complex, while two-
color Raman photoassociation is especially useful for probing the ground state
[20]. Optical methods are capable of tuning to bound states that are removed
even by hundreds of THz from threshold with a prcision determined by the
linewidths of the lasers involved.
Perhaps even more importantly than spectroscopic probing, a time-dependent
field can be used to associate an A $+$ B pair to make a stable weakly bound
AB complex that does not dissociate back to A $+$ B [9]. This stable complex
can then in turn be coupled to other bound states in the near-threshold
domain. Population can then be transferred coherently to other weakly bound
near-threshold states using time dependent magnetic [54, 55], radiofrequency
[56], or optical fields [39, 42]. Time-dependent optical fields can achieve
coherent population transfer to deeply bound states [40, 41]. We have every
reason to expect that these techniques will be extended to new species and
other domains of frequency, including microwave, THz, and infrared.
Thus, we see that a sample of ultracold atoms–and presumably now samples of
ultracold molecules–can be prepared in specific internal states at a precisely
defined energy near $E=0$. The collision of the prepared species then makes
available a large part of the entire bound state spectrum of the collision
complex for high resolution probing and coherent population transfer. Bound
states of the complex that can be brought into resonance with the near $E=0$
separated species then serve as ”gateway” states into the rich spectrum of the
complex. This permits very state-specific control over all degrees of freedom
of the complex, electronic, vibrational, rotational, spin, and translation. By
confining the stable species AB in a single trapping cell of an optical
lattice, even the energy of relative motion is quantized and even more sharply
defined. Furthermore, a molecule in a lattice cell is protected from
collisions with A or B atoms or other AB molecules, thus prolonging its
lifetime [38].
### 2.2 Importance of the long-range potential
Given the sensitivity of the near-threshold bound state spectrum to the
properties of the long-range form of the potential $V(R)$ between A and B,
much can be gained by trying to understand the states of the molecule AB
associated with the long range $V(R)$, which varies with some lead power of
$R$ as $1/R^{n}$. This contrasts with ”normal chemistry”, where one usually
seeks to understand the bound states from the ground state level up to the
dissociation limit. In the ultracold domain, collisions are normally much more
understandable and even quantitatively treated by starting with the states of
the separated species A and B and the states of their complex AB due to the
long range interaction between them. In this way, one does not need to know
the full spectrum of the AB molecule in order to characterize the near-
threshold domain quite precisely. This approach has been extraordinarily
successful with ultracold atoms [9, 10]; see Refs. [57, 58, 59] for some
examples.
For neutral S-state atoms, the long-range potential has the van der Waals form
with $n=6$. Molecules can have dipole or quadrupole moments, corresponding to
$n=$ 3 and 5 for two dipoles or two quadrupoles respectively. These potentials
are anisotropic, but have vanishing diagonal matrix elements for $s$-wave
interactions in free space. In the absence of an external field which breaks
the symmetry of free space, an isolated polar molecule in a definite state of
total angular momentum has a vanishing dipole moment. However, a strong
electric field can induce a dipole moment.
It is convenient to introduce a characteristic length and energy scale
associated with the long-range potential. For this purpose, we use the scale
length defined by Gribakin and Flambaum for a potential varying as
$-C_{n}/R^{n}$ [60],
$\bar{a}(n)=\cos\left(\frac{\pi}{n-2}\right)\left(\frac{2\mu
C_{n}}{\hbar^{2}(n-2)^{2}}\right)^{\frac{1}{n-2}}\Gamma\left(\frac{n-3}{n-2}\right)/\Gamma\left(\frac{n-1}{n-2}\right)\,,$
(1)
where $\mu$ is the reduced mass of the AB pair, $\Gamma$ is the Gamma
function, and $\hbar$ is Planck’s constant divided by $2\pi$. This length
defines a corresponding energy scale
$\bar{E}(n)=\frac{\hbar^{2}}{2\mu\bar{a}(n)^{2}}\,.$ (2)
For a van der Waals potential with $n=6$, this simplifies to
$\bar{a}=0.477989(2\mu C_{6}/\hbar^{2})^{1/4}$. Jones et al. [20] and Chin et
al. [10] review the properties of the van der Waals potential relevant to
ultracold physics, and Friedrich and Trost [61] adapt semiclassical theory to
obtain the threshold properties for $n=6$ and other cases of $n$.
### 2.3 Ultracold resonant scattering theory
Since molecules have more complex internal structure than atoms, subject to
electric as well as magnetic and electromagnetic field control, the collisions
of cold molecules should also have numerous scattering resonances; for
example, see the work of Refs. [62, 63, 64, 65]. Consequently, the theory of
resonant scattering is both necessary and useful for understanding cold
molecular collisions. As noted earlier, resonant scattering theory has been
extensively developed for photoassociative collisions, which represent optical
Feshbach resonances [49, 50, 51, 52, 53, 66, 67].
A particularly insightful set of articles on a resonant scattering viewpoint
of molecule association and chemical reactions has been given by Mies [68,
69]. He considered the role of molecular resonances in the association of two
atomic or molecular fragments A and B to form an AB molecule. He considered
both the case of radiative association, which is formally equivalent to the
resonant scattering theory of molecule formation by photoassociation, and the
case of collisonal association, where a third body deactivates a resonant
state of the complex to stabilize it. In the latter case, the decay if the
resonance is simulated by a complex energy with an imaginary term, just as
spontaneous emission is represented in radiative association. The formalism is
useful, since it gives a general $S$-matrix resonant scattering theory of
association and resonant-enhanced reactions, bounded by the unitarity limit of
the $S$-matrix, even when there is a complex set of overlapping resonances.
While the Mies theory was developed for a high temperature gas, where $k_{B}T$
is large compared to the spacing between the dense set of molecular
resonances, there is no reason why the formalism can not carry over directly
to the ultracold case, if the threshold properties of the $S$-matrix are
incorporated into the theory. This will be especially useful if the insights
of generalized multichannel quantum defect theory [70, 71] are brought to bear
in the ultracold regime [22, 72, 73, 74].
The thermally averaged expression from Mies [68, 69] for the inelastic
collision rate coefficient $K_{\mathrm{in}}(T)$ has a very instructive general
form. Assume species A and B are prepared in channel $\alpha$ in a gas
described by a Maxwellian thermal distribution of collision energies at
temperature $T$. Then
$K_{\mathrm{in}}(T)=\frac{1}{Q_{T}}\frac{k_{B}T}{h}f_{D}(T)\,,$ (3)
where the dimensionless dynamical factor $f_{D}(T)$ is
$f_{D}(T)=\sum_{\ell m_{\ell}}\left(1-|S_{\alpha\alpha}(\ell
m_{\ell})|^{2}\right)e^{-E/(k_{B}T)}dE/(k_{B}T)\,.$ (4)
Here $Q_{T}$ is the translational partition function, $1/Q_{T}=(2\pi\mu
k_{B}T/h^{2})^{-3/2}=\Lambda_{T}^{3}$ where $\Lambda_{T}$ is the thermal de
Broglie wavelength for relative motion of A and B. The sum defining $f_{D}(T)$
runs over the contributing partial waves $\ell$ and the $2\ell+1$ projections
$m_{\ell}$ of $\ell$. Inelastic collisions are those that remove A and B from
channel $\alpha\ell m_{\ell}$, such as loss to a different channel
$\beta\ell_{\beta}m_{\ell\beta}$, which could represent removal of the
resonant state through decay, relaxation, or reaction. If removal of A and B
results in formation of an AB molecule, the removal rate is the same as the
association rate. Note that in time-independent scattering theory, production
of a resonant quasibound state that only decays back to the entrance channel
is not a molecule formation process that can be represented by an $S$-matrix
element, since the quasibound state does not persist into the asymptotic
domain. However, if the quasibound resonant state is irreversibly removed to a
loss ”channel” so that it can not decay back to the entrance channel, then
this process can be represented by $S$-matrix scattering theory. Thus,
$S$-matrix resonant scattering theory can be used to represent two-color
photoassociation to bound states of the ground state potential, as long as
such states have some ”decay width” [52, 75].
The removal rate of the density $n_{A}$ of species A or density $n_{B}$ of
species B is $\dot{n}_{A}=\dot{n}_{B}=-K_{\mathrm{in}}n_{A}n_{B}$ (we assume
nonidentical species; otherwise identical particle properties would need to be
taken into account). We thus see that the respective removal rates
$K_{\mathrm{in}}n_{B}$ and $K_{\mathrm{in}}n_{A}$ of species A and B are
proportional to $n_{B}\Lambda_{T}^{3}$ and $n_{A}\Lambda_{T}^{3}$. One thus
finds that the removal rate of a species is proportional to the dimensionless
phase space density of its collision partner. The removal rate is also
proportional to a thermal factor $k_{B}T/h$, which sets a basic time scale for
the dynamics (e.g., 21 kHz at 1 $\mu$K). The only part that depends on the
specific collision dynamics of the particular system in encapsulated in the
dimensionless dynamical factor $\left(1-|S_{\alpha\alpha}(\ell
m_{\ell})|^{2}\right)$ that describes the loss of flux from the $\alpha\ell
m_{\ell}$ entrance channel. In the case of elastic scattering this factor is
replaced by $\left|1-S_{\alpha\alpha}(\ell m)\right|^{2}$ in a corresponding
definition of a dynamical $f_{D}(T)$ expression for elastic scattering.
The structure of the $S$-matrix elements in the expression for the $f_{D}$
term in the Mies theory is determined by a set of scattering resonances.
However, the dimensionless $f_{D}(T)$ factor for inelastic collisions is
bounded by the unitarity property of the $S$-matrix so that the contribution
from each $\ell m_{\ell}$ term in the sum has a maximum value of unity. Thus,
if $\ell_{\mathrm{max}}$ partial waves contribute to the sum, then the bounds
on $f_{D}(T)$ are $0\leq f_{D}(T)\leq(\ell_{\mathrm{max}}+1)^{2}$, so that
$0\leq f_{D}\leq 1$ for $s$-waves. The analogous $f_{D}$ factor for elastic
collisions has an upper bound due to unitarity that is 4 times larger,
$4(\ell_{\mathrm{max}}+1)^{2}$.
For $s$-waves in the $E\to 0$ threshold limit,
$S=[1-ik(a-ib)]/[1+ik(a-ib)]\approx\exp{[-2ik(a-ib)]}$ is represented by a
complex scattering length $a-ib$, where $\hbar k$ is the relative collision
momentum, $b$ is nonnegative, and the condition $k|a-ib|\ll 1$ applies. In
this limit, $1-|S|^{2}=4kb$, and $K_{\mathrm{in}}$ reduces to the usual
threshold law expression,
$K_{\mathrm{in}}=2(h/\mu)b=0.84\times
10^{-10}\frac{b[\mathrm{au}]}{\mu[\mathrm{amu}]}\,\,\mathrm{cm}^{3}/\mathrm{s}\,.$
(5)
Here $b[\mathrm{au}]$ is in atomic units and $\mu[\mathrm{amu}]$ is in atomic
mass units. Similarly, the $E\to 0$ $s$-wave elastic scattering cross section
reduces to the standard form $4\pi(a^{2}+b^{2})$. Given that $k|a-ib|\ll 1$,
the $f_{D}$ factor for elastic or inelastic collisions remains much less than
its upper bound with a value determined by $a$, $b$ and $T$. Threshold
resonance structure can also modify the value of $f_{D}(T)$, requiring the use
of full energy-dependent resonant scattering expressions [52, 53].
The expression for $K_{\mathrm{in}}$ in Eq. (3) is based on very general
thermodynamic and dynamical considerations at equilibrium and applies to
atomic or molecular collisions. We may expect Eq. (3) to give a guide to the
rate of molecular association even in more general nonequilibrium cases. It is
gratifying that the rate of association of a given species A is proportional
to the phase space density of its collision partner, as in semi-empirical
treatments of atom association to a Feshbach molecule with time-dependent
fields [42, 76]. This is to be expected for a fast process at constant
entropy. Also, time-dependent processes would not be expected to beat the
fundamental upper bound to $f_{D}$ set by the unitarity limit of the time-
independent $S$-matrix. If the dimensionless phase space and dynamical factors
in the association rate are both less than unity, as they would be for a
Maxwellian gas with an $s$-wave association process in the threshold law
domain, then the $k_{B}T/h$ factor in Eq. (3) sets a limiting upper bound to
the rate (and a lower bound to the time scale) for the association of A and B
pairs to form AB molecules.
Using the actual threshold resonance form for the $S$-matrix as a function of
collision energy [52, 53] would allow one to work out the general case of
resonant elastic and inelastic cross sections and rate constants, including
the effect of multiple or overlapping resonances if they are present. See
Machholm et al. [66], who develop a coupled channels $S$-matrix resonant
scattering theory for decaying resonances, including the effect of overlapping
resonances, using a form similar to the Mies theory. They also develop
approximations for describing isolated resonances, using the factorization of
matrix elements associated with quantum defect theory. Coupled channels
resonant scattering theory, in its various numerical or approximate analytical
representations, should prove to be a very powerful tool for characterizing
ultracold molecular collisions, especially if it can take advantage of the
long range properties of the interaction potentials to characterize near-
threshold states.
## 3 Multichannel Scattering of 40K87Rb
Since ultracold ground state 40K87Rb polar molecules have now been made using
resonant association followed by optical population transfer, we will
concentrate on using this system to illustrate the features of this process.
We calculate the bound and scattering states of 40K87Rb all the way from
threshold to the ground state, including the effect of the long range
potentials in determining the states near threshold. Fortunately, an excellent
set of adiabatic Born-Oppenheimer electronic potentials is available for the
X${}^{1}\Sigma^{+}$ and a${}^{3}\Sigma^{+}$ states of this species, based on
the high resolution spectroscopic analysis of Pashov et al. [77]. These are
the potentials needed to describe the interactions of two ground state atoms.
Using these potentials, and standard representations of the full Hamiltonian
of the interactions, including the angular momenta from electron and nuclear
spins and internuclear axis rotation, we have carried out full coupled
channels calculations [9, 10, 22, 59, 78] of the bound and scattering states
of this species.
Figures 3 and 4 show the Zeeman substructure of the 40K and 87Rb 2S ground
state atoms as a function of magnetic field. Each has an electron spin quantum
number of $S=1/2$ and respective nuclear spin quantum numbers of $I=$ 4 and
$3/2$. Thus, 40K is a composite fermion and 87Rb is a composite boson, and the
40K87Rb molecule is a fermion. The figures show how the two zero-field
hyperfine levels with total angular momentum $F=I-1/2$ and $F=I+1/2$ split
with increasing magnetic field strength $B$. We use an alphabetical notation
to designate each level of the Zeeman manifold by an italic Roman letter
$a,b,c\ldots$, starting with the lowest energy state at each $B$ and
increasing in order of energy. In the Figures, the zero of energy is that of
the spinless (nonrelativistic) atom. Thus, the lowest energy hyperfine level
at $B=0$ has an energy of $-4E_{\mathrm{hf}}(^{40}\mathrm{K})/9$ for 40K and
$-5E_{\mathrm{hf}}(^{87}\mathrm{Rb})/8$ for 87Rb, where
$E_{\mathrm{hf}}(^{40}\mathrm{K})/h=1.285790$ GHz and
$E_{\mathrm{hf}}(^{87}\mathrm{Rb})/h=6.8346826$ GHz are the hyperfine
splittings of the two atoms [79].
Figure 5 shows the adiabatic Born-Oppenheimer potentials for the
X${}^{1}\Sigma^{+}$ and a${}^{3}\Sigma^{+}$ electronic states that correlate
with the two ground state separated atoms [77]. The inset to the figure shows
the adiabatic potentials (i.e., those that diagonalize the full electronic
plus spin Hamiltonian) on an expanded scale showing the long range region.
Since the atoms are $S$-state atoms, all of the long range potentials have the
same van der Waals $C_{6}$ coefficient, which has a value of 4299.51 Eh
a${}_{0}^{6}$ [77], where E${}_{\mathrm{h}}=4.359744\times 10^{-18}$ J and
a${}_{0}=0.05291772$ nm. The characteristic van der Waals length for the long
range potential is $\bar{a}=68.8$ a0 and the characteristic energy is
$\bar{E}/h=13.9$ MHz.
Since we are concerned with $s$-wave resonances in the $aa$ channel, where
both atoms are in their lowest energy state, it is necessary to consider all
of the $s$-wave states with projection quantum number
$M_{\mathrm{tot}}=-9/2+1=-7/2$, where the $M_{\mathrm{tot}}$ quantum number
also includes the projection $m_{\ell}$ of partial wave ($m_{\ell}$ is
trivially zero for an $s$-wave). The projection quantum number is a conserved
quantum number at finite field, so that only states with the same value of
$M_{\mathrm{tot}}$ are coupled through terms in the Hamiltonian. While it is
possible to include other partial waves in the expansion basis that are
coupled to $s$-waves through anisotropic spin-dependent terms in the
Hamiltonian [59], such coupling terms are small and have a small effect here
and need not be included (but see below for the effect of coupling $d$-waves
to $s$-waves).
Figure 6 illustrates the 12 $s$-wave channels that have
$M_{\mathrm{tot}}=-7/2$. The dotted zero-field curves are the same as in
Figure 5, but these split into 12 different curves at finite $B$. These
channels separate themselves into four different groups A, B, C, and D
associated with the four different zero field hyperfine separated atom limits,
as described in the caption. Letting each channel be labeled by the quantum
numbers $\beta={ij}$, where $ij$ represents the alphabetic label of the two
atoms, with the 40K label first, the coupled channels expansion of the wave
function for atoms initially prepared in channel $\alpha=aa$ is
$\Psi_{\alpha}(R,E)=\sum_{\beta}f_{\beta\alpha}(R,E)|\beta\rangle/R\,.$ (6)
The coupled channels Schrödinger equation then determines the solutions with
scattering boundary conditions for $E>E_{\alpha}$ or the discrete set of bound
state solutions with bound state boundary conditions for $E<E_{\alpha}$. The
bound state solutions are found with the discrete variable method described in
Ref. [59], whereas a standard propagator method is used for the scattering
solutions.
Figure 7 shows the results of coupled channels scattering and bound state
calculations with the 12 $s$-wave basis functions with $M_{\mathrm{tot}}=-7/2$
in the expansion. The positions of the four scattering length singularities
are in excellent agreement with the measured positions [80, 81, 82]. The
resonance near 54.6 mT is well represented near its singularity by the
standard resonance formula [9, 10]
$a(B)=a_{\mathrm{bg}}\left(1-\frac{\Delta}{B-B_{0}}\right)\,,$ (7)
where the background scattering length is $a_{\mathrm{bg}}=-190.6$ a0, the
resonance width $\Delta=-0.3103$ mT, and the resonance position is
$B_{0}=54.6937$ mT, compared to measured positions of $54.69$ mT [80] and
$54.67$ mT [82].
The bound levels in Fig. 7 are labeled by the index $\beta$ of the dominant
spin component in the coupled channels expansion in Eq. (6) and by the
vibrational quantum number $n$ counting down from $n=-1$, which designates the
last bound state below the dissociation limit at $E=E_{\beta}$. Thus, the line
near $-0.4$ GHz parallel to the $E=0$ axis is the last $n=-1$ bound state in
the $aa$ entrance channel, whereas the bound state that crosses threshold to
make the 54.69 mT resonance is the next to last $n=-2$ bound state of the $rb$
channel, which is associated with the group B of Figure 6. The two $n=-3$
resonances in the Figure are associated with the highest energy group D.
Figure 8 shows an expanded view of Figure 7 near the 54.69 mT resonance. When
$d$-waves with $\ell=2$, $m_{\ell}=0$ are added to the basis set, the weak
coupling between the $s$\- and $d$-waves due to the spin-dipolar interaction
shifts $B_{0}$ from 54.694 mT to 54.687 mT (not shown). In addition, a new
narrow resonance appears on the shoulder of the 54.69 mT resonance at 54.8305
mT with a width of only 4.6 $\mu$T. The narrow resonance has not been
observed. It is due to the entrance channel $s$-wave being coupled to a bound
state of $d$-symmetry. The bound state is a $\beta(n\ell m_{\ell})=rf(-3d0)$
resonance with the same spin and $n$ character as the $rf(-3)$ $s$-wave
resonance in Fig. 7, but with $\ell=2$ units of axis rotation angular
momentum. To a good approximation its energy can be obtained by adding the
$d$-wave rotational energy for the $n=-3$ level of the ${}^{3}\Sigma^{+}$
potential to the energy of the $s$-wave $rf(-3)$ level. Finally, Figure 8
shows the universal energy that is derived from the scattering length [9],
$E=-\frac{\hbar^{2}}{2\mu a(B)^{2}}\,.$ (8)
This universal formula only applies very close to resonance where
$a(B)\gg\bar{a}$.
## 4 The Quantum Defect Approach with a Long-Range Potential
While coupled channels calculations have proven to be an excellent and
reliable tool for understanding the near threshold domain of bound and
scattering states of ultracold atoms, they are very computer intensive and
give a ”black box” representation of the physics. Therefore, it is also very
desirable to have alternative methods for analysis, understanding, and
developing approximations. In this regard, the general form of multichannel
quantum defect theory (MQDT) provides a very powerful set of tools and
concepts for the ultracold domain, which takes advantage of the separation
between the short-range and long-range aspects of the collision, and exploits
the analytic properties of the wavefunction as a function of interatomic
separation $R$ and energy $E$. The concepts of MQDT are implicit in the
accumulated phase method pioneered by the Eindhoven group for characterizing
ground state interactions of cold atoms [83, 84, 85, 86] and are explicitly
developed for neutral atoms in Refs [72, 73, 74, 87, 88] and for ion-atom
collisions in Ref. [89]. Gao has developed analytic MQDT for potentials having
the long-range form $-C_{n}/R^{n}$ [90, 91, 92, 93]. We will show here several
specific ways where MQDT theory has been applied or still needs to be
developed for application to the ultracold domain, especially when implemented
using the form of the long range potential,
### 4.1 The near-threshold spectrum
There is a close relation between the near-threshold bound state spectrum and
the near-threshold scattering properties of ultracold collisions. Not only is
there a general relation between the scattering length and the last bound
state binding energy, as illustrated by the limiting expression in Eq. (8),
but the number and properties of Feshbach resonance states are related to the
density and tuning properties of near-threshold bound states. In view of the
importance of the near threshold spectrum for precision spectroscopic probing
in Section 2.1 and collision dynamics in Section 2.3, simple ways for
understanding the spectrum is desirable.
Gao [92, 93] has worked out analytic bound state properties for the van der
Waals potential for $n=6$. We will characterize the long-range $n=6$ potential
by the length parameter $\bar{a}$ instead of the $\beta=2.092099\bar{a}$
parameter used by Gao, since the formulas of the theory are simpler when
$\bar{a}$ is used. The spectrum is completely determined for all partial waves
$\ell$ by specifying only three parameters: the reduced mass $\mu$, the
$C_{6}$ constant, and the short range QDT $K_{c}$-matrix element, or
equivalently, the $s$-wave scattering length in units of $\bar{a}$ that
uniquely determines the value of $K_{c}$. The value of $K_{c}$ is related to
the ”quantum defect” for the vibrational levels of the potential
Figure 9 shows the spectrum of the last two bound states below threshold for
the 40K87Rb molecule for different partial waves $\ell$. The Figure shows
several specific cases. The blue lines show the spectrum of bound states that
one gets for the special case that the $s$-wave scattering length $a=\infty$.
This corresponds to having an $s$-wave bound state at $E=0$. The Gao theory
shows in this case for a pure $-C_{6}/R^{6}$ potential that all partial waves
with $\ell$ divisible by 4 also have a bound state at $E=0$, as for example,
the $g$-wave in Fig. 9. In the $a=\infty$ case the first $s$-wave level with
$E<0$, labeled by $n=-1$ in the Figure, lies at $-36.1\bar{E}=-0.503$ GHz, and
the next $n=-2$ level lies at $-249\bar{E}=-3.47$ GHz. Varying the scattering
length over its full range between $-\infty$ and $+\infty$ will produce one
$n=-1$ and one $n=-2$ level in the ”bins” demarked by these values. Figure 9
also shows the actual energy levels for the X${}^{1}\Sigma^{+}$ and
a${}^{3}\Sigma^{+}$ potentials, relative to their nonrelativistic separated
atom limit at $V(\infty)=0$. These are near the bottom of their ”bins”, due to
their respective negative scattering lengths of $-111.8$ a0 and $-216.2$ a0
[77]. The actual binding energies for the $n=-1$ and $n=-2$ levels are
respectively about 1 percent and 2 percent larger than the same level
calculated for a pure van der Waals potential with the same scattering length.
This small difference indicates the small effect of other terms in the
potential that contribute to the binding energy of near threshold levels. It
also shows that the van der Waals theory alone is a very good approximate
theory, although real potentials need to be used in fitting binding energy
data that is more accurate than 1 percent [94].
The figure also shows the bound states for other partial waves. The energies
of these levels can be approximated by adding the rotational energy $\langle
n,s|\hbar^{2}\ell(2\ell+1)/(2\mu R^{2})|n,s\rangle$ to the energy of the
$s$-wave $|n,s\rangle$ level. In the angular-momemtum insensitive version of
the long range quantum defect theory [92, 93], the energies of the levels with
$\ell\geq 1$ are also determined from a knowledge of the $s$-wave scattering
length alone (plus $\mu$ and $C_{6}$, of course). However, there will always
be a small error since the short range form of the potential never corresponds
to $1/R^{6}$ all the way to $R=0$ but has a finite depth and inner turning
point. For any given entrance channel in the multichannel problem, the near-
threshold bound states can be calculated to a good approximation by using the
scattering length for that channel to get the $K_{c}$ matrix for that channel.
Figure 10 shows the vibrational wave function $f_{n}(R,E_{n})$ for the last
three $s$-wave bound states of the ${}^{3}\Sigma^{+}$ potential with energy
$E_{n}$ for $n=-1$, $-2$, and $-3$. The wave functions show distinct
differences for distances on the order of $\bar{a}$. However, the three wave
functions are remarkably similar in shape at small $R$, where the potential is
very deep and the local de Broglie wavelength, which governs the spacing
between oscillations, is small compared to $\bar{a}$. These three unit-
normalized wave functions have the same phase at small $R$ and differ only in
amplitude. In fact, these wave functions would have almost identical energy-
insensitive amplitudes at short range if they were given a ”quantum defect”
normalization per unit energy by multiplying $f_{n}(R,E_{n})$ by $|\partial
n/\partial E|^{1/2}$ for level $n$, where $n$ is viewed as a continuous
function of $E$ that takes on integer values at an eigenvalue $E=E_{n}$ and
$|\partial E/\partial n|$ is the mean vibrational spacing between levels at
level $n$ [70, 71, 72] (alternatively $|\partial n/\partial E|$ is the density
of states per unit energy). The wave function can be understood using its
energy-normalized semiclassical JWKB form in the short-range classical region,
namely $C\sin{\beta(R)}/\sqrt{k(R,E)}$, where $\beta(R)$ is the phase,
$\hbar^{2}k(R,E)^{2}=2\mu(E-V(R))$ determines the ”local” momentum wave number
$k(R,E)$, and $C=[2\mu/(\pi\hbar^{2})]^{1/2}$ is a constant. In the deep part
of the potential, where $E-V(R)\gg\bar{E}$ is very large and $k(R,E)\approx
k(R,0)$, the semiclassical phase and amplitude are not at all sensitive to the
value of $E\approx 0$. This insensitivity leads to the basic concept of MQDT
or the accumulated phase method that the short range physics, such as wave
function phases or coupling matrix elements due to short range interactions,
can be quite adequately represented by energy-insensitive quantities [70, 71,
72].
### 4.2 Feshbach resonance properties
The simple van der Waals form of MQDT also gives an excellent way to represent
near-threshold scattering and bound state properties of Feshbach resonances In
addition to $\bar{a}$ and $\bar{E}$, the only other quantities that are needed
in the near-threshold region are the background $s$-wave scattering length
$a_{\mathrm{bg}}$ for the open entrance channel, the resonance width $\Delta$,
and the difference $\delta\mu$ in magnetic moments between the separated atoms
in the entrance channel and the magnetic moment of the ”bare” resonance state
in the closed channel (the ”bare” state is the approximate eigenstate in the
closed channel before coupling to the entrance channel is turned on). Julienne
and Gao [74] have shown how the Mies version of MQDT [70, 71] can be can be
adapted for the Feshbach resonance scattering states. MQDT theory first
develops a set of uncoupled ”reference” states with which to analyze the
problem. Each ’reference’ channel $\beta$ for a given partial wave is
characterized by a single reference potential that dissociates to energy
$E_{\beta}$ as $R\to\infty$ and which has a reference scattering phase shift
$\eta_{\beta}(E)$ for $E\geq E_{\beta}$ and a reference bound state phase
shift $\nu_{\beta}(E)$ for $E<E_{\beta}$. Bound states exist in the reference
channel for a discrete set of energies for which $\tan\nu_{\beta}(E)=0$. Near
threshold, two auxiliary MQDT functions $C_{\beta}(E)$ and
$\tan\lambda_{\beta}(E)$ are needed for each reference channel to characterize
the quantum threshold behavior. These auxiliary functions can be physically
interpreted using semiclassical concepts [70, 71, 72] and have the property
that $C_{\beta}(E)\to 1$ and $\tan\lambda_{\beta}(E)\to 0$ when collision
energy is sufficiently large. For $s$-waves in a van der Waals reference
potential, this means $E-E_{\beta}\gg\bar{E}$; semicalssical connections
between the short and long range parts of the wave function break down when
$E-E_{\beta}<\bar{E}$, and the quantum connections expressed by the
$C_{\beta}(E)$ and $\tan\lambda_{\beta}(E)$ functions need to be used.
Even in a complex multichannel problem, it is usually sufficient to represent
the closed channel bound state as a single bound state whose properties do not
change with magnetic field detuning over a modest range of detuning [9, 22,
95]. Thus we can reduce the tunable Feshbach resonance problem to an effective
2-channel problem with a single ”bare”, or uncoupled, closed channel bound
state $|c\rangle$ and a single entrance channel we call the ”background”
channel, $\alpha=bg$. With this framework and using the background channel van
der Waals potential as the reference channel, the scattering phase shift for
the fully coupled problem, where the ”bare” closed channel with bound state
energy $E_{c}=\delta\mu(B-B_{c})$ interacts with the entrance channel with
background phase shift $\eta_{\mathrm{bg}}(E)$ to make a scattering resonance,
the phase shift $\eta(E,B)$ for $E>0$ is [74]
$\eta(E,B)=\eta_{\mathrm{bg}}(E)-\tan^{-1}\left(\frac{\frac{1}{2}\bar{\Gamma}_{c}C_{\mathrm{bg}}(E)^{-2}}{E-\delta\mu(B-B_{c})-\frac{1}{2}\bar{\Gamma}_{c}\tan\lambda_{\mathrm{bg}}(E)}\right)\,,$
(9)
where the coupling between the closed and background reference channels is
completely contained in the coupling parameter $\bar{\Gamma}_{c}$, which is
independent of energy $E$ and magnetic field $B$ as these vary near the
resonance over ranges on the order of $\bar{E}$ and $\Delta$ respectively. The
resonant phase shift is characterized by an energy-dependent width
$\Gamma_{c}(E)=\bar{\Gamma}C_{\mathrm{bg}}(E)^{-2}$ in the numerator and an
energy-dependent shift $\delta E_{c}(E)=\delta\mu\delta
B_{c}(E)=(\bar{\Gamma_{c}}/2)\tan{\lambda_{\mathrm{bg}}(E)}$ in the
denominator of the resonance term in Eq. (9). The general form of Eq. (9) has
been shown to be in excellent agreement with near-threshold coupled scattering
calculations for a number of examples of Feshbach resonances in the literature
[74]. Close to threshold the MQDT functions have the following $s$-wave
limiting forms, which permit analytic limiting expressions to be given for
Feshbach scattering as $E\to 0$ [87]: $C_{\mathrm{bg}}(E)^{-2}\to
k\bar{a}(1+(r-1)^{2}$, $\tan\lambda_{\mathrm{bg}}(E)\to 1-r$, where
$r=a_{\mathrm{bg}}/\bar{a}$ is the scattering length in dimensionless
$\bar{a}$ units.
In a similar way, the MQDT coupled channels bound state equation from Refs.
[70, 71] can be put in the form
$\left(E-\delta\mu(B-B_{c})\right)\tan\nu_{\mathrm{bg}}(E)=\frac{\bar{\Gamma}}{2}\,.$
(10)
where $\nu_{\mathrm{bg}}(E)$ is the background reference channel MQDT phase
function for $E<0$, which has the property near threshold [87] that
$\nu_{\mathrm{bg}}(E)\to\tan^{-1}[1/(r-1)]-\bar{a}\kappa$, where
$E=-\hbar^{2}\kappa^{2}/(2\mu)$ for $E<0$. Using this threshold property, it
is straightforward to show that the coupled channels bound state crosses
threshold at $B_{0}=B_{c}+\delta B_{c}$, where the shift $\delta
B_{c}=-\Delta\,r(r-1)/(1+(r-1)^{2})$ is the same shift that Eq. (9) gives for
the singularity at $B=B_{0}$ in the scattering length $a(B)$; see Eq. (7).
One advantage of the MQDT approach is the factorization of the resonance
coupling into a part associated with the long-range physics, $C(E)^{-2}$, and
a reduced coupling parameter, $\bar{\Gamma}$, associated with the short range
physics. The $\bar{\Gamma}$ factor is proportional to the dimensionless MQDT
energy-insensitive $Y_{\mathrm{c,bg}}$ matrix element which gives the strength
of the coupling:
$|Y_{\mathrm{c,bg}}|^{2}=\frac{\bar{\Gamma}}{2}\frac{\partial\nu_{c}(E)}{\partial
E}\,,$ (11)
where ${\partial\nu_{c}(E)}/{\partial E}\approx\pi/\Delta E_{c}$ and $\Delta
E_{c}$ is the mean vibrational spacing between adjacent vibrational levels
near $E=E_{c}$. For all ultracold atom Feshbach resonances in the literature,
$Y_{\mathrm{c,bg}}\ll 1$ represents weak coupling, that is, the width of the
resonance is small compared to the spacing between adjacent bound vibrational
levels in the closed channel (mixing sometimes occurs with other closed
channels; see Figure 8).
In order to more explicitly show the connection between the long-range
potential and Feshbach resonance properties, Chin et al. [10] introduced a
dimensionless resonant strength parameter $s_{\mathrm{res}}$ by expressing
$a_{\mathrm{bg}}$ and $\delta\mu\Delta$ in dimensionless units of $\bar{a}$
and $\bar{E}$ respectively:
$s_{\mathrm{res}}=\frac{a_{\mathrm{bg}}}{\bar{a}}\frac{\delta\mu\Delta}{\bar{E}}\,.$
(12)
This is the inverse of the $\eta$ parameter used by Ref. [9] to characterize
Feshbach resonances. The resonance width
$\Gamma_{c}(E)=(k\bar{a})(\bar{E}s_{\mathrm{res}})$ exhibits the threshold
energy-dependence near $E=0$, and the energy-insensitive MQDT resonance
strength $\bar{\Gamma}$ is
$\frac{\bar{\Gamma}}{2}=\bar{E}\frac{s_{\mathrm{res}}}{1+(r-1)^{2}}\,.$ (13)
This expression can be used in Eq. (10) to get a general dimensionless MQDT
equation for bound states. The bound state equation can be used to show that
as $\kappa\to 0$ the bound state energy is given by $\kappa=1/(a(B)-\bar{a})$,
which is the same relation for large positive scattering length as found by
Gribakin and Flambaum [60] for a single van der Waals potential, namely,
$\kappa_{\mathrm{bg}}=1/(a_{\mathrm{bg}}-\bar{a})$ when
$a_{\mathrm{bg}}\gg\bar{a}$. In the Feshbach resonance case, the relation
applies when $a(B)\gg\bar{a}$ when $s_{\mathrm{res}}\gg 1$, but only applies
over a much more restricted range where $a(B)\gg 4\bar{a}/s_{\mathrm{res}}$
when $s_{\mathrm{res}}\ll 1$. Solving Eq. (10) in general gives the right
coupled channels bound states; see Ref. [10]. When $B-B_{0}$ is sufficiently
far from resonance, the equation also recovers the ”bare” bound state energy
varying as $E=\delta\mu(B-B_{c})$. The variation with $B$ of the bound state
energy of the coupled Feshbach bound state can be used to determine the norm
$Z$ of the closed channel piece in the unit normalized coupled channels bound
state wave function: $Z=\delta\mu^{-1}\partial E/\partial B$ [9, 10].
The dimensionless parameter $s_{\mathrm{res}}$ permits the classification of
resonances into two basic types [9, 10], depending on how their character
changes as $B$ is tuned over a range on the order of $\Delta$ near $B=B_{0}$.
One type with $s_{\mathrm{res}}\gg 1$ is ”open channel dominated” and the
other type with $s_{\mathrm{res}}\ll 1$ is ”closed channel dominated”. In
either case this pertains to the character of the bound state over a tuning
range that is a significant fraction of $\Delta$. In the open channel
dominated case, the wave function takes on the spin character of the entrance
channel, has a universal energy given by Eq. (7), and closed channel norm
$Z\ll 1$. In contrast, the closed channel dominated case has only a very small
domain of universality very close to $B_{0}$ and takes on the spin character
of the closed channel bound state with $Z\approx 1$. Some of the more
interesting experimental resonances are open channel dominated, namely the 6Li
83.4 mT, 40K 20.2 mT, and 85Rb 15.5 mT resonances, but many others are closed
channel dominated [9, 10].
The 40K87Rb 54.69 mT resonance, for which $\delta\mu=2.40\mu_{\mathrm{B}}$ and
$s_{\mathrm{res}}=2.07$, is an example of a resonance tending to open channel
dominance. Thus, Figure 8 shows that it has a universal domain over about a
third of its width, and in fact, a calculation of $Z$ shows that $Z<0.4$ for
detuning up to 0.1 mT. At 54.6 mT near the field of the experiment of Ni et
al. [41] the energy is $E/h=-329$ kHz, $Z=0.28$, and the scattering length is
443 a0, predicting a ”universal” energy of -336 kHz. Figure 10 shows that the
open channel component $f_{aa,aa}(R)$ of the coupled channel wave function at
54.6 mT extends to very large $R\gg\bar{a}$. This entrance channel component
has a norm of 0.718 and is the dominant part of the wave function. The wave
function becomes even more ”open channel dominated” as $B$ increases towards
resonance at $B_{0}$. It will show more closed channel character as $B$ tunes
farther away from resonance.
Figure 10 shows the $f_{rb,aa}(R)$ closed channel component of the bound state
wave function at 54.2 mT, where the energy of the bound state is $E/h=-10.6$
MHz, the detuning from resonance is about 1.6 resonance widths $|\Delta|$, and
the scattering length is $-71$ a0. This level is far from resonance, the bound
state is far from universality, and the norm of the $f_{rb,aa}(R)$ closed
channel component is 0.90. It is evident that the $f_{rb,aa}(R)$ component has
a very strong overlap with the $n=-2$ bound vibrational level of the long
range potential, so that it is legitimate to characterize the level to a good
approximation as being an $n=-2$ bound state of the $rb$ channel. The other
$rf(-3d0)$ resonance in Figure 8 is a closed channel dominated resonance with
$s_{\mathrm{res}}=0.12$. It is a very narrow resonance with a very small
($\ll\mu$T) domain of universality, so the below-threshold bound state is very
strongly of $d$-wave $n=-3$ $rf$ channel character.
Using the full MQDT in its angular momentum insensitive form [73] should
permit the prediction of the positions and widths of the various Feshbach
resonances in all the channels of the problem. The only additional information
needed to develop this theory is the analytic basis set transformation (”frame
transformation”) between the short-range and long-range basis sets for
representing the approximate spin eigenstates of the system. This method is
similar to the asymptotic bound state (ABM) method used by Wille et al. [58]
to characterize 6Li40K resonances. The MQDT method is an ”on-the-energy-shell”
coupled channels method, whereas the ABM method relies on an expansion in a
basis set of bound states of the X${}^{1}\Sigma^{+}$ and a${}^{3}\Sigma^{+}$
potentials. Either are capable of giving an approximate coupled channels
representation of the bound states of the system, and finding the threshold
resonances within the framework of those approximations. Either method could
provide a useful alternative to full numerical coupled channels calculations.
### 4.3 Inelastic collision rates
Finally, there is an additional insight from MQDT that is very helpful in
estimating, and in some cases quantitatively calculating, the rate
coefficients for ultracold atomic or molecular collisions. The basic idea of
the method is given in Ref. [72] and was implemented for atomic collisions
with Penning ionization by Refs. [96, 97] and for molecular vibrational
quenching collisions by Ref. [98]. It takes advantage of Eqs. (3)-(4) together
with the MQDT concept of factoring the $S$-matrix into separate parts due to
short- and long-range interactions, as we saw with the resonant width in the
last section.
Let us assume that the probability for loss of the colliding species A and B
is unity, given that the species are close together and strongly interacting
at distances small compared to the scale length $\bar{a}$ of the long range
potential. If the collisional wave function were semiclassical at all $R$ and
not affected by quantum threshold effects, then one could calculate a
semiclassical rate coefficient in a Langevin model, summing over all
contributing partial waves that reach short distance, taking
$1-|S_{\alpha\alpha}|^{2}=1$ in Eq. (4) to represent maximal loss and no
reflection for the contributing partial waves. However, in the threshold
regime, the threshold law associated with the long De Broglie wavelength needs
to be taken into account. Let us assume that collision energy $E$ is low
enough and no electric field is present to induce an actual molecular dipole
moment so that only $s$-wave contribute. The total $s$-wave loss is calculated
by taking the loss probability $1-|S_{\alpha\alpha}|^{2}$ to be the
transmission probability for reaching short range, namely, $1-P_{r}$, where
$P_{r}$ is the quantum reflection probability from the long-range potential.
This gives the needed quantum correction to semiclassical theory as $E\to 0$.
The transmission factor can be calculated numerically [96, 98] or, in some
cases, analytically [61, 97, 99]. Given the analytic result
$1-P_{r}=4k\bar{a}$ for $s$-wave transmission through the long range van der
Waals potential [61, 99], the rate coefficient for collisional loss from Eq.
(5) with $b=\bar{a}$ is
$K_{\mathrm{in}}=2(h/\mu)\bar{a}$ (14)
for a ”strong” molecular loss collision with unit short range probability of
an inelastic event. This formula is an example of ”inelastic universality”,
where there is a universal rate constant depending only on the long-range
potential and not on the scattering length, since there is no ”back
reflection” from the short-range region. Typical values of $\mu$ and $\bar{a}$
give an order of magnitude of $10^{-10}$ cm${}^{3}/$s for $K_{\mathrm{in}}$
for such collisions.
Consequently, if the probability for a short-range loss process is very high,
then knowing the long range van der Waals coefficient is sufficient to make
quite accurate estimates of the threshold collision rate coefficient, as
demonstrated by applying the above model to vibrational quenching of excited
vibrational levels of the RbCs molecule [98]. This factorization procedure of
MQDT theory [72] can be generalized to include other partial waves or
resonance structure [66] and should be capable of giving much insight into
molecular collisions. If the probability $\bar{P}$ of short-range reaction or
loss is near unity and one has a way to calculate or estimate it, then that
factor can be included in the expression for the $s$-wave rate constant by
replacing $\bar{a}$ in Eq. (14) by $\bar{a}\bar{P}$. On the other hand, if
$\bar{P}\ll 1$ so there is significant reflection from the short range region,
$K_{\mathrm{in}}$ will depend on the scattering length $a_{\alpha}$ of the
entrance channel. In any case, we see that if a collision leads to a strong
highly probable short range loss event, the influence of the long-range
potential is critical in determining how threshold modifications to Langevin
theory occurs in the ultracold domain.
## 5 Deeply bound states of 40K87Rb
So far we have concentrated on bound states that are very close to threshold
with binding energies of less than 1 GHz. Figures 11 - 13 show an expanded
view of the $M=-7/2$ $s$-wave bound states with binding energies up to 30 GHz,
or about 1 cm-1, between $B=$ 0 and 120 mT. The character of the vibrational
levels is relatively easy to explain in this region of the spectrum, where the
splitting between adjacent vibrational levels of ${}^{1}\Sigma^{+}$ and
${}^{3}\Sigma^{+}$ symmetry is much smaller than the splitting of the atomic
hyperfine manifold. The singlet and triplet vibrational bound states have
similar binding energies because they (accidentally) have similar scattering
lengths. Each vibrational level, away from avoided crossings where different
levels mix, is characterized by the dominant spin character of one of the 12
separated atom channels $\beta$ shown in Figure 6. They also are characterized
by a vibrational wave function characteristic of $n=-1,-2,\dots$ long range
levels, so that levels are located at energies that are below the
corresponding separated atom limits by the binding energies of the $n$ levels
in these channels. Figure 11 indicates these groupings, and shows how the
levels in the close-up views in Figures 6 and 7 relate to neighboring levels.
Levels of different channels and $n$ are intermixed because of the similarity
in vibrational spacings and hyperfine spacings in some cases.
Figure 12 extends the broader view to a scale of 10 GHz. The A and B groups
cluster in the 10 GHz region, since the binding energy of the $n=-3$ levels
are on the order of 10 GHz. The $aa(-3)$ level parallel to the $E=0$ axis has
the spin character of the $aa$ entrance channel and is the level made by
Ospelkaus et al. [43] by 2-color STIRAP transfer of population from the
Feshbach molecule state at 54.6 mT.
Figure 13 gives an even more expanded view down to 30 GHz binding energy,
which includes the full manifold of the 12 $n=-4$ levels associated with each
of the separated atom channels $\beta$. The Figure also shows the energy
levels of the X${}^{1}\Sigma^{+}$ and a${}^{3}\Sigma^{+}$ potentials,
referenced to the nonrelativistic energy of the separated atoms at $B=0$. This
energy is 4.843439 GHz above the energy of two $a$ atoms in the $aa$ channel
at $B=0$. In this domain of binding energy, the Figure shows that the
splitting between the two adjacent singlet and triplet levels is much less
than the splittings of the atomic hyperfine manifold. Each molecular level,
except near avoided crossings of levels of different channels, thus has the
$\beta n$ character of two weakly bound separated atoms in channel $\beta$
with the long-range vibrational wave function of level $n$. If the singlet and
triplet potentials had very different scattering lengths, so the corresponding
singlet and triplet vibrational levels were not so close in energy as in the
Figure, then one would begin to see a breakdown in this atomic coupling scheme
in this range of energy.
Figures 14 and 15 examine the transition from the threshold domain, where the
singlet and triplet levels are mixed by spin-dependent interactions, and the
more deeply bound region, where the vibrational levels are to a good
approximation dominantly singlet or triplet in character. This transition
occurs for binding energies of around 200 GHz, where the splitting between
adjacent $n=-8$ singlet and triplet vibrational levels at $-205.987$ GHz and
$-192.174$ GHz becomes on the order of the atomic hyperfine splittings. Levels
with less binding tend to be of mixed singlet and triplet character. More
deeply bound levels are clearly identified as being dominantly singlet or
triplet, with the exception that the $n=-13$ singlet levels at $-922.325$ GHz
has an accidental near-degeneracy with the $n=-14$ triplet level at $-924.070$
GHz. This gives rise to a strong mixing of the two levels that is very local
in energy. Singlet-triplet mixing is very small for all levels with larger
binding energy.
Finally, Figure 16 shows the spin structure of the $v=0$ $J=0$
X${}^{1}\Sigma^{+}$ ground state molecule. Since the two active electrons are
paired into a singlet state with no net spin, and since the projection on the
molecular axis $\Lambda=0$ for a $\Sigma$ molecular state, there is no
coupling of the nuclear spins to the electrons or the molecular axis. If there
were no coupling to distant triplet electronic states, all nuclear spin
components of the $v=0$ $J=0$ X${}^{1}\Sigma^{+}$ level would be degenerate.
Figure 16 shows a slight zero-field splitting at $B=0$ of 41 kHz between the
lowest energy $I_{\mathrm{tot}}=11/2$ and highest energy
$I_{\mathrm{tot}}=5/2$ nuclear spin components, where $I_{\mathrm{tot}}$ is
the resultant of the 40K and 87Rb nuclear spins. This very small splitting in
our model is due to second-order coupling through the distant
a${}^{3}\Sigma^{+}$ state. This model calculation based on atomic spin
coupling constants may not be accurate in the molecular environment of the
spins in the $v=0$ level. However, the nuclear spin structure at the 54.6 mT
field of the experiment of Ref. [41] represents essentially uncoupled nuclear
spins in the large $B$ field, where the energies are very close to being the
energies of the isolated separate atoms in the same field. Thus, the 40K atom
splits into 9 Zeeman components and the 87Rb atoms splits into 4 components
with a total spread of 3.4 MHz across the manifold of levels at 54.6 mT.
It is a very interesting question to determine to what extent individual
nuclear spin states can be prepared and manipulated using the very precise
optical control that may be possible with the ultra high precision frequency
comb technology that went into the STIRAP experiment of Ni et al. [41]. This
may be possible using a combination of polarization and frequency control.
There also is the question of the collisional stability of the $v=0$ ground
state molecules. If reactive collisions are inhibited by a reaction barrier at
$\mu$K temperatures, then the only destructive collisions that change the
state are ones that depolarize or relax the nuclear spins. Even at zero field,
the spread of energy of 41 kHz corresponds to $E/k_{B}$ of 2 $\mu$K. Due to
the absence of coupling of the nuclear spins to the electrons or molecular
axis in the problem, spin depolarization due to molecule-molecule collisions
probably has a very small inelastic loss rate constant (perhaps too small to
be measurable). However, a collision with a 40K or 87Rb atom could have a non-
negligible inelastic collision rate coefficient, due to the coupling of the
nuclear spins in the molecule to the unpaired atomic electron during the
collision. It will be important to determine if the nuclear spins can be
controlled in a polar molecule gas or lattice. For example, the spins might
make good quantum memory, subject to weak decoherence, but capable of rapid
optical manipulation and measurement.
It is worth noting the excellent quality of the potentials in Ref. [77] that
were used in this calculation. We calculate the energy of the $v=0$ $J=0$
X${}^{1}\Sigma^{+}$ level relative to the energy of two $a$ state atoms at
54.588 mT to be $E/h=-125320.10$ GHz. This differs by only $0.4$ GHz from the
measured value of $-125319.703(1)$ GHz, which corresponds to an error of only
3 parts in $10^{6}$ in the absolute binding energy. The calculations for the
a${}^{3}\Sigma^{+}$ state are not as accurate. The lowest spin component of
the $v=0$ $N=0$ triplet state was measured [77] to be at $-7180.4180(5)$ GHz,
compared to a calculated value of $-7195.6$ GHz. The much larger error of 16
GHz is likely due to the lack of accurate spectroscopic data for the lowest
vibrational levels to constrain the minimum of the a${}^{3}\Sigma^{+}$
potential. The new very precise data on the absolute molecular binding
energies should aid the construction of much more accurate potentials for
future use.
## 6 Concluding remarks
This paper has presented some very general considerations for ultracold atomic
and molecular collisions and interactions as well as some specific results for
the 40K87Rb fermionic molecule. A general consideration is that ultracold
collisions very precisely prepare the collision complex of two colliding
species in a very sharp energy state, which then serves as a gateway for
various kinds of spectroscopic probing and manipulation of the complex. Thus,
one can manipulate the properties of an ultracold gas or lattice using tunable
scattering resonances, one can use such resonances to make a near-threshold
bound state, and one can probe or populate levels far from threshold by
optical manipulation. Thus, resonant scattering theory and an account of near-
threshold bound states are of crucial importance to ultracold atomic and
molecular physics.
Much progress can be made by understanding the spectrum and resonances
associated with the long-range potential. Good theoretical progress in
understanding the more complex world of molecular collisions can be made by
developing theories that explicate the near-threshold domain of resonant
scattering. Much of the physics associated with strong short-range
interactions can be parameterized by energy-insensitive parameters of
multichannel quantum defect theory. A simple example of applying these ideas
is in understanding the magnitude of rate constants for inelastic molecular
vibrational relaxation, where in the ultracold domain, one only needs to know
the quantum reflection off the long-range potential between the two colliding
species, given that relaxation has unit probability if the colliding species
reach the short-range region. Such ideas can be generalized to include
resonances and weak short range processes. It remains to be seen how
successful such an approach can be in the more complex environment of
molecular collisions.
The bound and threshold scattering states of the 40K87Rb molecule illustrate
some important aspects of molecular physics related to the formation and use
of a vibrational ground state molecule. The domain from near-threshold to the
most deeply bound levels can be understood quite precisely from coupled
channels calculations. These require the full Hamiltonian for the problem,
including accurate molecular adiabatic Born-Oppenheimer potentials and spin
coupling constants. In general these can only be obtained from a combination
of accurate ab initio calculations and precise spectroscopy. While excellent
quality potentials are available for KRb, these will need to be developed of
other systems of future interest. Much insight and practical calculation can
be done in the near-threshold domain from a combination of methods involving
the properties of the long-range potential and various implementations and
approximations based on the long-range potential. These methods require a
semi-empirical approach where parameters of the theoretical models, such as
scattering lengths for the potentials, need to be extracted from fitting
experimental data. Such methods are already highly developed for atomic
collisions, but still need to be developed for molecular collisions, if
possible. It is clear, however, that accurate theoretical models of the spin
structure of ground state molecules will be needed, including the effect of
external magnetic, electric, or electromagnetic fields. Ultracold resonant
scattering theory of colliding molecular species with structure must first
understand the various scattering channels associated with the internal
structure of the colliding species, and then try to understand the resonances
or other scattering properties associated with the long-range potential.
Finally, the threshold resonances and dynamics associated with strong short-
range interactions of the molecular collision complex will be difficult to
predict accurately from first principles, but may be accessible to a
combination of theoretical modeling and experimental probing of the complex.
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Figure 1: Schematic figure of the potential energy curves $V(R)$ of two
interacting atoms A and B versus interatomic distance $R$, with the zero of
energy $E$ set as the energy of two separated atoms in the states in which
they are prepared. The horizontal line just below threshold indicates the
energy of a weakly bound molecular level $|1\rangle$, which is prepared by
associating two cold atoms and then converted to the target deeply bound
vibrational level $|2\rangle$ by a Raman transition through an excited
intermediate molecular state $|i\rangle$. The Raman process could also be
initiated starting from a scattering state with $E>0$, although this has
proved difficult in practice. Schemes of this general type were proposed by
References [23, 24, 25] and realized in References [26, 40, 41] by starting
from associated atoms in a weakly bound state.
Figure 2: Schematic view of an ultracold collision to form a molecular
”complex”. The species A and B are each prepared in internal states labeled
collectively by $\alpha=1$ and with a very small relative collision energy
near the separated species energy $E_{1}=0$. Two closed channels 2 and 3 with
different internal energies $E_{2}$ and $E_{3}$ of the separated species are
also schematically indicated, along with the spectrum of associated bound
state levels (short horizontal lines) in channels 1, 2 and 3. The energy scale
for an ultracold collision is on the order of $k_{B}T=86$ peV or $k_{B}T/h=21$
kHz for $T=1$ $\mu$K. The energy scale for the short range part of the
potential, where $R\approx R_{\mathrm{bond}}$ in on the order of the chemical
bond length $R_{\mathrm{bond}}$, is given by the binding energy of the $v=0$
level of the ground state potential, on the order of $E/h=$5 THz for a weakly
bound van der Waals molecule (0.02 eV) to 1 PHz for a strong chemical bond (5
eV). The near-threshold bound and scattering states of the complex are
sensitive to the long range part of the potential that varies asymptotically
as $-C_{n}/R^{n}$ and has a characteristic length $\bar{a}$. Near-threshold
bound states associated with closed channels 2 and 3 of the collision form
threshold scattering resonances in channel 1 that can be tuned across
$E=E_{1}$ by varying magnetic, electric, or electromagnetic fields.
Figure 3: The energy of the magnetic Zeeman sublevels of the fermionic 40K
atom versus magnetic field $B$. The zero of energy is the nonrelativistic
energy center of gravity of the multiplet. The levels, labeled $a,b,\ldots$ in
order of increasing energy correlate at $B=0$ with the two hyperfine levels
with total electron spin plus nuclear spin angular momentum $F=9/2$ and
$F=7/2$. The $M$ quantum number specifies the angular momentum projection on
the magnetic field axis and, unlike $F$, remains a good quantum number as $B$
increases.
Figure 4: The energy of the magnetic Zeeman sublevels of the bosonic 87Rb
atoms versus magnetic field $B$. The zero of energy is the nonrelativistic
energy center of gravity of the multiplet. The levels, labeled $a,b,\ldots$ in
order of increasing energy correlate at $B=0$ with the two hyperfine levels
with total electron spin plus nuclear spin angular momentum $F=1$ and $F=2$.
The $M$ quantum number specifies the angular momentum projection on the
magnetic field axis and, unlike $F$, remains a good quantum number as $B$
increases.
Figure 5: Adiabatic Born-Oppenheimer X${}^{1}\Sigma^{+}$ and
a${}^{3}\Sigma^{+}$ potential energy curves $V(R)$ of the KRb molecule
correlating with 2S ground state K and Rb atoms. These curves dissociate to
$E=0$ at the nonrelativistic energy center of gravity of the $B=0$ atomic
hyperfine levels. The inset shows the long range adiabatic curves for the
$M=-7/2$ projection states of the 40K87Rb molecule at $B=0$. These curves
separate asymptotically to the atoms in one of their ground hyperfine levels.
In the inset, the energy zero is set as the energy of the lowest $9/2+1$ set
of levels at B=0. The characteristic van der Waals length is $\bar{a}=68.9$
a0.
Figure 6: Long range adiabatic curves for the 12 $M=-7/2$ projection states
of the 40K87Rb molecule. The dashed lines show the same curves as the inset of
Fig. 3 for $B=0$ and the solid lines show the 12 curves for $B=54$ mT, labeled
according to the Zeeman levels of each of the separated atoms. The zero of
energy is taken to be the energy of the lowest hyperfine levels $9/2+1$ at
zero field. The groupings into 4 sets of states labeled by $A$, $B$, $C$, and
$D$ correspond to the sets of states $(aa,bb,cc)$, $(rb,qc)$, $(dd,ce,bf,ag)$,
and $(pd,qe,rf)$. These respective groupings are associated with the zero
field separated atom hyperfine levels $9/2+1$, $7/2+1$, $9/2+2$ and $7/2+2$.
Figure 7: Scattering length (upper panel) and bound state energy (lower panel)
for the $M_{\mathrm{tot}}=-7/2$ $s$-wave channels of the 40K87Rb molecule. The
zero of energy is the energy $E_{\alpha}$ of the $aa$ channel at each $B$
field. Thus, the bound state energies give the binding energies of the levels
relative to the $aa$ separated atom energy. The dashed vertical lines show the
points of singularity $B_{0}$ of the scattering length, calculated to be at
46.239 mT, 49.563 mT, 54.694 mT and 65.969 mT, where the binding energy of a
molecular bound state becomes zero as the state reaches threshold. The solid
circles show the bound state energies calculated for a discrete set of $B$
values. The labels $\beta(n)$ show the dominant spin character of the bound
eigenstate, where $\beta$ indicates a separated atom closed channel and $n$
gives the vibrational quantum number of the vibrational level in that channel,
counting down from the dissociation limit of the channel at $E_{\beta}$.
Figure 8: Expanded view of the 54.69 mT resonance in Figure 7. The upper panel
shows the scattering length and the lower panel the bound state energies
versus $B$. The solid line in the upper panel shows the scattering length
calculated using an $s$-wave basis set only, the vertical dashed line marks
the resonance position $B_{0}$, and the double headed arrow shows the
magnitude of the resonance width $\Delta$. The dashed curve in the upper panel
shows an extra resonance that appears when $s$ and $d$ basis functions are
both used in the calculation. The bound state labeled $rb(-2s0)$ is the same
as in Figure 7. The bound state label includes the $\ell m_{\ell}=s0$
designation. The new bound state labeled $rf(-3d0)$ is a $d$-wave bound state
that appears when a $d$-wave basis set is added to the calculation with
$M_{\mathrm{tot}}=-7/2$. The dashed line in the lower panel indicates the
universal energy derived from the scattering length using Eq. (7).
Figure 9: Bound state spectrum of the last two vibrational levels of 40K87Rb
with quantum numbers $n=$ $-1$ and $-2$ for partial waves $\ell=0,1,2,3,4$
($s$,$p$,$d$,$f$,$g$). The horizontal lines show the levels for a pure van der
Waals potential with the $C_{6}$ constant of the 40K87Rb molecule and with an
infinite $s$-wave scattering length. The open and solid circles respectively
show the calculated energy levels for the X${}^{1}\Sigma^{+}$ and
a${}^{3}\Sigma^{+}$ potentials, for which the scattering lengths are $-111.8$
a0 and $-216.2$ a0 respectively [77]. The next lowest $n=-3$ levels of these
potentials are at $-10.24$ GHz and $-10.56$ GHz.
Figure 10: Calculated wave functions for near-threshold bound states of the
40K87Rb molecule. The solid lines labeled $n=-1,-2,-3$ show the unit-
normalized wave functions $f_{n}(R)$ for the last three levels of the
a${}^{3}\Sigma^{+}$ potential. The vertical line indicates the value of
$\bar{a}$. The dots and dashed lines show results from the coupled channels
calculation of the $aa$ channel bound state of Eq. 6 near the 54.69 mT
resonance, namely, the $rb$ component $f_{rb,aa}(R)$ at $B=54.2$ mT and the
$aa$ channel component $f_{aa,aa}(R)$ at $B=54.6$ mT. The latter represents a
”halo molecule” with open channel $aa$ spin character and an extension large
compared to $\bar{a}$. The former represents a closed channel molecule with
$rb$ spin character and a vibrational function very close to the $n=-2$
a${}^{3}\Sigma^{+}$ vibrational wave function.
Figure 11: Bound state energy for the $M_{\mathrm{tot}}=-7/2$ $s$-wave bound
states of the 40K87Rb molecule down to 4 GHz binding energy. The zero of
energy is the energy $E_{\alpha}$ of the $aa$ channel at each $B$ field. The
level are labeled according to the channel index of their dominant spin
component and by the vibrational quantum number $n$ counting down from the
dissociation limit of the channel. The Roman letters indicate the group in
Figure 6 with which the $B=0$ channels are associated.
Figure 12: Bound state energy for the $M_{\mathrm{tot}}=-7/2$ $s$-wave bound
states of the 40K87Rb molecule down to 12 GHz binding energy. The zero of
energy is the energy $E_{\alpha}$ of the $aa$ channel at each $B$ field.
Labels are the same as in Figure 11.
Figure 13: Bound state energy for the $M_{\mathrm{tot}}=-7/2$ $s$-wave bound
states of the 40K87Rb molecule down to 30 GHz binding energy. The zero of
energy is the energy $E_{\alpha}$ of the $aa$ channel at each $B$ field.
Labels are the same as in Figure 11. The short horizontal lines next to the
$B=0$ axis labeled by vibrational quantum number $n$ show the energies of the
levels of the X${}^{1}\Sigma^{+}$ and a${}^{3}\Sigma^{+}$ adiabatic Born-
Oppenheimer potentials relative to the energy center of gravity (CoG) of the
separated atom multiplet at $E/h=4.843439$ GHz. The X${}^{1}\Sigma^{+}$ and
a${}^{3}\Sigma^{+}$ potentials support $N=$ 100 and 32 vibrational levels
respectively. The normal vibrational quantum number $v$ counting up from $v=0$
as the lowest level is $v=N+n$.
Figure 14: Bound state energy for the $M_{\mathrm{tot}}=-7/2$ $s$-wave bound
states of the 40K87Rb molecule at $B=54.6$ mT. All energies are relative to
the energy center of gravity of the atomic hyperfine multiplets. The
horizontal scale gives a measure of the singlet or triplet character of the
eigenstate, with 0 representing a pure singlet state and 1 representing a pure
triplet state. The horizontal line at $E/h=-4.834$ GHz labeled $E_{aa}$ marks
the energy of two $a$ state atoms on this scale. The crosses labeled
”Uncoupled” mark the energies of the X${}^{1}\Sigma^{+}$ and
a${}^{3}\Sigma^{+}$ energy levels; $n=-5$ indicates the fifth level down in
each potential. The red dots labeled ”Coupled” locate the actual eigenstates
of the coupled channels calculation. No levels have pure singlet character in
this domain.
Figure 15: Bound state energy for the $M_{\mathrm{tot}}=-7/2$ $s$-wave bound
states of the 40K87Rb molecule at $B=54.6$ mT. The labeling is the same as in
Figure 14, but the energy scale extends to $E/h=1200$ GHz binding energy.
Uncoupled singlet levels with $|n|\leq 7$ are strongly mixed with triplet
levels of the same $n$. The eigenstates separate into two sets of eigenstates
of nearly pure singlet or triplet character when the binding energy becomes
larger than around 200 GHz. There is an accidental near-degeneracy of the
$n=-13$ $(v=87)$ singlet level and $n=-14$ $(v=18)$ triplet level near 900 GHz
binding that results in strong mixing between singlet and triplet levels in
that energy range. There are no more accidental degeneracies with strong
perturbations for any levels with lower energy.
Figure 16: Bound state energy for all $M_{\mathrm{tot}}$ $v=0$ $J=0$ bound
states of the 40K87Rb molecule at $B=0$ (x symbols) and 54.6 mT (diamond
symbols). Energy is relative to the energy center of gravity of the atomic
hyperfine multiplets. At $B=0$ the levels divide into four degenerate groups
with total nuclear spin quantum number $I_{\mathrm{tot}}=$ $11/2$, $9/2$,
$7/2$, and $5/2$ in order of increasing energy, with a spread of 41 kHz
between the highest and lowest energy. At $B=$ 54.6 mT the nuclear spins
become uncoupled from one another and represent two independent nuclear spins
in the strong $B$ field. The labels indicate the projection quantum numbers
for the four projection levels of the 87Rb nuclear spin and the nine levels of
the 40K nuclear spin.
|
arxiv-papers
| 2008-12-05T21:36:57 |
2024-09-04T02:48:59.245360
|
{
"license": "Public Domain",
"authors": "Paul S. Julienne",
"submitter": "Paul Julienne",
"url": "https://arxiv.org/abs/0812.1233"
}
|
0812.1279
|
# Species competition: coexistence, exclusion and clustering
Emilio Hernández-García1 Cristóbal López1 Simone Pigolotti2 Ken H. Andersen3
1IFISC (UIB-CSIC), Instituto de Física Interdisciplinar y Sistemas Complejos,
Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
2The Niels Bohr Institute, The Niels Bohr International Academy, Blegdamsvej
17 DK-2100, Copenhagen, Denmark
3National Institute of Aquatic Resources, Technical University of Denmark,
Charlottenlund slot, DK-2920 Charlottenlund, Denmark
###### Abstract
Competition, Lotka-Volterra, competitive exclusion, limiting similarity,
pattern formation We present properties of Lotka-Volterra equations describing
ecological competition among a large number of interacting species. First we
extend previous stability conditions to the case of a non-homogeneous niche
space, i.e. that of a carrying capacity depending on the species trait.
Second, we discuss mechanisms leading to species clustering and obtain an
analytical solution for a state with a lumped species distribution for a
specific instance of the system. We also discuss how realistic ecological
interactions may result in different types of competition coefficients.
Philosophical Transactions of the Royal Society A 367, 3183-3195 (2009)
http://dx.doi.org/10.1098/rsta.2009.0086
Published under the Creative Commons Attribution license.
## 1 Lotka-Volterra competition and species distribution
Competitive interactions occur when entities in a system grow by consuming
common finite resources. They are ubiquitous in many fields of science:
examples include biological species competing for food (MacArthur & Levins
1967; Roughgarden 1979; Case 1981), mode competition in nonlinear optical
systems (Benkert & Anderson 1991), or alternative technologies competing for a
market (Pistorius & Utterback 1997). An early, simple, but powerful model for
competitive interactions is the Lotka-Volterra (LV) set of competition
equations (Volterra 1926; Lotka 1932):
$\dot{N_{i}}=r_{i}N_{i}\left(1-\frac{1}{K_{i}}\sum_{j=1}^{m}G_{ij}N_{j}\right),\
\ i=1,...,m.$ (1)
where $m$ is the number of species, $N_{i}$ the population of species $i$,
$r_{i}$ its maximum growth rate, $K_{i}$ its carrying capacity, and $G_{ij}$
is the matrix characterizing the interaction among species $i$ and $j$, more
specifically the decreasing on the growth rate of species $i$ by the presence
of $j$. Competitive interactions are characterized by $G_{ij}\geq 0$, the
situation to be considered here, whereas negative interactions may model
situations of mutualism, predation or symbiosis.
In classical ecological niche theory, species are associated to points in an
abstract niche space. Coordinates in this space represent relevant phenotypic
characteristics, for example size of individuals in a species, or the size of
preferred prey, such that intensity of competition is larger if species are
closer in this space. We assume for simplicity this space to be one-
dimensional (multi-dimensional generalizations are straightforward, as briefly
mentioned later). If niche locations can be considered to be a continuum, we
can write Eq. (1) as:
$\partial_{t}\psi(u,t)=r(u)\psi(u,t)\left[1-\frac{1}{K(u)}\int
G(u,v)\psi(v,t)dv\right],\ \ \ \ \ $ (2)
where now $\psi(u,t)$ is the population density at niche location $u$. The
integral extends over the full niche space, which could be finite or infinity.
For most purposes, Eqs. (1) and (2) can be considered as equivalent, since the
second is obtained from the first in the limit of many close interacting
species, and (1) can be recovered from (2) for a discrete distribution of
species:
$\psi(u)=\sum_{i=1}^{m}N_{i}\delta(u-u_{i}),$ (3)
with $G_{ij}=G(u_{i},u_{j})$, $r_{i}=r(u_{i})$ and $K_{i}=K(u_{i})$.
It is widely believed that (1) or (2) predict a competitive exclusion leading
to a limiting similarity situation (Abrams 1983), in which a pair of species
too close in niche space can not coexist, and one of them would become
extinct. However it is known that the model allows for continuous coexistence
of species in some situations (Roughgarden 1979), and refinements on the
conditions for this coexistence have been developed, with emphasis on the
effect of the shape of the carrying capacity function $K(u)$ (Meszéna _et al._
2006; Szabó & Meszéna 2006). In this context, a particulary surprising result
was the finding by Scheffer & van Nes (2006) of a situation –for uniform
carrying capacity– which was neither of full coexistence nor of full
exclusion, but of clusters or lumps of tightly packed species which did not
exclude each other, but were well separated from other clusters so that there
was a type of limiting similarity leading to a minimum intercluster distance.
Clustering of individuals or entities under competitive interactions of the LV
type had been already observed in other contexts (Fuentes _et al._ 2003;
Hernández-García & López 2004, 2005; Ramos _et al._ 2008), where the mechanism
was the diffusive broadening of an otherwise zero-width species or entity. In
contrast, the lumps in Scheffer & van Nes (2006) appeared even in the absence
of diffusion in niche space, which is the situation also considered here.
The importance of the functional form of the interaction kernel $G_{ij}$ in
(1) or $G(u,v)$ in (2) was stressed by Pigolotti _et al._ (2007) for the case
of uniform carrying capacity and interactions depending only on differences of
niche positions, and found to be relevant in an evolutionary context by Leimar
_et al._ (2008). For that case the positive-definite character of the Fourier
transform of $G(u,v)=G(u-v)$ is a condition implying the absence of limiting
similarity. Species clustering was reported, but for interaction functions
rather different from the Gaussian used in Scheffer & van Nes (2006). In fact,
for the Gaussian interaction case most results are extremely sensitive to
details such as the implementation of the boundary conditions or weak
ecological second order effects (Pigolotti _et al._ 2008). Thus, a
clarification of the mechanisms leading to species clustering in LV models
would be desirable. In addition, the results in Pigolotti _et al._ (2007,
2008) were obtained under the unrealistic assumption of homogeneity in niche
space whereas the inhomogeneities in the carrying capacity are known to play
relevant roles (Szabó & Meszéna 2006). For simplicity we restrict our
description to the standard situation in which competition is stronger among
species closer in niche space. It is worth mentioning the existence of studies
of LV systems where non-local interactions are considered (Doebeli & Dieckmann
2000). That situation can also be described by the general formalism used here
of an integral kernel function, and our general results therefore also apply
to the situation with non-local interactions.
In this Paper we analyse some mathematical properties of the LV model (1) or
(2). In Sect. 2 we show that the positive-definiteness of the kernel $G$
remains a determining condition for stable coexistence even for non constant
$K(u)$. In Sect. 3, we discuss the mechanism producing lumped species
distributions and explicitly give an analytic expression for a particular
interaction kernel. In the Appendix we show that, in contrast with the
earliest characterizations of the interaction kernel $G$ (MacArthur & Levins
1967; Roughgarden 1979), both positive- and non-positive-definite kernels can
arise from more detailed ecological models which consider the dynamics of the
consumed resource. We use periodic boundary conditions in our numerical
simulations. We expect the effects of this simplifying but unrealistic
assumption to be unimportant at least when a non-constant carrying capacity
limits the presence of species to a limited region of niche space.
## 2 The stability of close coexistence
A simplifying assumption for the study of the LV model is that of homogeneity
in niche space. In this case, the carrying capacity and growth rate are
constants, $K_{0}$ and $r_{0}$, and the interaction kernel depends only on
differences of niche positions $G(u,v)=G(|u-v|)$. Niche space could be
infinite, but in the case in which it is finite, homogeneity can only be
achieved under periodic boundary conditions. Under these restrictions it is
easy to see that a steady solution to (2) which is homogeneous and everywhere
non-vanishing always exists: $\psi_{0}=K_{0}/\hat{G}_{0}$, where
$\hat{G}_{0}\equiv\int duG(u)$. This solution represents coexistence of all
possible species without a limit to their similarity. Its stability against
small perturbations can be analysed by linearization of the equation resulting
from substitution of $\psi(u,t)=\psi_{0}+\delta\psi(u,t)$ into (2). The
solution for the Fourier transform of the deviation from the homogeneous
state, $\delta\hat{\psi}_{q}(t)$, is
$\delta\hat{\psi}_{q}(t)=\delta\hat{\psi}_{q}(0)e^{\lambda_{q}t}\ ,{\rm with}\
\ \lambda_{q}=-r_{0}\frac{\hat{G}_{q}}{\hat{G}_{0}}\ .$ (4)
where $\hat{G}_{q}$ is the Fourier transform of $G(u)$. Thus, the homogeneous
solution $\psi_{0}$ is stable if $\hat{G}_{q}$ is positive $\forall q$, while
a instability leading to pattern formation occurs when $\hat{G}_{q}$ may take
negative values (Pigolotti _et al._ 2007; Fuentes _et al._ 2004; Hernández-
García & López 2004; López & Hernández-García 2004). We note that many steady
solutions to Eq. (2) exist besides $\psi_{0}$ (in particular, solutions of the
form (3)). This is so because dynamics preserves $\psi(u)=0$ at all places
where there is no initial population. Notice also that $\psi_{0}$ is the only
strictly positive solution. Among this multiplicity of solutions the ones that
will be more relevant are the ones which are stable under perturbations or
small immigration (Pigolotti _et al._ 2007).
An interesting class of functions to be used as kernels and carrying
capacities is the family $\\{g_{\sigma}^{p}\\}$ given by
$g_{\sigma}^{p}(u)\equiv\exp\left(-|u/\sigma|^{p}\right),$ (5)
which is parameterized by the value of $p$. The widely used Gaussian kernel is
obtained for $p=2$. When $p<2$ the functions are more peaked around $u=0$ (the
case $p=1$ is an exponential) and for $p>2$ they become more box-like
($g_{\sigma}^{\infty}(u)$ is the flat box with value $1$ in the interval
$[-\sigma,\sigma]$ and zero outside). The width of the kernel $\sigma$ gives
the competition range in niche space. We have positivity of the Fourier
transform if $p\leq 2$. This implies that the homogeneous solution is stable
under evolution with uniform $K$ and kernel $G$ of the form (5) if $p\leq 2$.
When $p>2$, the homogeneous solution is unstable and the system approaches
delta comb solutions of the type (3), with a spacing approximately $1.4\sigma$
(Pigolotti _et al._ 2007) which represent limiting similarity situations.
We now generalize the above stability analysis to the more realistic case in
which there is no homogeneity in niche space. First we consider the simpler
case of a symmetric kernel $G(u,v)=G(v,u)$, which in particular includes the
previous case of kernels depending only on species distance:
$G(u,v)=G(|u-v|)$. Note that in this symmetric case one can write Eq. (2) in
potential form:
$\partial_{t}\psi(u,t)=-r(u)\frac{\psi(u,t)}{K(u)}\frac{\delta
V[\psi]}{\delta\psi(u)},$ (6)
with the functional potential given by:
$V[\psi]=-\int K(u)\psi(u,t)du+\frac{1}{2}\int\int G(u,v)\psi(u,t)\psi(v,t)\
du\ dv.$ (7)
Stationary solutions of Eq. (2) are those for which the r.h.s of Eq. (6)
equals $0$. This has many possible solutions. We define the natural stationary
solution, $\psi^{N}(u)$, as the one which is positive and non vanishing for
all $u$, so that
$\left(\frac{\delta V}{\delta\psi}\right)_{\psi^{N}}=0,$ (8)
that is, the one satisfying:
$\int G(u,v)\psi^{N}(v)dv=K(u)\ ,\forall\ u$ (9)
The solution $\psi^{N}(u)$ can be considered the non-homogeneous
generalization of $\psi_{0}$ introduced in the homogeneous case. In the
particular case in which $G(u,v)=G(u-v)$ the natural solution can be
explicitly written in terms of Fourier transforms of the competition kernel
and the carrying capacity, either in an infinite system or in a finite one
with periodic boundary conditions:
$\hat{\psi}^{N}_{q}=\frac{\hat{K}_{q}}{\hat{G}_{q}}.$ (10)
This requires that these Fourier transforms and their inverses exist and lead
to positive populations densities. When this happens, a continuum species
coexistence is obtained, and its existence is generally robust against small
changes in $G$ or $K$. We show later that it is also an attractor of the
dynamics when $\hat{G}_{q}$ satisfy positivity requirements ($p\leq 2$, for
the family in (5), being $p=2$ the marginal case). For a uniform carrying
capacity, the natural solution (9) always exists and is uniform in phenotype
space $\psi^{N}(u)=\psi_{0}$. But the natural solution may lose positivity or
even cease to exist depending on the properties of $G$ and $K$. For example,
when both $G(u)$ and $K(u)$ are of the form (5) with $p=2$, the inverse
Fourier transform of (10) exists when the carrying capacity has a value of
$\sigma$ larger than the kernel $G$, but not in the opposite case.
Figure 1: Long-time solutions of (2) for different kernels and carrying
capacities. Left: $G=g_{\sigma}^{1}$, $K={\rm sech}(u/\sigma)$, with
$\sigma=0.1$. The natural steady solution ($\psi^{N}=a^{-1}{\rm
sech}^{3}(u/\sigma)$), which is positive and non-vanishing everywhere, is
reached at long times. Center: $G=g_{0.1}^{4}$, $K=g_{0.1}^{0.5}$. Under this
non-positive-definite competition kernel, the solution shown is still evolving
and approaches a singular delta comb of the type (3) at long times. Right:
$G=g_{0.1}^{0.5}$, $K=g_{0.1}^{1}$. A positive natural solution does not exist
and the system approaches a single hump solution which vanishes in part of
niche space.
Figure 1 shows stationary solutions attained at long times by the dynamics in
(2) illustrating the situations described above, starting from a smooth non-
vanishing initial condition. In the first case we choose a kernel and carrying
capacity functions ($G(u)=g_{\sigma}^{1}(u)$, $K(u)={\rm sech}(u/\sigma)$)
such that the natural solution exists and is positive everywhere. Thus it is
stable, and it is the steady state attained at long times. In fact it can be
analytically calculated:
$\psi^{N}(u)=a^{-1}{\rm sech}^{3}(u/\sigma)\ .$ (11)
In the second case the non-positiveness of the kernel used (with a carrying
capacity of the type Eq.(5)) breaks down the initial configuration into lumps,
which at long times approach zero-width delta functions with forbidden zones
in between. In the third case, despite $\hat{G}_{q}$ being positive, a
positive natural solution does not exist. Several outcomes are possible but
for the kernel and capacity used, the system approaches a single hump solution
which vanishes in part of the niche space.
More in general, but still in the symmetric $G$ case $G(u,v)=G(v,u)$, writing
the LV model in potential form (Eq. (6)) is of great use since one can show
that, provided $r(u)$ and $K(u)$ are positive, $dV/dt\leq 0$. This implies
that $V$ is a Lyapunov potential and dynamics proceeds towards its absolute
minimum, or if $\psi(u,t=0)=0$ for some $u$, towards the minimum of $V$ under
such constrain. Notice that, since the potential is a quadratic form,
$\psi^{N}$ is a global attractor (starting from non-vanishing initial
conditions) when the competition kernel is a positive definite quadratic form,
which means that $\int\int f(u)G(u,v)f(v)dudv\geq 0$, $\forall f$ (or
$\sum_{ij}x_{i}G_{ij}x_{j}\geq 0$, $\forall\\{x_{i}\\}$ in the discrete case).
This generalizes the previous stability condition on the Fourier transform
$\hat{G}_{q}>0$ to niche inhomogeneous cases, and shows that the stability
result was global indeed. In a multi-dimensional niche space the same analysis
shows that the positive-definiteness of the quadratic form remains the
condition for the global stability of the natural solution. In any case, the
important consequence is that the stability of the natural solution depends
uniquely on the competition kernel and not on the carrying capacity (provided
the relation kernel-capacity is such that the natural solution exists and is
positive). In particular, for competition kernels of the form (5), $\psi^{N}$
is always (if existing and positive) a globally stable solution of the
dynamics for $p\leq 2$, and unstable otherwise.
The crucial difference in the case of a non-symmetric competition kernel is
that there is no obvious Lyapunov potential for the system. This implies that
there are no available global stability results. However, local stability can
be investigated. Let us consider a small perturbation of the positive natural
solution $\psi^{N}(u)+\delta\psi(u,t)$. To linear order, the perturbation
evolves as:
$\frac{d\delta\psi(u,t)}{dt}=-r(u)\frac{\psi^{N}(u)}{K(u)}\int G(u,v)\
\delta\psi(v,t)\,dv.$ (12)
We now consider the functional $H(\delta\psi)\equiv\int\ du\
\left(A(u)K(u)/r(u)\psi^{N}(u)\right)\ (\delta\psi)^{2}$, where $A(u)$ is a
positive function so that $H\geq 0$ and $H(0)=0$. Let us compute its time
derivative:
$\frac{dH}{dt}=-2\int\delta\psi(u)\ A(u)\ G(u,v)\ \delta\psi(v)\,du\,dv.$ (13)
If for some choice of $A(u)$ one has that $A(u)G(u,v)$ is positive definite,
then $dH/dt<0$ and $\delta\psi=0$ will be approached. This shows that
$\psi^{N}$ is linearly stable in such case. We note that the case in which
$G(u,v)$ itself is positive-definite trivially guaranties the positivity of
$A(u)G(u,v)$, with a constant $A$. Thus, even in this more general
nonsymmetric case, it is the character of the interaction kernel $G$, and not
of the carrying capacity (provided it is such that the natural solution exists
and is positive), which determines the stability of the natural solution.
## 3 Lumped species distributions
Scheffer & van Nes (2006) found transient but long-lived solutions of Eq. (1)
consisting of periodically spaced lumps containing many close species. They
used a Gaussian interaction kernel which turned out to introduce an excessive
sensitivity of the results to the numerical implementation of the model and
boundary conditions (Pigolotti _et al._ 2008). They found however similar
solutions as steady configurations when adding an extra predation term acting
effectively only on species with high population. This can be thought as an
extra intraspecific competition since it decreases the growth of species with
many individuals. Exploiting this idea, Pigolotti _et al._ (2007) checked the
effect of using in (2) a kernel of the type (5) but with an enhanced
interaction at $u=0$, i.e. enhanced intraspecific competition. In particular,
they used a constant carrying capacity $K(u)=K_{0}$ and a flat box kernel with
an added delta function at the origin (see Fig. 2),
$G(u)=g_{\sigma}^{\infty}(u)+a\delta(u)\ .$ (14)
Lumped patterns were obtained numerically for $a=1$.
Because the dynamics of (2) usually involves very long transients, it is
interesting to calculate analytically the steady lumped solution in the simple
case of a kernel (14) and uniform carrying capacity $K_{0}$ (in the infinite
line).
Figure 2: The kernel in Eq. (14) (left), and the analytic steady solution
given by (18) and (21-22) for $a=K_{0}=1$, $\sigma=0.8$, $L=0.3$ and $d=1$
(right). Figure 3: The solution $\lambda$ (positive branch) of Eq. (20),
giving the inverse width of species lumps. The width is finite for
$d-\sigma<a$, which is favored by larger enhanced intraspecific competition
$a$.
We begin with the steady state condition
$\int G(u,v)\psi(v)dv=K(u)\ ,$ (15)
valid at $u$ such that $\psi(u)\neq 0$, that particularized to (14) and
constant $K$ reads:
$a\psi(u)+\int_{u-\sigma}^{u+\sigma}dv\psi(v)=K_{0}\ .$ (16)
This is transformed into a differential-difference equation after
differentiation with respect to $u$:
$a\psi^{\prime}(u)+\psi(u+\sigma)-\psi(u-\sigma)=0\ ,{\rm where}\ \psi(u)\neq
0\ .$ (17)
This steady equation has many solutions, including the natural one
$\psi_{0}=K_{0}/(a+2\sigma)$ which is non-vanishing everywhere, or delta combs
such as (3). We search for solutions of the type in Fig. 2, i.e. periodic
arrays of lumps, of period $d$, each one having a symmetric hump shape $f(u)$
of width $2L$ (i.e. $f(u)=0$ if $u\notin[-L,L]$):
$\psi(u)=\sum_{n=-\infty}^{\infty}f(x-nd)$ (18)
We are assuming that the lumps do not overlap, so that $d>2L$. We also note
that if $\sigma+2L<d$ there is no interaction between different lumps, so that
for $u\in[-L,L]$ Eq. (17) reduces to $f^{\prime}(u)=0$ and there is no lump
solution. Moreover, analysis is much simplified if each of the lumps interacts
only with its neighbors ($\sigma+2L<2d$). Thus we restrict to
$d<\sigma+2L<2d$, for which (17) with (18) and $u\in(-L,L)$ becomes:
$af^{\prime}(u)+f(u+\sigma-d)-f(u-\sigma+d)=0$ (19)
The general solution of this linear equation is obtained as a sum of
exponentials $\exp(\lambda u)$, with
$a\lambda=\sinh\left(\lambda(d-\sigma)\right)\ .$ (20)
$\lambda=0$ is always a solution, and if $d-\sigma<a$ there are two additional
solutions $\pm\lambda$, plotted in Fig. 3. For $d-\sigma>a$ the only solution
is the constant one, but in the opposite case (the situation favored by
enhanced intraspecific competition $a$) the solution is a linear combination
of three exponentials. Two of the constants of the combination are determined
from $f(L)=f(-L)=0$. The third one, which gives the overall normalization, can
be obtained by returning back to the original equation (16). Finally we get
$\displaystyle f(u)$ $\displaystyle=$ $\displaystyle
A\left(1-\frac{\cosh(\lambda u)}{\cosh(\lambda L)}\right)\ {\rm if}\
u\in[-L,L]$ (21) $\displaystyle=$ $\displaystyle 0\ \qquad{\rm elsewhere}$
with
$A=\frac{K_{0}}{a\left(1-{\rm sech}(\lambda
L)\right)+\frac{2}{\lambda}\left(\lambda L-\tanh(\lambda L)\right)}\ ,$ (22)
and the value of $\lambda$ which is plotted in Fig. 3. Figure 2 shows the
analytic solution (18) with (21)-(22). We have not studied the stability of
this configuration. But the numerical results in Pigolotti _et al._ (2007)
indicate that it is stable for some values of $L$ and $d$.
Figure 4: The kernel $G=g_{0.2}^{4}+0.8g_{0.02}^{1}$ (left), and the steady
solution obtained numerically from it at long times with constant $K_{0}=1$
(right).
We finally stress that the appearance of the lumped solution is not a
consequence of the singularity of the delta function in the kernel. In fact,
any kernel sufficiently peaked at the origin will favor the coexistence of
close species. If the behavior at larger distances of the kernel makes it not
positive-definite, then full coexistence will be unstable and the natural
solution will split into disjoint lumps. An example of the final steady state
in this situation is shown in Fig. 4, with a kernel
$G=g_{0.2}^{4}+0.8g_{0.02}^{1}$ which has the properties just described and
contains no delta singularity.
###### Acknowledgements.
C.L. and E.H-G. acknowledge support from project FISICOS (FIS2007-60327) of
MEC and FEDER and NEST-Complexity project PATRES (043268). K.H.A was supported
by the Danish Research Council, grant no. 272-07-0485
Models leading to LV competitive interactions
We have seen that the character of the interaction kernel $G$ is of major
importance to determine the qualitative outcome of LV competition. In the
original formulation of the niche model, however, only positive definite
kernels were allowed. The reason is that competition kernels were derived in
terms of utilization functions $u_{i}(x)$, describing how consumer $i$ uses
resource at niche location $x$ (assumed to be continuous) (MacArthur & Levins
1967; Roughgarden 1979):
$G_{ij}=\frac{\int u_{i}(x)u_{j}(x)\,dx}{\int u_{i}^{2}(x)\,dx}.$ (23)
When the resource is directly related to space, (23) can be justified by
considering the probability that consumer $i$ meets consumer $j$ (Roughgarden
1979). It is easy to see that $G_{ij}$ obtained from (23) is positive
definite. We show in the following, however, that relation (23) is by no means
general, and that a greater variety of kernels –positive or non-positive
definite, so that the natural solution representing coexistence can be either
stable or unstable– could be obtained from equations in which resources are
explicitly modelled. Related calculations could be found, for example, in
Schoener (1974).
We consider a set of predators (or consumers), with populations $N_{i}$,
$i=1,2,...m$, competing for different types of prey populations or resources,
$R_{\alpha}$, $\alpha=1,2,...n$, the later growing in a logistic way with
growth rate $\beta_{\alpha}$ and carrying capacity $Q_{\alpha}$ in the absence
of predators. Particular equations modelling this are
$\displaystyle\dot{R}_{\alpha}$ $\displaystyle=$ $\displaystyle-
R_{\alpha}\sum_{i}a_{\alpha
i}N_{i}+\beta_{\alpha}R_{\alpha}\left(1-\frac{R_{\alpha}}{Q_{\alpha}}\right)$
(24) $\displaystyle\dot{N}_{i}$ $\displaystyle=$ $\displaystyle
N_{i}\sum_{\alpha}S_{i\alpha}R_{\alpha}-d_{i}N_{i}$ (25)
$d_{i}$ is the death rate of species $i$. The interaction coefficients are
$a_{\alpha i}$, the depletion rate of resource $\alpha$ produced by species
$i$, and the sensitivity $S_{i\alpha}$, giving the growth rate of $i$ thanks
to resource $\alpha$. Lotka-Volterra type dynamics arises when the time scale
for resource evolution is much faster than that of the consumers (i.e.
$S_{i\alpha}$ and $d_{i}\rightarrow\infty$, but with their ratio finite). In
this case, adiabatic elimination of the resource can be done
($\dot{R}_{\alpha}\approx 0$, so that each prey is at each instant at the
equilibrium determined by their consumers), giving
$R_{\alpha}\approx Q_{\alpha}\left(1-\frac{1}{\beta_{\alpha}}\sum_{i}a_{\alpha
i}N_{i}\right)\ .$ (26)
for the non-vanishing resources. The impact matrix, $D_{\alpha i}$, describing
the depletion of resource $\alpha$ by species $i$ (Meszéna _et al._ 2006), is
$D_{\alpha i}=Q_{\alpha}a_{\alpha i}/\beta_{\alpha}$, which substituted into
the consumers equation leads to:
$\dot{N}_{i}=N_{i}\left(r_{i}-\sum_{j}C_{ij}N_{j}\right)\ ,$ (27)
where $r_{i}=\sum_{\alpha}S_{i\alpha}Q_{\alpha}$ is the maximum growth rate
and $C_{ij}=\sum_{\alpha}S_{i\alpha}D_{\alpha j}$. Thus, the result is an
effective interaction among the predators which is of Lotka-Volterra type. It
is customary to write (27) in terms of the carrying capacity $K_{i}$, defined
as the equilibrium population $N_{i}$ attained in the absence of the other
competitors, i.e. $K_{i}=r_{i}/C_{ii}$. In terms of it, Eq. (27) becomes
identical to (1), with
$G_{ij}=\frac{C_{ij}}{C_{ii}}=\frac{\sum_{\alpha}S_{i\alpha}D_{\alpha
j}}{\sum_{\alpha}S_{i\alpha}D_{\alpha i}}.$ (28)
Having a continuum $R(x)$ of resources instead of a discrete set $R_{\alpha}$
does not introduce important difficulties. Simply one should replace sums by
integrals, replacing the coefficients of Eq. (27) by:
$\displaystyle r_{i}$ $\displaystyle=$ $\displaystyle\int S_{i}(x)Q(x)dx\ ,$
(29) $\displaystyle C_{ij}$ $\displaystyle=$ $\displaystyle\int
S_{i}(x)D_{j}(x)dx\ ,$ (30) $\displaystyle G_{ij}$ $\displaystyle=$
$\displaystyle\frac{\int S_{i}(x)D_{j}(x)dx}{\int S_{i}(x)D_{i}(x)dx}\ ,$ (31)
One can also consider a continuum of species, labelled by their phenotypes
$u$, so that Eq. (1) is replaced by Eq. (2) with $K(u)=r(u)/C(u,u)$,
$G(u,v)=C(u,v)/C(u,u)$, and $r(u)$ and $C(u,v)$ given by obvious
generalizations of (29) and (30).
It is clear that the presence in the kernel $G_{ij}$ of two different
functions (compare with the most restrictive expression (23)) gives enough
freedom to obtain a variety of kernel behaviors under different circumstances.
A particularly popular choice is to assume that impact and sensitivity are
proportional: $S_{i\alpha}=\epsilon D_{\alpha i}$, with a constant efficiency
$\epsilon$. In the continuum resource case the functions can be written in
terms of a single utilization function $u_{i}(x)$ as $D_{i}(x)=u_{i}(x)$ and
$S_{i}(x)=\epsilon u_{i}(x)$, leading to the classical expression (23).
Slightly more general cases arise when the efficiency depends only on the
resource, $\epsilon=\epsilon_{\alpha}$, or on the consumer
$\epsilon=\epsilon_{i}$, or when dependence on the two types of species
factorizes, $\epsilon=v_{i}w_{\alpha}$. In all these cases (if the efficiency
is positive) one is lead to a kernel $G_{ij}$ which is positive definite. In
more general cases, one can have a kernel leading to instability of the
coexistence state.
We conclude with two instances of ecological interactions in (25) which allow
to tune the stability. First, a homogeneous discrete and infinite niche space
in which all resources have the same internal dynamics $Q_{\alpha}=Q$,
$\beta_{\alpha}=\beta$, $\forall\alpha$, as well as the consumers: $d_{i}=d$,
$\forall i$. The interactions are taken to be
$\displaystyle S_{i\alpha}$ $\displaystyle=$ $\displaystyle
g\delta_{i,\alpha}$ (32) $\displaystyle a_{\alpha,j}=\frac{\beta}{Q}D_{\alpha
j}$ $\displaystyle=$ $\displaystyle
a\delta_{\alpha,j}+b(\delta_{\alpha,j-1}+\delta_{\alpha,j+1}).$
This models a situation in which the consumer $k$ grows only by consuming its
optimal resource $R_{k}$, whereas it depletes also the neighboring resources,
$R_{k+1}$ and $R_{k-1}$. We have $r_{i}=Qg$, $K_{i}=\beta/a$,
$C_{ij}=(Qg/\beta)\left(a\delta_{i,j}+b(\delta_{i,j-1}+\delta_{i,j+1})\right)$,
and $G_{ij}=\delta_{ij}+(b/a)\left(\delta_{i,j-1}+\delta_{i,j+1}\right)$ so
that equation (1) is now
$\dot{N}_{i}=QgN_{i}\left[1-\frac{1}{\beta}\left(aN_{i}+b\left(N_{i+1}+N_{i-1}\right)\right)\right].$
(33)
The natural solution, i.e. the one in which all species have positive non zero
population, is $\overline{N}_{i}=\beta/(a+2b)$, $\forall i$. Its linear
stability can be studied by linearization, $N_{l}(t)=\overline{N}_{l}+\delta
N_{l}(t)$ and substitution of the ansatz $\delta N_{l}\approx
e^{\lambda_{q}t}e^{iql}$ (here $i=\sqrt{-1}$). We find
$\lambda_{q}=-(Qg/\beta)\left(a+2b\cos q\right)$, $q\in[-\pi,\pi]$.
$\lambda_{q}$ are the eigenvalues of $-C_{ij}$, and stability of
$\overline{N}_{i}$ requires all these eigenvalues to be negative, i.e.,
$C_{ij}$ to be positive definite. When $a>2b$, then $\lambda_{q}<0$ $\forall
q$, and the natural coexistence solution is globally stable (see results in
Sect. 2). It is unstable otherwise. In this example, there is no well-defined
single utilization function and the positivity properties of the interaction
kernel and thus the stability of the natural solution can be changed by
varying the parameters.
As a second example, with nonconstant carrying capacity, we consider a
continuous distribution of resources and species on the line, and we take
$\begin{array}[]{rclrcl}Q(x)&=&Qg(x),&\beta(x)&=&\beta g(x),\\\
S_{u}(x)&=&s\delta(u-x),&a_{u}(x)&=&f(u-x).\end{array}$
which implies that the consumers of phenotype $u$ grow only from the resource
at location $x=u$, but they deplete a wider range characterized by $f$. This
leads (in the continuous formalism) to $r(u)=sQg(u)$ ,
$C(u,v)=sQf(u-v)/\beta$, $K(u)=\beta g(u)/f(0)$ and $G(u,v)=f(u-v)/f(0)$. In
this example, by choosing the functions $g$ and $f$, we can impose any desired
combinations of carrying capacity and interaction kernel. Gaussianity or
positive definiteness are particular cases, no more natural in this
generalization than alternative choices leading to non-positiveness,
instability, and thus exclusion zones between clumps of species.
## References
* (1)
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* Schoener (1974) Schoener, T. W., 1974 Some methods for calculating competition coefficients from resource-utilization spectra. _Am. Nat._ 108, 332. 10.1086/282911.
* Szabó & Meszéna (2006) Szabó, P. & Meszéna, G., 2006 Limiting similarity revisited. _Oikos_ 112, 612–619. 10.1111/j.0030-1299.2006.14128.x.
* Volterra (1926) Volterra, V., 1926 Variazioni e fluttuazioni del numero d’individui in specie animali conciventi (Variations and fluctuations of the number of individuals in animal species living together). _Memoria della R. Accademia Nazionale dei Lincei, Ser. VI_ 2, 31–113. Translated in Chapman, R. (1931) Animal Ecology (McGraw Hill, New York), pp. 409–448.
|
arxiv-papers
| 2008-12-06T12:09:02 |
2024-09-04T02:48:59.260919
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Emilio Hernandez-Garcia, Cristobal Lopez (IFISC), Simone Pigolotti\n (NBI), Ken H. Andersen (AQUA)",
"submitter": "Emilio Hernandez-Garcia",
"url": "https://arxiv.org/abs/0812.1279"
}
|
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